International Congress of Mathematicians

Beijing 2002August 20-28Proceedings of theInternationalCongress ofMathematiciansVol-H: Invited LecturesH igne- Education ProswWorkls«** 1 * 0: (on

International Congress of Mathematicians (2002, Beijing)Proceedings of the International Congress of MathematiciansAugust 20^28, 2002, BeijingEditor: LI Tatsien (LI Daqian)dqliQfudan.edu.cnEditorial Assistants: Cai Zhijie, Lu Fang, Xue Mi, Zhou ChunlianThis volume is the first part of the collection of manuscripts of the lectures given bythe invited speakers of the ICM2002. The second part of this collection is publishedin Volume III.The manuscripts of the invited lectures are ordered by sections and, in each section,alphabetically by author's names. In case of several authors for one manuscript,the name of invited speaker is written in boldface type.The electronic version of this volume will be published on the international MathArXiv with the addresshttp://front.math.ucdavis.edu/This work is subject to copyright. All rights are reserved,whether the whole or part of the material is concerned,specially the rights of translation, reprinting, reuse ofillustrations, broadcasting, reproduction on microfilms orin other ways, and storage in data banks.©2002 Higher Education Press55 Shatan Houjie, Beijing 100009, Chinahttp://www.hep.com.cn http://www.hep.edu.enCopy Editors: Li Rui, Li Yanfu, Wang YuISBN 7-04-008690-5Set of 3 Volumes

ContentsSection 1. LogicE. Bouscaren: Groups Interprétable in Theories of Fields 3J. Denef, F. Loeser: Motivic Integration and the Grothendieck Group ofPseudo-Finite Fields 13D. Lascar: Automorphism Groups of Saturated Structures; A Review 25Section 2. AlgebraS. Bigelow: Representations of Braid Groups 37A. Bondal, D. Orlov: Derived Categories of Coherent Sheaves 47M. Levine: Algebraic Cobordism 57Cheryl E. Praeger: Permutation Groups and Normal Subgroups 67Markus Rost: Norm Varieties and Algebraic Cobordism 77Z. Sela: Diophantine Geometry over Groups and the Elementary Theoryof Free and Hyperbolic Groups 87J. T. Stafford: Noncommutative Projective Geometry 93Dimitri Tamarkin: Deformations of Chiral Algebras 105Section 3. Number TheoryJ. W. Cogdell, I. I. Piatetski-Shapiro: Converse Theorems,Functoriality, and Applications to Number Theory 119H. Cohen: Constructing and Counting Number Fields 129Jean-Marc Fontaine: Analyse p-adique et Représentations Galoisiennes 139A. Huber, G. Kings: Equivariant Bloch-Kato Conjecture and Non-abelianIwasawa Main Conjecture 149Kazuya Kato: Tamagawa Number Conjecture for zeta Values 163Stephen S. Kudla: Derivatives of Eisenstein Series and ArithmeticGeometry 173Barry Mazur, Karl Rubin: Elliptic Curves and Class Field Theory 185Emmanuel Ullmo: Théorie Ergodique et Géométrie Arithmétique 197Trevor D. Wooley: Diophantine Methods for Exponential Sums, andExponential Sums for Diophantine Problems 207

Section 4. Differential GeometryB. Andrews: Positively Curved Surfaces in the Three-sphere 221Robert Bartnik: Mass and 3-metrics of Non-negative Scalar Curvature 231P. Biran: Geometry of Symplectic Intersections 241Hubert L. Bray: Black Holes and the Penrose Inequality in GeneralRelativity 257Xiuxiong Chen: Recent Progress in Kahler Geometry 273Weiyue Ding: On the Schrödinger Flows 283P. Li: Differential Geometry via Harmonic Functions 293Yiming Long: Index Iteration Theory for Symplectic Paths withApplications to Nonlinear Hamiltonian Systems 303Anton Petrunin: Some Applications of Collapsing with BoundedCurvature 315Xiaochun Rong: Collapsed Riemannian Manifolds with Bounded SectionalCurvature 323Richard Evan Schwartz: Complex Hyperbolic Triangle Groups 339Paul Seidel: Fukaya Categories and Deformations 351Weiping Zhang: Heat Kernels and the Index Theorems on Even and OddDimensional Manifolds 361Section 5. TopologyMladen Bestvina: The Topology of Out(F n ) 373Yu. V. Chekanov: Invariants of Legendrian Knots 385M. Furuta: Finite Dimensional Approximations in Geometry 395Emmanuel Giroux: Géométrie de Contact: de la Dimension Trois vers lesDimensions Supérieures 405Lars Hesselholt: Algebraic K-theory and Trace Invariants 415Eleny-Nicoleta Ionel: Symplectic Sums and Gromov- Witten Invariants 427Peter Teichner: Knots, von Neumann Signatures, and Grope Cobordism 437Ulrike Tillmann: Strings and the Stable Cohomology of Mapping ClassGroups 447Shicheng Wang: Non-zero Degree Maps between 3-Manifolds 457Section 6. Algebraic and Complex GeometryHélène Esnault: Characteristic Classes of Flat Bundles and Determinantof the Gauss-Manin Connection 471L. Göttsche: Hilbert Schemes of Points on Surfaces 483Shigeru Mukai: Vector Bundles on a K3 Surface 495

R. Pandharipande: Three Questions in Gromov-Witten Theory 503Miles Reid: Update on 3-folds 513Vadim Schechtman: Sur les Algèbres Vertex Attachées aux VariétésAlgébriques 525B. Totaro: Topology of Singular Algebraic Varieties 533Section 7. Lie Group and Representation TheoryPatrick Delorme: Harmonic Analysis on Real Reductive SymmetricSpaces 545Pavel Etingof: On the Dynamical Yang-Baxter Equation 555D. Gaitsgory: Geometric Langlands Correspondence for GL n 571Michael Harris: On the Local Langlands Correspondence 583Alexander Klyachko: Vector Bundles, Linear Representations, and SpectralProblems 599Toshiyuki Kobayashi: Branching Problems of Unitary Representations 615Vikram Bhagvandas Mehta: Representations of Algebraic Groups andPrincipal Bundles on Algebraic Varieties 629E. Meinrenken: Clifford Algebras and the Duflo Isomorphism 637Maxim Nazarov: Representations of Yangians Associated with SkewYoung Diagrams 643Freydoon Shahidi: Automorphic L-Functions and Functoriality 655Marie-France Vignéras: Modular Representations of p-adic Groups andof Affine Hecke Algebras 667Section 8. Real and Complex AnalysisA. Eremenko: Value Distribution and Potential Theory 681Juha Heinonen: The Branch Set of a Quasiregular Mapping 691Carlos E. Kenig: Harmonic Measure and "Locally Flat" Domains 701Nicolas Lerner: Solving Pseudo-Differential Equations 711C. Thiele: Singular Integrals Meet Modulation Invariance 721S. Zelditch: Asymptotics of Polynomials and Eigenfunctions 733Xiangyu Zhou: Some Results Related to Group Actions in SeveralComplex Variables 743Section 9. Operator Algebras and Functional AnalysisSemyon Alesker: Algebraic Structures on Valuations, Their Properties andApplications 757P. Biane: Free Probability and Combinatorics 765D. Bisch: Subfactors and Planar Algebras 775Ill

Liming Ge: Free Probability, Free Entropy and Applications to vonNeumann Algebras 787V. Lafforgue: Banach KK-theory and the Baum-Connes Conjecture 795R. Latala: On Some Inequalities for Gaussian Measures 813Author Index 823

Section 1. LogicE. Bouscaren: Groups Interprétable in Theories of Fields 3J. Denef, F. Loeser: Motivic Integration and the Grothendieck Group ofPseudo-Finite Fields 13D. Lascar: Automorphism Groups of Saturated Structures; A Review 25

ICM 2002 • Vol. II • 3^12Groups Interprétablein Theories of FieldsE. Bouscaren*AbstractWe survey some results on the structure of the groups which are definablein theories of fields involved in the applications of model theory to Diophantinegeometry. We focus more particularly on separably closed fields of finite degreeof imperfection.2000 Mathematics Subject Classification: 03C60, 03C45, 12L12.Keywords and Phrases: Groups, Fields, Definability, Algebraic groups.1. IntroductionIn the last ten years, the model theory of fields has seen striking new developments,with applications in particular to differential algebra and Diophantinegeometry. One of the main ingredients in these applications is the analysis of thestructure of groups definable in fields with added "definable structure".Model theory studies structures with a family of distinguished subsets of theirCartesian products, the family of definable subsets, which is requested to be closedunder finite Boolean operations and projections. In the case of algebraically closedfields, the definable sets are exactly the constructible sets in the Zariski topology(finite Boolean combinations of Zariski closed sets). If one considers fields whichare not algebraically closed (for example, fields of positive characteristic which areseparably closed and not perfect) or algebraically closed fields with new operators(differentially closed fields, fields with a generic automorphism), then the family ofdefinable sets is much richer than the family of Zariski constructible sets. In eachof the above cases, one can generalize the classical geometric notions, by using thetools developed by model theory (abstract notion of independence, of dimensions...).For example:1. One can define "good" topologies which strictly contain the Zariski topology-* University Paris 7 - CNRS, Department of Mathematics, Case 7012, 2 Place Jussieu, 75251Paris Cedex 05, France. E-mail: elibou@logique.jussieu.fr

4 E. Bouscaren2. Different notions of dimensions can be attached to definable sets (or infiniteintersections of definable sets, which we call infinitely definable, or oo-definable,sets). In the case of algebraically closed fields, all such possible notions of abstractdimension must coincide and be equal to the classical algebraic dimension. In theother cases, these dimensions may be different, some may take infinite ordinal valuesor may be defined only for some special classes of definable (or oo-definable) sets.3. If K is any of the above mentioned fields, and if H is an algebraic groupdefined over K, then the group H(K) of the if-rational points of H is a definablegroup. But there are "new" families of definable groups which are not of this form.In fact, it is precisely the study of certain specific families of "new" definablegroups of finite dimension which are at the center of the applications to Diophantinegeometry. We will not attempt here to explain how the model theoretic analysis ofthe finite rank definable groups yields these applications. There have been in recentyears many surveys and presentations of the subject to which we refer the reader(see for example, [4],[5], [14], [22] or [28]). We will come back to this subject, butvery briefly, at the end in Section 3.5..The first general question raised by the existence of these new definable groupsis that of their relation to the classical algebraic groups. Remark that this questionalready makes sense in the context of "pure" algebraically closed fields, about theclass of definable (= constructible) groups. In that case, it is true that any constructiblegroup in an algebraically closed field K is constructibly isomorphic to theif-rational points of an algebraic group defined over K (see for example [3] or [23]).Let us now consider briefly the case of a field K of characteristic p > 0 whichis separably closed and not perfect. Then the class of constructible sets is nolonger closed under projection and there are many definable groups which are notconstructible, the most obvious one being K p . There are also some groups which areproper intersections of infinite descending chains of definable groups: for example,K p ( = fi n K p )j the field of infinitely p-divisible elements of the multiplicativegroup, or f] n p n A(K), for A an Abelian variety defined over K.It is nevertheless true, as we will see, that every definable group in K isdefinably isomorphic to the if-rational points of an algebraic group defined overK. Furthermore, as in the classical case of one-dimensional algebraic groups, it ispossible to give a complete description, up to definable isomorphism, of the onedimensionalinfinitely definable groups.There are results of similar type for the other classes of enriched fields mentionedabove. In this short paper, we will concentrate mainly on the case of separablyclosed fields (in Section 3.). Before this, in Section 2., we will only very brieflypresentthe model theoretic setting for two other examples of "enriched" fields, incharacteristic zero, differentially closed fields and generic difference fields. We hopethis will give the reader an idea of what the common features and the differencesmight be in the model theoretic analysis of these different classes of fields.Finally, there are of course many other classes of fields whose model theoryhasbeen extensively developed in the past years with many connections to algebra,semi-algebraic or subanalytic geometry, and which we are not going to mentionhere: for example, valued fields, ordered fields, "o-minimal" expansions of the real

Groups Interprétable in Theories of Fields 5field...2. Two short examplesWe will just very briefly describe the two characteristic zero examples mentionedabove.2.1. Differentially closed fields of characteristic zeroWe consider a field K of characteristic zero, with a derivation Ö, that is, anadditive map from K to K which satisfies that for all x, y in K, 8(xy) = x5(y) +yö(y). We define the ring -?C$[X] of differential polynomials over K to be the ringof polynomials in infinitely many variables K[X, ö(X),ö 2 (X), • • • , ö n (X), • • •]. Theorder of the differential polynomial f(X) in -?C$[X] is —1 if / G K and otherwisethe largest n such that ö n (X) occurs in f(X) with non zero coefficient. We saythatK is differentially closed if for any non-constant differential polynomials f(X)and g(X), where the order of g is strictly less than the order of /, there is a z suchthat f(z) = 0 and g(z) ^ 0. In model theoretic terms, this means exactly that Kis existentially closed.From now on we suppose that (K, Ö) is a large differentially closed field (auniversal domain).We say that F C K n is a ö-closed set, if there are /i, • • • , f r G Kg [X\, • • • , X„]suchthat F = {(cu,-•• ,a n ) € K n ; /i(cn, • • • ,a n ) = ••• = / r (cn,--- ,a n ) = 0}. Thering Kg [X\, • • • , X„] is of course not Noetherian but the ^-closed sets (which correspondto radical differential ideals) form the closed sets of a Noetherian topologyon K, the ö-topology.We now consider the ö-constructible sets, that is, the finite Boolean combinationsof ^-closed sets. This class is closed under projection (this is quantifierelimination for the theory), hence the definable sets (we call them ö-definable sets)are exactly the ^-constructible sets. To every ^-definable set one can associate adimension (the Morley rank) which can take infinite countable ordinal values.There are "new" definable groups, which are not of the form H(K) for any algebraicgroup H. In particular, any H(K) will have infinite dimension. In contrast,the field of constants of K, Cons(K) = {a G K; ö(x) = 0}, is a ^-closed set which isnot constructible; it is an algebraically closed subfield of K and has dimension one.Nevertheless the following is true:Proposition 1 ([21]) Let G be a 5-definable group in K. Then there is an algebraicgroup H, defined over K, such that G is definably isomorphic to a (5-definable)subgroup of H(L).For the many more existing results on ^-definable groups, we refer the readerto [20], or from the differential algebra point of view, to [8].2.2. Generic difference fields

6 E. BouscarenWe now consider an algebraically closed field K with an automorphism a. Wesay that (K, a) is a generic difference field if every difference equation which has asolution in an extension of K has a solution in K. The theory of generic differencefields has been extensively studied in [9] and [10].Let us suppose that (K,a) is a generic difference field in characteristic zero.We consider the ring of a-polynomials,K a [Xi,--- ,X n ] = K[X lr -- ,X„,CT(XI),: ••• ,a(X n ),a 2 (X 1 ), • • • ,a 2 (X,We say that F C K n is a fa-closed sei if there are /i, • • • , f r G K a [Xi, • • • ,X„] suchthat F = {(ai,--- ,a n ) G K n : /i(cn,--- ,a n ) = ••• = / r (cn,--- ,a n ) = 0}. TheCT-closed sets form the closed sets of a Noetherian topology on K, the a-topology.The class of a-definable sets is the closure under finite Boolean operations andprojections of the a-closed sets.Again there are "new" a-definable groups. For example, the field Fix(K) ={a G K : a(a) = a}, the fixed field of a in K, is a a-closed set of dimension one.Here the best result possible for arbitrary a-definable groups is the following:Proposition 2 ([18]) Let G be a group definable in (K,a). Then there are analgebraic group H defined over K, a finite normal subgroup N\ of G, a a-definablesubgroup Hi of H(K) and a finite normal subgroup N 2 of Hi, such thatG/Ni andHi/N 2 are a-definably isomorphic.The analysis of groups of finite dimension is one of the main tools in Hrushovski'sproof of the Manin-Mumford conjecture in [15].3. Separably closed fields of finite degree of imperfectionSeparably closed fields are particularly interesting from the model theoreticpoint of view for many reasons, in addition to the fact that they form the frameworkfor Hrushovski's proof of the Mordell-Lang conjecture in charactersitic p. Let usjust mention one reason here: they are the only fields known to be stable and nonsuperstable, and in fact it is conjectured that they are the only existing ones.We will just focus on the main properties of the groups that are definable in aseparably closed field of finite degree of imperfection, but we need first to introducesome notation and recall some basic facts (see [11]).3.1. Some basic facts and notationLet L be a separably closed field of charcteristic p > 0 and of finite degree ofimperfection which is not perfect, i.e., L has no proper separable algebraic extension,and \L : L p \ = p v , with 0 < v. In order to avoid confusion we denote the Cartesianproduct of k copies of L by L xk .A subset B = {61, • • • ,b v } of L is called a p-basis of L if the set ofp-monomialsof B, {Mj := nr=i ^ > i e P v ] forms a linear basis of L over LP. Each element x

Groups Interprétable in Theories of Fields 7in L can be written in a unique way as x = Xê P " xfMj. From now on we fix ap-basis B of L and the Mj's, with j G p", always denote the p-monomialsof B. We suppose that L is large (a universal domain, or in model theoretic terms,saturated) and we fix some small separably closed subfield K of L, containing Band of same degree of imperfection v.We let fj denote the map which to x associates Xj. The Xj's are called thep-components of x of level one. More generally, one can associate to i a tree ofcountable height indexed by (p v )

8 E. Bouscarendimension would need to be invariant under definable bijections, but for every nthe map A„, defined by X n (x) := (avêCp")", is a definable bijection between L andL xpVn . But some oo-definable sets will have a well-defined dimension, for examplethe field L p °° := P| n L p ", which is the biggest algebraically closed subfield of L, hasdimension one. In fact, L p °° is the unique (up to definable isomorphism) infinitelydefinablefield of dimension one ([19], [13]).3.3. Definable groupsAgain, amongst the definable groups, one finds the "classical" ones, that isgroups of the form H(L) for H any algebraic group defined over L. These groupshave certain specific properties which are not true of all the definable groups inL. Recall that a definable subset X of G is said to be generic if G is covered bya finite number of translates of X, and an element of G is generic for the groupif every definable set which contains it is generic. In an algebraic group, genericsin the topological sense coincide with generics for the algebraic group. Recall alsothat a definable group is said to be connected if it has no proper definable subgroupof finite index, and connected-by-finite if it has a definable connected subgroup offinite index.Proposition 5 ([6], [13]) Let H be an algebraic group defined over K. Then H(L)is connected-by-finite. If H is connected (hence irreducible as an algebraic group),then H(L) is connected (and irreducible for the X-topology) and if a £ H(L) is ageneric point, then the ideal 1(a) = {/ G iffX^] : /(a œ ) = 0} is minimal amongstthe ideals 1(h), for h G H(L).The above says that in the group H(L), the generics in the topological sensecoincide with generics for the group. In an arbitrary group defined in L, this neednot be the case.Consider the definable bijection / from L to L defined in the following way:if x G L \ L p , f(x) = x", iîx£LP\ LP\ f(x) = X X /P, if a: G IA>', f(x) = x.Transporting addition through /, one gets a group on L again, G := (L,*),definably isomorphic to (L,+), hence connected. The set L itself is of course A-closed and irreducible with associated ideal I(L) = I°(X). The ideal associated tothe (group) generic of (L,*) is generated by I°(X) and {Xj = 0 : i G p",i ^ 0},and strictly contains I°(X).This question of the uniqueness of the notion of generic is not the only oneposing problems for arbitrary definable groups in L. For example, there is no reason,coming from general properties of stable (non superstable) theories, which a prioriforces all these definable groups to be connected-by-finite.Nevertheless, one can in fact show that the situation is as close to the classicalone as it could be:Proposition 6 [6] Every definable group G in L is connected-by-finite and is definablyisomorphic to the group of L-rational points of an algebraic group H definedover L.

Groups Interprétable in Theories of Fields 9One more remark, in the case of algebraic groups, by Prop. 5, irreducibilitytransfers down to the set of L-rational points. But this is not the case for anarbitrary variety: if one considers for example the irreducible variety defined by theequation Y p X + Z p = 0, for m > 1, then the A-closed set V(L) is no longerirreducible in the sense of the A-topology.3.4. Minimal groupsThe previous result enables us to give a complete description of groups ofdimension one, and more generally of some classes of commutative groups.We say that an oo-definable set D is minimal if any definable subset of G isfinite or co-finite. If D is actually definable, then we say that D is strongly minimal.The minimal groups are exactly the connected groups of dimension (U-rank)equal to one. A minimal group must be commutative.From the basic properties of commutative algebraic groups over an algebraicallyclosedfield of characteristic p and Proposition 6, one can deduce:Lemma 7 Let G be a minimal group oo-definable in L, then G has exponent p orG is divisible.We first consider the commutative groups of exponent p:Proposition 8 [7] Let G be a commutative oo-definable group of exponent p definablein L. Then G is definably isomorphic to a X-closed subgroup of the additivegroup (L,+). Furthermore, if G is definable, then it is definably isogenous to thegroup of L-rational points of a vector group.Note that even when G is connected it is not necessarily definably isomorphicto the group of rational points of a vector group.Then we consider the commutative divisible groups, which we show to beexactly the ones that were considered by Hrushovski in [13]:Proposition 9 [7] 1. Let G be any oo-definable commutative divisible group inL. Then G is definably isomorphic to some p°°A(L) := f] n p n A(L), for A a semi-Abelian variety defined over L.2. If A is a semi-Abelian variety defined over L, p°°A(L), which is the maximaldivisible subgroup of A(L) is also the smallest oo-definable subgroup of A(L) whichis Zariski dense in A.Finally, this analysis, together with some results from [11] and [13], yields thefull description of minimal groups.Before stating the actual result, let us give some last definitions. The groupG is said to be of linear type if for every n, every definable subgroup of G xn is afinite Boolean combination of translates of definable subgroups of G xn . We definethe transcendence rank over if of a group G, defined over K, to be the maximumof {tr.degree(if(#oo))iQ : g G G}.Proposition 10 Let G be an oo-definable minimal group in L.

10 E. Bouscaren1. Either G is not of linear type and then,• G is definably isomorphic to the multiplicative group ((L p ) ,•),• or G is definably isomorphic to E(L P ) for E an elliptic curve definedover L p ,• or G is definably isogenous to (L p °°,+). (isogenous here cannot be replacedby isomorphic).2. Or G is of linear type and then,• G is divisible and G is definably isomorphic to p°°A(L) for some simpleAbelian variety A defined over K which is not isogenous to an Abelianvariety defined over L p ,• or G is of exponent p and is definably isomorphic to a minimal X-closedsubgroup of (L,+).In the divisible case G has finite transcendence rank; in the exponent p case,all transcendence ranks are possible.The induced module-type structure on the minimal groups of exponent p andof linear type is analyzed in [2].A short word about some of the tools involved in the proofs of Propositions6 and 10: the proofs of 6, 1 and 2 all involve at some point the classical theoremof Weil's constructing an algebraic group from a generic group law on a variety, orsome generalizations of this theorem to an abstract model theoretic context. In thespecific case of separably closed fields, another fundamental tool is the analysis ofthe properties of the A n -functors, naturally associated to the maps A„: for each n,A n is a covariant functor from the category of varieties V defined over K to itself,with the property that the L-rational points of the variety A n V are exactly theimage by the map A„ of the L-rational points of V. In the case of an algebraicgroup defined over K, Ai is equal to the composition of the inverse of the Frobeniusand of the classical Weil restriction of scalars functor from K X IP to K.Finally, the way we have stated Proposition 10 uses the fact that if a minimalgroup is not of linear type, then it is non orthogonal to L p °° (and hence definablyisogenous to the L p -rational points of some definable group over LP ). The onlyknownproof of this so far uses the powerful abstract machinery of Zariski structuresfrom [16]. This dichotomy result, for the particular case of groups of theform p°°A(L), is essential in Hrushovski's proof of the Mordell-Lang conjecture incharacteristic p, which is still the only existing proof for the general case. In arecent paper Pillay and Ziegler ([24]), show that, with some extra assumptions onA, one can replace in this proof the heavy Zariski structure argument by a muchmore elementary one. These extra assumptions are satisfied when A is an ordinarysemi-Abelian variety (i.e. A has the maximum possible number of p"-torsion pointsfor every n), case which was already covered by previous non model-theoretic proofs(see [1]).3.5. Final remarks and questionsAs we have already mentioned earlier, the groups of finite dimension definablein these "enriched" theories of fields play a major role in the applications of

Groups Interprétable in Theories of Fields 11model theory to Diophantine geometry. In the characteristic zero case, the relevantgroups are the definable subgroups of the group of rational points of Abelianvarieties in differentially closed fields (Mordell-Lang conjecture for function fields[13]), in generic difference fields (the Manin-Mumford conjecture [15], [5] and theTate-Voloch conjecture for semi-Abelian varieties defined over Q p [25], [26]). In thecharacteristic p case, the relevant groups are: the oo-definable divisible subgroupsof the group of rational points of semi-Abelian varieties in separably closed fields(the Mordell-Lang conjecture for function fields [13]) and the definable subgroups ofthe additive groups in generic difference fields of characteristic p (Drinfeld modules[27]).One should note that, in fact, separably closed fields are just another instanceof a field with extra operators (derivations or automorphisms): one can equip anyseparably closed field L of finite degree of imperfection, with an infinite family ofHasse derivations in such a way that the resulting structure is bi-definably equivalentwith L considered as a structure in the language described in section 3.2.. Thereare many interesting other possible types of "enriched" fields in this sense wherethe complete analysis of the model theoretic structure remains to be done.Finally, one crucial step towards possible further applications of the fine studyof finite rank definable sets to geometry would be an understanding of the structureinduced on the so-called trivial or disintegrated definable (or infinitely definable)minimal sets, that is the minimal sets such that the induced pregeometry is disintegrated.This condition immediately rules out definable groups. The absence ofany well-understood algebraic structure living on these "trivial" sets makes themvery difficult to analyze. The only results obtained so far are in the context ofdifferentially closed fields of characteristic 0: Hrushovski ([12]), building on someresults of Jouanolou ([17]), showed that in any trivial strongly minimal set definedby a differential equation of order one, the induced pregeometry is locally finite.The question of whether this is true for higher order equations is still open.References[1] D. Abramovic & F. Voloch, Towards a proof of the Mordell-Lang conjecture incharacteristic p, Intern. Math. Research Notices (IMRN), 2 (1992), 103-115.[2] T. Blossier, Ensembles minimaux localement modulaires, Thèse de Doctorat,Université Paris 7, 2001.[3] E. Bouscaren, Model-theoretic versions of Weil's theorem on pre-groups, inThe Model Theory of Groups, (A. Nesin & A. Pillay, editors), Notre DameUniversity Press, 1989.[4] E. Bouscaren, Proof of the Mordell-Lang conjecture for function fields, in Modeltheory and algebraic geometry (E. Bouscaren, editor), Lecture Notes in Mathematics,Vol. 1696, Springer-Verlag, 1998.[5] E. Bouscaren, Théorie des modèles et conjecture de Manin-Mumford (d'aprèsEhud Hrushovski), Séminaire Bourbaki, Vol. 1999/2000, Astérisque No. 276(2002), 137^159.[6] E. Bouscaren & Françoise Delon, Groups definable in separably closed fields,Transactions of the A.M.S., 354 (2002), 945^966.

12 E. Bouscaren[7] E. Bouscaren & Françoise Delon, Minimal groups in separably closed fields,The Journal of Symbolic Logic, 67 (2002), 239^259.[8] A. Buium, Differential Algebra and Diophantine Geom., Hermann, Paris, 1994.[9] Z. Chatzidakis & E. Hrushovski, The model theory of difference fields, Transactionsof the A.M.S, Vol. 351 (1999), 2997-3071.[10] Z. Chatzidakis, E. Hrushovski & Y. Peterzil, The model theory of differencefields II, Proceedings of the London Math. S oc. (to appear).[11] F. Delon, Separably closed fields, in Model Theory and Algebraic Geometry, E.Bouscaren (Ed.), Lecture Notes in Mathematics 1696, Springer-Verlag, 1998.[12] E. Hrushovski, ODE's of order 1 and a generalisation of a theorem ofJouanolou's, Manuscript, 1995.[13] E. Hrushovski, The Mordell-Lang conjecture for function fields, Journal of theA.M.S., 9 (1996), 667^690.[14] E. Hrushovski, Geometric model theory, in Proceedings of the InternationalCongress of Mathematicians, Berlin, Vol. I (1998), Doc. Math., 281^302.[15] E. Hrushovski, The Manin-Mumford conjecture and the model theory of differencefields,Annals of Pure and Applied Logic, 112 (2001), 43^115.[16] E. Hrushovski & B. Zilber, Zariski Geometries, Journal of the A.M.S., 9 (1996),1-56.[17] J.P. Jouanolou, Hypersurfaces solutions d'une équation de Pfaff analytique,Mathematische Annalen, 232 (1978), 239^245.[18] P. Kowalski & A. Pillay, A note on groups definable in difference fields, preprint,2000.[19] M. Messmer, Groups and fields interprétable in separably closed fields, Transactionsof the A.M.S., 344 (1994), 361^377.[20] A. Pillay, Differential algebraic groups and the number of countable differentiallyclosed fields, in Model Theory of Fields, D. Marker, M. Messmer & A.Pillay, Lecture Notes in Logic 5, Springer, 1996.[21] A. Pillay, Some foundational questions concerning differential algebraic groups,Pacific Journal of Math., 179 (1997), 179-200.[22] A. Pillay, Model Theory and Diophantine geometry, Bulletin of the A.M.S., 34(1997), 405-422.[23] A. Pillay, Model theory of algebraically closed fields, in Model theory and algebraicgeometry (E. Bouscaren, editor), Lecture Notes in Mathematics, Vol.1696, Springer-Verlag, 1998.[24] A. Pillay & M. Ziegler, Jet spaces of varieties over differential and differencefields, preprint, 2002.[25] T. Scanlon, p-adic distance from torsion points of semi-Abelian varieties, Journalfür dir Reine und Angewandte Mathematik, 499 (1998), 225-236.[26] T. Scanlon, The conjecture of Tate & Voloch on p-adic proximity to torsion,Intern. Math. Research Notices (IMRN), 17 (1999), 909-914.[27] T. Scanlon, Diophantine geometry of the torsion of a Drinfeld module, preprint1999.[28] T. Scanlon, Diophantine geometry from model theory, Bulletin of SymbolicLogic, 7 (2001), 37^57.

ICM 2002 • Vol. II • 13^23Motivic Integration and the GrothendieekGroup of Pseudo-Finite FieldsJ. Denef* F. Loeser 1 'AbstractMotivic integration is a powerful technique to prove that certain quantitiesassociated to algebraic varieties are birational invariants or are independent ofa chosen resolution of singularities. We survey our recent work on an extensionof the theory of motivic integration, called arithmetic motivic integration. Wedeveloped this theory to understand how p-adic integrals of a very general typedepend on p. Quantifier elimination plays a key role.2000 Mathematics Subject Classification: 03C10, 03C98, 12E30, 12L12,14G15, 14G20, 11G25, 11S40, 12L10, 14F20.Keywords and Phrases: Motivic integration, p-adic integration, Quantifierelimination.1. IntroductionMotivic integration was first introduced by Kontsevich [20] and further developedby Batyrev [3] [4], and Denef-Loeser [8] [9] [12]. It is a powerful technique toprove that certain quantities associated to algebraic varieties are birational invariantsor are independent of a chosen resolution of singularities. For example, Kontsevichused it to prove that the Hodge numbers of birationally equivalent projectiveCalabi-Yau manifolds are equal. Batyrev [3] obtained his string-theoretic Hodgenumbers for canonical Gorenstein singularities by motivic integration. These arethe right quantities to establish several mirror-symmetry identities for Calabi-Yauvarieties. For more applications and references we refer to the survey papers [11] and[21]. Since than, several other applications to singularity theory were discovered,see e.g. Mustafa [24].In the present paper, we survey our recent work [10] on an extension of thetheory of motivic integration, called arithmetic motivic integration. We developed* Department of Mathematics, University of Leuven, Celestijnenlaan 200 B, 3001 Leuven, Belgium.E-mail: Jan.Denef@wis.kuleuven.ac.bet Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d'Ulm,75230 Paris Cedex 05, France (UMR 8553 du CNRS). E-mail: Francois.Loeser@ens.fr

14 J. Denef F. Loeserthis theory to understand how p-adic integrals of a very general type depend on p.This is used in recent work of Hales [18] on orbital integrals related to the Langlandsprogram. Arithmetic motivic integration is tightly linked to the theory of quantifierelimination, a subject belonging to mathematical logic. The roots of this subjectgo back to Tarski's theorem on projections of semi-algebraic sets and to the workof Ax-Kochen-Ersov and Macintyre on quantifier elimination for Henselian valuedfields (cf. section 4). We will illustrate arithmetic motivic integration startingwith the following concrete application. Let X be an algebraic variety given byequationswith integer coefficients. Denote by N P: „ the cardinality of the imageof the projection X(Z P ) —t X(Z/p n+1 ), where Z p denotes the p-adic integers. Aconjecture of Serre and Oesterlé states that P P (T) := ^N P:n T n is rational. Thisnwas proved in 1983 by Denef [7] using quantifier elimination, expressing P P (T) asa p-adic integral over a domain defined by a formula involving quantifiers. Thisgave no information yet on how P p (T) depends on p. But recently, using arithmeticmotivic integration, we proved:Theorem 1.1. There exists a canonically defined rational power series P(T)over the ring K°*(VarQ) ® Q, such that, for p >• 0, P P (T) is obtained from P(T)by applying to each coefficient of P(T) the operator N p .Here K 0 (VarQ) denotes the Grothendieck ring of algebraic varieties over Q,and K°*(VarQ) is the quotient of this ring obtained by identifying two varietiesif they have the same class in the Grothendieck group of Chow motives (this isexplained in the next section). Moreover the operator N p is induced by associatingto a variety over Q its number of rational points over the field with p elements, forp>0.As explained in section 8 below, this theorem is a special case of a much moregeneral theorem on p-adic integrals. There we will also see how to canonicallyassociate a "virtual motive" to quite general p-adic integrals. A first step in theproof of the above theorem is the construction of a canonical morphism from theGrothendieck ring K 0 (PFFQ) of the theory of pseudo-finite fields of characteristiczero, to K°*(VarQ) ® Q. Pseudo-finite fields play a key role in the work of Ax[1] that leads to quantifier elimination for finite fields [19] [14] [5]. The existenceof this map is interesting in itself, because any generalized Euler characteristic,such as the topological Euler characteristic or the Hodge-Deligne polynomial, canbe evaluated on any element of K°*(VarQ) ® Q, and hence also on any logicalformula in the language of fields (possibly involving quantifiers). All this will beexplained in section 2. In section 3 we state Theorem 3.1, which is a stronger versionof Theorem 1.1 that determines P(T). A proof of Theorem 3.1 is outlined in section7, after giving a survey on arithmetic motivic integration in section 6.2. The Grothendieck group of pseudo-finite fieldsLet k be afield of characteristic zero. We denote by K 0 (Varj;) the Grothendieckring of algebraic varieties over k. This is the group generated by symbols [V] withV an algebraic variety over k, subject to the relations [Vi] = [V2] if Vi is isomorphicto Vii an d [V \ W] = [V] — [W] if W is a Zariski closed subvariety of W. The ring

Motivic Integration 15multiplication on K 0 (Varj;) is induced by the cartesian product of varieties. Let Lbe the class of the affine line over k in K 0 (Varj;). When V is an algebraic varietyoverQ, and p a prime number, we denote by N P (V) the number of rational pointsover the field F p with p elements on a model V of V over Z. This depends on thechoice of a model V, but two different models will yield the same value of N P (V),when p is large enough. This will not cause any abuse later on. For us, an algebraicvariety over k does not need to be irreducible; we mean by it a reduced separatedscheme of finite over k.To any projective nonsingular variety over k one associates its Chow motiveover k (see [27]). This is a purely algebro-geometric construction, which is made insuch a way that any two projective nonsingular varieties, V\ and V2, with isomorphicassociated Chow motives, have the same cohomology for each of the known cohomologytheories (with coefficients in a field of characteristic zero). In particular,when k is Q, N p (Vi) = N p (\~2), for p >• 0. For example two elliptic curves definethe same Chow motive iff there is a surjective morphism from one to the other.We denote by K°*(Varj;) the quotient of the ring K 0 (Varj;) obtained by identifyingany two nonsingular projective varieties over k with equal associated Chowmotives. From work of Gillet and Soulé [15], and Guillen and Navarro Aznar[17], itdirectly follows that there is a unique ring monomorphism from K°*(Varj;) to theGrothendieck ring of the category of Chow motives over k, that maps the class of aprojective nonsingular variety to the class of its associated Chow motive. What isimportant for the applications, is that any generalized Euler characteristic, whichcan be defined in terms of cohomology (with coefficients in a field of characteristiczero), factors through K°*(Varj;). With a generalized Euler characteristic we meanany ring morphism from K 0 (Varj;), for example the topological Euler characteristicand the Hodge-Deligne polynomial when k = C. For [V] in K°*(Varj;), with k =Q, we put iV p ([V]) = N P (V); here again this depends on choices, but two differentchoices yield the same value for iV p ([V]), when p is large enough.With a ring formula ip over k we mean a logical formula build from polynomialequations over k, by taking Boolean combinations and using existential and universalquantifiers. For example, (3x)(x 2 + x + y = 0 and Ay ^ 1) is a ring formulaover Q. The mean purpose of the present section is to associate in a canonical wayto each such formula (p an element Xe(M) of K°*(Varj;) ® Q. One of the requiredproperties of this association is the following, when k = Q: If the formulas ipi and ipiare equivalent when interpreted in F p , for all large enough primes p, then Xe([

16 J. Denef F. Loeserwill require much more, namely that the association ip i—y Xc(M) factors throughthe Grothendieck ring K 0 (PFFj;) of the theory of pseudo-finite fields containing k.This ring is the group generated by symbols [tp], where tp is any ring formula over k,subject to the relations [tpi or ip 2 ] = [ipi] + {^2} — [

Motivic Integration 17linearity, we can hence associate to any Q-central function a on G (i.e. a Q-linearcombination of characters of representations of G over Q), an element Xc(Y,a) ofthat Grothendieck group tensored with Q. Using Emil Artin's Theorem, that any Cicentralfunction a on G is a Q-linear combination of characters induced by trivialrepresentations of cyclic subgroups, one shows that Xc(Y,a) G K°*(Varj;) ® Q.For X := Y/G and C any cyclic subgroup of G, we define XC([-¥ Xc(Y, ct)satisfies the nice compatibility relations stated in Proposition 3.1.2 of loc. cit. Thiscompatibility (together with the above mentioned quantifier elimination) is used,exactly as in loc. cit., to prove that the above definition of XC([

18 J. Denef F. Loeserseries P P (T), for p^>0, is obtained from P(T) by applying the operator N p to eachcoefficient of the numerator and denominator of P(T).In particular we see that the degrees of the numerator and the denominatorof P P (T) remain bounded for p going to infinity. This fact was first proved byMacintyre [23] and Pas [26].4. Quantifier elimination for valuation ringsLet R be a ring and assume it is an integral domain. We will define the notionof a DVR-formula over R. Such a formula can be interpreted in any discretevaluation ring A D R with a distinguished uniformizer n. It can contain variablesthat run over the discrete valuation ring, variables that run over the valuegroup Z, and variables that run over the residue field. A DVR-formula over R isbuild from quantifiers with respect to variables that run over the discrete valuationring, or over the value group, or over the residue field, Boolean combinations,and expressions of the following form: gi(x) = 0, oid(gi(xj) < oid(g2(xj) + L(a),oid(gi(xj) = L(a) mod d, where gi(x) and #2(x) are polynomials over R in severalvariables x running over the discrete valuation ring, where L(a) is a polynomialof degree < 1 over Z in several variables a running over the value group, and dis any positive integer (not a variable). Moreover we also allow expressions of theform tp(~äc(hi(xj), ...,~äc(h r (x)j), where ip is a ring formula over R, to be interpretedin the residue field, hi(x),...,h r (x) are polynomials over R in several variables xrunning over the discrete valuation ring, and äc(w), for any element v of the discretevaluation ring, is the residue of the angular component ac(w) := virôrdv . For thediscrete valuation rings Z p and Ä" [[£]], we take as distinguished uniformizer n theelements p and t.Theorem 4.1 (Quantifier Elimination of Pas [26]). Suppose that R hascharacteristic zero. For any DVR-formula 9 over R there exists a DVR-formulaip over R, which contains no quantifiers running over the valuation ring and noquantifiers running over the value group, such that(1) 9

Motivic Integration 195. Definable subassignements and truncationsLet h : C —¥ Sets be a functor from a category C to the category of sets. Weshall call the data for each object C of C of a subset h'(C) of h(C) a subassignementof h. The point in this definition is that h' is not assumed to be a subfunctor of h.For h' and h" two subassignements of h, we shall denote by h' n h" and h' U h", thesubassignements C H> /i'(C) n h"(C) and C H> /i'(C) U h"(C), respectively.Let k be a field of characteristic zero. We denote by Fields the category of fieldswhich contain k. For X a variety over k, we consider the functor hx '• K >-¥ X(K)from Fields to the category of sets. Here X(K) denotes the set of if-rational pointson X. When X is a subvariety of some affine space, then a subassignement h ofhx is called definable if there exists a ring formula ip over k such that, for any fieldK containing k, the set of tuples that satisfy the interpretation of ip in K, equalsh(K). Moreover we define the class [h] of h in K 0 (PFFj;) as [ip]. More generally,for any algebraic variety X over k, a subassignement h of hx is called definable ifthere exists a finite cover (XJ)J G J of X by affine open subvarieties and definablesubassignements hj of hxi, for i £ I, such that h = Uj G j/i,. The class [h] of h inK 0 (PFFj;) is defined by linearity, reducing to the affine case.For any algebraic variety X over k we denote by hc{x) the functor K H>X(if[[i]]) from Fields to the category of sets. Here X(if[[i]]) denotes the set ofif[[i]]-rational points on X. When X is a subvariety of some affine space, thena subassignement h of hc(x) is called definable if there exists a DVR-formula tpover k such that, for any field K containing k, the set of tuples that satisfy theinterpretation of tp in if[[t]], equals h(K). More generally, for any algebraic varietyX over k, a subassignement h of hc{x) is called definable if there exists a finitecover (Xj)j G j of X by affine open subvarieties and definable subassignements hj ofhc(Xi); f° r i £ I, such that h = Uj G j/i,. A family of definable subassignementsh n , n £ Z, of hc(x) is called a definable family of definable subassignements if oneach affine open of a suitable finite affine covering of X, the family h n is given bya DVR-formula containing n as a free variable running over the value group.Let X be a variety over k. Let h be a definable subassignement of hc{x),and n a natural number. The truncation of h at level n, denoted by n n (h), is thesubassignement of hc n (x) that associates to any field K containing k the imageof h(K) under the natural projection map from X(if[[i]]) to £ n (X)(K). Usingthe Quantifier Elimination Theorem of Pas, we proved that n n (h) is a definablesubassignement of hc„(x), so that we can consider its class [TT„(/I)] in K 0 (PFFj;).Using the notion of truncations, we can now give an alternative (but equivalent)definition of the motivic Poincaré series P(T), which works for any algebraic varietyX over k, namely P(T) := EXe([7Tn(h C (x))])T n .nA definable subassignement h of hc(x) is called weakly stable at level n if forany field K containing k the set h(K) is a union of fibers of the natural projectionmap from X(if[[i]]) to £ n (X)(K). If X is nonsingular, with all its irreduciblecomponents of dimension d, and h is a definable subassignement of hc(x)-, which isweakly stable at level n, then it is easy to verify that[n n (h)]L- nd = [n m (h)]L- md

20 J. Denef F. Loeserfor all m> n. Indeed this follows from the fact that the natural map from £ m (X)to £ n (X) is a locally trivial fibration for the Zariski topology with fiber A^n) ,when X is nonsingular.6. Arithmetic motivic integrationHere we will outline an extension of the theory of motivic integration, calledarithmetic motivic integration. If the base field k is algebraically closed, then itcoincides with the usual motivic integration.We denote by Kg^Var^fL -1 ] the completion of Kg^Var^fL -1 ] with respectto the filtration of K°*(Varj;)[L _1 ] whose ro-th member is the subgroupgenerated by the elements [V']L _ * with i — dimF > m. Thus a sequence [Vi]L _ *converges to zero in K°*(Varj;)[L _1 ], for i H> +00, if * — dimV; H> +00.Definition-Theorem 6.1. Let X be an algebraic variety of dimension d overa field k of characteristic zero, and let h be a definable subassignement of hc(x) •Then the limitp(h) := lim Xc(M/0])L-(" +1)rfexists in K 0 *(Varj;)[L _1 ] ® Q and is called the arithmetic motivic volume of h.We refer to [10, §6] for the proof of the above theorem. If X is nonsingularand h is weakly stable at some level, then the theorem follows directly from whatwe said at the end of the previous section. When X is nonsingular affine, but hgeneral, the theorem is proved by approximating h by definable subassignementshj of hc(x); i € N, which are weakly stable at level n(i). For hj we take thesubassignement obtained from h by adding, in the DVR-formula tp defining h, thecondition ordg(x) < i, for each polynomial g(x) over the valuation ring, that appearsin tp. (Here we assume that tp contains no quantifiers over the valuation ring.) Itremains to show then that Xc([K n (ordg(x) > i)])L^^n+1^dgoes to zero when both iand n >• i go to infinity, but this is easy.Theorem 6.2. Let X be an algebraic variety of dimension d over a field k ofcharacteristic zero, and let h, hi and I12 be definable subassignements of hc(x) •(1) If hi(K) = fi2(K) for any pseudo-finite field K D k, then v(hi) = v(fi2)-(2) v(hi U h 2 ) = v(hi) + v(h 2 ) - v(hi n h 2 )(3) If S is a subvariety of X of dimension < d, and if h a hc(s)> then v(h) = 0.(4) Let h n , n £ N, be a definable family of definable subassignements of hc(x)- Vh n C\h m = 0j for all n ^ m, then J2 v {hn) * s convergent and equals v(\J h n ).nn(5) Change of variables formula. Let p:Y—¥Xbea proper birational morphism ofnonsingular irreducible varieties over k. Assume for any field K containing k thatthe jacobian determinant of p at any point of p^1(h(K))in F(if[[i]]) has t-orderequal to e. Then i>x(fo) = L - Vy(p -1 (^))- Here vx, VY denote the arithmeticmotivic volumes relative to X, Y, andp^1(K)is the subassignement ofhc(y) givenby K^ p-^-^K)) n Y(K[[t]]).Assertion (1) is a direct consequence of the definitions. Assertions (2) and (4)are proved by approximating the subassignements by weakly stable ones. Moreover

Motivic Integration 21for (4) we also need the fact that h n = 0 for all but a finite number of n's, when allthe h n , and their union, are weakly stable (at some level depending on n). Assertion(5) follows from the fact that for n >• e the map £ n (Y) —¥ £ n (X) induced by p isa piecewise trivial fibration with fiber A| over the image in £ n (X) of the points of£(Y) where the jacobian determinant of p has i-order e. See [10] for the details.7. About the proof of Theorem 3.1We give a brief sketch of the proof of Theorem 3.1, in the special case that Xis a hypersurface in A^ with equation f(x) = 0. Actually, here we will only explainwhy the image P(T) of P(T) in the ring of power series over K 0 *(Varj;)[L _1 ] ® Q isrational. The rationality of P(T) requires additional work. Let tp(x, n) be the DVRformula(3y)(f(y) = 0 and ord(a: — y) > n), with d free variables x running overthe discrete valuation ring, and one free variable n running over the value group.That formula determines a definable family of definable subassignements h^_ tH ) ofh c , A dy Since h^_ tH ) is weakly stable at level n, unwinding our definitions yieldsthat the arithmetic motivic volume on h C / A d^ of h v^^ equals J J^( n + 1^d times then-th coefficient of P(T). To prove that P(T) is a rational power series we have toanalyze how the arithmetic motivic volume of h v {-^n) depends on n. To study this,we use Theorem 4.1 (quantifier elimination of Pas) to replace the formula tp(x, n) bya DVR-formula ip(x,n) with no quantifiers running over the valuation ring and noquantifiers over the value group. We take an embedded resolution of singularitiesIT : Y —y Af of the union of the loci of the polynomials over the valuation ring, thatappear in ip(x,n). Thus the pull-backs to Y of these polynomials, and the jacobiandeterminant of n, are locally a monomial times a unit. Thus the pull-back of theformula ip(x, n) is easy to study, at least if one is not scared of complicated formulain residue field variables. The key idea is to calculate the arithmetic motivic volumeof /fy(_ jn ), by expressing it as a sum of arithmetic motivic volumes on hc(Y)-, usingthe change of variables formula in Theorem 6.2. These volumes can be computedexplicitly, and this yields the rationality of P(T).To prove that P(T) specializes to the Serre Poincaré series P P (T) for p >• 0, werepeat the above argument working with Z d instead of £(Af). The p-adic volumeof the subset of Z p defined by the formula tp(x, n) equals p-( n + 1 ) d times the n-th coefficient of P P (T). Because of Theorem 4.1.(2), we can again replace tp(x,n)by the formula ip(x,n) that we obtained already above. That p-adic volume canbe calculated explicitly by pulling it back to the p-adic manifold Y(Z P ), and oneverifies a posteriory that it is obtained by applying the operator N p to the arithmeticmotivic volume that we calculated above. This verification uses the last assertionin Theorem 2.1.8. The general settingWe denote by M the image of Kg^Var^fL -1 ] in Kg^Var^fL -1 ], and byMioc the localization of M ® Q obtained by inverting the elements L* — 1, for all

22 J. Denef F. Loeseri > 1. One verifies that the operator N p can be applied to any element of Mi oc ,for p >• 0, yielding a rational number. The same holds for the Hodge-Delignepolynomial which now belongs to Q(«,w). By the method of section 7, we provedin [10] the followingTheorem 8.1. Let X be an algebraic variety over a field k of characteristiczero, let h be a definable subassignement of hc(x)tan d h n a definable family ofdefinable subassignements of hc(x) •(1) The motivic volume v(h) is contained in Mi oc -(2) The power series ^v(h n )T n £ -Mi 0C [[T]] is rational, with denominator a prodnuct of factors of the form 1 — L _a T 6 , with a, b £ N, 6 ^ 0.Let X be a reduced separable scheme of finite type over Z, and let A = (.4 P ) P>0be a definable family of subsets of X(Z P ), meaning that on each affine open, of asuitable finite affine covering of X, A p can be described by a DVR-formula overZ. (Here p runs over all large enough primes.) To A we associate in a canonicalway, its motivic volume V(ìIA) £ Mioc, in the following way: Let h f \ be a definablesubassignement of /i£(x®Q)> given by DVR-formulas that define A. Because theseformulas are not canonical, the subassignement h f \ is not canonical. But by theAx-Kochen-Ersov Principle (see 4.2), the set ìIA(K) is canonical for each pseudofinitefield K containing Q. Hence V(ìIA) £ Mioc is canonical, by Theorem 6.2.(1).By the method of section 7, we proved in [10] the following comparison result:Theorem 8.2. With the above notation, for all large enough primes p,N P (V(ìIA)) equals the measure of A p with respect to the canonical measure on X(Z P ).When X ® Q is nonsingular and of dimension d, the canonical measure onX(Z P ) is defined by requiring that each fiber of the map X(Z P ) —t X(Z p /p m ) hasmeasure p^md whenever m >• 0. For the definition of the canonical measure in thegeneral case, we refer to [25].The above theorem easily generalizes to integrals instead of measures, but thisyields little more because quite general p-adic integrals (such as the orbital integralsappearing in the Langlands program) can be written as measures of the definablesets we consider. For example the p-adic integral J \f(x)\dx on Z p equals the p-adicmeasure of {(x, t) £ Z d+1 : oid p (f(xj) < ord p (t)}.References[1] J. Ax, The elementary theory of finite fields, Ann. of Math, 88 (1968), 239-271.[2] S. del Bario Rollin, V. Navarro Aznar, On the motive of a quotient variety,Collect. Math., 49 (1998), 203-226.[3] V. Batyrev, Stringy Hodge numbers of varieties with Gorenstein canonicalsingularities, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997),1-32, World Sci. Publishing, River Edge, NJ, 1998.[4] V, Batyrev, Non-Archimedean integrals and stringy Euler numbers of logterminalpairs, J. Eur. Math. Soc. (JEMS), 1 (1999), 5-33.[5] Z. Chatzidakis, L. van den Dries, A. Macintyre, Definable sets over finite fields,J. Reine Angew. Math., 427 (1992), 107-135.

Motivic Integration 23[6] F. Delon, Quelques propriétés des corps values, Thèse d' État, Université ParisVII (1981).[7] J. Denef, The rationality of the Poincaré series associated to the p-adic pointson a variety, Invent. Math., 77 (1984), 1-23.[8] J. Denef, F. Loeser, Germs of arcs on singular algebraic varieties and motivicintegration, Invent. Math., 135 (1999), 201-232.[9] J. Denef, F. Loeser, Motivic exponential integrals and a motivic Thom-Sebastiani theorem, Duke Math. J., 99 (1999), no. 2, 285-309.[10] J. Denef, F. Loeser, Definable sets, motives and p-adic integrals, J. Amer.Math. Soc, 14 (2001), 429-469.[11] J. Denef, F. Loeser, Geometry on arc spaces of algebraic varieties, Proceedingsof the Third European Congress of Mathematics, Volume 1, Progress inMathematics 201, Birkhauser 2001, ISBN 3-7643-6417-3.[12] J. Denef, F. Loeser, Motivic integration, quotient singularities and the McKaycorrespondence,Compositio Math., 131 (2002), 267-290.[13] M. Fried, M. Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrerGrenzgebiete (3), Springer-Verlag, Berlin, 1986. ISBN: 3-540-16640-8.[14] M. Fried, G. Sacerdote, Solving diophantine problems over all residue classfields of a number field and all finite fields, Ann. Math., 100 (1976), 203-233.[15] H. Gillet, C. Soulé, Descent, motives and if-theory, J. Reine Angew. Math.,478 (1996), 127-176.[16] M. Greenberg, Rational points in Henselian discrete valuation rings, Inst.Hautes Études Sci. Pubi. Math., 31 (1966), 59-64.[17] F. Guillen, V. Navarro Aznar, Un critère d'extension d'un foncteur défini surles schémas lisses, preprint (1995), revised (1996).[18] T.C. Hales, Can p-adic integrals be computed? (to appear).[19] C. Kiefe, Sets definable over finite fields: their zeta-functions, Trans. Amer.Math. Soc, 223 (1976), 45-59.[20] M. Kontsevich, Lecture at Orsay (December 7, 1995).[21] E. Looijenga, Motivic measures, Séminaire Bourbaki, Vol. 1999/2000,Astérisque 276 (2002), 267-297.[22] A. Macintyre, Twenty years of p-adic model theory, Logic colloquium '84, 121-153, Stud. Logic Found. Math., 120, North-Holland, Amsterdam, 1986.[23] A. Macintyre, Rationality of p-adic Poincaré series: Uniformity in p, Ann. PureAppi. Logic, 49 (1990), 31-74.[24] M. Mustafa, Jet schemes of locally complete intersection canonical singularities,Invent. Math., 145 (2001), 397-424.[25] J. Oesterlé, Réduction modulo p n des sous-ensembles analytiques fermés deZ*, Invent. Math., 66 (1982), 325-341.[26] J. Pas, Uniform p-adic cell decomposition and local zeta functions, J. ReineAngew. Math., 399 (1989) 137-172.[27] A.J. Scholl, Classical motives, Motives, Seattle, WA, 1991, 163-187, Proc.Sympos. Pure Math., 55, Part 1, Amer. Math. Soc, Providence, RI, 1994.

ICM 2002 • Vol. II • 25-33Automorphism Groups of SaturatedStructures; A ReviewD. Lascar*AbstractWe will review the main results concerning the automorphism groups ofsaturated structures which were obtained during the two last decades. Themain themes are: the small index property in the countable and uncountablecases; the possibility of recovering a structure or a significant part of it fromits automorphism group; the subgroup of strong automorphisms.2000 Mathematics Subject Classification: 03C50, 20B27.Keywords and Phrases: Automorphism groups, Small index property,Strong automorphisms.1. IntroductionSaturated models play an important role in model theory. In fact, when studyingthe model theory of a complete theory T, one may work in a large saturatedmodel of T with its definable sets, and forget everything else about T. This large saturatedstructure is sometimes called the "universal domain", sometimes the "monstermodel".A significant work has been done the last twenty years on the automorphismgroups of saturated models. It is this work that I want to review here. There isa central question that I will use as a "main theme" to organize the paper: whatinformation about M and its theory are contained in its group of automorphisms?In the best case, M itself is "encoded" in some way in this group; recovering M fromit is known as "the reconstruction problem". A possible answer to this problem is atheorem of the form: If Mi and M 2 are structures in a given class with isomorphicautomorphism groups, then Mi and M 2 are isomorphic.Throughout this paper, T is supposed to be a countable complete theory. Thecountability of T is by no means an essential hypothesis. Its purpose is only tomake the exposition smoother, and most of the results generalize without difficulty*CNRS, Université Denis Diderot Paris 7, 2 Place Jussieu, UFR de mathématiques, case 7012,75521 Paris Cedex 05, France. E-mail: lascar@logique.jussieu.fr

26 D. Lascarto uncountable theories. We will denote by Aut(M) the group of automorphisms ofthe structure M, and if A is a subset of M, AUìA(M) will be the pointwise stabilizerof A:Aut A (M) = {f £ Aut(M) ; Va G A f(a) = a} .When we say "definable", we mean "definable without parameters".2. The countable caseAs a preliminary remark, let us say that the automorphism group of a saturatedmodel is always very rich: if M has cardinality A, then its automorphism group hascardinality 2 A .I do not know who was the first to introduce the small index property. As wewill see, it is crucial in the subject.Definition 1 Let M be a countable structure. We say that M (or Aut(M)) hasthe small index property if for any subgroup H of Aut(M) of index less than 2 N °,there exists a finite set A c M such that AUìA(M) C H.Remark that the converse is true: any subgroup containing a subgroup of theform AUìA(M) where A is finite, has a countable index in Aut(M). Moreover,the subgroups containing a subgroup of the form AUìA(M) are precisely the openneighborhoods of the identity for the pointwise convergence topology. In otherwords, the small index property allows us to recover the topological structure ofAut(M) from its pure group structure.The small index property has been proved for a number of countable saturatedstructures:1. The infinite set without additional structure [23], [5].2. The linear densely ordered sets [25].3. The vector spaces over a finite field [6].4. The random graph [10].5. Various other classes of graphs [9].6. Generic relational structures [8].7. w-categorical w-stable structures [10].The small index property has also been proved for some countable structureswhich are not saturated: for the free group with w-generators ([2]), for arithmeticallysaturatedmodels of arithmetic ([17]).There are examples of countable saturated structures which fail to have thesmall index property. The simplest may be an algebraically closed field of characteristic0 of infinite countable transcendence degree: Let Q be the algebraic closureof the field of rational numbers. There is an obvious homomorphism tp from Aut(M)onto Aut(Q) (the restriction map). Now, it is well known that there is a subgroupH of Aut(Q) of countable index (in fact of finite index) which is not closed for theKrull topology, which is nothing else that the pointwise convergence topology. Then

Automorphism Groups of Saturated Structures 27As we will see later, the small index property is particularly relevant for OJcategoricalstructures. Evans and Hewitt have produced an example of such a structurewithout the small index property ([7]).With the pointwise convergence topology, Aut(M) is a topological polishgroup. So, we may use the powerful tools of descriptive set theory. In many cases(for example for structures 1-6 above), it can be shown that there is a (necessarilyunique)conjugacy class which is generic, that is, is the countable intersection ofdense open subsets. The elements of this class are called generic automorphisms,and they play an important role in the proof of the small index property.Another possible nice property of these automorphism groups which is sometimesobtained as a bonus of the proof of the small index property, is the fact thatits cofinality is not countable, that is, Aut(M) is not the union of a countable chainof proper subgroups. This is proved in particular for the full permutation group of acountable set ([21]), for the random graph and for w-categorical w-stable structures([IO])-I would like to mention here the work of Rubin ([24]). He has shown how toreconstruct a certain number of structures from their automorphism group using asomewhat different method. His methods apply essentially to "combinatorial structures"such as the random graph, the universal homogeneous poset, the generictournament (a structure for which the small index property is not known), etc.3. Subgroups and imaginary elementsRecall that an imaginary element of M is a class of a tuple of M n modulo adefinable equivalence relation on M n . For instance, if G is a group and H a definablesubgroup of G n , then any coset of H in G n is an imaginary element. When we addall these imaginary elements to a saturated structure M, we obtain the structureM eq , and we can consider M eq as a saturated structure (in a larger language).It is clear that M and M eq have canonically the same automorphism group:every automorphism of M extends uniquely to an automorphism of M eq . Thisshows a limitation to the reconstruction problem: If M and N are two structureswhich are such thatu M eq and N eq are isomorphic", then Aut(M) and Aut(N)are isomorphic via a bicontinuous isomorphism. The condition "M eq and N eq areisomorphic" may seem weird, but in fact, it is natural. Roughly speaking, it meansthat M can be interpreted in N, and conversely (a little more in fact, see [1] formore details). In this case, we say that M and N are bi-interpretable.Consider now the case of an w-categorical structure M. It is not difficult tosee that any open subgroup of Aut(M) is the stabilizer Aut a (M) of an imaginaryelementa. Moreover, Aut(M) acts by conjugation on the set of its open subgroups,and this action is (almost) isomorphic to the action of Aut(M) on M eq (almostbecause two different imaginary elements a and ß may have the same stabilizer).So, from the topological group Aut(M) we can (almost) reconstruct its action onM eq . We can do better:Theorem 2 [1] Assume that M and N are countable u-categorical structures.the following two conditions are equivalent:Then

28 D. Lascar1. there is a bicontinuous isomorphism from Aut(M) onto Aut(N);2. M and N are bi-interpretable.In fact, these conditions are also equivalent to: there exists a continuous isomorphismfrom Aut(M) onto Aut(N) (see [15]). Thus, if one of the structure Mor N has the small index property and Aut(M) is isomorphic to Aut(N) (as puregroups), then M and N are bi-interpretable.Now, if M is not necessarily w-categorical (but still saturated), the situationis a bit more complicated. We need to introduce new elements.Definition 3 1. An ultra-imaginary element of M is a class modulo E, where Eis an equivalence relation on M n (n < u) which is invariant under the action ofAut(M). An ultra-imaginary element is finitary if n < u.2. A hyperimaginary element of M is a class modulo E, where E is an equivalencerelation on M n (n < u) which is defined by a (possibly infinite) conjunction of firstorder formulas.An imaginary element is hyperimaginary, and a hyperimaginary element isultra-imaginary. A hyperimaginary element is a class modulo an equivalence relationE defined by a formula of the formwhere the tpi are first-order formulas (without parameters) and whose free variablesare among the xu for k < n. An ultra-imaginary element is a class modulo anequivalence relation E defined by a formula of the formVA*'«jeJ teiwhere the tpij are first order-formulas (without parameters) and whose free variablesare among the xu for k < n.If M is a countable saturated structure, the stabilizer of a finitary ultraimaginaryelement is clearly an open subgroup, and it is not difficult to see thatif H an open subgroup of Aut(M), then there exists a finitary ultra-imaginary elementa such that H is the stabilizer of a. In the w-categorical case, any finitaryultra-imaginary is in fact imaginary, and this explain why this case is so simple.In some cases, for example for w-stable theories (see [18]), it is possible tocharacterize, among all open subgroups, those which are of the form Aut a (M)with a imaginary. Something similar has been done for countable arithmeticallysaturatedmodels of arithmetic in [11], and in [13], it is proved that if two suchmodels have isomorphic automorphism groups, then they are isomorphic.4. Strong automorphismsIt is now time to introduce the group of strong automorphisms.

Automorphism Groups of Saturated Structures 29Definition 4 [14] The group of strong automorphisms of M is the group generatedby the setand is denotedAutf(M).\J{Aut N (M);N

30 D. LascarThere is a natural topology on Gal(T) (see [19] for details). It can be definedin two different ways.My favorite one is via the ultraproduct construction. Let (7, ; i £ I) be afamily of elements of Gal(T) and U an ultrafilter on J. Choose a saturated modelM and, for each i £ L an automorphism fi £ Aut(M) lifting 7$. Consider theultrapower M' = Yl i€U M. We can define the automorphism Y\ i€U fi on M'. Thisautomorphism acts on Bdd(M') = Bdd(T), so defines an element of Gal(T), say ß.This element ß should be considered as a limit of the family (7$ ; i £ I) along U.A subset X of Gal(T) is closed for the topology we are defining if it is closed forthis limit operation. You should be aware that the element ß may depend on thechoices of the /,'s, because the topology we are defining is not Hausdorff in general.The other way to define a topological structure on Gal(T) is to define a topologyon Bdd(T). If, as it is the case when T is stable, Gal(T) can be identified with agroup of permutation on the set of imaginary elements, then we just endow Gal(T)with the pointwise convergence topology (that is we consider the set of imaginaryelementswith the discrete topology). Otherwise, it is more complicated, and hereis what should be done in general:For each n ofM n /E is closed if and only if tp^(X) is the intersection of a family of subsetsdefinable with parameters. Gal(T) acts on M n /E and the topology on Gal(T) isdefined as the coarsest topology which makes all these actions (with various n andE) continuous.Now, we can prove:Theorem 5 1. Gal(T) is a topological compact group.2. It is Hausdorff if and only if it acts faithfully on the set of bounded hyperimaginaryelements, if and only if it acts faithfully on the set of finitary boundedhyperimaginary elements.3. It is profinite if and only if it acts faithfully on the set of bounded imaginaryelements.There is a Galois correspondence between the subgroups of Gal(T) and thebounded ultra-imaginary elements: every subgroup of Gal(T) is the stabilizer ofan ultra-imaginary element. The hyperimaginary elements correspond to the closedsubgroups and the imaginary elements correspond to the clopen subgroups of Gal (T).Let H 0 be the topological closure of the identity. Then H 0 is a normal subgroupof Gal(T). If we consider Gal(T) as a permutation group on Bdd(T), H 0is exactly the pointwise stabilizer of the set of bounded hyperimaginary elements.So, if we set GOIQ(T) = Gal(T)/H 0 , GOIQ(T) acts faithfully on the set of boundedhyperimaginary elements. As a quotient group, GOIQ(T) is canonically endowedwith a topology. This way, we get a compact Hausdorff group.Recently, L. Newelski ([22]) has proved that H 0 is either trivial or of cardinality2 N °.I would like to conclude this section by a conjecture. In all the known examplesof countable saturated structures where the small index property is false, there is a

Automorphism Groups of Saturated Structures 31non open subgroup of G al (T) of countable index (and, its preimage by the canonicalhomomorphism from Aut(M) onto Gal(T) is a non open subgroup of Aut(M) ofcountable index). If A has a cardinality strictly less than card(M), define AUì/A(M)as the subgroup of AUìA(M) generated by\J{Aut N (M) ; AÇN - K 0 . The smallindex property has a natural generalization. If we assume that A

32 D. Lascar1. Let M and M' be two saturated models of the same theory. Then we canreconstruct (Aut(M'),T a ) from (Aut(M),T a ).2. Let M and M' be two models of the same theory, and assume card(M) = A

Automorphism Groups of Saturated Structures 33saturated model of Peano arithmetic, J. London Math. Soc. (2) 52 (1995), no.2, 235-244.[14] D. Lascar, On the category of models of a complete theory, J. Symbolic Logic47 (1982), 249-266.[15] D. Lascar, Autour de la propriété du petit indice, Proc. London Math. Soc. (3)62 no.l (1991), 25-53.[16] D. Lascar, Les automorphismes d'un ensemble fortement minimal J. SymbolicLogic 57 no. 1 (1992) , 238-251.[17] D. Lascar, The small index property and recursively saturated models of Peanoarithmetic, Automorphisms of first-order structures, Oxford Sci. Pubi., OxfordUniv. Press, New York, 1994, 281-292.[18] D. Lascar, Recovering the action of an automorphism group, Logic: from Foundationsto Application, European logic Colloquium, edited by W. Hodges andall, Clarendon Press (1996), 313-326.[19] D. Lascar & A. Pillay, Hyperimaginaries and automorphism groups, J. SymbolicLogic 66, no. 1 (2001), 127-143.[20] D. Lascar & S. Shelah, Uncountable saturated structures have the small indexproperty, Bull. London Math. Soc. 25, no. 2 (1993), 125—131.[21] H. D. Macpherson & P. M. Neumann, Subgroups of infinite symmetric groups,J. London Math. Soc. (2) 42 (1990), no. 1, 64-84.[22] L. Newelski, The diameter of a Lascar strong type, preprint, (2002).[23] S. W. Semmes, Endomorphisms of infinite symmetric groups, Abstracts of theAm. Math. Soc. 2 (1981), 426.[24] M. Rubin, On the reconstruction of H 0 -categorical structures from their automorphismgroups, Proc. London Math. Soc. (3) 69 no. 2 (1994), 225-249.[25] J. K. Truss, Infinite permutation group; subgroups of small index, J. Algebra120 (1989), 494-515.

Section 2. AlgebraS. Bigelow: Representations of Braid Groups 37A. Bondal, D. Orlov: Derived Categories of Coherent Sheaves 47M. Levine: Algebraic Cobordism 57Cheryl E. Praeger: Permutation Groups and Normal Subgroups 67Markus Rost: Norm Varieties and Algebraic Cobordism 77Z. Sela: Diophantine Geometry over Groups and the Elementary Theoryof Free and Hyperbolic Groups 87J. T. Stafford: Noncommutative Projective Geometry 93Dimitri Tamarkin: Deformations of Chiral Algebras 105

ICM 2002 • Vol. II • 37-45Representations of Braid GroupsS. Bigelow*AbstractIn this paper we survey some work on representations of B n given by theinduced action on a homology module of some space. One of these, called theLawrence-Krammer representation, recently came to prominence when it wasshown to be faithful for all n. We will outline the methods used, applying themto a closely related representation for which the proof is slightly easier. Themain tool is the Blanchfield pairing, a sesquilinear pairing between elementsof relative homology. We discuss two other applications of the Blanchfieldpairing, namely a proof that the Burau representation is not faithful for largen, and a homological definition of the Jones polynomial. Finally, we discusspossible applications to the representation theory of the Hecke algebra, andultimately of the symmetric group over fields of non-zero characteristic.2000 Mathematics Subject Classification: 20F36, 20C08.Keywords and Phrases: Braid groups, Configuration spaces, Homologicalrepresentations, Blanchfield pairing.1. IntroductionArtin's braid group B n was originally defined as a group of geometric braidsin R 3 . Representations of B n have been studied for their own intrinsic interest, andalso in connection to other areas of mathematics, most notably to knot invariantssuch as the Jones polynomial.We will use the definition of B n as the mapping class group of an n-timespunctured disk D n . A rich source of representations of B n is the induced action onhomology modules of spaces related to D n . The Burau representation, one of thesimplest and best known representations of braid groups, is most naturally definedas the induced action of B n on the first homology of a cyclic covering space of D n .Lawrence [9] extended this idea to configuration spaces in D n , and was able toobtain all of the so-called Temperley-Lieb representations.* Department of Mathematics & Statistics, University of Melbourne, Victoria 3010, Australia.E-mail: bigelow@unimelb.edu.au

38 S. BigelowLawrence's work seems to have received very little attention until one of her homologicalrepresentations was shown to be faithful, thus proving that braid groupsare linear and solving a longstanding open problem. Two independent and verydifferentproofs of this have appeared in [1] and [8]. In this paper we will outlinethe former, emphasising the importance of the Blanchfield pairing. We then discusstwo other applications of the Blanchfield pairing, namely the proof that theBurau representation is not faithful for large n, and a homological definition ofthe Jones polynomial of a knot. We conclude with some speculation on possiblefuture applications to representations of Hecke algebras when q is a root of unity.These are related to representations of the symmetric group S n over fields of badcharacteristic, that is, fields in which n\ = 0.2. The Lawrenee-Krammer representationLet D be the unit disk centred at the origin in the complex plane. Fix arbitraryrealnumbers —l

3. The Blanchfield pairingRepresentations of Braid Groups 39Let e > 0 be small. We define P, B c C as follows. Suppose {x, y} is a pointin C. We say {x, y} e F if either |x — y| < e, or there is a puncture point pi suchthat \x — Pì\ < e or \y — p t \ < e. We say {x,y} £ B if x £ dD or y £ dD.For u £ H2(C,P) and v £ H2(C,B) let (u • v) £ Z denote the standardalgebraic intersection number. We define an intersection pairing(•,•): H 2 (C,P) xH 2 (C,É)^Z[q ±1 ,t ±1 ]by(u,v) = 2_. ( u '(fVvitft 3 •For a proof that these are well-defined, see [7, Appendix E], where the followingproperties are also proved.For u £ H2(C,P), v £ H 2 (C,B), a £ B n , and A G Z[q ±x ,t ±x ], we haveand(au,av) = (u,v),(Xu,v) = X(u,v) = (u,Xv),where A is the image of A under the automorphism of Z[q ±1 ,t ±x ] taking q to q^1and t to £ _1 .4. A faithful representationThe aim of this section is to outline a proof of the following.Theorem. Let C and P be as above. The induced action of B n on H2(C,P)is faithful.For the details, the reader is referred to [1], where the same techniques areused to prove that B n acts faithfully on H 2 (C). Our use of relative homology hereactually simplifies the argument somewhat.There is a slight technical difficulty in defining the action of B n on H 2 (C, P).Namely, the action of a braid on C need not preserve the set P. Thus we shouldreally take a limit as e approaches 0. The representation obtained is very similarto the Lawrenee-Krammer representation, but has a slightly different modulestructure, as discussed in [3].Let E be the straight edge from pi to P2 • Let E' be the set of points in C ofthe form {x,y}, where x,y £ E. Let E' be a lift of E' to C. This represents anelement of H 2 (C, P), which we will call e. Let Fi and F 2 be parallel vertical edgeswith endpoints on dD, passing between p 2 and p$. Let F' be the set of points inC of the form {x,y}, where x £ Fi and y £ F 2 . Let F' be a lift of F' to C. Thisrepresents an element of H 2 (C, B), which we will call /. Note that(e,/)=0,

40 S. Bigelowsince E' and F' are disjoint in C.Suppose the action of B n on H2(C,P) is not faithful. It is not hard to showthat there must be a braid a in the kernel of this representation such that a(E) isnot isotopie to E relative to endpoints. Now a(e) = e, so(a(e),f)=0.From this, we will derive a contradiction.By applying an isotopy, we can assume a(E) intersects Fi and F 2 transverselyat a minimal number of points k > 0. Let x\,...,xu be the points of a(E) n Fi,and let J/1,..., y* be the points of a(E) n F 2 , numbered from top to bottom in bothcases.For i,j £ {l,...,k}, let a,j and bjj be the unique integers such that a(E')intersects q ai -Jt bi -'F' at a point in the fibre over {xj,yj}, and let e,j be the sign ofthat intersection. Thenkkj=li=lTo calculate a,j and bjj, it is necessary to specify choices of lift for E' andF'. We will not do this since we only need to calculate differences a,/j/ — a,j andbjiji — bjj. To do this, let 7 be a path in C that goes from {xj,yj} to {ar,/,j/j/} inCT(F'), and then back to {#», t/j} in F'. Thenç(Oi'.j'-Oi.3)f(6i'.j'-6i.3)= $( bj : i then aiji > a,,, for some j' = 1,..., k,• if bjj > bjj then a,/j > a,jj for some i' = 1,..., k.The first of these is [1, Lemma 2.1], and the second and third follow from theproof of [1, Claim 3.4]. We now sketch the proof of the second in the special casewhere y, lies between x» and %jj along a(E).Let a be the path from y t to %jj along a(E). Let ß be the path from %jj to%ji along F 2 . Then bjj — 6,,, is two times the winding number of aß around x». Inparticular, this winding number is positive.Let D\ be the once punctured disk D \ {#»}, and let D\ be its universal cover.Let äß be a lift of aß to D\. This is a path from a point in the fibre over y t to a"higher" point in the fibre over y t .Let F2 be the the segment of F 2 going from y t upwards to dD. Let F^ be thelift of F2 to D\ that has an endpoint at ä(0). In order to reach a higher sheet inTJ>i, à must intersect F^ • Let 7 be the loop in D\ that follows à to the first pointof intersection with F^ , and then follows F^ back to â(0).Let 7 be the projection of 7 to D\. This travels along a(E) from y t to somepoint yji £ F 2 , then along F 2 back to j/,. Then ajji — a,,, is the total winding

Representations of Braid Groups 41number of 7 around the puncture points. We must show that this winding numberis positive.By construction, 7 is a simple closed curve in D\. By the Jordan curve theorem,it must bound a disk B. Let B be the projection of B to D\. This is animmersed disk in D, whose boundary is 7. Note that 7 passes anticlockwise aroundB, since the puncture x» lies to its right. Thus the total winding number of 7 aroundthe puncture points is equal to the total number of puncture points contained in B,counted with multiplicities.It remains to show that B intersects at least one puncture point. Supposenot. Then B is an immersed disk in D n . Using an "innermost disk" argument, onecan find an embedded disk B' in D n whose boundary consists of a subarc of a(E)and a subarc of F 2 . Using B', one can isotope a(E) so as to have fewer points ofintersection with F 2 , thus contradicting our assumptions.This completes the proof of the second part of the lemma in the case where%ji lies between x» and %jj along a(E). The remaining case, where x» lies between%ji and t/j, is only slightly trickier. The third part of the lemma is similar to thesecond. The first part of the lemma is much easier.We now return to the proof of the theorem. Let a be the maximum of allüij. Let 6 be the maximum of {bjj : a,j = a}. Suppose i, j £ {1,... ,k} are suchthat aij = a and bjj = b, and also i',j' £ {1,... ,k} are such that a,/j/ = a andbjiji = 6. I claim that e,j = e,/j/. From this claim, it follows that all occurrencesof q a t b in the expressionkk»=i i=ioccur with the same sign, so the coefficient of q a t b is non-zero in (a(e),f). Thisprovides our desired contradiction, and completes the proof of the theorem. Itremains to prove that e,/j/ = e,j.Using the above lemma, it is not hard to show that a,,, = a,jj = a andbi,i = bjj = b. Similarly, a,/,,/ = ay ji = a and 6,/,,/ = by ^y = b. We will only needthe equalitiesVi :i — Vi J — VjJ — Vi' ,£' — u i'J'— Oj' J' •In fact, we only need these modulo two.Orient a(E) so that it crosses Fi from left to right at x». Let 7 be the path inC which goes from {x,, j/,} to {x,/,j/,/} in E' and then back to {x,,t/j} in F'. Nowbi',i' — bj : i is the exponent of t in $(7). The fact that this is an even number meansthat the pair of points in D n do not "switch places" when they go around this loop.Thus a(E) crosses Fi from left to right at x\. By similar arguments,• ß(E) intersects Fi with the same sign at x, and x,/,• ß(E) intersects F 2 with the same sign at %jj and %jy,• x, occurs before %jj and x,/ occurs before %jy with respect to the orientationoîa(E).It is now intuitively clear that E' must intersect F' with the same signs at {x,,t/j}and {xji, yj'}. This can be proved rigorously by careful consideration of orientations

42 S. Bigelowof these surfaces, as discussed in [1, Section 2.1]. It follows that e,j = e,/j/, whichcompletes the proof of the theorem.5. The Burau representationThe proof that the Lawrenee-Krammer representation is faithful basically reducesto proving that the Blanchfield pairing detects whether corresponding edgesin the disk can be isotoped to be disjoint. A converse to this idea leads to a proofthat the Burau representation is not faithful for large n.The Burau representation can be defined by a similar but simpler constructionto that of the Lawrenee-Krammer representation. Let#: Tii(D n )^(q)be the homomorphism that sends each of the obvious generators to q. Let D n bethe corresponding covering space. The Burau representation is the induced actionof B n on Hi(D n ) by Z[g ±1 ]-module automorphisms.Let F be an e-neighbourhood of the puncture points, and let F be the preimageof F in D n . The Blanchfield pairing in this context is a sesquilinear pairing(•,•): Hi(D n ,P) x Hi(D n ,dD n ) ^ Z^1].Let E be the straight edge from pi to p2- Let F be a lift of F to D n . Thisrepresents an element of Hi(D n ,P), which we will call e. Let F be a vertical edgewith endpoints on dD, passing between p n -i and p n . Let F be a lift of F to D n .This represents an element of Hi(D n ,dD n ), which we will call /. The following is[2, Theorem 5.1].Theorem. Let E, e, F and f be as above. The Burau representation of B nis unfaithful if and only if there exists a £ B n such that (a(e),f) = 0, but a(E) isnot isotopie relative to endpoints to an edge that is disjoint from F.Using this theorem, one can show that the Burau representation of B n is notfaithful simply by exhibiting the required edges a(E) and F. Such edges can befound by hand in the case n = 6. In the case n = 5, they can be found by acomputer search, and then laboriously checked by hand. The case n = 4 seems tobe beyond the reach of any known computer algorithm. This is the last open case,since the Burau representation is known to be faithful for n < 3.6. The Jones polynomialIn this section, we use the Blanchfield pairing to give a homological definitionof the Jones polynomial of a knot or link. The Jones polynomial was defined in [6]using certain algebraically defined representations of braid groups. No satisfactorygeometricdefinition is known, but some insight might be offered by defining the representationshomologically and using the Blanchfield pairing. This was the originalmotivation for Lawrence to study homological representations of braid groups.

Representations of Braid Groups 43A geometric braid a £ B n is a collection of n disjoint edges inCxR withendpoints {1,... ,n} x {0,1}, such that each edge goes from C x {0} to C x {1}with a constantly increasing R component. The correspondence between geometricbraids and elements of the mapping class group is described in [1], and in manyotherintroductory expositions on braids. The plat closure of a geometric braid

44 S. BigelowWe conclude with some speculation about possible applications of the Blanchfieldpairing to the representation theory of Hecke algebras. We first give a verybriefoverview of the basic theory of Hecke algebras.Let q £ C \ {0}. The Hecke algebra H n (q), or simply H n , is the C-algebragiven by generators gi, • • • ,g n -i and relations• Qigj = g,gi if \i-j\ > i,if• gtgj gì = 9ì9J9ì \i-j\ = 1 ,• (9i-l)(9i + q) = 0.It is an n!-dimensional C-algebra. We are restricting to the ring C for convenience,although other rings can be used.Note that H n (l) is the group algebra CS n of the symmetric group S n . TheHecke algebra is called a "quantum deformation" of CS n . The representation theoryof CS n is well understood except when working over a field of finite characteristicin which n! = 0. This is because the classical theory sometimes requires one todivide by n!, the order of the group. When studying H n it turns out to be usefulto be able to divide by(1 + q + h q n^)(l + q+ h q n^2) ...(1 + q).This is sometimes written as [n]!, and can be thought of as a "quantum deformation"of n\. Note that [n]! = n! if q = 1. A generic value of q is one for which [n]! ^ 0.The non-generic values are the primitive fcth roots of unity for k = 2,..., n. Therepresentation theory of H n is well understood for generic values of q, but thenon-generic values are the subject of ongoing active research.One of the most important papers on this subject is Dipper and James [5].For every partition A of n, they define a F%-module S x called the Specht module.They then define a bilinear pairing on S x , which we denote (•, -)DJ- Let S x denotethe set of u £ S x such that («,W)DJ = 0 for all v £ S x . Let D x be the quotientmodule S x /S x . Dipper and James show that the non-zero D x form a complete listof all distinct irreducible representations of H n . For generic values of q we haveD x = S x . For non-generic values of q, the D x are not well understood.Lawrence [10] gave a homological definition of the Specht modules. The constructionbegins with the action of B n on a homology module of a configurationspace. The variable t is then specialised to —q^1,and a certain quotient module istaken. A detailed treatment of the case A = (n — 2,2) is given in [3].There is a Blanchfield pairing on the Specht modules as defined by Lawrence.It would be nice if this were the same as the pairing defined by Dipper and James.Unfortunately the Blanchfield pairing is sesquilinear, whereas the pairing definedby Dipper and James is bilinear. This problem can be overcome as follows. Letp: D n —t D n be the conjugation map. Let p be the induced map on Hf.(C,B).Then the pairing(u,v)' = (u,p(v))can be shown to give a bilinear pairing on the Specht module.For generic values of q, this topologically defined pairing is the same as thealgebraically defined pairing of Dipper and James, up to some renormalisation.

Representations of Braid Groups 45There is some evidence that this can be made to work at non-generic values of q. Ifso, it would give rise to a new homological definition of the modules D x , and newtopological tools for studying them. In any case, it would be interesting to betterunderstand the behaviour of this Blanchfield pairing at roots of unity.References[i[2:[3;[4;[5;[6;[7;[9[10:[11S. Bigelow, Braid groups are linear, J. Amer. Math. Soc, 14 (2001), 471^486.S. Bigelow, Does the Jones polynomial detect the unknot? J. Knot TheoryRamifications, (to appear).S. Bigelow, The Lawrenee-Krammer representation, Proceedings, GeorgiaTopology Conference, 2001, (to appear).S. Bigelow, A homological definition of the Jones polynomial, Proceedings,RIMS, Kyoto, 2001, (to appear).R. Dipper & G. James, Representations of Hecke algebras of general lineargroups, Proc London Math. Soc. (3), 52 (1986), 20^52.V. Jones, A polynomial invariant for knots via von Neumann algebras, Bull.Amer. Math. Soc. (N.S.), 12 (1985), 103-111.A. Kawauchi, A survey of knot theory, Birkhäuser Verlag, 1996.D. Krammer, Braid groups are linear, Ann. of Math. (2), 155 (2002), 131-156.R. Lawrence, Homological representations of the Hecke algebra, Comm. Math.Phys., 135 (1990), 141-191.R. Lawrence, Braid group representations associated with sl m , J. Knot TheoryRamifications, 5 (1996), 637^660.L. Paoluzzi & L. Paris, A note on the Lawrence-Krammer-Bigelow representation,Algebr. Geom. Topol, 2 (2002), 499^518.

ICM 2002 • Vol. II • 47^56Derived Categories of Coherent SheavesA. Bondal* D. Orlov 1ÂbstractWe show how derived categories build bridges across the current mathematicalmainstream, linking geometric and algebraic, commutative and noncommutative,local and global banks. Arches in these bridges are pieces ofsemiorthogonal decompositions of triangulated categories.2000 Mathematics Subject Classification: 18E30, 14F05.Keywords and Phrases: Derived categories, Coherent sheaves, Fully faithfulfunctors, Noncommutative geometry.1. IntroductionThis paper is devoted to studying the derived categories V b (X) of coherentsheaves on smooth algebraic varieties X and on their noncommutative counterparts.Derived categories of coherent sheaves proved to contain the complete geometricinformation about varieties (in the sense of the classical Italian school of algebraicgeometry) as well as the related homological algebra.The situation when there exists a functor V b (M) —y V b (X) which is fullyfaithfulis of special interest. We are convinced that any example of such a functoris both algebraically and geometrically meaningful.A particular case of a fully faithful functor is an equivalence of derived categoriesV b (M) -^ V b (X).We show that smooth projective varieties with ample canonical or anticanonicalbundles are uniquely determined by their derived categories. Hence the derivedequivalences between them boil down to autoequivalences. We prove that for sucha variety the group of exact autoequivalences is the semidirect product of the groupof automorphisms of the variety and the Picard group plus translations.Equivalences and autoequivalences for the case of varieties with non-ample(anti) canonical sheaf are now intensively studied. The group of autoequivalences* Algebra Section, Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkin St.,GSP-1, Moscow 117966, Russia. E-mail: bondal@mi.ras.rutAlgebra Section, Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkin St.,GSP-1, Moscow 117966, Russia. E-mail: orlov@mi.ras.ru

48 A.Bondal D.Orlovis believed to be closely related to the holonomy group of the mirror-symmetricfamily.We give a criterion for a functor between derived categories of coherent sheaveson two algebraic varieties to be fully faithful. Roughly speaking, it is in orthogonalityof the images under the functor of the structure sheaves of distinct closedpoints of the variety. If a functor $ : V b (M) —y V b (X) is fully faithful, thenit induces a so-called semiorthogonal decomposition of V b (X) into V b (M) and itsright orthogonal category.It turned out that derived categories have nice behavior under special birationaltransformations like blow ups, flips and flops. We describe a semiorthogonaldecomposition of the derived category of the blow-up of a smooth variety X in asmooth center Y c X. It contains one component isomorphic to V b (X) and severalcomponents isomorphic to V b (Y).We also consider some flips and flops. Examples support the conjecture that forany generalized flip X —•* X + there exists a fully faithful functor V b (X + ) —t V b (X)and it must be an equivalence for generalized flops. This suggests the idea that theminimal model program of the birational geometry can be viewed as a 'minimization'of the derived category V b (X) in a given birational class of X.Then we widen the categorical approach to birational geometry by including inthe picture some noncommutative varieties. We propose to consider noncommutativedesingularizations and formulate a conjecture generalizing the derived McKaycorrespondence.We construct a semiorthogonal decomposition for the derived category of thecomplete intersections of quadrics. It is related to classical questions of algebraicgeometry, like 'quadratic complexes of lines', and to noncommutative geometricversion of Koszul quadratic duality.2. Equivalences between derived categoriesThe first question that arises in studying algebraic varieties from the pointof view of derived categories is when varieties have equivalent derived categoriesof coherent sheaves. Examples of such equivalences for abelian varieties and K3surfaces were constructed by Mukai [Mul], [Mu2], Polishchuk [Po] and the secondauthor in [Or2], [Or3]. See below on derived equivalences for birational maps.Yet we prove that a variety X is uniquely determined by its category V b (X),if its anticanonical (Fano case) or canonical (general type case) sheaf is ample. Tothis end, we use only the graded (not triangulated) structure of the category. Bydefinitiona graded category is a pair (V,Tx>) consisting of a category T> (which wealways assume to be fc-linear over a field k) and a fixed equivalence Tp : T> —y T> ,called translation functor. For derived categories the translation functor is definedto be the shift of grading in complexes.Of crucial importance for exploring derived categories are existence and propertiesof the Serre functor, defined in [BK].Definition 1 [BK] [B02] Let V be a k-linear category with finite-dimensionalHorn's. A covariant additive functor S : T> —t T> is called a Serre functor if it

Derived Categories of Coherent Sheaves 49is an equivalence and there are given bi functorial isomorphisms for any A, B £ T>:, if it exists, is unique up to a graded naturalisomorphism.If X is a smooth algebraic variety, n = dimX, then the functor (•) ®wj[n] isthe Serre functor in V b (X). Thus, the Serre functor in V b (X) can be viewed as acategorical incarnation of the canonical sheaf OJX •Theorem 2 [B02] Let X be a smooth irreducible projective variety with amplecanonical or anticanonical sheaf. IfV b (X) is equivalent as a graded category toT> b (X r ) for some other smooth algebraic variety X', then X is isomorphic to X'.The idea of the proof is that for varieties with ample canonical or anticanonicalsheaf we can recognize the skyscraper sheaves of closed points in V b (X) by meansof the Serre functor. In this way we find the variety as a set. Then we reconstructone by one the set of line bundles, Zariski topology and the structural sheaf of rings.This theorem has a generalization to smooth orbifolds related to projectivevarieties with mild singularities, as it was shown by Y. Kawamata [Kaw].Now consider the problem of computing the group Aut£> 6 (X) of exact (i.e.preserving triangulated structure) autoequivalences of V b (X) for an individual X.Theorem 3 [B02] Let X be a smooth irreducible projective variety with amplecanonical or anticanonical sheaf. Then the group of isomorphism classes of exactautoequivalences V b (X) —t V b (X) is generated by the automorphisms of the variety,twists by all invertible sheaves and translations.In the hypothesis of Theorem 3 the group Aut£> 6 (X) is the semi-direct productof its subgroups PicX® Z and AutX, Z being generated by the translation functor:Aut£> 6 (X) ~ AutX x (PicX e Z).3. Fully faithful functors and semiorthogonal decompositionsAn equivalence is a particular instance of a fully faithful functor. This is afunctor F : C ^tT> which for any pair of objects X, Y £ C induces an isomorphismHom(X , Y) ~ Hom(FX , FY). This notion is especially useful in the context oftriangulated categories.If a functor $ : V b (M) —y V b (X) is fully faithful, then it induces a so-calledsemiorthogonal decomposition of V b (X) into V b (M) and its right orthogonal.Let B be a full triangulated subcategory of a triangulated category T>. Theright orthogonal to B is the full subcategory B 1 - c T> consisting of the objects C suchthat Hom(B , C) = 0 for all B £ B. The left orthogonal ± B is defined analogously.The categories ± B and B L are also triangulated.

50 A.Bondal D.OrlovDefinition 4 [BK] A sequence of triangulated subcategories (BQ,...,B U ) in a triangulatedcategory T> is said to be semiorthogonal if Bj C B(- whenever 0 < j and denote this as follows:V=(B 0 ,....,B rExamples of semiorthogonal decompositions are provided by exceptional sequencesof objects [Bo]. These arise when all B,'s are equivalent to the derived categoriesof finite dimensional vector spaces V b (k — mod). Objects which correspond tothe 1-dimensional vector space under a fully faithful functor F :V b (k — mod) —¥ T>can be intrinsically defined as exceptional, i.e. those satisfying the conditionsHom'(B , E) = 0, when i ^ 0, and Hom°(ii', E) = k. There is a natural actionof the braid group on exceptional sequences [Bo] and, under some conditions,on semiorthogonal sequences of subcategories in a triangulated category [BK].We propose to consider the derived category of coherent sheaves as an analogueof the motive of a variety, and semiorthogonal decompositions as a tool for simplificationof this 'motive' similar to splitting by projectors in Grothendieck motivictheory.Let X and M be smooth algebraic varieties of dimension n and m respectivelyand E an object in V b (X x M). Denote by p and IT the projections of M x X to Mand X respectively. With E one can associate the functor # : V b (M) —y V b (X)defined by the formula: E (-):=B,iT*(E(êp*(-)).It happens that any fully faithful functor is of this form.Theorem 5 [Or2] Let F : V b (M) -+ V b (X) be an exact fully faithful functor,where M and X are smooth projective varieties. Then there exists a unique up toisomorphism object E £ V b (M x X) such that F is isomorphic to the functor #.The assumption on existence of the right adjoint to F, which was originally in[Or2], can be removed in view of saturatedness of V b (M) due to [BK], [BVdB].This theorem is in conjunction with the following criterion.Theorem 6 [BOI] Let M and X be smooth algebraic varieties and E £ V b (M x X).Then # is fully faithful functor if and only if the following orthogonality conditionsare verified:i) E.om x ($ E (O tl ) , $ E (O t2 )) = 0 for every i andti^t 2 .it) Eom x (Ê(O t ),Ê(O t )) = k,Eom x (Ê(O t ) , & E (O t )) = 0, for i

Derived Categories of Coherent Sheaves 51The criterion is a particular manifestation of the following important principle:suppose M is realized as an appropriate moduli space of pairwise homologicallyorthogonal objects in a triangulated category V taken 'from real life', then one canexpect a sheaf of finite (noncommutative) algebras AM over OM and a fully faithfulfunctor from the derived category V b (coh(AM)) of coherent modules over AM to V.There are also strong indications that this principle should have a generalization,at the price of considering noncommutative DG moduli spaces, to the casewhen the orthogonality condition is dropped.4. Derived categories and birational geometryBehavior of derived categories under birational transformations shows thatthey can serve as a useful tool in comprehending various phenomena of birationalgeometry and play possibly the key role in realizing the minimal model program.First, we give a description of the derived category of the blow-up of a varietywith smooth center in terms of the categories of the variety and of the center. Let Ybe a smooth subvariety of codimension r in a smooth algebraic variety X. Denote Xthe smooth algebraic variety obtained by blowing up X along the center Y. Thereexists a fibred square:Y AXp 4- 7T 4-Y A xwhere i and j are smooth embeddings, and p:Y—¥ Y is the projective fibration ofthe exceptional divisor Y in X over the center Y. Recall that Y = W(N X / Y ) is theprojective normal bundle. Denote by 0 Y (1) the relative Grothendieck sheaf.Proposition 7 [Ori] Let L be any invertible sheaf on Y. The functorsLTT* :V b (X)—yV b (X),are fully faithful.Hj4L®p*(-j): V b (Y) —• V b (X)Denote by D(X) the full subcategory of V b (X) which is the image of V b (X)with respect to the functor hn* and by D(Y)k the full subcategories of V b (X) whichare the images of V b (Y) with respect to the functors Rj»(öy(fc) ®p*(-)).Theorem 8 [Orl][B01] We have the semiorthogonal decomposition of the categoryof the blow-up:V b (X) = (D(Y)^r+ i,...,D(Y)î,D(X)).Now we consider the behavior of the derived categories of coherent sheaveswith respect to the special birational transformations called flips and flops.

52 A.Bondal D.OrlovLet Y be a smooth subvariety of a smooth algebraic variety X such thatY =ë P* and N x/Y =* 0(^\)® {l+1) with I < k.If X is the blow-up of X along Y, then the exceptional divisor Y = W k x W l isthe product of projective spaces. We can blow down X in such a way that Y projectsto the second component P' of the product. As a result we obtain a smooth varietyX+, which for simplicity we assume to be algebraic, with subvariety Y + = P'. Thisis depicted in the following diagram:The birational map X —•* X+ is the simplest instance of flip, for I < k. If I = k,this is a flop.Theorem 9 [BOI] In the above notations, the functor R7r»L7r + * : V b (X + )V b (X) is fully faithful for I < k. If I = k, it is an equivalence.—•This theorem has an obvious generalization to the case when Y is isomorphicto the projectivization of a vector bundle E of rank k on a smooth variety W,q : Y —y W, and N x / Y = q*F(E) 0 E ( — 1) where F is a vector bundle on W of rankI < k. Then the blow-up with a smooth center can be viewed as the particular caseof this flip when Y is a divisor in X. Kawamata [Kaw] generalized the theorem tothose flips between smooth orbifolds which are elementary (Morse type) cobordismsin the theory of birational cobordisms due to Wlodarczyk et al. [Wl], [AKMW].fiLet X and X+ be smooth projective varieties. A birational map X —+ X+will be called a generalized flip if for some (and consequently for any) its smoothresolutionthe difference D = n*K x —jr + *K x + between the pull-backs of the canonical divisorsis an effective divisor on X. The particular case when D = 0 is called generalizedflop.Theorem 9 together with calculations of 3-dimensional flops with centers in(—2)-curves [BOI] lead us to the following conjecture.fiConjecture 10 For any generalized flip X —•* X+ there is an exact fully faithfulfunctor F : V b (X + ) —• V b (X). It is an equivalence for generalized flops.

Derived Categories of Coherent Sheaves 53This conjecture was recently proved in dimension 3 by T. Bridgeland [Br].The functor R7r»L7r + * : V b (X + ) —• V b (X) is not always fully faithful underconditions of the conjecture, but we expect that it is such when X is replaced bythe fibred product of X and X+ over some common singular contraction of X andX+. Namikawa proved that this is the case for Mukai symplectic flops [Na].A fully faithful functor V b (X + ) —• V b (X) induces a semiorthogonal decompositionof V b (X) into V b (X + ) and its right orthogonal (which is trivial for flops).Hence, passing from X to X+ for generalized flips has the categorical meaning ofbreaking off semiorthogonal summands from the derived category. This suggeststhe idea that the minimal model program of birational geometry should be interpretedas a minimization of the derived category V b (X) in a given birational classof X. Promisingly, chances are that the very existence of flips can be achieved byconstructingX+ as an appropriate moduli space of objects in V b (X), in accordancewith the principle of the previous section (this is done by T. Bridgeland for flops indimension 3 [Br]).5. Noncommutative resolutions of singularitiesIn this section we will give a perspective for categorical interpretation of theminimal model program by enriching the landscape with the derived categories ofnoncommutative varieties.Let IT : X —t X be a proper birational morphism, where X has rational singularities.Then RTT* : V b (X) -+ V b (X) identifies V b (X) with the quotient of V b (X)by the kernel of RîT*. For this reason, let us call by a categorical desingularizationof a triangulated category T> a pair (C, K.) consisting of an abelian category Cof finite homological dimension and of K., a thick subcategory in V b (C) such thatT> = V b (C)/K.. We expect that for T> = V b (X) there exists a minimal desingularization,i.e. such one that V b (C) has a fully faithful embedding in T> b (C) for anyothercategorical desingularization (C',K. r ) of X>. Such a desingularization is uniqueup to derived equivalence of C.For the derived categories of singular varieties one may hope to find the minimaldesingularizations in the spirit of noncommutative geometry.Let X be an algebraic variety. We call by noncommutative (birational) desingularizationof X a pair (p, A) consisting of a proper birational morphism p : Y —¥ Xand an algebra A = £nd(T) on Y, the sheaf of local endomorphisms of a torsionfree coherent öy-module T, such that the abelian category of coherent „4-moduleshas finite homological dimension.When / : Y —t X is a morphism from a smooth Y onto an affine X withfibres of dimension < 1 and R/»(öy) = Ox, M. Van den Bergh [VdB] has recentlyconstructeda noncommutative desingularization of X, which is derived equivalenttoV b (Y).Conjecture 11 Let X has canonical singularities andq :Y—¥X a finite morphismwith smooth Y. Then the pair (idx,£nd(q*ÖY)) is a minimal desingularization ofX.

54 A.Bondal D.OrlovIn particular, we expect that T> b (coh(£nd(q*ÖY))) has a fully faithful functorinto V b (X) for any (commutative) resolution of X. Moreover, if the resolution iscrêpant then the functor has to be an equivalence.Let X be the quotient of a smooth Y by an action of a finite group G. Ifthe locus of the points in Y with nontrivial stabilizer in G has codimension > 2,then the category of coherent £nd(g»öy )-modules is equivalent to the category ofG-equivariant coherent sheaves on Y. Therefore the conjecture is a generalizationof the derived McKay correspondence due to Bridgeland-King-Reid [BKR].6. Complete intersection of quadrics and noncommutativegeometryThis section is related to the previous one by Grothendieck slogan that projectivegeometry is a part of theory of singularities.Let X be a smooth intersection of two projective quadrics of even dimensiond over an algebraically closed field of characteristic zero. It appears that if weconsider the hyperelliptic curve C which is the double cover of P 1 that parameterizesthe pencil of quadrics, with ramification in the points corresponding to degeneratequadrics, then V b (C) is embedded in V b (X) as a full subcategory [BOI]. This givesa categorical explanation for the classical description of moduli spaces of semistablebundles on the curve C as moduli spaces of (complexes of) coherent sheaves on X.The orthogonal to V b (C) in V b (X) is decomposed into an exceptional sequence(of line bundles ). More precisely, we have a semiorthogonal decompositionV b (X) = (öxhd + 3),...,öx,'D b (C)). (6.1)When a greater number of quadrics is intersected, objects of noncommutativegeometry naturally show up: instead of coherent sheaves on hyperelliptic curveswe must consider modules over a sheaf of noncommutative algebras. More aboutnoncommutative geometry is in the talk of T. Stafford at this Congress.Consider a system of m quadrics in V(V), i.e. a linear embedding U ^y S 2 V*,where dimU = m, dimV = n, 2m < n. Let X, the complete intersection ofthe quadrics, be a smooth subvariety in W(V) of dimension n — m — 1. Let A =® H 0 (X,O(ij) be the coordinate ring of X. This graded quadratic algebra isKoszul due to Tate [Ta]. The quadratic dual algebra B = A is the generalizedhomogeneous Clifford algebra. It is generated in degree 1 by the space V, therelations being given by the kernel of the dual to map S 2 V —¥ U*, viewed as asubspace in V V. The center of B is generated by U* (a subspace of quadraticelements in B) and an element d, which satisfies the equation d 2 = f where / isthe equation of the locus of degenerate quadrics in U. Algebra B is finite over thecentral subalgebra S = S'U*. The Veronese subalgebra B ev = (BB 2 ì is finite overthe Veronese subalgebra S ev = ®S 2% U*. Since Proj S ev is isomorphic to V(U),the sheafification of B ev over Proj S ev is a sheaf B of finite algebras over ö V ( V yConsider the derived category T> b (coh(Bj) of coherent right B-modules.

Derived Categories of Coherent Sheaves 55Theorem 12 Let X be the smooth intersection of m quadrics in P" _1 , 2m < n.Then there exists a fully faithful functor T> b (coh(Bj) ^yV b (X). Moreover,(i) if 2m < n, we have a semiorthogonal decompositionV b (X) = (Ox(^n + 2m+i),...,O x ,T) b (coh(B)) s j,(ii) i/2m = n, there is an equivalence T> b (coh(Bj) ^y V b (X).For m = 0, i.e. when there is no quadrics, the theorem coincides with Beilinson'sdescription of the derived category of the projective space [Be]. For m = 1,this is Kapranov's description of the derived category of the quadric [Kap].For odd n, the element d generates the center of B over ö V ( V y Hence thespectrum of the center of B is a ramified double cover Y over W(U). Also B yieldsa coherent sheaf of algebras B' over Y, such that coh(B') is equivalent to coh(B).For the above case of two even dimensional quadrics, B' is an Azumaya algebraover Y = C. Since Brauer group of Y (taken over an algebraically closed field ofcharacteristic zero) is trivial, the category coh(B') is equivalent to COìI(OY). Hence(6.1) follows from the theorem.Furthermore, when X is a K3 surface, the smooth intersection of 3 quadricsin P 5 , then the double cover Y is also a K3 surface, but B' is in general a nontrivialAzumaya algebra over Y. The theorem states an equivalence V b (X) ~ T> b (coh(B'j).This theorem illustrates the principle from section 3. The fully faithful functoris related to the moduli space of vector bundles on X, which are the restrictions toX of the spinor bundles on the quadrics. The (commutative) moduli space involvedis either W(U) or Y, depending on parity of n.Algebraically, the fully faithful functor in the theorem is given by an appropriateversion of Koszul duality. Theorem 12 has a generalization to a class of KoszulGorenstein algebras, which includes the coordinate rings of superprojective spaces.References[AKMW] Abramovich D., Karu K., Matsuki K., Wlodarczyk J., Torification andFactorization of Birational Maps preprint math. AG/9904135.[Be] Beilinson A., Coherent sheaves on P" and problems of linear algebra,Funkcionalnyi analiz i ego pril. 12 (1978), 68^69.[Bo] Bondal A., Representations of associative algebras and coherent sheaves,Izv. Akad. Nauk SSSR, Ser.Mat. 53 (1989), 25^44; English transi, inMath. USSR Izv. 34 (1990).[BK] Bondal A., Kapranov M., Representable functors, Serre functors, andmutations, Izv. Akad. Nauk SSSR, Ser.Mat, 53 (1989), 1183^1205; Englishtransi, in Math. USSR Izv., 35 (1990), 519-541.[BOI] Bondal A., Orlov D., Semiorthogonal decomposition for algebraic varieties,preprint MPIM 95/15 (1995), preprint math.AG/9506012.[B02] Bondal A., Orlov D., Reconstruction of a variety from the derived categoryand groups of autoequivalences, Compositio Mathematica, v.125(2001) N.3, 327^344.

56 A.Bondal D.Orlov[BVdB] Bondal A., Van den Bergh M., Generators and representabilityof functors in commutative and noncommutative geometry, preprintmath.AG/0204218.[Br][BKR][Kap]Bridgeland T., Flops and derived categories, preprint math. AG/0009053.Bridgeland T., King A., Reid M., Mukai implies McKay: the McKaycorrespondenceas an equivalence of derived categories, J. Amer. Math.Soc, 14 (2001), 535^554, preprint math.AG/9908027.Kapranov M., On the derived categories of coherent sheaves on somehomogeneous spaces, Invent. Math., 92 (1988), 479^508.[Kaw] Kawamata Y., Francia's flip and derived categories, preprintmath.AG/0111041.[Mul] Mukai S., Duality between D(X) and D(X) with its application toPicard sheaves, Nagoya Math. J. 81 (1981), 153^175.[Mu2] Mukai S., On the moduli space of bundles on a K3 surface I, Vectorbundles on algebraic varieties, Tata Institute of Fundamental Research,Oxford University Press, Bombay and London, 1987.[Na] Namikawa Y., Mukai flops and derived categories, preprintmath.AG/0203287.[Ori] Orlov D., Projective bundles, monoidal transformations and derivedcategories of coherent sheaves, Izv. Akad. Nauk SSSR Ser.Mat. 56 (1992),852-862; English transi, in Math. USSR Izv. 38 (1993), 133-141.[Or2] Orlov D., Equivalences of derived categories and K3 surfaces, J.of Math. Sciences, Alg. geom.-S, 84, N5, (1997), 136H381, preprintmath.AG/9606006.[Or3][Po][Ta][VdB][Wl]Orlov D., On derived categories of coherent sheaves on abelian varietiesand equivalences between them, Izv. RAN, Ser. Mat., 66 (2002) N3, 131-158, (see also math.AG/9712017).Polishchuk A., Symplectic biextensions and a generalization of theFourier-Mukai transform, Math. Res. Let., v.3 (1996), 813^828.Tate J., Homology of Noetherian rings and local rings, Illinois J. Math.,1 (1957) Nl, 14^27.Van den Bergh M., Three-dimensional flops and non-commutative ring,paper in preparation.Wlodarczyk J., Birational cobordisms and factorization of birationalmaps, preprint math. AG/9904074.

ICM 2002 • Vol. II • 57^66Algebraic CobordismM. Levine*AbstractTogether with F. Morel, we have constructed in [6, 7, 8] a theory of algebraiccobordism,., an algebro-geometric version of the topological theory ofcomplex cobordism. In this paper, we give a survey of the construction andmain results of this theory; in the final section, we propose a candidate for atheory of higher algebraic cobordism, which hopefully agrees with the cohomologytheory represented by the P 1 -spectrum MGL in the Morel-Voevodskystable homotopy category.2000 Mathematics Subject Classification: 19E15, 14C99, 14C25.Keywords and Phrases: Cobordism, Chow ring , _R"-theory.1. Oriented cohomology theoriesFix a field k and let Seh/, denote the category of separated finite-type k-schemes. We let Sm^ be the full subcategory of smooth quasi-projective fc-schemes.We have described in [7] the notion of an oriented cohomology theory on Smj.Roughly speaking, such a theory A* consists of a contravariant functor from Sm^to graded rings (commutative), which is also covariantly functorial for projectiveequi-dimensional morphisms f :Y —¥ X (with a shift in the grading):/, :A*(Y)Â*- dimxY (X).The pull-back g* and push-forward /» satisfy a projection formula and commute intransverse cartesian squares. If L —¥ X is a line bundle with zero-section s : X —t L,we have the first Chern class of L, defined byCi(L):=s*(s*(l x ))£A 1 (X),where lx £ A°(X) is the unit. A* satisfies the projective bundle formula:* Department of Mathematics, Northeastern University, Boston, MA 02115, USA. E-mail:marc@neu.edu

58 M. Levine(PB) Let £ be a rank r + 1 locally free coherent sheaf on X, with projective bundleq : W(£) —t X and tautological quotient invertible sheaf q*£ —¥ 0(1). Let£ = ci(0(l)). Then A*(¥(£)) is a free .4*(X)-module with basis 1,£,.. • ,f •Finally, A* satisfies a homotopy property: if p : V —¥ X is an affine-space bundle(i.e., a torsor for a vector bundle over X), then p* : A*(X) —t A*(V) is anisomorphism.Examples 1.1. (1) The theories CH* and H?^(—, Z/n(*)) on Sm/. (also with Z/(*)or Qi(*) coefficients).(2) The theory K 0 [ß, ß^1]on Smj. Here ß is an indeterminant of degree —1, usedto keep track of the relative dimension when taking projective push-forward.Remarks 1.2. (1) In [8], we consider a more general (dual) notion, that of anoriented Borel-Moore homology theory .4». Roughly, this is a functor from a fullsubcategory of Seh/, to graded abelian groups, covariant for projective maps, andcontravariant (with a shift in the grading) for local complete intersection morphisms.In addition, one has external products, and a degree -1 Chern class endomorphismci(L) : A*(X) —¥ -A»_i(X) for each line bundle L on X, defined by ci(L)(n) =S*(S»(JJ)), s : X —t L the zero-section. As for an oriented cohomology theory,there are various compatibilities of push-forward and pull-back, and .4» satisfies aprojective bundle formula and a homotopy property.This allows for a more general category of definition for .4», e.g., the categorySeh/.. As we shall see, the setting of Borel-Moore homology is often more naturalthan cohomology. On Smj, the two notions are equivalent: to pass from Borel-Moore homology to cohomology, one re-grades by setting A n (X) := .4„_dim fe x(X)and uses the l.c.i. pull-back for .4» to give the contravariant functoriality of A*,noting that every morphism of smooth fc-schemes is an l.c.i. morphism. We willstate most of our results for cohomology theories on Sm/., but they extend to thesetting of Borel-Moore homology on Seh/, (see [8] for details).(2) Our notion of oriented cohomology is related to that of Panin [10], but is notthe same.2. The formal group lawLet .4» be an oriented cohomology theory on Smj. As noticed by Quillen [11],a double application of the projective bundle formula (PB) yields the isomorphismof ringsA*(k)[[u,v]] =* lim.4*(P" x P ro ),the isomorphism sending u to ci(p\0(t)) and v to ci(plö(lj). The class ofci(p*0(t) ®P2Ö(1)) thus gives a power series FA(U,V) £ A*(k)[[u,v]] withci(plO(l) ®p* 2 0(l)) = F A (ci(plO(l)),ci(plO(l))).

Algebraic Cobordism 59By the naturality of c\, we have the identity for X £ Sm/. with line bundles L, M,ci(L® M) =F A (ci(L),ci(M)).In addition, FA(U,V) = U + V mod uv, FA(U,V) = FA(V,U), and FA(FA(U,V), W) =FA(U,FA(V,WJ). Thus, FA gives a formal group law with coefficients in A*(k).Remark 2.3. Note that Ci : Pic(X) —t A 1 (X) is a group homomorphism if andonly if FA(U,V) = U + V. If this is the case, we call A* ordinary, if not, A* isextraordinary. If FA(U,V) = u + v — auv with a a unit in A*(k), we call A*multiplicative and periodic.Examples 2.4. For A* = CH* or H 2 *, FA = U + V, giving examples of ordinarytheories.For the theory A = if 0 [/5,/5 _1 ], ci(L) = (1 — L v )ß^1,and FA(U,V) =u + v — ßuv, giving an example of a multiplicative and periodic theory.Remark 2.5. Let L* = Z[ay | i, j > 1], where we give ay degree —i — j + 1, andlet F £ L* [[«,v]] be the power series F = u + v + ^•. ay«*w J . LetL* = L* /F(u, v) = F(v, u), F(F(u, v),w) = F(u, F(v, w)),and let FL £ L* [[«, v]] be the image of F. Then (F^, L* ) is the universal commutativedimension 1 formal group; L* is called the Lazard ring (cf. [5]).Thus, if A* is an oriented cohomology theory on Sm/., there is a canonicalgraded ring homomorphism

60 M. Levine2. Let 1Z d%m (X) be the subgroup of Z*(X) generated by cobordism cycles ofthe form (/ : Y -ï X,n*Li,... ,n*L r , Mi,..., M s ), where TX : Y -ï Z is asmooth morphism in Smj, the L, are line bundles on Z, and r > dim/. Z. LetZ,(X) = Z4X)/TZ dim (X).3. Add the Gysin isomorphism: If L —¥ Y is a line bundle and s : Y —¥ Lis a section transverse to the zero-section with divisor i : D —¥ Y, identify(/ : Y -• X,Li,...,L r ,L) with (/o»:D4 X,i*L u ... ,i*L r ). We letn„(X) denote the resulting quotient of Z_^(X). Note that on n„(X) we have,for each line bundle L —¥ X, the Chern class operatorCi(L) :£,(*)-• 0,-1 (*)(/ : F -+ X,Li,...,L r ) ^ (f : F -+ X,L U . ..,L r ,fL)as well as push-forward maps /» : Ü»(X) —¥ Q*(X') for / : X —^ X' projective.4. Impose the formal group law: Regrade L by setting L„ := L _n . Let 0»(X) bethe quotient of L» ®Q» (X) by the imposing the identity of maps L» ®I2» (F) —^L» ®Q,(X)(id ® /») o F L (ci(L),ci(M)) = id ® (fi oci(L®Mj)for f :Y —¥ X projective, and L, M line bundles on Y. Note that, having imposedthe relations in 1Z d%m , the operators ci(L), ci(M) are locally nilpotent,so the infinite series FL(CI(L), ci(Mj) makes sense.As the notation suggests, the most natural construction of 0 is as an orientedBorel-Moore homology theory rather than an oriented cohomology theory; the transitionto an oriented cohomology theory on Sm/. is given as in remark 1.2(1). Theproof of theorem 3.6 uses resolution of singularities [4] and the weak factorizationtheorem [1] in an essential way.Remark 3.7. In addition to the properties of 0» listed in theorem 3.6, 0»(X) isgenerated by the classes of "elementary" cobordism cycles (/ : Y —t X).4. Degree formulasIn the paper [12], Rost made a number of conjectures based on the theoryof algebraic cobordism in the Morel-Voevodsky stable homotopy category. Manyof Rost's conjectures have been proved by homotopy-theoretic means (see [3]); ourconstruction of algebraic cobordism gives an alternate proof of these results, andsettles many of the remaining open questions as well. We give a sampling of someof these results.4.1. The generalized degree formulaAll the degree formulas follow from the "generalized degree formula". We firstdefine the degree map Q*(X) —t Q*(k).

Algebraic Cobordism 61Definition 4.8. Let k be a field of characteristic zero and let X be an irreduciblefinite type fc-scheme with generic point i : x —¥ X. For an element n of 0*(X), definedegn £ ii*(k) to be the element mapping to i*r\ in Q*(k(xj) under the isomorphismsn*(k) =* L* =ë n*(k(x)) given by theorem 3.6(2).Theorem 4.9(generalized degree formula). Let k be a field of characteristiczero. Let X be an irreducible finite type k-scheme, and let n be in 0»(X). Let/o : B 0 —¥ X be a resolution of singularities of X, with B 0 quasi-projective over k.Then there are a, £ 0»(fc), and projective morphisms fi : B t —t X such that1. Each Bi is in Snij, fi : B t —t /(!?,) is birational and f(Bi) is a proper closedsubset of X (for i > 0).2.ÌÌ- (degn)[/ 0 : B 0 -+ X] = E[ = i ««[/« : B t "• x i «" °* W-Proof. It follows from the definitions of 0* that we haveÜ*(k(x))=limÜ*(U),uwhere the limit is over smooth dense open subschemes U of X, and ii*(k(xj) is thevalue at Specfc(ar) of the functor Q* on finite type fc(a:)-schemes. Thus, there is asmooth open subscheme j : U —¥ X of X such that j*n = (degn)fid^] in Q*(U).Since U x x B 0 = U, it follows that j*(n - (degn)[/ 0 ]) = 0 in Q*(U).Let W = X \U. From the localization sequencen.(w) A o,(x) A 04c/) -• 0,we find an element % £ Q»(W) with »*(%) = n — (degn)[/o], and noetherianinduction completes the proof.DRemark 4.10. Applying theorem 4.9 to the class of a projective morphism / :Y —¥ X, with X, F £ Sni/., we have the formular[f : Y -+ X] - (deg/)[idx] = $>[/« : A ^ X]in Q*(X). Also, if dim^X = dim/. Y, deg/ is the usual degree, i.e., the fieldextension degree [k(Y) : k(X)] if / is dominant, or zero if / is not.4.2. Complex cobordismFor a differentiable manifold M, one has the complex cobordism ring MU* (M).Given an embedding a : k —¥ C and an X e Sni/., we let X er (C) denote the complexmanifold associated to the smooth C-scheme IxjC. Sending X to MU 2 *(X cr (C))defines an oriented cohomology theory on Sni/.; by the universality of 0*, we havea natural homomorphismi=l&„ : 0*(X) -• MU 2 *(X a (

62 M. LevineNow, if P = P(ci,. • • ,Cd) is a degree d (weighted) homogeneous polynomial,it is known that the operation of sending a smooth compact d-dimensional complexmanifold M to the Chern number deg(F(ci,... , C• Z. If X is smooth and projectiveof dimension dover k, we have F([X]) = deg(P(ci,... ,Cd)(Q x «(c)))', P([X]) is infact independent of the choice of embedding a.Let Sd(ci,... ,Cd) be the polynomial which corresponds to ^ £f, where £1,...are the Chern roots. The following divisibility is known (see [2]): if d = p n — 1 forsome prime p, and dimX = d, then Sd(X) is divisible by p.In addition, for integers d = p n — 1 and r > 1, there are mod p characteristicclasses td, r , with td,i = Sd/p mod p. The Sd and the td, r have the followingproperties:(4.1)1. Sd(X) £ pZ is defined for X smooth and projective of dimension d = p n — 1.td,r(X) £ Z/p is defined for X smooth and projective of dimension rd =r(p n - 1).2. Sd and td, r extend to homomorphisms Sd '• ii^d(k)—¥ pZ, td, r '• ii^rd (k) —¥Z/p.3. If X and Y are smooth projective varieties with dim X, dim Y > 0, dim X +dim F = d, then Sd(X x Y) = 0.4. If Xi,...,X s are smooth projective varieties with ^-dimXj = rd, thentd,r(Y\i x i) = 0 unless d\ dimXj for each i.We can now state Rost's degree formula and the higher degree formula:Theorem 4.11 (Rost's degree formula). Let f : Y —¥ X be a morphism ofsmooth projective k-schemes of dimension d, d = p n — 1 for some prime p. Thenthere is a zero-cycle n on X such thats d (Y) - (deg f)sd(X)= p • deg(n).Theorem 4.12(Rost's higher degree formula). Let f :Y —¥ X be a morphismof smooth projective k-schemes of dimension rd, d = p n — 1 for some prime p.Suppose that X admits a sequence of surjective morphismssuch that:X = X 0 —¥ Xi —t ... —t X r _i —t X r = Spec k,1. dimXj = d(r — i).2. Let n be a zero-cycle on X t Xx i+1 Specfc(Xj + i). Then p\ deg(n).Thent d ,r(Y) =deg(f)t d ,r(X).Proof. These two theorems follow easily from the generalized degree formula.Indeed, for theorem 4.11, take the identity of remark 4.10 and push forward to

0*(fc). Using remark 3.7, this gives the identityAlgebraic Cobordism 63r[Y]^(degf)[X] = Y,m{AiXB i ]in Q*(X), for smooth, projective fc-schemes Ay By and integers rrij, where eachBi admits a projective morphism fi : B t —t X which is birational to its image andnot dominant. Since Sd vanishes on non-trivial products, the only relevant part ofthe sum involves those Bj of dimension zero; such a Bj is identified with the closedpoint bj := fj(Bj) of X. Applying Sd, we haves d (Y) -deg(f)s d (X) = J2 m 3 s d( A j)àeg k (bj).Since Sd(Aj) = prij for suitable integers rij, we haves d (Y) - deg(f)sd(X) = pdeg(^m i n i 6 i ).Taking n = V. rnyrijbj proves theorem 4.11.The proof of theorem 4.12 is similar: Start with the decomposition of [/ :F —t X] — (deg/)[idx] given by remark 4.10. One then decomposes the mapsBi —¥ X = X 0 further by pushing forward to X\ and using theorem 4.9. Iteratingdown the tower gives the identity in Q»(fc)[F] - (deg f)[X] = J2m i [B t 0x...x B«] ;ithe condition (2) implies that, if d\ dim/. B l - for all j = 0,..., r, then p\rrij. Applyingtd, r and using the property (4.1)(4) yields the formula.Di=ljj5. Comparison resultsSuppose we have a formal group (f,R),Q : L* —t Q*(k). Theuniversal property of 0* gives the analogous universal property for QT. R,.In particular, let 0^j_ be the theory with (f(u,v),R) = (« + w,Z),and let 0^ bethe theory with (/(«, v),R) = (u + v — ßuv, Z[ß, ß^1]).We thus have the canonicalnatural transformations of oriented theories on Sni/.O; -• CH*; Q* x -• K^ßJ- 1 ]. (5.2)Theorem 5.13. Letk be a field of characteristic zero. The natural transformations(5.2) are isomorphisms, i.e., CH* is the universal ordinary oriented cohomologytheory and K^ßjß^1]is the universal multiplicative and periodic theory.

64 M. LevineProof. For CH*, this uses localization, theorem 4.9 and resolution of singularities.For Ko, one writes down an integral Chern character, which gives the inverseisomorphism by the Grothendieck-Riemann-Roch theorem.D6. Higher algebraic cobordismThe cohomology theory represented by the P 1 -spectrum MGL in the Morel-Voevodsky Â 1 -stable homotopy category [9, 13] gives perhaps the most natural algebraicanalogue of complex cobordism. By universality, Q"(X) maps to MGL 2n ' n (X);to show that this map is an isomorphism, one would like to give a map in the otherdirection. For this, the most direct method would be to extend 0* to a theory ofhigher algebraic cobordism; we give one possible approach to this construction here.The idea is to repeat the construction of 0», replacing abelian groups withsymmetric monoidal categories throughout. Comparing with the Q-construction,one sees that the cobordism cycles in 1Z. d%m (X) should be homotopic to zero, butnot canonically so. Thus, we cannot impose this relation directly, forcing us tomodify the group law by taking a limit.Start with the category Z(X) 0 , with objects (/ : F —t X, L\,..., L r ), whereF is irreducible in Sni/., / is projective, and the L, are line bundles on F. Amorphism (/ : F —t X, L\,..., L r ) —t (/' : F' —t X, L[,..., L' r ) in Z(X) 0 consist ofa tuple (*L' via ), let i' : D' -t Y' be the map induced by , s' : Y' -t V thesection induced by s, andip D : (/ o i : D -> X, i*Li,..., i*L r ) -> (/' o i' : D' -> X, i'*L[,..., i'*L' r ),the morphism induced by tp. We impose the relation V°7L,S = 1L', S ' °'

Algebraic Cobordism 65is universal for symmetric monoidal functors C —¥ C such that C admits an actionof R via natural transformations. In case R = Z, Z ®N C is the standard groupcompletion C~ 1 C. In general, if {e a | a £ A} is a Z-basis for R, thenR® N C = ]JC- 1 C,awith the A-action given by expressing xx : R —¥ R in terms of the basis {e a }.For each integer n > 0, let L» be the quotient of L» by the ideal of elementsof degree > n. We thus have the formal group (F L („),li" ).We form the category li") ®nÛ.(X), which we grade by total degree. For eachf :Y —¥ X projective, with F £ Sni/., and line bundles L, M, L\,...,L r on F, weadjoin an isomophism PL,M/»(F L(n) (ci(L), ci(M))(id Y ,Li,..., L r )) ^ /»(id ®ëi(L® M)(id Y ,Li,..., L r )).We impose the condition of naturality with respect to the maps in li") ®N I2„(F),in the evident sense; the Chern class transformations extend in the obvious manner.We impose the following commutativity condition: We have the evident isomorphismìL,M '• F L {n)(ci(L),ci(M)) —¥ F h („) (ci(M), ci(Lj) of natural transformations,as well as TL,M '• c\(L® M) —t ci(M ® L), the isomorphism induced by thesymmetry L®M = M®L. Then we impose the identity TL,M°PL,M = PM,L°tL,M-We impose a similar identity between the associativity of the formal group law andthe associativity of the tensor product of line bundles.We also adjoin a • TL,M for all a £ li"), with similar compatibilities as above,respecting the li") -action and sum. This forms the symmetric monoidal categoryQ(«) (X), which inherits a grading from O(X). We have the inverse system of gradedsymmetric monoidal categories:... -• Q(" +1) (X) -• Q(") (X)^ ... .Definition 5.14. Set 0^r(X) := n^BÜ^(X)) and iì m ,r( x )At present, we can only verify the following:Theorem 5.15. There is a natural isomorphism O TOj0 (X) = Q TO (X).: = u 0 ,^)-Proof. First note that iro(Z m (Xj) is a commutative monoid with group completionZ m (X). Next, the natural map 7ro(0» i (X)) + —t 0*(X) i s surjective with kernelgenerated by the classes generating TZ dtm (X). Given such an element ip := (/ :F —t X,n*Li,... ,n*L r , Mi,..., M s ), with n : Y —¥ Z smooth, and r > dim/. Z,suppose that the L, are very ample. We may then choose sections «j : Z —¥ L t withdivisors Di all intersecting transversely. Iterating the isomorphisms 7L ; ,Sì givesa path from tp to 0 in BÙ r _(X). Passing to Bum (X), the group law allows usto replace an arbitrary line bundle witha difference of very ample ones, so all theclasses of this form go to zero in Q^0(X). This shows that the natural mapü^Q(X)^(h^® h ü4X)) mn

66 M. Levineis an isomorphism. Since (li") ®L ^»(X)) TO = Q TO (X) for m > n, we are done. DThe categories 0 (X) are covariantly functorial for projective maps, contravariantfor smooth maps (with a shift in the grading) and have first Chern classnatural transformations ci(L) : flm (X) —^ Q^tl1 (X) for L —^ X a line bundle.We conjecture that the inverse system used to define 0 TOjr (X) is eventuallyconstant for all r, not just for r = 0. If this is true, it is reasonable to define thespace BO TO (X) as the homotopy limitBÜm(X) := holimBQ^)(X).nOne would then have 0 TOjr (X) = 7r r (TiO TO (X),0) for all m,r; hopefully the propertiesof 0» listed in theorem 3.6 would then generalize into properties of the spacesBflm(X).References[i[2[3;[4;[6;[r[9[io;[H[12:[is;D. Abramovich, K. Karu, K. Matsuki, J. Wlodarczyk, Torification and factorizationof birational morphisms, preprint 2000, AG/9904135.J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures inMathematics. University of Chicago Press, Chicago, Ill.-London, 1974.S. Borghesi, Algebraic Morava K-theories and the higher degree formula,preprint May 2000, www.math.uiuc.edu/K-theory/0412/index.html.H. Hironaka, Resolution of singularities of an algebraic variety over a field ofcharacteristic zero. I, II, Ann. of Math., (2) 79 (1964), 109^203; ibid. 205^326.M. Lazard, Sur les groupes de Lie formels à un paramètre, Bull. Soc. Math.France, 83 (1955), 251-274.M. Levine et F. Morel, Cobordisme algébrique I, II, C.R. Acad. Sci. Paris, SérieI, 332 (2001), 723-728; ibid. 815-820.M. Levine et F. Morel, Algebraic cobordism, I, preprint Feb. 2002,www.math.uiuc.edu/K-theory/0547/index.html.M. Levine, Algebraic cobordism, II, preprint June 2002,www.math.uiuc.edu/K-theory/0577/index.html.F. Morel, V. Voevodsky, Â 1 homotopy of schemes, Publications Mathématiquesde PI.H.E.S, volume 90.I. Panin, Push-forwards in oriented cohomology theories of algebraic varieties,preprint Nov. 2000, www.math.uiuc.edu/K-theory/0459/index.html.D. Quillen, Elementary proofs of some results of cobordism theory using Steenrodoperations, Advances in Math., 7 (1971), 29^56.M. Rost, Construction of splitting varieties, preprint, 1998.V. Voevodsky, Â 1 -homotopy theory, Proceedings of the International Congressof Mathematicians, Vol. I (1998). Doc. Math. Extra Vol. I (1998), 579^604.

ICM 2002 • Vol. II • 67^76Permutation Groups andNormal SubgroupsCheryl E. Praeger*AbstractVarious descending chains of subgroups of a finite permutation group canbe used to define a sequence of 'basic' permutation groups that are analoguesof composition factors for abstract finite groups. Primitive groups have beenthe traditional choice for this purpose, but some combinatorial applicationsrequire different kinds of basic groups, such as quasiprimitive groups, that aredefined by properties of their normal subgroups. Quasiprimitive groups admitsimilar analyses to primitive groups, share many of their properties, and havebeen used successfully, for example to study s-arc transitive graphs. Moreoverinvestigating them has led to new results about finite simple groups.2000 Mathematics Subject Classification: 20B05, 20B10 20B25, 05C25.Keywords and Phrases: Automorphism group, Simple group, Primitivepermutation group, Quasiprimitive permutation group, Arc-transitive graph.1. IntroductionFor a satisfactory understanding of finite groups it is important to study simplegroups and characteristically simple groups, and how to fit them together to formarbitrary finite groups. This paper discusses an analogous programme for studyingfinite permutation groups. By considering various descending subgroup chains offinite permutation groups we define in §2 sequences of 'basic' permutation groupsthat play the role for finite permutation groups that composition factors or chieffactors play for abstract finite groups. Primitive groups have been the traditionalchoice for basic permutation groups, but for some combinatorial applications largerfamilies of basic groups, such as quasiprimitive groups, are needed (see §3).Application of a theorem first stated independently in 1979 by M. E. O'Nanand L. L. Scott [4] has proved to be the most useful modern method for identifyingthe possible structures of finite primitive groups, and is now used routinely for their* Department of Mathematics & Statistics, University of Western Australia, 35 Stirling Highway,Crawley, Western Australia 6009, Australia. E-mail: praeger@maths.uwa.edu.au

68 Cheryl E. Praegeranalysis. Analogues of this theorem are available for the alternative families of basicpermutation groups. These theorems have become standard tools for studyingfinite combinatorial structures such as vertex-transitive graphs and examples aregiven in §3 of successful analyses for distance transitive graphs and «-arc-transitivegraphs. Some characteristic properties of basic permutation groups, including thesestructure theorems are discussed in §4.Studying the symmetry of a family of finite algebraic or combinatorial systemsoften leads to problems about groups of automorphisms acting as basic permutationgroups on points or vertices. In particular determining the full automorphism groupof such a system sometimes requires a knowledge of the permutation groups containinga given basic permutation group, and for this it is important to understandthe lattice of basic permutation groups on a given set. The fundamental problemhere is that of classifying all inclusions of one basic permutation group in another,and integral to its solution is a proper understanding of the factorisations of simpleand characteristically simple groups. In §3 and §4 we outline the current status ofour knowledge about such inclusions and their use.The precision of our current knowledge of basic permutation groups dependsheavily on the classification of the finite simple groups. Some problems aboutbasic permutation groups translate directly to questions about simple groups, andanswering them leads to new results about simple groups. Several of these resultsand their connections with basic groups are discussed in the final section §5.In summary, this approach to analysing finite permutation groups involvesan interplay between combinatorics, group actions, and the theory of finite simplegroups. One measure of its success is its effectiveness in combinatorial applications.2. Defining basic permutation groupsLet G be a subgroup of the symmetric group Sym(Q) of all permutations of afinite set 0. Since an intransitive permutation group is contained in the direct productof its transitive constituents, it is natural when studying permutation groupsto focus first on the transitive ones. Thus we will assume that G is transitive on Q.Choose a point a £ ii and let G a denote the subgroup of G of permutations thatfix a, that is, the stabiliser of a. Let Sub(G,G a ) denote the lattice of subgroupsof G containing G a . The concepts introduced below are independent of the choiceof a because of the transitivity of G. We shall introduce three types of basic permutationgroups, relative to C\ := Sub(G,G a ) and two other types of lattices £2and £3, where we regard each £ t as a function that can be evaluated on any finitetransitive group G and stabiliser G a .For G a < H < G, the If-orbit containing a is a H = {a h | h £ H}. IfG a < H < K < G, then the FJ-images of a H form the parts of a FJ-invariant partitionV(K, H) of a K , and K induces a transitive permutation group Comp(F', H)on V(K,H) called a component of G. In particular the component Comp(G,G a )permutes V(G,G a ) = {{ß} \ß £ 0} in the same way that G permutes 0, and wemay identify G with Comp(G,G a ).For a lattice £ of subgroups of G containing G a , we say that K covers H

Permutation Groups and Normal Subgroups 69in £ if K,H £ £, H < K, and there are no intermediate subgroups lying in £.The basic components of G relative to £ are then defined as all the componentsComp(F', H) for which K covers H in £. Each maximal chain G Q = Go < Gi 2).A finite graph F = (Q, E) consists of a finite set 0 of points, called vertices, anda subset E of unordered pairs from Q called edges. For s > 1, an s-arc of F is a vertexsequence (ao,ai,... ,a s ) such that each {ai,ai + i} is an edge and a,_i ^ a, + i forall i. We usually call a 1-arc simply an arc. Automorphisms of F are permutationsof 0 that leave E invariant, and a subgroup G of the automorphism group Aut(F) is

70 Cheryl E. Praegers-arc-transitive if G is transitive on the «-arcs of F. If F is connected and is regularof valency k > 0 so that each vertex is in k edges, then an «-arc-transitive subgroupG < Aut(F) is in particular transitive on Q and also, if « > 2, on (« — l)-arcs. It isnatural to ask which of the components of this transitive permutation group G on0 act as «-arc-transitive automorphism groups of graphs related to F.For G a < H < G, there is a naturally defined quotient graph T'H with vertexset the partition of 0 formed by the G-images of the set a H , where two such G-images are adjacent in T'H if at least one vertex in the first is adjacent to at least onevertex of the second. If F is connected and G is arc-transitive, then T'H is connectedand G induces an arc-transitive automorphism group of T'H, namely the componentComp(G, H). If IT is a maximal subgroup of G, then Comp(G, H) is both vertexprimitiveand arc-transitive on T'H- This observation enables many questions aboutarc-transitive graphs to be reduced to the vertex-primitive case.Perhaps the most striking example is provided by the family of finite distancetransitive graphs. The distance between two vertices is the minimum number ofedges in a path joining them, and G is distance transive on F if for each i, G istransitive on the set of ordered pairs of vertices at distance i. In particular if Gis distance transitive on F then F is connected and regular, of valency k say. Ifk = 2 then F is a cycle and all cycles are distance transitive, so suppose that k > 3.If T'H has more than two vertices, then Comp(G, H) is distance transitive on T'H,while if T'H has only two vertices then H is distance transitive on a smaller graphF 2 , namely F 2 has a H as vertex set with two vertices adjacent if and only if theyare at distance 2 in F (see for example [12]). Passing to T'H or F 2 respectively andrepeating this process, we reduce to a vertex-primitive distance transitive graph.The programme of classifying the finite vertex-primitive distance transitive graphsis approaching completion, and surveys of progress up to the mid 1990's can befound in [12, 31]. The initial result that suggested a classification might be possibleis the following. Here a group G is almost simple if T < G < Aut(T) for somenonabelian simple group T, and a permutation group G has affine type if G has anelementary abelian regular normal subgroup.Theorem 3.1 [28] If G is vertex-primitive and distance transitive on a finite graphT, then either T is known explicitly, or G is almost simple, or G has affine type.In general, if G is «-arc-transitive on F with « > 2, then none of the componentsComp(G, H) with G Q < H < G is «-arc-transitive on T'H, so there is no hope thatthe problem of classifying finite «-arc-transitive graphs, or even giving a usefuldescription of their structure, can be reduced to the case of vertex-primitive «-arctransitivegraphs. However the class of «-arc transitive graphs behaves nicely withrespect to normal quotients, that is, quotients T'H where H = G a N for some normalsubgroup N of G. For such quotients, the vertex set of T'H is the set of iV-orbits,G acts «-arc-transitively on T'H, and if T'H has more than two vertices then F is acover of T'H in the sense that, for two iV-orbits adjacent in T'H, each vertex in oneiV-orbit is adjacent in F to exactly one vertex in the other iV-orbit. We say that F isa normal cover of T'H- If in addition N is a maximal intransitive normal subgroupof G with more than two orbits, then G is both vertex-quasiprimitive and «-arctransitiveon T'H, see [24]. If some quotient T'H has two vertices then F is bipartite,

Permutation Groups and Normal Subgroups 71and such graphs require a specialised analysis that parallels the one described here.On the other hand if F is not bipartite then F is a normal cover of at least oneT'H on which the G-action is both vertex-quasiprimitive and «-arc-transitive. Thewish to understand quasiprimitive «-arc transitive graphs led to the developmentof a theory for finite quasiprimitive permutation groups similar to the theory offinite primitive groups. Applying this theory led to a result similar to Theorem 3.1,featuring two additional types of quasiprimitive groups, called twisted wreath typeand product action type. Descriptions of these types may be found in [24] and [25].Theorem 3.2 [24] If G is vertex-quasiprimitive and s-arc-transitive on a finitegraph T with « > 2, then G is almost simple, or of affine, twisted wreath or productaction type.Examples exist for each of the four quasiprimitive types, and moreover thisdivision of vertex-quasiprimitive «-arc transitive graphs into four types has resultedin a better understanding of these graphs, and in some cases complete classifications.For example all examples with G of affine type, or with T < G < Aut(T) andT = PSL 2 (y),Sz(g) or Ree(q) have been classified, in each case yielding new «-arctransitive graphs, see [13, 25]. Also using Theorem 3.2 to study the normal quotientsof an «-arc transitive graph has led to some interesting restrictions on the numberof vertices.Theorem 3.3 [15, 16] Suppose that T is a finite s-arc-transitive graph with « > 4.Then the number of vertices is even and not a power of 2.The concept of a normal quotient has proved useful for analysing many familiesof edge-transitive graphs, even those for which a given edge-transitive group isnot vertex-transitive. For example it provides a framework for a systematic studyof locally «-arc-transitive graphs in which quasiprimitive actions are of central importance,see [11].We have described how to form primitive arc-transitive quotients of arc-transitivegraphs, and quasiprimitive «-arc-transitive normal quotients of non-bipartite«-arc-transitive graphs. However recognising these quotients is not always easywithout knowing their full automorphism groups. To identify the automorphismgroup of a graph, given a primitive or quasiprimitive subgroup G of automorphisms,it is important to know the permutation groups of the vertex set that contain G,that is the over-groups of G. In the case of finite primitive arc-transitive andedge-transitive graphs, knowledge of the lattice of primitive permutation groups onthe vertex set together with detailed knowledge of finite simple groups led to thefollowing result. The socle of a finite group G, denoted soc(G), is the product ofits minimal normal subgroups.Theorem 3.4 [22] Let G be a primitive arc- or edge-transitive group of automorphismsof a finite connected graph T. Then either G and Aut(F) have the samesocle, or G < H < Aut(F) where soc(G) ^ soc(H) and G, H are explicitly listed.In the case of graphs F for which a quasiprimitive subgroup G of Aut(F) isgiven, it is possible that Aut(F) may not be quasiprimitive. However, even in this

72 Cheryl E. Praegercase a good knowledge of the quasiprimitive over-groups of a quasiprimitive groupis helpful, for if N is a maximal intransitive normal subgroup of Aut(F) then both Gand Aut(F) induce quasiprimitive automorphism groups of the normal quotient T'H,where H = Aut(F) Q iV, and the action of G is faithful. This approach was used,for example, in classifying the 2-arc transitive graphs admitting Sz(q) or Ree(q)mentioned above, and also in analysing the automorphism groups of Cayley graphsof simple groups in [8].Innately transitive groups, identified in § as a third possibility for basic groups,have not received much attention until recently. They arise naturally when investigatingthe full automorphism groups of graphs. One example is given in [7] forlocally-primitive graphs F admitting an almost simple vertex-quasiprimitive subgroupG of automorphisms. It is shown that either Aut(F) is innately transitive, orG is of Lie type in characteristic p and Aut(F) has a minimal normal p-subgroupinvolving a known G-module.4. Characteristics of basic permutation groupsFinite primitive permutation groups have attracted the attention of mathematiciansfor more than a hundred years. In particular, one of the central problemsof 19th century Group Theory was to find an upper bound, much smaller than n!,for the order of a primitive group on a set of size n, other than the symmetric groupS n and the alternating group A n . It is now known that the largest such groupsoccur for n of the form c(c — l)/2 and are S c and A c acting on the unordered pairsfrom a set of size c. The proofs of this and other results in this section depend onthe finite simple group classification.If G is a quasiprimitive permutation group on Q, a £ 0, and IT is a maximalsubgroup of G containing G Q , then the primitive component Comp(G, H) isisomorphic to G since the kernel of this action is an intransitive normal subgroupof G and hence is trivial. Because of this we may often deduce information aboutquasiprimitive groups from their primitive components, and indeed it was found in[29] that finite quasiprimitive groups possess many characteristics similar to thoseof finite primitive groups. This is true also of innately transitive groups. We statejust one example, concerning the orders of permutation groups acting on a set ofsize n, that is, of degree n.Theorem 4.1 [4, 29] There is a constant c and an explicitly defined family T offinite permutation groups such that, if G is a primitive, quasiprimitive, or innatelytransitive permutation group of degree n, then either G £ T, or \G\ < n clog ".The O'Nan-Scott Theorem partitions the finite primitive permutation groupsinto several disjoint types according to the structure or action of their minimalnormal subgroups. It highlights the role of simple groups and their representationsin analysing and using primitive groups. One of its first successful applications wasthe analysis of distance transitive graphs in Theorem 3.1. Other early applicationsinclude a proof [6] of the Sims Conjecture, and a classification result [18] for maximalsubgroups of A n and S n , both of which are stated below.

Permutation Groups and Normal Subgroups 73Theorem 4.2 [6] There is a function f such that if G is primitive on a finite setii, and for a £ ii, G a has an orbit of length d in ii\ {a}, then \G a \ < f(d).Theorem 4.3 [18] Let G = A n or S n with M a maximal subgroup. Then either Mbelongs to an explicit list or M is almost simple and primitive. Moreover if H < Gand H is almost simple and primitive but not maximal, then (H, n) is known.This is a rather curious way to state a classification result. However it seemsalmost inconceivable that the finite almost simple primitive groups will ever belisted explicitly. Instead [18] gives an explicit list of triples (H,M,n), where H isprimitive of degree n with a nonabelian simple normal subgroup T not normalisedby M, and H < M < HA n . This result suggested the possibility of describing thelattice of all primitive permutation groups on a given set, for it gave a descriptionof the over-groups of the almost simple primitive groups. Such a description wasachieved in [23] using a general construction for primitive groups called a blow-upconstruction introduced by Kovacs [14]. The analysis leading to Theorem 3.4 wasbased on this theorem.Theorem 4.4 [23] All inclusions G < H < S n with G primitive are either explicitlydescribed, or are described in terms of a blow-up of an explicitly listed inclusionGi < Hi < S ni with n a proper power of rii.Analogues of the O'Nan-Scott Theorem for finite quasiprimitive and innatelytransitivegroups have been proved in [3, 24] and enable similar analyses to be undertakenfor problems involving these classes of groups. For example, the quasiprimitiveversion formed the basis for Theorems 3.2 and 3.3. It seems to be the most usefulversion for dealing with families of vertex-transitive or locally-transitive graphs. Adescription of the lattice of quasiprimitive subgroups of S n was given in [2, 26] andwas used, for example, in analysing Cayley graphs of finite simple groups in [8].Theorem 4.5 [2, 26] Suppose that G < H < S n with G quasiprimitive and imprimitive,and H quasiprimitive but H ^ A n . Then either G and H have equal soclesand the same O'Nan-Scott types, or the possibilities for the O'Nan-Scott types ofG, H are restricted and are known explicitly.In the latter case, for most pairs of O'Nan-Scott types, explicit constructionsare given for these inclusions. Not all the types of primitive groups identified by theO'Nan-Scott Theorem occur for every degree n. Let us call permutation groups ofdegree n other than A n and S n nontrivial. A systematic study by Cameron, Neumannand Teague [5] of the integers n for which there exists a nontrivial primitivegroup of degree n showed that the set of such integers has density zero in the naturalnumbers. Recently it was shown in [30] that a similar result holds for the degreesof nontrivial quasiprimitive and innately transitive permutation groups. Note that2-2 < 2^d=i 'dJÏSj < 2-23-Theorem 4.6 [5, 30] For a positive real number x, the proportion of integers n < xfor which there exists a nontrivial primitive, quasiprimitive, or innately transitivepermutation group of degree n is at most (1 + o(l))c/ Ioga:, where c = 2 in the caseof primitive groups, or c= 1 + X^dLi déid) f or îe °îer cases -

74 Cheryl E. Praeger5. Simple groups and basic permutation groupsMany of the results about basic permutation groups mentioned above relyonspecific knowledge about finite simple groups. Sometimes this knowledge wasalready available in the simple group literature. However investigations of basicpermutation groups often raised interesting new questions about simple groups.Answering these questions became an integral part of the study of basic groups,and the answers enriched our understanding of finite simple groups. In this finalsection we review a few of these new simple group results. Handling the primitivealmost simple classical groups was the most difficult part of proving Theorem 4.3,and the following theorem of Aschbacher formed the basis for their analysis.Theorem 5.1 [1] Let G be a subgroup of a finite almost simple classical group Xsuch that G does not contain soc(X), and let V denote the natural vector spaceassociated with X. Then either G lies in one of eight explicitly defined families ofsubgroups, or G is almost simple, absolutely irreducible on V and the (projective)representation of soc(G) on V cannot be realised over a proper subfield.A detailed study of classical groups based on Theorem 5.1 led to Theorem 5.2,a classification of the maximal factorisations of the almost simple groups. Thisclassification was fundamental to the proofs of Theorems 3.4 and 4.3, and has beenused in diverse applications, for example see [9, 17].Theorem 5.2 [19, 20] Let G be a finite almost simple group and suppose that G =AB, where A,B are both maximal in G subject to not containing soc(G). ThenG, A, B are explicitly listed.For a finite group G, let n(G) denote the set of prime divisors of \G\. Formany simple groups G there are small subsets of n(G) that do not occur in theorder of any proper subgroup, and it is possible to describe some of these preciselyas follows.Theorem 5.3 [21, Theorem 4, Corollaries 5 and 6] Let G be an almost simple groupwith socle T, and let M be a subgroup of G not containing T.(a) If G = T then for an explicitly defined subset H Ç n(T) with |TI| < 3, ifn C n(M) then T,M are known explicitly, and in most cases n(T) = n(M).(b) If n(T) C n(M) then T,M are known explicitly.Theorem 5.3 was used in [10] to classify all innately transitive groups having nofixed-point-free elements of prime order, settling the polycirculant graph conjecturefor such groups. Another application of Theorems 5.2 and 5.3 is the followingfactorisation theorem that was used in the proof of Theorem 4.5. It implies inparticular that, if G is quasiprimitive of degree n with nonabelian and non-simplesocle, then S n and possibly A n are the only almost simple over-groups of G.Theorem 5.4 [26, Theorem 1.4] Let T,S be finite nonabelian simple groups suchthat T has proper subgroups A, B with T = AB and A = S e for some £ > 2. ThenT = A n , B = A n -i, where n = \T : B\, and A is a transitive group of degree n.

Permutation Groups and Normal Subgroups 75Finally we note that Theorem 4.6 is based on the following result about indicesof subgroups of finite simple groups.Theorem 5.5 [5, 30] For a positive real number x, the proportion of integers n < xof the form n = \T : M\, where T is a nonabelian simple group and M is eithera maximal subgroup or a proper subgroup, and (T,M) ^ (A n ,A nî), is at most(1 + o(lj)c/Ioga:, where c = 1 or c= X^dLi 11(d)res P ec tively.We have presented a framework for studying finite permutation groups byidentifyingand analysing their basic components. The impetus for extending thetheory beyond primitive groups came from the need for an appropriate theory ofbasic permutation groups for combinatorial applications. Developing this theoryrequiredthe answers to specific questions about simple groups, and the power ofthe theory is largely due to its use of the finite simple group classification.References[i[2[3;[4;[6;[r[9[io;[H[12:[is;[w:M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent.Math. 76 (1984), 469^514.R. Baddeley and C. E. Praeger, On primitive overgroups of quasiprimitivepermutation groups, Research Report No. 2002/3, U. Western Australia, 2002.J. Bamberg and C. E. Praeger, Finite permutation groups with a transitiveminimal normal subgroup, preprint, 2002.P. J. Cameron, Finite permutation groups and finite simple groups, Bull. LondonMath. Soc. 13 (1981), 1-22.P. J. Cameron, P. M. Neumann and D. N. Teague, On the degrees of primitivepermutation groups, Math. Z. 180 (1982), 14H49.P. J. Cameron, C. E. Praeger, J. Saxl, and G. M. Seitz, On the Sims conjectureand distance transitive graphs, Bull. London Math. Soc. 15 (1983), 499^506.X. G. Fang, G. Havas, and C. E. Praeger, On the automorphism groups ofquasiprimitive almost simple graphs, J. Algebra 222 (1999), 271^283.X. G. Fang, C. E. Praeger and J. Wang, On the automorphism groups of Cayleygraphs of finite simple groups, J. London Math. Soc. (to appear).M. D. Fried, R. Guralnick and J. Saxl, Schur covers and Carlitz's conjecture,Israel J. Math. 82 (1993), 157^225.M. Giudici, Quasiprimitive groups with no fixed point free elements of primeorder, J. London Math. Soc, (to appear).M. Giudici, C. H. Li and C. E. Praeger, Analysing finite locally «-arc-transitivegraphs, in preparation.A. A. Ivanov, Distance-transitive graphs and their classification, in Investigationsin algebraic theory of combinatorial objects, Kluwer, Dordrecht, 1994,283^378.A. A. Ivanov and C. E. Praeger, On finite affine 2-arc transitive graphs, EuropeanJ. Combin. 14 (1993), 421-144.L. G. Kovacs, Primitive subgroups of wreath products in product action, ProcLondon Math. Soc. (3) 58 (1989), 306^322.

76 Cheryl E. Praeger[15] C. H. Li, Finite s-arc transitive graphs of prime-power order, Bull. LondonMath. Soc. 33 (2001), 129-137.[16] C. H. Li, On finite «-are transitive graphs of odd order, J. Combin. TheorySer. B 81 (2001), 307-317.[17] C. H. Li, The finite vertex-primitive and vertex-biprimitive «-transitive graphsfor « > 4, Trans. Amer. Math. Soc. 353 (2001), 3511-3529.[18] M. W. Liebeck, C. E. Praeger and J. Saxl, A classification of the maximalsubgroups of the finite alternating and symmetric groups, Proc London Math.Soc. 55 (1987), 299-330.[19] M. W. Liebeck, C. E. Praeger and J. Saxl, The maximal factorisations of thefinite simple groups and their automorphism groups, Mem. Amer. Math. Soc.No. 432, Vol. 86 (1990), 1-151.[20] M. W. Liebeck, C. E. Praeger and J. Saxl, On factorisations of almost simplegroups, J. Algebra 185 (1996), 409-119.[21] M. W. Liebeck, C. E. Praeger and J. Saxl, Transitive subgroups of primitivepermutation groups, J. Algebra 234 (2000), 291-361.[22] M. W. Liebeck, C. E. Praeger and J. Saxl, Primitive permutation groups witha common suborbit, and edge-transitive graphs, Proc. London Math. Soc. (3)84 (2002), 405-138.[23] C. E. Praeger, The inclusion problem for finite primitive permutation groups,Proc. London Math. Soc. (3) 60 (1990), 68-88.[24] C. E. Praeger, An O'Nan-Scott theorem for finite quasiprimitive permutationgroups and an application to 2-arc transitive graphs, J. London Math. Soc. (2)47 (1993), 227-239.[25] C. E. Praeger, Quasiprimitive graphs. In Surveys in combinatorics, 1997 (London),65-85, Cambridge University Press, Cambridge, 1997.[26] C. E. Praeger, Quotients and inclusions of finite quasiprimitive permutationgroups, Research Report No. 2002/05, University of Western Australia, 2002.[27] C. E. Praeger, Seminormal and subnormal subgroup lattices for transitive permutationgroups, in preparation.[28] C. E. Praeger, J. Saxl and K. Yokoyama, Distance transitive graphs and finitesimple groups, Proc. London Math. Soc. (3) 55 (1987), 1-21.[29] C. E. Praeger and A. Shalev, Bounds on finite quasiprimitive permutationgroups, J. Austral Math. Soc. 71 (2001), 243-258.[30] C. E. Praeger and A. Shalev, Indices of subgroups of finite simple groups andquasiprimitive permutation groups, preprint, 2002.[31] J. van Bon and A. M. Cohen, Prospective classification of distance-transitivegraphs, in Combinatorics '88 (Ravello), Mediterranean, Rende, 1991, 25-38.

ICM 2002 • Vol. II • 77-85Norm Varieties and Algebraic CobordismMarkus Rost*AbstractWe outline briefly results and examples related with the bijectivity of thenorm residue homomorphism. We define norm varieties and describe someconstructions. We discuss degree formulas which form a major tool to handlenorm varieties. Finally we formulate Hubert's 90 for symbols which is thehard part of the bijectivity of the norm residue homomorphism, modulo atheorem of Voevodsky.IntroductionThis text is a brief outline of results and examples related with the bijectivityof the norm residue homomorphism—also called "Bloch-Kato conjecture" and, forthe mod 2 case, "Milnor conjecture".The starting point was a result of Voevodsky which he communicated in 1996.Voevodsky's theorem basically reduces the Bloch-Kato conjecture to the existenceof norm varieties and to what I call Hilbert's 90 for symbols. Unfortunately thereis no text available on Voevodsky's theorem.In this exposition p is a prime, k is a field with charfc ^ p and K^fk denotesMilnor's n-th FJ-group of k [15], [19].Elements in K^fk/p of the formu = {cti,..., a n } mod pare called symbols (modp, of weight n).A field extension F of k is called a splitting field of u if u E = 0 inLetKffF/p.be the norm residue homomorphism.h {n}P y.K^k/p^Hg(k,pf n ),{ai,...,a„} H> (ai,...,a n )* Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH43210, USA. E-mail: rost@math.ohio-state.edu, URL: http://www.math.ohio-state.edu/~rost

78 Markus Rost1. Norm varietiesAll successful approaches to the Bloch-Kato conjecture consist of an investigationof appropriate generic splitting varieties of symbols. This goes back to thework of Merkurjev and Suslin on the case n = 2 who studied the FJ-cohomology ofSeveri-Brauer varieties [12]. Similarly, for the case p = 2 (for n = 3 by Merkurjev,Suslin [14] and the author [18], for all n by Voevodsky [23]) one considers certainquadrics associated with Pfister forms. For a long time it was not clear which sort ofvarieties one should consider for arbitrary n, p. In some cases one knew candidates,but these were non-smooth varieties and desingularizations appeared to be difficultto handle. Finally Voevodsky proposed a surprising characterization of the necessaryvarieties. It involves characteristic numbers and yields a beautiful relationbetween symbols and cobordism theory.Definition. Let u = {cti,... ,a n } modp be a symbol. Assume that u ^ 0. Anorm variety for u is a smooth proper irreducible variety X over k such that(1) The function field k(X) of X splits u.(2) 6àmX = d:=p n - 1 -l.(3) ^ i ^ 0 mod p.Here Sd(X) £ Z denotes the characteristic number of X given by the d-thNewton polynomial in the Chern classes of TX. It is known (by Milnor) that indimensions d = p n — 1 the number Sd(X) is p-divisible for any X. If k C C onemay rephrase condition (3) by saying that X(C) is indecomposable in the complexcobordism ring mod p.We will observe in section 2. that the conditions for a norm variety are birationalinvariant.The name "norm variety" originates from some constructions of norm varieties,see section 3..We conclude this section with the "classical" examples of norm varieties.Example. The case n = 2. Assume that k contains a primitive p-th root ( ofunity. For a, b £ k* let Aç (a, 6) be the central simple fc-algebra with presentationAç(a,b) = (u,v | u p = a,v p = b,vu = (uv).The Severi-Brauer variety X(a,6) of Aç(a,6) is a norm variety for the symbol{a, 6} mod p.Example. The case p = 2. For a,\, ..., a n £ k* one denotes by((ai,...,a n ))n=ÇQ{l,-a,i),lthe associated n-fold Pfister form [9], [21]. The quadratic formip= ((ai,...,a n -i)) ± {-a n )is called a Pfister neighbor. The projective quadric Q(*p) defined by (p = 0 is a normvariety for the symbol {cti,..., a n } mod 2.

2. Degree formulasNorm Varieties and Algebraic Cobordism 79The theme of "degree formulas" goes back to Voevodsky's first text on theMilnor conjecture (although he never formulated explicitly a "formula") [22]. Inthis section we formulate the degree formula for the characteristic numbers Sd- Itshows the birational invariance of the notion of norm varieties.The first proof of this formula relied on Voevodsky's stable homotopy theoryof algebraic varieties. Later we found a rather elementary approach [11], which isin spirit very close to "elementary" approaches to the complex cobordism ring [16],[4]-For our approach to Hilbert's 90 for symbols we use also "higher degree formulas"which again were first settled using Voevodsky's stable homotopy theory [3].These follow meanwhile also from the "general degree formula" proved by Moreland Levine [10] in characteristic 0 using factorization theorems for birational mapsWe fix a prime p and a number d of the form d = p n — 1.For a proper variety X over k letI(X) = deg(CH 0 (X)) C Zbe the image of the degree map on the group of 0-cycles. One has L(X) = i(X)Zwhere i(X) is the "index" of X, i. e., the gcd of the degrees [k(x) : k] of the residueclass field extensions of the closed points a: of X. If X has a fc-point (in particular ifk is algebraically closed), then L(X) = Z. The group L(X) is a birational invariantof X. We putJ(X) = L(X)+pZ.Let X, Y be irreducible smooth proper varieties over k with dim F = dimX =d and let / : Y —t X be a morphism. Define deg / as follows: If dim f(Y) < dim X,then deg/ = 0. Otherwise deg/ £ N is the degree of the extension k(Y)/k(X) ofthe function fields.Theorem (Degree formula for Sd)-MI) = ( deg/ )M^lmodJ(X).Corollary. The classSd(X)Pmod J(X) G Z/ J(X)is a birational invariant.Remark. If X has a fc-rational point, then J(X) = Z and the degree formulais empty. The degree formula and the birational invariants Sd(X)/p mod J(X) arephenomena which are interesting only over non-algebraically closed fields. Over thecomplex numbers the only characteristic numbers which are birational invariant arethe Todd numbers.

80 Markus RostWe apply the degree formula to norm varieties. Let « be a nontrivial symbolmod p and let X be a norm variety for u. Since k(X) splits u, so does any residueclass field k(x) for a: G X. As « is of exponent p, it follows that J(X) = pZ.Corollary (Voevodsky). Let u be a nontrivial symbol and let X be a normvariety of u. Let further Y be a smooth proper irreducible variety with dim F =dim X and let f: Y —¥ X be a morphism. Then Y is a norm variety for u if andonly if deg / is prime to p.It follows in particular that the notion of norm variety is birational invariant.Therefore we may call any irreducible variety U (not necessarily smooth or proper)a norm variety of a symbol u if U is birational isomorphic to a smooth and propernorm variety of u.3. Existence of norm varietiesTheorem. Norm varieties exists for every symbol u £ Kffh/p for every pand every n.As we have noted, for the case n = 2 one can take appropriate Severi-Brauervarieties (if k contains the p-th roots of unity) and for the case p = 2 one can takeappropriate quadrics.In this exposition we describe a proof for the case n = 3 using fix-point theoremsof Conner and Floyd in order to compute the non-triviality of the characteristicnumbers. Our first proof for the general case used also Conner-Floyd fix-point theory.Later we found two further methods which are comparatively simpler. Howeverthe Conner-Floyd fix-point theorem is still used in our approach to Hilbert's 90 forsymbols.Let u = {a, b,c} mod p with a, b, c £ k*. Assume that k contains a primitivep-th root ( of unity, let A = AQ (a, 6) and letMS(A,c) = {x £ A | Nrd(ar) = c}.We call MS(A,c) the Merkurjev-Suslin variety associated with A and c. The symbol«is trivial if and only if MS (A, c) has a rational point [12]. The variety MS(,A, c)is a twisted form of SL(p).Theorem. Suppose ujÔ. Then MS(A,c) is a norm variety for u.Let us indicate a proof for a subfield k C C (and for p > 2). Let U = MS (A, c).It is easy to see that k(U) splits u. Moreover one has dim U = dim A — 1 = p 2 — 1.It remains to show that there exists a proper smooth completion X of U withnontrivial characteristic number.LetÜ = {[x,t]£ P(A e k) | Nrd(ar) = ct p }be the naive completion of U. We let the group G = Z/p x Z/p act on the algebra Avia(r, «) • u = ( r u, (r, «) • v = ( s v.This action extends to an action on P(.4 ® k) (with the trivial action on k) whichinduces a G-action on Ü. Let Fix(C7) be the fixed point scheme of this action. One

z = { [ E L ' = I s « t - V , t] £ P ( A e k ) I E f J = i 4 =tP} •Norm Varieties and Algebraic Cobordism 81finds that Tix(Ü) consists just of the p isolated points [1, Ç], i = 1, • • •, P, whichare all contained in U.The variety U is smooth, but Ü is not. However, by equivariant resolutionof singularities [2], there exists a smooth proper G-variety X together with a G-morphism X —t Ü which is a birational isomorphism and an isomorphism over U.It remains to show thatSd(X)5E 0 mod p.PFor this we may pass to topology and try to compute Sd(X(C)). We note thatfor odd p, the Chern number Sd is also a Pontryagin number and depends only onthe differentiable structure of the given variety. Note further that X has the sameG-fixed points as Ü since the desingularization took place only outside U.Consider the varietyThis variety is a smooth hypersurface and it is easy to check— 5É 0 mod p.PAs a G-variety, the variety Z has the same fixed points as X ("same" means thatthe collections of fix-points together with the G-structure on the tangent spacesare isomorphic). Let M be the differentiable manifold obtained from X(C) and—Z(C) by a multi-fold connected sum along corresponding fixed points. Then Mis a G-manifold without fixed points. By the theory of Conner and Floyd [5], [7]applied to (Z/p) 2 -manifolds of dimension d = p 2 — 1 one has= 0 mod p.PThusSd(X) _ Sd(Z)mod pP Pand the desired non-triviality is established.The functions #„. We conclude this section with examples of norm varietiesfor the general case.Let cti, Ü2, ...be a sequence of elements in k*. We define functions $ n =$oi,...,a„ m P n variables inductively as follows.o(t)=t p ,P-i* n (To,...,T p _i) = * n _i(T 0 ) IKl-a^n-iCTO).i=lHere the T t stand for tuples of p" _1 variables. Let U(ai,...,a n ) be the varietydefinedby§a 1 ,...,a n - 1 ( T ) = a„.Theorem. Suppose that the symbol u = {ai,... ,a n } modp is nontrivial.Then U(ai,..., a n ) is a norm variety of u.

82 Markus Rost4. Hubert's 90 for symbolsThe bijectivity of the norm residue homomorphisms has always been consideredas a sort of higher version of the classical Hilbert's Theorem 90 (which establishesthe bijectivity for n = 1). In fact, there are various variants of the Bloch-Katoconjecture which are obvious generalizations of Hilbert's Theorem 90: The Hilbert'sTheorem 90 for Kff of cyclic extensions or the vanishing of the motivic cohomologygroup H n+1 (k, Z(n)). In this section we describe a variant which on one hand isvery elementary to formulate and on the other hand is the really hard part of theBloch-Kato conjecture (modulo Voevodsky's theorem).Let u = {cti,... ,a n } £ Kffh/p be a symbol. Consider the norm mapA4 = ^2 N F/k • 0FFKiF -• Kikwhere F runs through the finite field extensions of k (contained in some algebraicclosure of k) which split u. Hilbert's Theorem 90 for u states that ker A4 is generatedby the "obvious" elements.To make this precise, we consider two types of basic relations between thenorm maps N F / k .Let Fi, F 2 be finite field extensions of k. Then the sequenceir IT? o,r\ ( N F 1 IS,F 2 /F 1 ,-N FIIS!F2/F2 ) N Fl/k +Np 2/kKi(Li®i

Norm Varieties and Algebraic Cobordism 83Hilbert's 90 for symbols. For every symbol u the norm mapA4: H 0 (u,Ki)^Kikis injective.Example. If u = 0, then it is easy to see that N u is injective. In fact, it is atrivial exercise to check that A4 is injective.Example. The case n = 1. The splitting fields F of « = {a} modp areexactly the field extensions of k containing a p-th root of a. It is an easy exercise toreduce the injectivity of N u (in fact of A4) to the classical Hilbert's Theorem 90,i. e., the exactness ofKiL ^ KiL ^ > Kikfor a cyclic extension L/k of degree p with a a generator of Gal(F/fc).Example. The case n = 2. Assume that k contains a primitive p-th root ( ofunity. The splitting fields F of « = {a, 6} mod p are exactly the splitting fields ofthe algebra Aç(a, 6). One can show thatH 0 (u,Ki) = KiA ç (a,b)with N u corresponding to the reduced norm map Nrd [13]. Hence in this caseHilbert's 90 for u reduces to the classical fact SKiA = 0 for central simple algebrasof prime degree [6].Example. The case p = 2. The splitting fields F of « = {cti,... ,a n } mod2 are exactly the field extensions of k which split the Pfister form ((ai,..., a n ))or, equivalently, over which the Pfister neighbor ((cti,..., a n _i)) _L (—a n ) becomesisotropic. Hilbert's 90 for symbols mod 2 had been first established in [17]. Thistext considered similar norm maps associated with any quadratic form (which arenot injective in general). A treatment of the special case of Pfister forms is containedin [8].Remark. One can show that the group H 0 (u,Ki) as defined above is alsothe quotient of (B F KiF by the F-trivial elements in ker A4. This is quite analogousto the description of KiA of a central simple algebra A: The group KiA is thequotient of A* by the subgroup of F-trivial elements in the kernel of Nrd : A* —t F*.Similarly for the case p = 2: In this case the injectivity of N u is related with thefact that for Pfister neighbors ip the kernel of the spinor norm SO(ip) —¥ k*/(k*) 2is F-trivial.In our approach to Hilbert's 90 for symbols one needs a parameterization ofthe splitting fields of symbols.Definition. Let u = {ai,... ,a n } modp be a symbol. A p-generic splittingvariety for u is a smooth variety X over k such that for every splitting field Fof u there exists a finite extension F'/F of degree prime to p and a morphismSpecF'^X.Theorem. Suppose charfc = 0. Let m > 3 and suppose for n < m and everysymbol u = {cti,... ,a n } mod p over all fields over k there exists a p-generic splittingvariety for u of dimension p" _1 — 1. Then Hilbert's 90 holds for such symbols.The proof of this theorem is outlined in [20].

84 Markus RostFor n = 2 one can take here the Severi-Brauer varieties and for n = 3 theMerkurjev-Suslin varieties. Hence we have:Corollary. Suppose charfc = 0. Then Hilbert's 90 holds for symbols ofweight < 3.References[i[2:[3;[4;[*.[6;[r[8[9[io;[n[12:[is;[w:[15[16D. Abramovich, K. Karu, K. Matsuki, and J. Wlodarczyk, Tarification andfactorization of birational maps, J. Amer. Math. Soc. 15 (2002), no. 3, 531^572 (electronic).E. Bierstone and P. D. Milman, Canonical desingularization in characteristiczero by blowing up the maximum strata of a local invariant, Invent. Math. 128(1997), no. 2, 207^302.S. Borghesi, Algebraic Morava K-theories and the higher degree formula, thesis,2000, Evanston, http://www.math.uiuc.edu/K-theory/0412S. Buoncristiano and D. Hacon, The geometry of Chern numbers, Ann. of Math.(2) 118 (1983), no. 1, 1-7.P. E. Conner and E. E. Floyd, Differentiable periodic maps, Academic PressInc., Publishers, New York, 1964, Ergebnisse der Mathematik und ihrer Grenzgebiete,N. F., Band 33.P. K. Draxl, Skew fields, London Mathematical Society Lecture Note Series,vol. 81, Cambridge University Press, Cambridge, 1983.E. E. Floyd, Actions of (Z p ) k without stationary points, Topology 10 (1971),327^336.B. Kahn, La conjecture de Milnor (d'après V. Voevodsky), Astérisque (1997),no. 245, Exp. No. 834, 5, 379^418, Séminaire Bourbaki, Vol. 1996/97.T. Y. Lam, The algebraic theory of quadratic forms, Benjamin/Cummings PublishingCo. Inc. Advanced Book Program, Reading, Mass., 1980, Revised secondprinting, Mathematics Lecture Note Series.M. Levine and F. Morel, Algebraic cobordism I, preprint, 2002,http://www.math.uiuc.edu/K-theory/0547A. S. Merkurjev, Degree formula, notes, May 2000,http://www.math.ohio-state.edu/~rost/chain-lemma.htmlA. S. Merkurjev and A. A. Suslin, K-cohomology of Severi-Brauer varietiesand the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46(1982), no. 5, 1011-1046, 1135-1136 (Russian), [Math. USSR Izv. 21 (1983),307-340]., The group of K\ -zero-cycles on Severi-Brauer varieties, Nova J. AlgebraGeom. 1 (1992), no. 3, 297^315., Norm residue homomorphism of degree three, Izv. Akad. Nauk SSSRSer. Mat. 54 (1990), no. 2, 339^356 (Russian), [Math. USSR Izv. 36 (1991),no. 2, 349-367], also: LOMI-preprint (1986).J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1970),318^344.D. Quillen, Elementary proofs of some results of cobordism theory using Steen-

Norm Varieties and Algebraic Cobordism 85rod operations, Advances in Math. 7 (1971), 29^56 (1971).[17] M. Rost, On the spinar norm and A 0 (X,Ki) for quadrics, preprint, 1988,http://www.math.ohio-state.edu/~rost/spinor.html[18] , Hilbert 90 for FJ 3 for degree-two extensions, Preprint, 1986.[19] , Chow groups with coefficients, Doc. Math. 1 (1996), No. 16, 319^393(electronic).[20] , Chain lemma for splitting fields of symbols, preprint, 1998,http://www.math.ohio-state.edu/~rost/chain-lemma.html[21] W. Scharlau, Quadratic and Hermitian forms, Grundlehren der mathematischenWissenschaften, vol. 270, Springer-Verlag, Berlin, 1985.[22] V. Voevodsky, The Milnor conjecture, preprint, 1996, Max-Planck-Institute forMathematics, Bonn, http://www.math.uiuc.edu/K-theory/0170/[23] , On 2-torsion in motivic cohomology, preprint, 2001,http://www.math.uiuc.edu/K-theory/0502/

ICM 2002 • Vol. II • 87^92Diophantine Geometry over Groupsand the Elementary Theory ofFree and Hyperbolic Groups*Z. SelatAbstractWe study sets of solutions to equations over a free group, projections ofsuch sets, and the structure of elementary sets defined over a free group. Thestructre theory we obtain enable us to answer some questions of A. Tarski's,and classify those finitely generated groups that are elementary equivalent toa free group. Connections with low dimensional topology, and a generalizationto (Gromov) hyperbolic groups will also be discussed.2000 Mathematics Subject Classification: 14, 20.Sets of solutions to equations defined over a free group have been studiedextensively, mostly since Alfred Tarski presented his fundamental questions on theelementary theory of free groups in the mid 1940's. Considerable progress in thestudy of such sets of solutions was made by G. S. Makanin, who constructed analgorithm that decides if a system of equations defined over a free group has asolution [Mai], and showed that the universal and positive theories of a free groupare decidable [Ma2]. A. A. Razborov was able to give a description of the entire setof solutions to a system of equations defined over a free group [Ra], a descriptionthat was further developed by O. Kharlampovich and A. Myasnikov [Kh-My].A set of solutions to equations defined over a free group is clearly a discreteset, and all the previous techniques and methods that studied these sets are combinatorialin nature. Naturally, the structure of sets of solutions defined over a freegroup is very different from the structure of sets of solutions (varieties) to systems ofequations defined over the complexes, reals or a number field. Still, perhaps surprisingly,concepts from complex algebraic geometry and from Diophantine geometrycanbe borrowed to study varieties defined over a free group.*Partially supported by an Israel academy of sciences fellowship, an NSF grant DMS9729992through the IAS, and the IHES.fHebrew University, Jerusalem 91904, Israel. E-mail: zlil@math.huji.ac.il

88 Z. SelaIn this work we borrow concepts and techniques from geometric group theory,low dimensional topology, and Diophantine geometry to study the structure ofvarieties defined over a free (and hyperbolic) group. Our techniques and point ofview on the study of these varieties is rather different from any of the pre-existingtechniques in this field, though, as one can expect, some of our preliminary resultsoverlap with previously known ones. The techniques and concepts we use enablethe study of the structure of varieties defined over a free group and their projections(Diophantine sets), and in particular, give us the possibility to answer somequestions that seem to be essential in any attempt to understand the structure ofelementary sentences and predicates defined over a free (and hyperbolic) group.In this note we summarize the main results of our work, that enable one toanswer affirmatively some of A. Tarski 's problems on the elementary theory of a freegroup, and classify those finitely generated groups that are elementary equivalent toa (non-abelian) free group, we further survey some of our results on the elementarytheoryof a (torsion-free) hyperbolic group, that generalize the results on free groups.The work itself appears in [Sel]-[Se8].We start with what we see as the main result on the elementary theory of afree group we obtained - quantifier elimination. Quantifier elimination and its proofis behind all the other results presented in this note.Theorem 1 ([Se7],l). Let F be a non-abelian free group, and let Q(p) be a definableset over F. Then Q(p) is in the Boolean algebra of AE sets over F.In fact it is possible to give a strengthening of theorem 1 that specifies asubclass of AE sets that generates the Boolean algebra of definable sets, a morerefined description that is essential in studying other model-theoretic properties ofthe elementary theory of a free group.Theorem 1 proves that every definable set over a free group is in the Booleanalgebra of AE sets. To answer Tarski's questions on the elementary theory of afree group, i.e., to show the equivalence of the elementary theories of free groupsof various ranks, we need to show that for coefficient free predicates, our quantifierelimination procedure does not depend on the rank of the coefficient group.Theorem 2 ([Se7],2). Let Q(p) be a set defined by a coefficient-free predicate overa group. Then there exists a set L(p) defined by a coefficient-free predicate whichis in the Boolean algebra of AE predicates, so that for every non-abelian free groupF, the sets Q(p) and L(p) are equivalent.Theorem 2 proves that in handling coefficient-free predicates, our quantifierelimination procedure does not depend on the rank of the coefficient (free) group.This together with the equivalence of the AE theories of free groups ([Sa],[Hr]) impliesan affirmative answer to Tarski's problem on the equivalence of the elementarytheoriesof free groups.Theorem 3 ([Se7],3). The elementary theories of non-abelian free groups areequivalent.

Diophantine Geometry over Groups and the Elementary Theory • • • 89Arguments similar to the ones used to prove theorems 2 and 3, enable us toanswer affirmatively another question of Tarski's.Theorem 4 ( [Se7] ,4). Let F k , Fi be free groups for 2 < k < I. Then the standardembedding F k —t Fi is an elementary embedding.More generally, let F, Fi be non-abelian free groups, let F 2 be a free group, andsuppose that F = Fi * F 2 . Then the standard embedding Fi —t F is an elementaryembedding.Tarski's problems deal with the equivalence of the elementary theories of freegroups of different ranks. Our next goal is to get a classification of all the f.g. groupsthat are elementary equivalent to a free group.Non-abelian w-residually free groups (limit groups) are known to be the f.g.groups that are universally equivalent to a non-abelian free group. If a limit groupcontains a free abelian group of rank 2, it can not be elementary equivalent to a freegroup. Hence, a f.g. group that is elementary equivalent to a non-abelian free groupmust be a non-elementary (Gromov) hyperbolic limit group. However, not everynon-elementary hyperbolic limit group is elementary equivalent to a free group.To demonstrate that we look at the following example. Suppose that G = F * F =< 6i, 62 > * < bz, 64 > is a double of a free group of rank 2, suppose that whas no roots in F, and suppose that the given amalgamated product is the abelianJSJ decomposition of the group G. By our assumptions, G is a hyperbolic limitgroup (see [Sel], theorem 5.12).Claim 5 ([Se7],5). The group G = F * F is not elementary equivalent to thefree group F.In section 6 of [Sel] we have presented w-residually free towers, as an exampleof limit groups (the same groups are presented in [Kh-My] as well, and are calledthere NTQ groups).A hyperbolic w-residually free tower is constructed in finitely many steps. Inits first level there is a non-cyclic free product of (possibly none) (closed) surfacegroups and a (possibly trivial) free group, where each surface in this free productis a hyperbolic surface (i.e., with negative Euler characteristic), except the nonorientablesurface of genus 2. In each additional level we add a punctured surfacethat is amalgamated to the group associated with the previous levels along itsboundary components, and in addition there exists a retract map of the obtainedgroup onto the group associated with the previous levels. The punctured surfacesare supposed to be of Euler characteristic bounded above by -2, or a puncturedtorus.The procedure used for eliminating quantifiers over a free group enables us toshow that every hyperbolic w-residually free tower is elementary equivalent to a freegroup. The converse is obtained by using basic properties of the JSJ decompositionand the (canonical) Makanin-Razborov diagram of a limit group ([Se7], theorem 6).

90 Z. SelaTherefore, we are finally able to get a classification of those f.g. groups that areelementary equivalent to a free group.Theorem 6 ([Se7],7). A f.g. group is elementary equivalent to a non-abelian freegroup if and only if it is a non-elementary hyperbolic oj-residually free tower.So far we summarized the main results of our work, that enable one to answeraffirmatively some of A. Tarski's problems on the elementary theory of a free group,and classify those finitely generated groups that are elementary equivalent to a (nonabelian)free group. In the rest of this note we survey some of our results on theelementary theory of a (torsion-free) hyperbolic group, that generalize the resultspresented for a free group.In the case of a free group, we have shown that every definable set is in theBoolean algebra of AE sets. The same holds for a general hyperbolic group.Theorem 7 ([Se8],6.5). LetT be a non-elementary torsion-free hyperbolic group,and let Q(p) be a definable set over T. Then Q(p) is in the Boolean algebra of AEsets over T.Furthermore, ifQ(p) is a set defined by a coefficient-free predicate defined overT, then Q(p) can be defined by a coefficient-free predicate which is in the Booleanalgebra of AE predicates.The procedure used for quantifier elimination over a free group enabled usto get a classification of those f.g. groups that are elementary equivalent to a freegroup (theorem 6). In a similar way, it is possible to get a classification of those f.g.groups that are elementary equivalent to a given torsion-free hyperbolic group.We start with the following basic fact, that shows the elementary invariance ofnegative curvature in groups.Theorem 8 ([Se8],7.10). Let T be a torsion-free hyperbolic group, and let G bea f.g. group. If G is elementary equivalent to T, then G is a torsion-free hyperbolicgroup.Theorem 8 restricts the class of f.g. groups that are elementary equivalent to agiven hyperbolic group, to the class of hyperbolic groups. To present the elementaryclassificationof hyperbolic groups we start with the following basic fact.Proposition 9 ([Se8],7.1). Let Fi,F 2 be non-elementary torsion-free rigid hyperbolicgroups (i.e., Fi and F 2 are freely-indecomposable and do not admit anynon-trivial cyclic splitting). Then Fi is elementary equivalent to F 2 if and only ifFi is isomorphic to F 2 .Proposition 9 implies that, in particular, a uniform lattice in a real rank 1semi-simple Lie group that is not SL 2 (R) is elementary equivalent to another suchlattice if and only if the two lattices are isomorphic, hence, by Mostow's rigiditythe two lattices are conjugate in the same Lie group. By Margulis 's normality andsuper-rigidity theorems, the same hold in higher rank (real) Lie groups.

Diophantine Geometry over Groups and the Elementary Theory • • • 91Theorem 10 ([Se8],7.2). Let Li,L 2 be uniform lattices in real semi-simple Liegroups that are not SL 2 (R). Then Li is elementary equivalent to F 2 if and only ifLi and F 2 are conjugate lattices in the same real Lie group G.Proposition 9 shows that rigid hyperbolic groups are elementary equivalentif and only if they are isomorphic. To classify elementary equivalence classes ofhyperbolic groups in general, we associate with every (torsion-free) hyperbolic groupF, a subgroup of it, that we call the elementary core of F, and denote EC(T). Theelementary core is a retract of the ambient hyperbolic group F, and although it isnot canonical, its isomorphism type is an invariant of the ambient hyperbolic group.The elementary core is constructed iteratively from the ambient hyperbolic groupas we describe in definition 7.5 in [Se8].The elementary core of a hyperbolic group is a prototype for its elementary theory.Theorem 11 ([Se8],7.6). LetT be a non-elementary torsion-free hyperbolic groupthat is not a oj-residually free tower, i.e., that is not elementary equivalent to a freegroup. Then T is elementary equivalent to its elementary core EC(T). Furthermore,the embedding of the elementary core EC(T) in the ambient group T is anelementary embedding.Finally, the elementary core is a complete invariant of the class of groups thatare elementary equivalent to a given (torsion-free) hyperbolic group.Theorem 12 ([Se8],7.9). Le£Fi,F 2 be two non-elementary torsion-free hyperbolicgroups. Then Fi and F 2 are elementary equivalent if and only if their elementarycores EC(T'i) and FC(F 2 ) are isomorphic.Theorem 12 asserts that the elementary class of a torsion-free hyperbolic groupis determined by the isomorphism class of its elementary core. Hence, in order to beable to decide whether two torsion-free hyperbolic groups are elementary equivalentone needs to compute their elementary core, and to decide if the two elementarycoresare isomorphic. Both can be done using the solution to the isomorphismproblem for torsion-free hyperbolic groups.Theorem 13 ([Se8],7.11). LetT'i,T2 be two torsion-free hyperbolic groups. Thenit is decidable if T\ is elementary equivalent to F 2 .References[Hr] E. Hrushovski, private communication.[Kh-My] O. Kharlampovich and A. Myasnikov, Irreducible affine varieties over afree group II, Jour, of Algebra 200 (1998), 517^570.[Mai] G. S. Makanin, Equations in a free group, Math. USSR Izvestiya 21(1983), 449-169.[Ma2], Decidability of the universal and positive theories of a free group,Math. USSR Izvestiya 25 (1985), 75^88.

92 Z. Sela[Ra] A. A. Razborov, On systems of equations in a free group, Ph.D. thesis,Steklov Math, institute, 1987.[Sa] G. S. Sacerdote, Elementary properties of free groups, Transactions Amer.Math. Soc. 178 (1973), 127^138.[Sel-Se8] Z. Sela, Diophantine geometry over groups I-VIII, preprints,www.ma.huji.ac.il/ zlil.

ICM 2002 • Vol. II • 93^103Noneommutative Projective Geometry*J. T. StaffordAbstractThis article describes recent applications of algebraic geometry to noncommutativealgebra. These techniques have been particularly successful indescribing graded algebras of small dimension.2000 Mathematics Subject Classification: 14A22, 16P40, 16W50.Keywords and Phrases: Noneommutative projective geometry, Noetheriangraded rings, Deformations, Twisted homogeneous coordinate rings.1. IntroductionIn recent years a surprising number of significant insights and results in noncommutativealgebra have been obtained by using the global techniques of projectivealgebraic geometry. This article will survey some of these results.The classical approach to projective geometry, where one relates a commutativegraded domain C to the associated variety X = Proj C of homogeneous,nonirrelevant prime ideals, does not generalize well to the noneommutative situation,simply because noneommutative algebras do not have enough ideals. However,there is a second approach, based on a classic theorem of Serre: If C is generatedin degree one, then the categories coh(X) of coherent sheaves on X and qgrC offinitely generated graded C-modules modulo torsion are equivalent.Surprisingly, noneommutative analogues of this idea work very well and havelead to a number of deep results. There are two strands to this approach. First,since X can be reconstructed from coh(X) [21] we will regard coh(X) rather thanX as the variety since this is what generalizes. Thus, given a noneommutativegraded fc-algebra R = (J) F, generated in degree one we will consider qgr R as thecorresponding "noneommutative variety" (the formal definitions will be given in amoment). In particular, we will regard qgr F as a noneommutative curve, respectivelysurface, if dim/. Ri grows linearly, respectively quadratically. This analogyworkswell, since there are many situations in which one can pass back and forth*The author is supported in part by the National Science Foundation under grant DMS-9801148.^Department of Mathematics, The University of Michigan, Ann Arbor, MI 48109-1109, USA.E-mail: jts@umich.edu

94 J. T. Staffordbetween R and qgrF [8] and, moreover, substantial geometric techniques can beapplied to study qgrF. A survey of this approach may be found in [25].The second strand is more concrete. In order to use algebraic geometry tostudy noneommutative algebras we need to be able to create honest varieties fromthose algebras. This is frequently possible and such an approach will form the basisof this survey. Once again, the idea is simple: when R is commutative, the pointsof Proj R correspond to the graded factor modules M = R/I = @ i>0 M t for whichdim*, Mi = 1 for all i. These modules are still defined when R is noneommutativeand are called point modules. In many circumstances the set of all such modules isparametrized by a commutative scheme and that scheme controls the structure ofR.This article surveys significant applications of this idea. Notably:• If R = (J) Ri is a domain such that dim/. F, grows linearly, then qgr R ~coh(X) for a curve X and R can be reconstructed from data on X. Thus,noneommutative curves are commutative (see Section 4).• The noneommutative analogues qgr R of the projective plane can be classified.In this case, the point modules are parametrized by either P 2 (in which caseqgr R ~ P 2 ) or by an cubic curve E c P 2 , in which case data on E determinesR (see Section 2).• For strongly noetherian rings, as defined in Section 5, the point modulesare always parametrized by a projective scheme. However there exist manynoetherian algebras R for which no such parametrization exists. This hasinteresting consequences for the classification of noneommutative surfaces.We now make precise the definitions that will hold throughout this article.All rings will be algebras over a fixed, algebraically closed base field k (althoughmost of the results actually hold for arbitrary fields). A fc-algebra R is calledconnected graded (eg) if F is a finitely generated N-graded fc-algebra R = @ i>0 F,with RQ = k. Note that this forces dim?. F, < oo for all i. Usually, we will assumethat F is generated in degree one in the sense that F is generated by R\ as a k-algebra. If F = ® iGN F, is a right noetherian eg ring then define gr F to be thecategory of finitely generated, Z-graded right F-modules, with morphisms beinggraded homomorphisms of degree zero. Define the torsion subcategory, tors R, tobe the full subcategory of gr F generated by the finite dimensional modules andwrite qgrF = grF/torsF. We write n for the canonical morphism grF —¥ qgrFand set TZ = n(R).One can—and often should—work more generally with all graded F-modulesand all quasi-coherent sheaves of Ox-modules, but two categories are enough.In order to measure the growth of an algebra we use the following dimensionfunction: For a eg ring F = © i>0 Ri, the Gelfand-Kirillov dimension of F isdefined to be GKdim F = inf {a £ R : dinij;(^" =0 F,) < n a for all n >• 0} . Basicfacts about this dimension can be found in [17]. If F is a commutative eg algebrathen GKdim F equals the Krull dimension of F and hence equals dim Proj F + 1.Thus a noneommutative curve, respectively surface, will more formally be definedas qgrF for a eg algebra F with GKdim F = 2, respectively 3.

Noncommutative Projective Geometry 952. Historical backgroundWe begin with a historical introduction to the subject. It really started withthe work of Artin and Schelter [2] who attempted to classify the noneommutativeanalogues F of a polynomial ring in three variables (and therefore of P 2 ). The firstproblem is one of definition. A "noneommutative polynomial ring" should obviouslybe a eg ring of finite global dimension, but this is too general, since it includes thefree algebra. One can circumvent this problem by requiring that dim/. F, growspolynomially, but this still does not exclude unpleasant rings like k{x,y}/(xy) thathas global dimension two but is neither noetherian nor a domain. The solution isto impose a Gorenstein condition and this leads to the following definition:Definition 1 A eg algebra R is called AS-regular of dimension d if gl dim R = d,GKdim F < oo and R is AS-Gorenstein; that is, Ext*(fc,F) = 0 for i ^ d butExt rf (fc, R) = k, up to a shift of degree.One advantage with the Gorenstein hypothesis, for AS-regular rings of dimension3, is that the projective resolution of k is forced to be of the form0 —•+ F —• R (n) —• R (n) —• R —•+ jfc —•+ 0for some n and, as Artin and Schelter show in [2], this gives strong informationon the Hilbert series and hence the defining relations of R. In the process theyconstructed one class of algebras that they were unable to analyse:Example 2 The three-dimensional Sklyanin algebra is the algebraSkl 3 = Skl 3 (a, 6,c) = k{xo,xi,a; 2 }/'(aXjXj+i + bxj+ix» + cx 2 +2 : i £ Z 3 ),where (a, 6, c) £ P 2 \ F, for a (known) set F.The original Sklyanin algebra Skl 4 is a 4-dimensional analogue of Skl 3 discoveredin [23]. Independently of [2], Odesskii and Feigin [18] constructed analogues ofSkl 4 in all dimensions and coined the name Sklyanin algebra. See [13] for applicationsof Sklyanin algebras to another version of noneommutative geometry.In retrospect the reason Skl 3 is hard to analyse is because it depends upon anelliptic curve and so a more geometric approach is required. This approach camein [6] and depended upon the following simple idea. Assume that F is a eg algebrathat is generated in degree one. Define a point module to be a cyclic graded (right)F-module M = @ i>0 M t such that dim?. M t = 1 for all i > 0. The notation isjustified by the fact that, if F were commutative, then such a point module Mwould be isomorphic to k[x] and hence equal to the homogeneous coordinate ring ofa point in Proj R. Point modules are easy to analyse geometrically and this providesan avenue for using geometry in the study of eg rings.We will illustrate this approach for S = Skl 3 . Given a point module M =(J) Mi write Mi = m,fc for some m, £ M t and suppose that the module structureis defined by m»Xj = Ayro, + i for some Ay £ k. If / = Yl fìj' x ì' x j ' 1S > one °f t nerelations for S, then necessarily mo/ = (^/yAojAij)m 2 , whence ^fijôiîj = 0.

96 J. T. StaffordThis defines a subvariety F Ç P(S , J) x PfSJ) = P 2 x P 2 and clearly F parametrizesthe truncated point modules of length three: cyclic F-modules M = M 0 © Afi © Af 2with dimAfj = 1 for 0 of an automorphism a of an ellipticcurve F c P 2 . It follows easily that F also parametrizes the point modules. As amorphism of point modules, a is nothing more than the shift functor M = (J) M t H>M>! [1] = Mi © M 2 © • • • .The next question is how to use E and a to understand Skl 3 . Fortunately,one can create a noneommutative algebra from this data that is closely connectedto Skl 3 . This is the twisted homogeneous coordinate ring of E and is defined asfollows. Let X be a fc-scheme, with a line bundle £ and automorphism a. Setn —1£ n = £ ® £" ® • • • ® £" , where £ T = T*£ denotes the pull-back of £ along anautomorphism r. Then the twisted homogeneous coordinate ring is defined to bethe graded vector space B = B(X,£,a) = k + ® n>1 B n where B n = H°(X, £ n ).The multiplication on B = B(Y,£,a) is defined by the natural mapB n ® k B m = E 0 (X,£ n )® k a n E°(X,£m)— H (X,£ n ) ® k H (X,£^n ) —y H (X, £ n+m ) = B n+m -The ring B has two significant properties. First, the way it has been constructedensures that the natural isomorphism S\ = H°(P 2 ,Ö P 2(1)) = Fi induces aring homomorphism : S —ï B. With a little more work using the Riemann-Rochtheorem one can even show that B = S/gS for some g £ S3. Secondly—and thiswill be explained in more detail in the next section—qgrF = coh(F). The latterfact allows one to obtain a detailed understanding of the structure of B and theformer allows one to pull this information back to S.To summarize, the point modules over the Sklyanin algebra Skl 3 are determinedby an automorphism of an elliptic curve F and the geometry of F allows oneto determine the structure of Skl 3 . As is shown in [6] this technique works moregenerally and this leads to the following theorem.Theorem 3 [6, 26, 27] The AS-regular rings R of dimension 3 are classified. Theyare all noetherian domains with the Hilbert series of a weighted polynomial ringk[x,y,z]; thus the (x,y,z) can be given degrees (a,b,c) other than (1,1,1).Moreover, R always maps homomorphically onto a twisted homogeneous coordinatering B = B(X,£,a), for some scheme X. Thus coh(X) ~ qgrF ^y qgrF.In this result, Artin, Tate and Van den Bergh [6] classified the algebras generatedin degree one, while Stephenson [26, 27] did the general case.There are strong arguments (see [11] or [25, Section 11]) for saying that thenoneommutative analogues of the projective plane are precisely the categories qgr F,where F is an AS-regular ring with the Hilbert series 1/(1 — t) 3 of the unweightedpolynomial ring k[x, y, z]. So consider this class, which clearly includes the Sklyaninalgebra. The second paragraph of the theorem can now be refined to say that eitherX = P 2 , in which case qgrF ~ coh(P 2 ), or X = E is a cubic curve in P 2 . Thus,the theorem can be interpreted as saying that noneommutative projective planes areeither equal to P 2 or contain a commutative curve E.

Noncommutative Projective Geometry 973. Twisted homogeneous coordinate ringsThe ideas from [6] outlined in the last section have had many other applications,but before we discuss them we need to analyse twisted homogeneous coordinaterings in more detail. The following exercise may give the reader a feel for theconstruction.Exercise 4 Perhaps the simplest algebra appearing in the theory of quantumgroups is the quantum (affine) plane k q [x,y] = k{x,y}/(xy — qyx), for q £ k*.Prove that k q [x,y] = B(W 1 ,Opi(l),a) where a is defined by a(a : 6) = (a : qb), for(a:b) £ P 1 .For the rest of the section, fix a fc-scheme X with an in verüble sheaf £ and automorphisma. When a = 1, the homogeneous coordinate ring B(X,£) = B(X,£, 1)is a standard construction and one has Serre's fundamental theorem: If £ is amplethen coh(X) ~ qgr(F). As was hinted in the last section, this does generalize tothe noneommutative case, provided one changes the definition of ampleness. Define£ to be a-ample if, for all T £ coh(X), one has H g (X, T ® £ n ) = 0 for all q > 0and all n >• 0. The naïve generalization of Serre's Theorem then holds.Theorem 5 (Artin-Van den Bergh [7]) Let X be a projective scheme with an automorphisma and let £ be a a-ample invertible sheaf. Then B = B(X,£,a) is aright noetherian eg ring such that qgr(F) ~ coh(X).This begs the question of precisely which line bundles are a-ample. A simpleapplication of the Riemann-Roch Theorem shows thatif X is a curve, then any ample invertible sheaf is a-ample, (3.1)and the converse holds for irreducible curves. This explains why Theorem 5 couldbe applied to the factor of the Sklyanin algebra in the last section.For higher dimensional varieties the situation is more subtle and is describedby the following result, for which we need some notation. Let X be a projectivescheme and write Aûm (X) for the set of Cartier divisors of X modulo numericalequivalence. Let a be an automorphism of X and let P a denote its induced actionon .4jj um (X). Since Aj ium (X) is a finitely generated free abelian group, P a mayberepresented by a matrix and P a is called quasi-unipotent if all the eigenvalues ofthis matrix are roots of unity.Theorem 6 (Keeler [15]) If a be an automorphism of a projective scheme X then:(1) X has a a-ample line bundle if and only if P a is quasi-unipotent. If P a isquasi-unipotent, then all ample line bundles are a-ample.(2) In Theorem 5, B is also left noetherian.There are two comments that should be made about Theorem 6. First, it isstandard that GKdimB(X,£) = 1 + dimX, whenever £ is ample. However, itcan happen that GKdim F (X, £, a) > 1 + dimX. Secondly, one can still constructB(X,£,a) when £ is ample but P a is not quasi-unipotent, but the resulting algebrais rather unpleasant. Indeed, possibly after replacing £ by some £® n , B(X,£,a)will be a non-noetherian algebra of exponential growth. See [15] for the details.

98 J. T. Stafford4. Noneommutative curves and surfacesAs we have seen, twisted homogeneous coordinate rings are fundamental to thestudy of noneommutative projective planes. However, a more natural starting placewould be eg algebras of Gelfand-Kirillov dimension two since, as we suggested in theintroduction, these should correspond to noneommutative curves. Their structureis particularly simple.Theorem 7 [4] Let R be a eg domain of GK-dimension 2 generated in degree one.Then there exists an irreducible curve Y with automorphism a and ample invertiblesheaf £ such that R embeds into the twisted homogeneous coordinate ring B(Y,£,a)with finite index. Equivalently, R n = H°(Y, £ ® £" ® • • • ® £" n ) for n >• 0.By (3.1) we may apply Theorem 5 to obtain part (1) of the next result.Corollary 8 Let R be as in Theorem 7. Then:(1) R is a noetherian domain with qgrF ~ coh(Y). In particular, qgrF ~ qgrCfor the commutative ring C = B(Y,£,M).(2) If \a\ < oo then R is a finite module over its centre. If \a\ = oo, then R is aprimitive ring with at most two height one prime ideals.If F is not generated in degree one, then the analogue of Theorem 7 is moresubtle, since more complicated algebras appear. See [4] for the details. One shouldreally make a further generalization by allowing F to be prime rather than a domainand to allowing k to be arbitrary (since this allows one to consider the projectiveanalogues of classical orders over Dedekind domains). Theorem 7 and Corollary 8do generalize appropriately but the results are more technical. The details can befound in [5].Although these results are satisfying they are really only half of the story.As in the commutative case one would also like to define noneommutative curvesabstractly and then show that they can indeed be described by graded rings of theappropriate form. Such a result appears in [19] but to state it we need a definition.Let C be an Ext-finite abelian category of finite homological dimension withderived category of bounded complexes D b (C). Recall that a cohomological functorH : D b (C) —¥ mod(fc) is of finite type if, for A £ D b (C), only a finite numberof the H(A[n]) are non-zero. The category C is saturated if every cohomologicalfunctor H : D b (C) —¥ mod(fc) of finite type is of the form Hom(A, —) (that is, H isrepresentable). If X is a smooth projective scheme, then coh(X) is saturated [10],so it is not unreasonable to use this as part of the definition of a "noneommutativesmooth curve."Theorem 9 (Reiten-Van den Bergh [19, Theorem V.l.2]) Assume thatC is a connectedsaturated hereditary noetherian category. Then C has one of the followingforms:(1) mod(A) where A is an indecomposable finite dimensional hereditary algebra.(2) coh(ö) where Ö is a sheaf of hereditary Ox-orders over a smooth connectedprojective curve X.

Noncommutative Projective Geometry 99It is easy to show that the abelian categories appearing in parts (1) and (2) ofthis theorem are of the form qgr F for a graded ring F with GKdim F < 2, and sothis result can be regarded as a partial converse to Theorem 7. A discussion of thesaturation condition for noneommutative algebras may be found in [12].If one accepts that noneommutative projective curves and planes have beenclassified, as we have argued, then the natural next step is to attempt to classify allnoneommutative surfaces and this has been a major focus of recent research. Thisprogram is discussed in detail in [25, Sections 8-13] and so here we will be very brief.For the sake of argument we will assume that an (irreducible) noneommutativesurface is qgrF for a noetherian eg domain F with GKdim F = 3, although theprecise definition is as yet unclear. For example, Artin [1] demands that qgrFshould also possess a dualizing complex in the sense of Yekutieli [30]. Neverthelessin attempting to classify surfaces it is natural to mimic the commutative proof:(a) Classify noneommutative surfaces up to birational equivalence; equivalentlyclassify the associated graded division rings of fractions for graded domains Fwith GKdim F = 3. Artin [1, Conjecture 4.1] conjectures that these divisionrings are known.(b) Prove a version of Zariski's theorem that asserts that one can pass from anysmoothsurface to a birationally equivalent one by successive blowing up anddown. Then find minimal models within each equivalence class.Van den Bergh has created a noneommutative theory of blowing up and down[28, 29] and used this to answer part (b) in a number of special cases. A keyfactin his approach is that (after minor modifications) each known example of anoneommutative surface qgr F contains an embedded commutative curve C, just asqgr(Skl 3 ) ^ coh(F) = F in Section 2. This is important since he needs to blow uppoints on that subcategory. In general, define a point in qgrF to be n(M) for apoint module M £ gr F. Given such a point p, write p = n(R/L) = TZ/1. Mimickingthe classical situation we would like to writeB = K®1®1 2 ®--- , (4.1)and then define the blow-up of qgr F to be the category qgr B of finitely generatedgraded B-modules modulo those that are right bounded. However, there are twoproblems. A minor one is that 1 needs to be twisted to take into account the shiftfunctor on qgr F. The major one is that J is only a one-sided ideal of F, and sothere is no natural multiplication on B. To circumvent these problems, Van denBergh [28] has to define B in a more subtle category so that it is indeed an algebra.It is then quite hard to prove that qgr B has the appropriate properties.5. Hilbert schemesSince point modules and twisted homogeneous coordinate rings have provedso useful, it is natural to ask how generally these techniques can be applied. Inparticular, one needs to understand when point modules, or other classes of modules,can be parametrized by a scheme. Indeed, even for point modules over surfaces the

100 J. T. Staffordanswer was unknown until recently and this is obviously rather important for theprogram outlined in the last section.The best positive result is due to Artin, Small and Zhang [3, 9], for which weneed a definition. A fc-algebra F is called strongly noetherian if R® k C is noetherianfor all noetherian commutative fc-algebras C.Theorem 10 (Artin-Zhang [9, Theorems E4.3 and E4.4]) Assume that R is astrongly noetherian, eg algebra and fix h(t) = VJ/ijt* £ k[[t]]. Let C denote theset of cyclic R-modules M = R/L with Hilbert series ìIM(ì) = Ydìm k (Mi)t % equalto h(t). Then:(1) C is naturally parametrized by a (commutative) projective scheme.(2) There exists an integer d such that, if M = R/L £ C, then L is generated indegrees < d as a right ideal of R.In particular, if F is a strongly noetherian eg algebra generated in degree one,then the set of point modules is naturally parametrized by a projective scheme V.In this case one can further show that the shift functor M H> Af>i[l] induces anautomorphism aofV. Thus one can form the corresponding twisted homogeneouscoordinate rings B = B(V, £, a) and for an appropriate line bundle £ there will exista homomorphism

Noncommutative Projective Geometry 101(4) The category qgr S is not Extfinite; indeed if S = TT(S) £ qgr S, then H 1 ( 2 such that C is criticallydense. Then B = B(a,c) is noetherian. Moreover qgr(B) ~ qgrSâ, c).Thus, qgr S(a, c) is nothing more than the (noneommutative) blow-up of P" ata point! The differences between this blow-up and Van den Bergh's are illustrative.Van den Bergh had to work hard to ensure that the analogue of the exceptionaldivisor really looks like a curve. Indeed much of his formalism is required for just

102 J. T. Staffordthis reason. In contrast, in Theorem 12 the analogue of the exceptional divisor(which in this case equals B/(X C _ 1 )B) is actually a point. This neatly explains thestructure of the points in qgr S (a, c); they are indeed parametrized by P" althoughthe point corresponding to c (and hence the shifts of this point, which are nothingmore than the points corresponding to the e,) are distinguished.References[1] M. Artin, Some problems on three-dimensional graded domains, Representationtheory and algebraic geometry, London Math. Soc. Lecture Note Ser., vol. 238,Cambridge Univ. Press, Cambridge, 1995, 1-19.[2] M. Artin and W. Schelter, Graded algebras of global dimension 3, Adv. inMath., 66 (1987), 171-216.[3] M. Artin, L. W. Small and J. J. Zhang, Generic flatness for strongly noetherianalgebras. J. Algebra, 221 (1999), 579^610.[4] M. Artin and J. T. Stafford, Noneommutative graded domains with quadraticgrowth, Invent. Math., 122 (1995), 231^276.[5] M. Artin and J. T. Stafford, Semiprime graded algebras of dimension two,J. Algebra, 277 (2000), 68^123.[6] M. Artin, J. Tate, and M. Van den Bergh, Some algebras associated to automorphismsof elliptic curves, The Grothendieck Festschrift, vol. 1, Birkhäuser,Boston, 1990, 33^85.[7] M. Artin and M. Van den Bergh, Twisted homogeneous coordinate rings, J.Algebra, 133 (1990), 249-271.[8] M. Artin and J. J. Zhang, Noneommutative projective schemes, Adv. in Math.,109 (1994), 228^287.[9] M. Artin and J. J. Zhang, Abstract Hilbert schemes, Algebr. Represent. Theory,4 (2001), 305^394.[10] A. I. Bondal and M. M. Kapranov, Representable functors, Serre functors, andreconstructions, Math. USSR-Izv., 35 (1990), 519^541.[11] A. I. Bondal and A. E. Polishchuk, Homological properties of associative algebras:the method of helices, Russian Acad. Sei. Izv. Math., 42 (1994), 219^260.[12] A. I. Bondal and M. Van den Bergh, Generators and representability of functorsin commutative and noneommutative geometry; math.AG/0204218 (toappear).[13] A. Connes and M. Dubois-Violette, Noneommutative finite-dimensional manifolds.I. Spherical manifolds and related examples; math.QA/0107070 (to appear).[14] D. A. Jordan, The graded algebra generated by two Eulerian derivatives, Algebr.Represent. Theory, 4 (2001), 249^275.[15] D. S. Keeler, Criteria for a-ampleness, J. Amer. Math. Soc, 13 (2000), 517^532.[16] D. S. Keeler, D. Rogalski and J. T. Stafford, work in progress.[17] G. R. Krause and T. H. Lenagan, Growth of algebras and Gelfand-Kirillovdimension. Research Notes in Mathematics, vol. 116, Pitman, Boston, 1985.

Noncommutative Projective Geometry 103[18] A. V. Odesskii and B. L. Feigin, Sklyanin's elliptic algebras, Functional Anal.Appi., 23 (1989), no. 3, 207^214.[19] I. Reiten and M. Van den Bergh, Noetherian hereditary categories satisfyingSerre duality, J. Amer. Math. Soc, 15 (2002), 295^366.[20] D. Rogalski, Examples of generic noneommutative surfaces; math.RA/0203180(to appear).[21] A. L. Rosenberg, The spectrum of abelian categories and reconstruction ofschemes, Rings, Hopf algebras, and Brauer groups, Lecture Notes in Pure andAppi. Math., vol. 197, Marcel Dekker, New York, 1998, 257^274.[22] B. Shelton and M. Vancliff, Schemes of line modules I, J. London Math. Soc;www.uta.edu/math/vancliff/R/ (to appear).[23] E. K. Sklyanin, Some algebraic structures connected to the Yang-Baxter equation,Functional Anal. Appi., 16 (1982), 27^34.[24] J. T. Stafford and J. J. Zhang, Examples in noneommutative projective geometry,Math. Proc. Cambridge Philos. Soc, 116 (1994), 415^433.[25] J. T. Stafford and M. Van den Bergh, Noneommutative curves and noneommutativesurfaces, Bull. Amer. Math. Soc, 38 (2001), 171^216.[26] D. R. Stephenson, Artin-Schelter regular algebras of global dimension three, J.Algebra, 183 (1996), 55^73.[27] D. R. Stephenson, Algebras associated to elliptic curves, Trans. Amer. Math.Soc, 349 (1997), 2317^2340.[28] M. Van den Bergh, Blowing up of noneommutative smooth surfaces, Mem.Amer. Math. Soc, 154 (2001), no. 734.[29] M. Van den Bergh, Abstract blowing down, Proc. Amer. Math. Soc, 128(2000), 375-381.[30] A. Yekutieli, Dualizing complexes over noneommutative graded algebras, J.Algebra, 153 (1992), 41^84.

ICM 2002 • Vol. II • 105-116Deformations of Chiral AlgebrasDimitri Tamarkin*AbstractWe start studying chiral algebras (as defined by A. Beilinson and V. Drinfeld)from the point of view of deformation theory. First, we define the notionof deformation of a chiral algebra on a smooth curve X over a bundle of localartinian commutative algebras on X equipped with a flat connection (whereas'usual' algebraic structures are deformed over a local artinian algebra) and weshow that such deformations are controlled by a certain *-Lie algebra g. Thenwe try to contemplate a possible additional structure on g and we conjecturethat this structure up to homotopy is a chiral analogue of Gerstenhaber algebra,i.e. a coisson algebra with odd coisson bracket (in the terminology ofBeilinson-Drinfeld). Finally, we discuss possible applications of this structureto the problem of quantization of coisson algebras.2000 Mathematics Subject Classification: 14, 18.1. IntroductionChiral algebras were introduced in [1]. In the same paper the authors introducedthe classical limit of a chiral algebra which they call a coisson algebra andposed the problem of quantization of coisson algebras. The goal of this paper is toshow how the theory of deformation quantization (=the theory of deformations ofassociative algebras of a certain type) in the spirit of [3] can be developed in thissituation.Central object in the theory of deformations of associative algebras is thedifferential graded Lie algebra of Hochschild cochains. It turns out that in oursituation it is more appropriate to use what we call pro-*-Lie-algebras rather thanusual Lie algebras (the notion of *-Lie algebra was also introduced in [1]). Next, wecompute the cohomology of the pro-*-Lie-algebra controlling chiral deformations ofa free commutative F>x-algebra SK, where K is a locally free F>x-module.Next, we state an analogue of Gerstenhaber theorem which says that the cohomologyof the deformation complex of an associative algebra carries the structure ofa Gerstenhaber algebra. We give a definition of a chiral analogue of Gerstenhaber*Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA 02138,USA. E-mail: tamarkin@math.harvard.edu

106 Dimitri Tamarkinalgebra and define the operations of this structure on deformation pro-*-Lie algebraof a chiral algebra.Finally, mimicking Kontsevich's formality theorem, we formulate the formalityconjecturefor the deformation pro-*-Lie algebra of the chiral algebra SK mentionedabove and claim that this conjecture implies a 1-1 correspondence between deformationsof SK and coisson brackets on SK.2. Chiral algebras and their deformations2.1. Chiral operationsIn [1] chiral operations are defined as follows. Let X be a smooth curve andM t ,N F>x-modules. Denote by i n : X —t X n the diagonal embedding and byin '• U n —¥ X n the open embedding of the complement to all diagonals in X n . SetIn the case n = 0 setP ch (Mi,..., M„; N) = homi, x „ (j,j*(Mi M • • • M M n ), i w N). (1)Let M be a fixed F^-module. WriteP ch (M) = H°(M ® Vx O x ).F chM(«) = F ch( M ' M '---' M ; M )-It is explained in [1] that PM is an operad.2.1.1. Chiral algebrasLet lie be the operad of Lie algebras. A chiral algebra structure on M is ahomomorphism lie —t PM • We have a standard chiral algebra structure on M = OJX •A chiral algebra M is called unital if it is endowed with an injection OJX —ï M ofchiral algebras.2.2. Deformations2.2.1. AgreementsTo simplify the exposition, we will only consider unital chiral algebras M withthe following restrictions: we assume that X is affine and the F^-module M can berepresented as M = OJX ® N, where N = F ®o x ®x for some locally free coherentsheaf F.2.2.2. Nilpotent X> x -algebrasLet F be a left T>x -module equipped with a commutative associative unitalproduct F ® E —t E. Let u : Ox —*• E be the unit embedding. Call F nilpotent ifthere exists a Dx-module splitting s : F = M ® Ox and a positive integer N suchthat the iV-fold product vanishes on M. M is then a unique maximal T>x-ideal inF.

Deformations of Chiral Algebras 1072.2.3. Deformations over a nilpotent X>x-algebraLet F be a nilpotent T>x -algebra with maximal ideal M. We have a notionof F-module and of an F-linear chiral algebra . For any T>x -module M, M F :=1S anM ®Ox E ' F-module.Let M be a chiral algebra. An F-linear unital chiral algebra structure on M Fis called deformation of M over E if the induced structure on M F /M.M F — Mcoincides with the one on M. Denote by GM(E) the set of all isomorphism classesof such deformations.2.3. The functor GM and its representabilityIt is clear that F H> GM(E) is a functor from the category of nilpotent Dxalgebrasto the category of sets. In classical deformation theory one usually has afunctor from the category of (usual) local Arminian (=nilpotent and finitely dimensional)algebras to the category of sets and one tries to represent it by a differentialgraded Lie algebra. In this section we will see that in our situation a natural substitutefor a Lie algebra is a so-called *-Lie algebra in the sense of [1]. More precise,given a *-Lie algebra g, we are going to construct a functor F B from the category ofnilpotent T>x-algebras to the category of sets. In the next section we will show thatthe functor GM is 'pro-representable' in this sense. We will construct a pro-*-Liealgebra dei M (exact meaning will be given below) and an isomorphism of functorsGM and F^f^.2.3.1. *-Lie algebras[1] Let Qi, N be right ©^-modules. SetP*(9i,---,9n,N) := homD x „(ßi M • • • M g n ,i n *N),and F» B (n) := P(g,... ,g;g). It is known that F» B is an operad. A *-Lie algebrastructure on g is by definition a morphism of operads / : lie —t F» B . Let 6 £ lie(2)be the element corresponding to the Lie bracket. We call /(6) £ F» B (2) the *-Liebracket.2.3.2.Let g be a *-Lie algebra and A be a commutative T>x -algebra, introduce avector space g(A) = g ®x> x A. This space is naturally a Lie algebra. Indeed, wehave a *-Lie bracket gij-> Ì2*g- Multiply both parts by A M A:(g m g) ® Vxxx (A m A) -+ i 2 »g ®v xxx (A m A). (*)The left hand side is isomorphic to g(A) ® g(A). The right hand side is isomorphicto g ®T> X (A ®o x -4). Thus, (*) becomes:g(A) ® g(A) -• g ® Vx (A ® 0xA).

108 Dimitri TamarkinThe product on A gives rise to a mapB ®v x (A ® 0x A) -+ g ® Vx A ~ g(A),and we have a map g(A) ® g(A) —t g(A). It is straightforward to check that thismap is a Lie bracket.2.3.3.Let now g be a differential graded *-Lie algebra and let A be a differentialgraded commutative F>x-algebra. Then g(A) := g ®x> x A is a differential gradedLie algebra.2.3.4.Let A be a nilpotent T>x algebra and MA C A be the maximal nilpotent ideal.Then Q(MA) is a nilpotent differential graded Lie algebra.2.3.5.Recall that given a differential graded nilpotent Lie algebra n, one can constructthe so called Deligne groupoid Q n . Its objects are all x £ n 1 satisfyingdx + [x,x]/2 = 0 (so called Maurer-Cartan elements). The group exp(n°) acts onthe set of Maurer-Cartan elements by gauge transformations. Q n is the groupoidof this action. Denote by T> n the set of isomorphism classes of this groupoid. If/ : n —¥ m is a map of differential graded Lie algebras such that the induced map oncohomology H l (f) is an isomorphism for all i > 0, then the induced map T> n —t T> mis a bijection. If n, m are both centered in non-negative degrees, then the inducedmap Q n —¥ Q m is an equivalence of categories. Since in our situation we will deal withLie algebras centered in arbitrary degrees, we will use T> n rather than groupoids.2.3.6.Set F M (A) = X> B (MA)- It ' 1S a functor from the category of nilpotent T>xalgebrasto the category of sets.2.4. Pro-*-Lie- algebras*-Lie algebras are insufficient for description of deformations of chiral algebras.We will thus develop a generalization. We need some preparation2.4.1. ProcategoryFor an Abelian category C consider the category pro C whose objects arefunctors I —¥ C, where J is a small filtered category. Let F k : L k —t C, k = 1,2 beobjects. Sethom(Fi,F 2 ) := liminv , 2G / 2 limdir iiei 1 (Fi(ii),F 2 (î2))-The composition of morphisms is naturally defined. One can show that pro C isan Abelian category. Objects of pro C are called pro-objects.

2.4.2. Direct image of pro-F>-modulesDeformations of Chiral Algebras 109Let M : L —t Vy — mod be a pro-object, where Y is a smooth algebraicvariety and let / : Y —¥ Z be a locally closed embedding. Denote the compositionfioM : L —t T>z — mod simply by fi M. We will get a functor /„ : pro Vy — mod —¥pro T>z — mod.2.4.3. Chiral and *-operationsFor N, Mi £ pro V x -mod we define F*(Mi,..., M n , N), P ch (A/i,..., M n , N)by exactly the same formulas as for usual ©^-modules.2.4.4. pro-*-Lie algebras*-Lie algebra structure on a pro-î^x-module is defined in the same way as forusual T>x-modules.2.4.5.For a pro-right T>x -module I —¥ M and a left T>x -module F define a vectorspace M ®v x L = liminv i(M ®v x L). For a *-Lie algebra g and a commutativeT>x -algebra a, g ®x> x a is a Lie algebra. Construction is the same as for usual *-Liealgebras. Similarly, we can define the functor F B from the category of nilpotentT>x -algebras to the category of sets.2.5. Representability of G M by a pro-*-Lie algebraWe are going to construct a differential graded *-pro-Lie algebra g such thatF B is equivalent to GM . We need a couple of technical lemmas.2.5.1.Let Y be a smooth affine algebraic varieties and U, V be right X>y-modules.Let U a ,a £ A be the family of all finitely generated submodules of U. Denoteprohom([7, V) = liminv a(U a ,V) the corresponding pro-vector space.2.5.2.Let i : X —t Y be a closed embedding, let F be a X>y-module and M be aF>x-module. Thenprohom VY (B,i*(M ®o x T>x))is a pro-F>x-module. Denote it by P(B,M). Let now Y = X n .Lemma 2.1 Assume that B = j n *j n (E ®o X n F'x»), where E is locally free andcoherent. For any left T>x -module L we haveprohom(B,i m (M® 0x L)) =* P(B,M) ® Vx L.

110 Dimitri TamarkinProof. Let F = j m .j*E. We have B = F ® öx „ î>x»- Note that F =limdir F Q , where F Q runs through the set of all free coherent submodules of F.We haveP(B,M) = liminv hom^» (F Q ® 0x „ V x^,i n *(M® 0x L))~ liminv F* cgi 0x „ t„,(M ® 0x D x ) ®i, x F=* liminv hom 0x „ (F a ,i m (M ® 0x V x )) ®v x L=* prohom(B,z n »(M cgi 0x T> x )) ® Px F.2.5.3.Let F, Af be as above. We have a natural morphismi : i m .P(B, M) ~ P(B, M) ® Vx vf >x -+ prohom(F, M ®v x ®(V x f°x ).The above lemmas imply that i is an isomorphism.2.5.4.Let M be a right F> x -module. SetU M (n) = V ch (M, M,..., M ; M ® V x ) •= prohom(j„,j;M H ", t„,(M ® 2> x )),it is a right pro-F> x -module. We will endow the collection UM with the structureof an operad in *-pseudotensor category. This means that we will define thecomposition mapsOj G Px(UM(n),UM(m);U M (n + m - 1)),z = l,...,n + m — 1, satisfying the operadic axioms. We need a couple of technicaltacts.2.5.5.Let i n : X —t X n be the diagonal embedding and p l n : X n -+ I be theprojections. Lemma 2.5.3. implies thatLemma 2.2i m U M (k) = V ch (M,...,M;M® Vx Vf n ).Lemma 2.3 For any T>x -modules M,S we have an isomorphismin*(M)®p j n*Sî m .(M®S).

2.5.6.Deformations of Chiral Algebras 111We are now ready to define the desired structure. In virtue of 2.3 we havenatural maps:V ch (Mi,..., M„; N) -+ V ch (Mi,..., Mi ® V x , • • •, M„; (N ® V x ))-Thus, we have maps:V ch (Mi,..., M„; (Ni ® Vx)) M V ch (Ni, ...N n ;,(K® V x ))-• V ch (Mi,...,M n ;(Ni®D x ))mV ch (Ni,...,Ni®Dx,...,Nm;(K®Dx)®D x )-> V^N!,... Niî,Mi,..., M n , N i+1 , ...,N m ;K®Vx®V x )= Ì2*V ch (Ni,... Ni- lt Mi,..., M n , N i+1 ,. ..,N m ,K® V x ).By substituting M instead of all Ni, Mj, K, we get the desired insertion mapOj : UM(n) H UM (m) -ï Ì2*UM(^ + m — 1).2.5.7.and2.5.8.Similarly, we have insertion mapsOj : U M (n) ® ^ch M ( m ) "^ U M (n + m - 1),°i : -P c hM( n ) ® u M(m) -^ U M (n + m - 1).Let O be a differential graded operad. Set3o,n-=0(n) s %andßo = © n flo, n [l - n].Let p n : 0(n) —¥ go,n be the natural projection, which is the symmetrizationmap. Define the brace (x, y) >-¥ x{y}, go, n ® 9o,m, —* 9o,n+m-i as follows.and the bracketx{y\ = np n (oi(x,y)) (2)[x,y] = x{y}-(-l)WMy{x}. (3)We see that [, ] is a Lie bracket. Thus, go is a differential graded Lie algebra. For anoperad O denote by O' the shifted operad such that the structure of an C-algebraon a complex V is equivalent to the structure of an ö-algebra on a complex V[l].Thus, 0'(n) = 0(n) ® e n [l — n], where e n is the sign representation of S n .Let O be an operad of vector spaces. The set of Maurer-Cartan elements ofgo> is in 1-1 correspondence with maps of operads lie —t O.Assume that 0(1) is a nilpotent algebra (x n = 0 for any x £ 0(1)). Let A be0(1) with adjoined unit and let A x be the group of invertible elements. A x actson O by automorphisms. Therefore, A x acts on the set of maps lie —t O. Thegroupoid of this action is isomorphic to the Deligne groupoid of go> •

112 Dimitri Tamarkin2.5.9.Similarly, let A be a *-operad. Then formula 3 defines a Lie-* algebra g A- W ehave natural action of a usual pro-Lie algebra g-p ( M ) on a pro *-Lie algebra gu Mby derivations. The chiral bracket b £ gip, M -, satisfies [6,6] = 0. Therefore, thebracket with 6 defines a differential on gu M . Denote this differential graded *-Liealgebra by VM-2.5.10.To avoid using derived functors, we will slightly modify BM • Recall that M =OJX © X, where N is free. LetV^d(M,...,M;M® V x ) C V cil (M, ...,M;M® V x )be the subset of all operations vanishing under all restrictionsV ch (M, ...,M;M®Vx)-> ^ch( M > • • • ,M,u x ,M,..., M ; M ® V x ).Let defM C BM be the submodule such thatdef M = ®„(^g d (Af, ...,M;M®V x ) ® e„) s " [1 - n].We see that def^f is a *-Lie differential subalgebra of 5M-2.5.11.Proposition 2.4 The functors GM and Fêfare canonically isomorphic.2.6. ExampleLet K be a free left F> x -module. Let T X K = K °x i . The symmetric groupSi acts on the Vx -module T % K; let S % K = (T l K) Si be the submodule of invariantsand SK = ®^L 0 S l K. SK is naturally a free commutative F> x -algebra and, hence,SK r := SK ® OJX is a chiral algebra. We will compute the cohomology of theF> x -module deign*. Let SQK = (B < n L 1 S l K. We have:defSK* = ® n (P ch (SoK r [l],..., S 0 K r [l]; SK r ® V x )[l]f".On the other hand, denote by 0 := SK®K. Consider Q as an SK-Vx-moduleof differentials of SK. We have the de Rham differential D : SQK —ï Q. We have athrough mapc„ : P ch (K[l] r ,..., K[l] r , SK[l] r ) s » = F|f (Ü[l] r ,..., Ü[l] r , SK[l] r ) s »Dp ch (s 0 K[iY,...,s 0 K[i] r , SK[iywhere P s £ stands for SFJ-linear chiral operations. Denote by the same letter theinduced mapc, : P ch(K[l] r ,..., K[1Y, SK[lYf- -+ def Sif .

Proposition 2.5 (1) dc n = 0;(2) c n induces an isomorphismDeformations of Chiral Algebras 113P ch (K[l] r ,..., K[1Y, SK[\] r ) s » -+ F"- 1 (def sx )[l - n].2.6.1.For a chiral algebra M denote by H M the graded Lie algebra of cohomologyof defM-2.6.2.Assume that K is finitely generated. Letbe the dual module. ThenK* = hom(K,V x )®(ux)~ 1H SK *^ ® n (P*(K r ,... ,K r ;SK r ®V x )e n ) s »[l-n] = ® n (A n K^' ® 0x SK) r [l-n\.2.6.3.We will postpone the calculation of the *-Lie bracket on H$K>• until we showin the next section that H M has in fact a richer structure.3. Algebraic structure on the cohomology of thedeformation pro-*-Lie algebraWe will keep the agreements and the notations from 2.2.1..3.1. Cup productWe will define a chiral operation U £ F^ def r_ 1 -i(2) and then we will studythe induced map on cohomology.3.1.1.whereRecall thatdef M [-l] = ® n (a r ,a n = P ch (N[l],...,N[l];M®V x ).Let i n : X —t X n be the diagonal embedding and let p % : X n —t X U n C X n bethe complement to the union of all pairwise diagonals p % x = p>>x and j n : U n —¥ X nbe the open embedding. Let U n>m C X n+m be the complement to the diagonalsp l x = pix, where 1 < i < n, n+l

114 Dimitri TamarkinComputeJ2,jl(a n H a m ) = hom(j m j n (N[lf n ) H jm*Jm(N[lf m ),(in X Ìm)*(Ì2*Ì2*(M ®V X MM® Vx)))- hom(j m j n (N[lf n ) mjm*Jm(N[lf m ) ® j,0(Un, m ),(in X Ìm)*(h*J2 (M ®V X^M®V X )® j*0(U n , m )))^hom(j n+m *j n+ m(N[l]f n+m ,(in x i m )*(h*Ul(M m M)) ®V x^x)) •Taking the composition with the chiral operation on M, we obtain a chiraloperationto.fe(a n &a m ) -+ hom(j n+ m*j n+ m(X[l}f n+m ,in+m*(M ®V X ®V X )) ~ Ì2*a n+m ,which induces a chiral operation from F c | 1 (af",a^m;a r^^*) and, hence, an operationU G (F ch (defMH],defMH];def M H])) S2 .3.1.2.To investigate the properties of this operation, consider the brace *-operation•{} G F»(defM,defM;defM) defined by formula (2). Letbe the natural mapProposition 3.1 d(-{\) = r(U).r : P ch (Ai,A2;A 3 ) -+ P,(Ai,A 2 ;A 3 )Let Uft be the induced operation on HM[^1]-that r(Uft) = 0. In virtue of exact sequenceThe above proposition implies0 -+ hom((Ai) 1 ® (A 2 ) l ,(A 3 ) 1 ) -> P ch (A 1 ,A 2 ;A 3 ) 4 P.(A 1 ,A 2 ;A 3 ),Uft defines a Vx -commutative product I/M[-1] ® HM[—1] —t HM[—1], denoted bythe same letter.3.1.3.Proposition 3.2 U^ is associative.3.1.4. Leibnitz ruleWe are going to establish a relation between U and •{}. This relation is similarto the one of coisson algebras. Our exposition will mimic the definition of coissonalgebras from [1].

3.1.5.Deformations of Chiral Algebras 115iLet Ai be right F> x -modules. Write A ± ® A 2 := (A[ ®A l 2) r ; Pi(Ai,A 2 ;A 3 ) :=hom(.4i ® A 2 ,A 3 ).We have a mapWe have (.4 ® B) = i* 2 (B M C);ii 2 *(A® B) -+i 2 *A®p* 2 (B l ).c : P r .(Ai,A 2 ;B) ® P,(B, C; D) -+ P»(Ai,A 2 ® C; D)1defined as follows. Let u : Ai M A 2 —¥ Ì2*B and m : B ® C —¥ D. Putc(u,v) : AiM(A 2 ® C) = (-4 1 H.4 2 )cgi_ P ;(C') -^ i 2 *B®p* 2 (C l ) ^ i 2 .(B ® C) -+ i 2 .D.3.1.6.Denoteie = c(0,U ft ) G P*(HM,H M ® H M ;H M ).1 1Let T : HM ® HM —ï H M ® HM be the action of symmetric group and let e Tthe composition with T. Let / G P*(HM,HM® HM', HM) be defined by:beHM E3 (HM® HM) -^ HM E3 HM ~^ ì*HM-Proposition 3.3 We have f = e + e T .3.1.7.In other words, the cup product and the bracket satisfy the Leibnitz identity.We see that HM has a pro-*-Lie bracket, (HM)'[1] has a commutative Vxalgebrastructure, and these structures satisfy the Leibnitz identity. Call this structurea c-Gerstenhaber algebra structure. Thus, our findings can be summarized asfollows.Theorem 3.4 The cohomology of the deformation pro-*-Lie algebra of a chiralalgebra is naturally a pro-c-Gerstenhaber algebra.3.2. Example M = (SK) rWe come back to our example 2.6.. For simplicity assume K is finitely generatedfree Vx -module. We have seen in this case that(H M ) l [-l] ~ (Bi A* K v ® S K [î] ~ S(K V [^1] e K).Proposition 3.5 The cup product on HM coincides with the natural one on thesymmetric power algebra.

116 Dimitri Tamarkin3.2.1.To describe the bracket it suffices to define it on the submodule of generatorsG = (K v [—1] ® K) r . Define [] G P*(G,G;HM) to be zero when restricted ontoK r M K r and K Vr [^l] H Ä" Vr [-l]. Restriction onto K M K v [^l] takes values inu)x C H M and is given by the canonic *-pairing from [1](K v mK) r î 2 *üj x .Recall the definition. We have K Vr = hom(K r ,Vx ® w x ). For open U,V C X wehave the composition mapK(U) ® K V (V) -^V x ®oj x (Un V) = i 2 *oj x (U x V)which defines the pairing. This uniquely defines the *-Lie bracket.4. Formality ConjectureFollowing the logic of Kontsevich's formality theorem, one can formulate aformality conjecture in this situation.4.1. Quasi-isomorphismsA map / : g —¥ I) of differential graded pro-*-Lie algebras is called quasiisomorphismif it induces an isomorphism on cohomology. Call a pro-*-Lie algebraperfect if such is its underlying complex of pro-vector spaces. The morphism / iscalled perfect quasi-isomorphism if it is a quasi-isomorphism and both g and f) areperfect.Two perfect pro-*-lie algebras are called perfectly quasi-isomorphic if thereexists a chain of perfect quasi-isomorphisms connecting g and h.Conjecture 4.1 deign and H$K are perfectly quasi-isomorphic.The importance of this conjecture can be seen from the following theorem:Theorem 4.2 Any chain of perfect quasi-isomorphisms between deign* and H$K*establishes a bijection between the set of isomorphism classes of A-linear coissonbrackets on SK r ® A which vanish modulo the maximal ideal MA and the set ofisomorphism classes of all deformations of the chiral algebra SK r over A.References[1] A. Beilinson, V. Drinfeld, Chiral Algebras.[2] W. Goldman, J. Millson, Deformations of Flat Bundles over Kahler Manifolds,Geometry and Topology, 129^145, Lect. Notes in Pure and Applied Math. 105Dekker NY (1987).[3] M. Kontsevich, Quantization of Poisson Manifolds, preprint.

Section 3. Number TheoryJ. W. Cogdell, I. I. Piatetski-Shapiro: Converse Theorems,Functoriality, and Applications to Number Theory 119H. Cohen: Constructing and Counting Number Fields 129Jean-Marc Fontaine: Analyse p-adique et Représentations Galoisiennes 139A. Huber, G. Kings: Equivariant Bloch-Kato Conjecture and Non-abelianIwasawa Main Conjecture 149Kazuya Kato: Tamagawa Number Conjecture for zeta Values 163Stephen S. Kudla: Derivatives of Eisenstein Series and ArithmeticGeometry 173Barry Mazur, Karl Rubin: Elliptic Curves and Class Field Theory 185Emmanuel Ullmo: Théorie Ergodique et Géométrie Arithmétique 197Trevor D. Wooley: Diophantine Methods for Exponential Sums, andExponential Sums for Diophantine Problems 207

ICM 2002 • Vol. II • 119-128Converse Theorems, Fune tor iality,and Applications to Number TheoryJ. W. Cogdell* I. I. Piatetski-ShapiroÂbstractThere has been a recent coming together of the Converse Theorem forGL n and the Langlands-Shahidi method of controlling the analytic propertiesof automorphic L-functions which has allowed us to establish a number of newcases of functoriality, or the lifting of automorphic forms. In this article wewould like to present the current state of the Converse Theorem and outlinethe method one uses to apply the Converse Theorem to obtain liftings. We willthen turn to an exposition of the new liftings and some of their applications.2000 Mathematics Subject Classification: 11F70, 22E55.Keywords and Phrases: Automorphic forms, L-functions, Converse theorems,Functoriality.1. IntroductionConverse Theorems traditionally have provided a way to characterize Dirichletseries associated to modular forms in terms of their analytic properties. Mostfamiliar are the Converse Theorems of Hecke and Weil. Hecke first proved that F-functions associated to modular forms enjoyed "nice" analytic properties and thenproved "Conversely" that these analytic properties in fact characterized modularF-functions. Weil extended this Converse Theorem to F-functions of modular formswith level.In their modern formulation, Converse Theorems are stated in terms of automorphicrepresentations of GL n (A) instead of modular forms. Jacquet, Piatetski-Shapiro, and Shalika have proved that the F-functions associated to automorphicrepresentations of GL n (A) have nice analytic properties via integral representationssimilar to those of Hecke. The relevant "nice" properties are: analytic continuation,boundedness in vertical strips, and functional equation. Converse Theorems in thiscontext invert these integral representations. They give a criterion for an irreducible* Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA. E-mail:cogdell@math.okstate.edu^Department of Mathematics, Yale University, New Haven, CT 06520, USA, and School ofMathematics, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: ilya@math.yale.edu

120 J. W. Cogdell I. I. Piatetski-Shapiroadmissible representation II of GL„ (A) to be automorphic and cuspidal in terms ofthe analytic properties of Rankin-Selberg convolution F-functions L(s, II x n') of IItwisted by cuspidal representations n' of GL TO (A) of smaller rank groups.To use Converse Theorems for applications, proving that certain objects areautomorphic, one must be able to show that certain F-functions are "nice". However,essentially the only way to show that an F-function is nice is to have it associatedto an automorphic form. Hence the most natural applications of ConverseTheorems are to functoriality, or the lifting of automorphic forms, to GL n . Moreexplicit number theoretic applications then come as consequences of these liftings.Recently there have been several applications of Converse Theorems to establishingfunctorialities. These have been possible thanks to the recent advances inthe Langlands-Shahidi method of analysing the analytic properties of general automorphicF-functions, due to Shahidi and his collaborators [21]. By combiningour Converse Theorems with their control of the analytic properties of F-functionsmany new examples of functorial liftings to GL„ have been established. These aredescribed in Section 4 below. As one number theoretic consequence of these liftingsKim and Shahidi have been able to establish the best general estimates overa number field towards the Ramamujan-Selberg conjectures for GL 2 , which in turnhave already had other applications.2. Converse Theorems for GL nLet k be a global field, A its adele ring, and ip a fixed non-trivial (continuous)additive character of A which is trivial on k. We will take n > 3 to be an integer.To state these Converse Theorems, we begin with an irreducible admissiblerepresentation II of GL n (A). It has a decomposition II = C^'n^, where n^ is anirreducible admissible representation of GL n (fc„). By the local theory of Jacquet,Piatetski-Shapiro, and Shalika [9, 11] to each n^ is associated a local F-functionL(s,U v ) and a local e-factor e(s,U v ,ip v ). Hence formally we can formL(s,U) = JJ_L(s,U v ) and e(s,U,ip) = JJ_e(s,U v ,'ip v ).We will always assume the following two things about II:(1) L(s,U) converges in some half plane Re(s) >> 0,(2) the central character un of II is automorphic, that is, invariant under k x .Under these assumptions, e(s, II, ip) = e(s, II) is independent of our choice of ip [4].As in Weil's case, our Converse Theorems will involve twists but now by cuspidalautomorphic representations of GL TO (A) for certain m. For convenience, letus set A(m) to be the set of automorphic representations of GL TO (A), Ao(m) theset of (irreducible) cuspidal automorphic representations of GL TO (A), and T(m) =UrfLi Ao(d). If S is a finite set of places, we will let T s (m) denote the subset ofrepresentations n £ T with local components n v unramified at all places v £ S andlet Ts(m) denote those n which are unramified for all v $ S.

Converse Theorems, Functoriality, and Applications 121Let ir' = ®'n' v be a cuspidal representation of GL TO (A) with m < n. Thenagain we can formally defineL(s,U x n') = JjL(s,II„ x n' v ) and e(s,U x n') = JJ_e(s,U v x n' v ,tp v )since the local factors make sense whether II is automorphic or not. A consequenceof (1) and (2) above and the cuspidality of n' is that both L(s, II x n') and L(s, II xn') converge absolutely for Re(s) » 0, where II and n' are the contragredientrepresentations, and that e(s,Il x n') is independent of the choice ofip.We say that L(s, II x n') is nice if it satisfies the same analytic properties itwould if II were cuspidal, i.e.,1. L(s,U x IT') and L(s, II x n') have continuations to entire functions of s,2. these entire continuations are bounded in vertical strips of finite width,3. they satisfy the standard functional equationL(s,U x IT') = e(s,U x n')L(l - s,ft x n').The basic converse theorem for GL„ is the following.Theorem 1. [6] Let II be an irreducible admissible representation of GL„(A)as above. Let S be a finite set of finite places. Suppose that L(s,U x n') is nicefor all n' £ T s (n — 2). Then II is quasi-automorphic in the sense that there is anautomorphic representation II' such that n^ ~ Yl' v for all v $ S. If S is empty, thenin fact II is a cuspidal automorphic representation of GL„ (A).It is this version of the Converse Theorem that has been used in conjunctionwith the Langlands-Shahidi method of controlling analytic properties of F-functionsin the new examples of functoriality explained below.Theorem 2. [4] Let II be an irreducible admissible representation o/GL„(A)as above. Let S be a non-empty finite set of places, containing Soo, such that theclass number of the ring og of S-integers is one. Suppose that L(s,H x n') is nicefor all IT 1 £ Ts(n — 1). Then II is quasi-automorphic in the sense that there is anautomorphic representation II' such that n^ ~ Yl' v for all v £ S and all v ^ S suchthat both n^ and Yl' v are unramified.This version of the Converse Theorem was specifically designed to investigatefunctoriality in the cases where one controls the F-functions by means of integralrepresentations where it is expected to be more difficult to control twists.The proof of Theorem 1 with S empty and n — 2 replaced by n — 1 essentiallyfollows the lead of Hecke, Weil, and Jacquet-Langlands. It is based on the integralrepresentations of F-functions, Fourier expansions, Mellin inversion, and finally ause of the weak form of Langlands spectral theory. For Theorems 1 and 2 where wehave restricted our twists either by ramification or rank we must impose certain localconditions to compensate for our limited twists. For Theorem 1 are a finite numberof local conditions and for Theorem 2 an infinite number of local conditions. Wemust then work around these by using results on generation of congruence subgroupsand either weak approximation (Theorem 1) or strong approximation (Theorem 2).As for our expectations of what form the Converse Theorem may take in thefuture, we refer the reader to the last section of [6].

122 J. W. Cogdell I. I. Piatetski-Shapiro3. Functoriality via the Converse TheoremIn order to apply these theorems, one must be able to control the analyticproperties of the F-function. However the only way we have of controlling global F-functions is to associate them to automorphic forms or representations. A minute'sthought will then convince one that the primary application of these results will beto the lifting of automorphic representations from some group H to GL n .Suppose that H is a reductive group over k. For simplicity of exposition we willassume throughout that H is split and deal only with the connected component of itsF-group, which we will (by abuse of notation) denote by L H [1]. Let n = ®'n v be acuspidal automorphic representation of H and p a complex representation of L H. Tothis situation Langlands has associated an F-function L(s,n,p) [1]. Let us assumethat p maps L H to GL n (C). Then by Langlands' general Principle of Functorialityto n should be associated an automorphic representation II of GL n (A) satisfyingL(s,U) = L(s,n,p), e(s,U) = e(s,n,p), with similar equalities locally and for thetwisted versions [1]. Using the Converse Theorem to establish such liftings involvesthree steps: construction of a candidate lift, verification that the twisted F-functionsare "nice", and application of the appropriate Converse Theorem.1. Construction of a candidate lift: We construct a candidate lift II = C^'nôn GL n (A) place by place. We can see what n^ should be at almost all places.Since we have the arithmetic Langlands (or Hecke-Frobenius) parameterization ofrepresentations n v of H(k v ) for all archimedean places and those non-archimedeanplaces where the representations are unramified [1], we can use these to associateto n v and the map p v : L H V —¥ L H —t GL n (C) a representation n^ of GL n (fc„). Thiscorrespondence preserves local F- and e-factorsL(s,U v ) = L(s,n v ,p v ) and e(s,U v ,ip v ) = e(s,n v ,p v ,ip v )along with the twisted versions. If H happens to be GL TO or a related group thenwe in principle know how to associate the representation n^ at all places now thatthe local Langlands conjecture has been solved for GL TO . For other situations, wemay not know what n^ should be at the ramified places. We will return to thisdifficulty momentarily and show how one can work around this with the use of ahighly ramified twist. But for now, let us assume we can finesse this local problemand arrive at a global representation II = C^'n^ such thatL(s,U) = JjL(s,n„) = Y[L(s,n v ,p v ) = L(s,n,p)and similarly e(s,U) = e(s,n,p) with similar equalities for the twisted versions. IIshould then be the Langlands lifting of n to GL n (A) associated to p.2. Analytic properties of global L-functions: For simplicity of exposition, let usnow assume that p is simply a standard embedding of L H into GL„(C), such as willbe the case if we consider H to be a split classical group, so that L(s, n, p) = L(s, n)is the standard F-function of n. We have our candidate II for the lift of n to GL„from above. To be able to assert that the II which we constructed place by placeis automorphic, we will apply a Converse Theorem. To do so we must control thetwisted F-functions L(s, II x n') = L(s,n x n') for n' £ T with an appropriate

Converse Theorems, Functoriality, and Applications 123twisting set T from Theorem 1 or 2. In the examples presented below, we haveused Theorem 1 above and the analytic control of L(s,n x n') achieved by the socalledLanglands-Shahidi method of analyzing the F-functions through the Fouriercoefficients of Eisenstein series [21]. Currently this requires us to take k to be anumber field. The functional equation L(s, n xn') = e(s, n x n')L(l — s, n x n') hasbeen proved in wide generality by Shahidi [18]. The boundedness in vertical stripshas been proved in close to the same generality by Gelbart and Shahidi [7]. As forthe entire continuation of L(s,n x n'), a moments thought will tell you that oneshould not always expect a cuspidal representation of H (A) to necessarily lift to acuspidal representation of GL„(A). Hence it is unreasonable to expect all L(S, / KX / K')to be entire. We had previously understood how to work around this difficulty fromthe point of view of integral representations by again using a highly ramified twist.Kim realized that one could also control the entirety of these twisted F-functions inthe context of the Langlands-Shahidi method by using a highly ramified twist. Wewill return to this below. Thus in a fairly general context one has that L(s,n x n')is entire for n' in a suitably modified twisting set T'.3. Application of the Converse Theorem: Once we have that L(s,n x n') isnice for a suitable twisting set T' then from the equalitiesL(s,U x n') = L(s,n x n') and e(s,U x n') = e(s,n x n')we see that the L(s, II x n') are nice and then we can apply our Converse Theoremsto conclude that II is either cuspidal automorphic or at least that there is an automorphicII' such that n^ = Yl' v at almost all places. This then effects the (possiblyweak)automorphic lift of n to II or II'.4. Highly ramified twists: As we have indicated above, there are both localand global problems that can be finessed by an appropriate use of a highly ramifiedtwist. This is based on the following simple observation.Observation. Let II be as in Theorem 1 or 2. Suppose that r\ is a fixedcharacter ofk x \A x . Suppose that L(s, II x n') is nice for all n' £ T' = T®r], whereT is either of the twisting sets of Theorem 1 or 2. Then II is quasi-automorphic asin those theorems.The only thing to observe is that if n' £ T then L(s, II x (n' ® r]j) = L(s, (II ®if) x IT') so that applying the Converse Theorem for II with twisting set T ® r\ isequivalent to applying the Converse Theorem for II ® r\ with the twisting set T. So,by either Theorem 1 or 2, whichever is appropriate, II ®r] is quasi-automorphic andhence II is as well.If we now begin with n automorphic on H (A), we will take T to be the set offinite places where n v is ramified. For applying Theorem 1 we want S = T and forTheorem 2 we would want S n T = 0. We will now take r\ to be highly ramified atall places v £ T, so that at v £ T our twisting representations are all locally of theform (unramified principal series)®(highly ramified character).In order to finesse the lack of knowledge of an appropriate local lift, we needto know the following two local facts about the local theory of F-functions for H.Multiplicativity of 7-factors. If n' v = Ind(ir'i v ® n' 2 v ), with 7r- V and irreducibleadmissible representation of GL ri (k v ), then we have ^(s,ir v x n' v ,ip v ) =7(S,7T„ X ir'i v,Xp v Yf(s,Tï v X TT 2 v ,1pv)-

124 J. W. Cogdell I. I. Piatetski-ShapiroStability of 7-factors. If ni :V and TT2, V are two irreducible admissible representationsofH(k v ) with the same central character, then for every sufficiently highlyramified character r] v of GL\(k v ) we have ^(s,ni^v X r) v ,ip v ) = 7(s,7r 2jt , x r h,'ipv)-Both of these facts are known for GL„, the multiplicativity being found in[9] and the stability in [10]. Multiplicativity in a fairly wide generality useful forapplications has been established by Shahidi [19]. Stability is in a more primitivestate at the moment, but Shahidi has begun to establish the necessary results in ageneral context in [20].To utilize these local results, what one now does is the following. At the placeswhere n v is ramified, choose n^ to be arbitrary, except that it should have the samecentral character as n v . This is both to guarantee that the central character of nis the same as that of n and hence automorphic and to guarantee that the stableforms of the 7-factors for n v and n^ agree. Now form II = 'II„. Choose ourcharacter r\ so that at the places v £ T we have that the F- and 7-factors forboth n v ® r) v and n^ ® r) v are in their stable form and agree. We then twist byT' = T ® T) for this fixed character r\. If n' £ T', then for v £ T, n' v is of theform n' v = Ind(\ \ Sl ® • • • ® | | Sm ) ® r) v . So at the places v £ T, applying bothmultilplcativity and stability, we have7(s,7T„ x n' v ,ip v ) = Jj7(s + Sj,7r„ ®i} v ,ip v )= Y[>y(s + Si,Yl v ®i} v ,ip v ) =7(s,II„ x n' v ,ip v )from which one deduces a similar equality for the F- and e-factors. From this itwill then follow that globally we will have L(s, n xn') = L(s, II x n') for all n' £ T'with similar equalities for the e-factors. This then completes Step 1.To complete our use of the highly ramified twist, we must return to the questionof whether L(s,n x n') can be made entire. In analysing F-functions via theLanglands-Shahidi method, the poles of the F-function are controlled by those of anEisenstein series. In general, the inducing data for the Eisenstein series must satisfya type of self-contragredience for there to be poles. The important observation ofKim is that one can use a highly ramified twist to destroy this self-contragredienceat one place, which suffices, and hence eliminate poles. The precise condition willdepend on the individual construction. A more detailed explanation of this can befound in Shahidi's article [21]. This completes Step 2 above.4. New examples of functorialityNow take k to be a number field. There has been much progress recently inutilizing the method described above to establish global liftings from split groupsH over k to an appropriate GL n . Among them are the following.1. Classical groups. Take H to be a split classical group over k, more specifically,the split form of either S0 2 „ + i, Sp 2 „, or S0 2 „. The the F-groups L H arethen Sp 2n (C), S0 2 „ + i(C), or S0 2 „(C) and there are natural embeddings into thegeneral linear group GL 2 „(C), GL 2 „+i(C), or GL 2 „(C) respectively. Associated toeach there should be a lifting of admissible or automorphic representations from

Converse Theorems, Functoriality, and Applications 125H (A) to the appropriate GLjv(A). The first lifting that resulted from the combinationof the Converse Theorem and the Langlands-Shahidi method of controllingautomorphic F-functions was the weak lift for generic cuspidal representations fromS0 2 „ + i to GL 2 „ over a number field k obtained with Kim and Shahidi [2]. We cannow extend this to the following result.Theorem. [2, 3] Let H be a split classical group over k as above and n aglobally generic cuspidal representation o/H(A). Then there exists an automorphicrepresentation n of GLJV (A) for the appropriate N such that n^ is the local Langlandslift of n v for all archimedean places v and almost all non-archimedean placesv where n v is unramified.In these examples the local Langlands correspondence is not understood at theplaces v where n v is ramified and so we must use the technique of multiplicativityand stability of the local 7-factors as outlined in Section 3. Multiplicativity hasbeen established in generality by Shahidi [19] and in our first paper [2] we reliedon the stability of 7-factors for S0 2 „ + i from [5]. Recently Shahidi has establishedan expression for his local coefficients as Mellin transforms of Bessel functions insome generality, and in particular in the cases at hand one can combine this withthe results of [5] to obtain the necessary stability in the other cases, leading to theextension of the lifting to the other split classical groups [3].2. Tensor products. Let H = GL ro x GL„. Then L H = GL ro (C) x GL„(C).Then there is a natural simple tensor product map from GL TO (C) x GL n (C) toGL TOn (C). The associated functoriality from GL„ x GL TO to GL TO „ is the tensorproduct lifting. Now the associated local lifting is understood in principle since thelocal Langlands conjecture for GL„ has been solved. The question of global functorialityhas been recently solved in the cases of GL 2 x GL 2 to GL 4 by Ramakrishnan[17] and GL 2 x GL 3 to GL 6 by Kim and Shahidi [15, 16].Theorem. [17, 15] Let m be a cuspidal representation o/GL 2 (A) and 7r 2 acuspidal representation o/GL 2 (A) (respectively GL 3 (A)J. Then there is an automorphicrepresentation II o/GL 4 (A) (respectively GL 6 (A)) such that Il v is the localtensor product lift of m, v x 7r 2jt , at all places v.In both cases the authors are able to characterize when the lift is cuspidal.In the case of Ramakrishnan [17] n = m x 7r 2 with each 7r, cuspidal representationof GL 2 (A) and II is to be an automorphic representation of GL 4 (A). Toapply the Converse Theorem Ramakrishnan needs to control the analytic propertiesof L(s,U x n') for n' cuspidal representations of GLi(A) and GL 2 (A), that is,the Rankin triple product F-functions L(s,II x n') = L(s,ni x 7r 2 x n'). This hewas able to do using a combination of results on the integral representation for thisF-function due to Garrett, Rallis and Piatetski-Shapiro, and Ikeda and the work ofShahidi on the Langlands-Shahidi method.In the case of Kim and Shahidi [15, 16] 7r 2 is a cuspidal representation ofGL 3 (A). Since the lifted representation II is to be an automorphic representationof GL 6 (A), to apply the Converse Theorem they must control the analytic propertiesof L(s, II x n') = L(s, 7Ti x 7T 2 x n') where now n' must run over appropriate cuspidalrepresentations of GL TO (A) with m = 1,2,3,4. The control of these triple productsis an application of the Langlands-Shahidi method of analysing F-functions and

126 J. W. Cogdell I. I. Piatetski-Shapiroinvolves coefficients of Eisenstein series on GL 5 , Spin 10 , and simply connected E 6and E 7 [15, 21]. We should note that even though the complete local lifting theoryisunderstood, they still use a highly ramified twist to control the global propertiesof the F-functions involved. They then show that their lifting is correct at all localplaces by using a base change argument.3. Symmetric powers. Now take H = GL 2 , so L H = GL 2 (C). For each n > 1there is the natural symmetric n-th power map sym n : GL 2 (C) —¥ GL n+ i(C). Theassociated functoriality is the symmetric power lifting from representations of GL 2to representations of GL n+ i. Once again the local symmetric powers liftings areunderstood in principle thanks to the solution of the local Langlands conjecture forGL n . The global symmetric square lifting, so GL 2 to GL 3 , is an old theorem ofGelbart and Jacquet. Recently, Kim and Shahidi have shown the existence of theglobal symmetric cube lifting from GL 2 to GL 4 [15] and then Kim followed withthe global symmetric fourth power lifting from GL 2 to GL 5 [14].Theorem. [15, 14] Letn be a cuspidal automorphic representation o/GL 2 (A).Then there exists an automorphic representation II of GL 4 (A) (resp. GL 5 (A)) suchthat n^ is the local symmetric cube (resp. symmetric fourth power) lifting of n v .In either case, Kim and Shahidi have been able to give a very interestingcharacterization of when the image is in fact cuspidal [15, 16].The original symmetric square lifting of Gelbart and Jacquet indeed used theconverse theorem for GL 3 . For Kim and Shahidi, the symmetric cube was deducedfrom the functorial GL 2 x GL 3 tensor product lift above [15, 16] and did not requirea new use of the Converse Theorem. For the symmetric fourth power lift, Kim firstused the Converse Theorem to establish the exterior square lift from GL 4 to GL 6by the method outlined above and then combined this with the symmetric cube liftto deduce the symmetric fourth power lift [14].5. ApplicationsThese new examples of functoriality have already had many applications. Wewill discuss the primary applications in parallel with our presentation of the examples.k remains a number field.1. Classical groups: The applications so far of the lifting from classical groupsto GL„ have been "internal" to the theory of automorphic forms. In the case of thelifting from S0 2 „ + i to GL 2 „, once the weak lift is established, then the theory ofGinzburg, Rallis, and Soudry [8] allows one to show that this weak lift is indeed astrong lift in the sense that the local components n^ at those v £ S are completelydeterminedand to completely characterize the image locally and globally. This willbe true for the liftings from the other classical groups as well. Once one knowsthat these lifts are rigid, then one can begin to define and analyse the local lift forramified representations by setting the lift of n v to be the n^ determined by theglobal lift. This is the content of the papers of Jiang and Soudry [12, 13] for the caseof H = SC>2n+i • In essence they show that this local lift satisfies the relations onF-functions that one expects from functoriality and then deduce the local Langlandsconjecture for S0 2 „ + i from that for GL 2 „. We refer to their papers for more detail

Converse Theorems, Functoriality, and Applications 127and precise statements.2. Tensor product lifts: Ramakrishnan's original motivation for establishingthe tensor product lifting from GL 2 x GL 2 to GL 4 was to prove the multiplicity oneconjecture for SL 2 of Langlands and Labesse.Theorem. [17] In the spectral decompositionZ4. p (SL 2 (*)\SL 2 (A))=0 m^ninto irreducible cuspidal representations, the multiplicities m* are at most one.This was previously known to be true for GL„ and false for SL„ for n > 3.For further applications, for example to the Tate conjecture, see [17].The primary application of the tensor product lifting from GL 2 x GL 3 to GL 6 ofKim and Shahidi was in the establishment of the symmetric cube lifting and throughthis the symmetric fourth power lifting, so the applications of the symmetric powerliftings outlined below are applications of this lifting as well.3. Symmetric powers: It was early observed that the existence of the symmetricpower liftings of GL 2 to GL„ + i for all n would imply the Ramanujan-Peterssonand Selberg conjectures for modular forms. Every time a symmetric power lift isobtained we obtain better bounds towards Ramanujan. The result which followsfrom the symmetric third and fourth power lifts of Kim and Shahidi is the following.Theorem. [16] Let n be a cuspidal representation of GL 2 (A) such that thesymmetric cube lift of'n is again cuspidal. Let diag(a v ,ß v ) be the Satake parameterfor an unramified local component. Then \a v \, \ß v \ < qj . If in addition the fourthsymmetric power lift is not cuspidal, the full Ramanujan conjecture is valid.The corresponding statement at infinite places, i.e., the analogue of the Selbergconjecture on the eigenvalues of Mass forms, is also valid [14]. Estimatestowards Ramanujan are a staple of improving any analytic number theoretic estimatesobtained through spectral methods. Both the 1/9 non-archimedean and1/9 archimedean estimate towards Ramanujan above were applied in obtaining theprecise form of the exponent in our recent result with Sarnak breaking the convexitybound for twisted Hilbert modular F-series in the conductor aspect, which inturn was the key ingredient in our work on Hilbert's eleventh problem for ternaryquadraticforms. Similar in spirit are the applications by Kim and Shahidi to thehyperbolic circle problem and to estimates on sums of shifted Fourier coefficients[15].In addition Kim and Shahidi were able to obtain results towards the Sato-Tateconjecture.Theorem. [16] Let n be a cuspidal representation o/GL 2 (A) with trivial centralcharacter. Let diag(a v ,ß v ) be the Satake parameter for an unramified localcomponent and let a v = a v + ß v . Assuming n satisfies the Ramanujan conjecture,there are sets T ± of positive lower density for which a v > 2cos(27r/ll) — e for allv £ T+ and a v < ^2COS(2TT/11) + e for all v £ T~. [Note: 2COS(2TT/11) = 1.68.../Kim and Shahidi have other conditional applications of their liftings suchas the conditional existence of Siegel modular cusp forms of weight 3 (assumingArthur's multiplicity formula for Sp 4 ). We refer the reader to [15] for details onthese applications and others.

128 J. W. Cogdell I. I. Piatetski-ShapiroReferences[1] A. Borei, Automorphic F-functions. Proc. Symp. Pure Math. 33, Part 2, (1979),27-61.[2] J.W. Cogdell, H. Kim, I.I. Piatetski-Shapiro, and F. Shahidi, On lifting fromclassical groups to GL n . Pubi. Math. IHES 93 (2001), 5^30.[3] J.W. Cogdell, H. Kim, I.I. Piatetski-Shapiro, and F. Shahidi, On lifting fromclassical groups to GL„, II, in preparation.[4] J.W. Cogdell and I.I. Piatetski-Shapiro, Converse theorems for GL„, I. Pubi.Math. IHES 79 (1994), 157^214.[5] J.W. Cogdell and I.I. Piatetski-Shapiro, Stability of gamma factors for S0 2 „ + i.manuscripta math. 95 (1998) 437-461.[6] J.W. Cogdell and I.I. Piatetski-Shapiro, Converse theorems for GL„, IL J.reine angew. Math. 507 (1999), 165^188.[7] S. Gelbart and F. Shahidi, Boundedness of automorphic F-functions in verticalstrips. Journal of the AMS 14 (2001), 79^107.[8] D. Ginzburg, S. Rallis, and D. Soudry, Generic automorphic forms on S0 2n +i:functorial lift to GL 2 „, endoscopy, and base change. Internat. Math. Res. Notices.no. 14(2001), 729^764.[9] H. Jacquet, I.I. Piatetski-Shapiro, and J. Shalika, Rankin-Selberg convolutions,Amer. J. Math., 105 (1983), 367-164.[10] H. Jacquet and J. Shalika, A lemma on highly ramified e-factors, Math. Ann.,271 (1985), 319^332.[11] H. Jacquet and J. Shalika, Rankin-Selberg convolutions: Archimedean theory,in Festschrift in Honor of I.I. Piatetski-Shapiro, Part I, Weizmann SciencePress, Jerusalem, 1990, 125^207.[12] D. Jiang and D. Soudry, The local converse theorem for S0 2n +i and applications.Ann. of Math, to appear.[13] D. Jiang and D. Soudry, Generic representations and local Langlands reciprocitylaw for p-adic S0 2n +i. Preprint (2001).[14] H. Kim, Functoriality for the exterior square of GL 4 and the symmetric fourthofGL 2 . Preprint (2000).[15] H. Kim and F. Shahidi, Functorial products for GL 2 x GL 3 and symmetriccube for GL 2 . Ann. of Math., to appear.[16] H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications.Duke Math. J., 112 (2002), to appear.[17] D. Ramakrishnan, Modularity of the Rankin-Selberg F-series, and multiplicityonefor SL 2 . Ann. of Math. (2) 152 (2000), 45-111.[18] F. Shahidi, A proof of Langlands' conjecture on Plancherel measures; complementaryseries for p-adic groups. Ann. of Math. (2) 132 (1990), 273^330.[19] F. Shahidi, On multiplicativity of local factors. Festschrift in honor of I. I.Piatetski-Shapiro, Part II (Ramat Aviv, 1989), 279^289. Israel Math. Conf.Proc, 3, Weizmann, Jerusalem, 1990.[20] F. Shahidi, Local coefficients as Mellin transforms of Bessel functions; Towardsa general stability. Preprint (2002).[21] F. Shahidi, Automorphic F-Functions and Functoriality. These Proceedings.

ICM 2002 • Vol. II • 129^138Constructing and CountingNumber FieldsH. Cohen*AbstractIn this paper we give a survey of recent methods for the asymptotic andexact enumeration of number fields with given Galois group of the Galoisclosure. In particular, the case of fields of degree up to 4 is now almostcompletely solved, both in theory and in practice. The same methods alsoallow construction of the corresponding complete tables of number fields withdiscriminant up to a given bound.2000 Mathematics Subject Classification: 11R16, 11R29, 11R45, 11Y40.Keywords and Phrases: Discriminants, Number field tables, Kummer theory.1. IntroductionLet K be a number field considered as a fixed base field, K an algebraicclosure of K, and G a transitive permutation group on n letters. We considerthe set TK,U(G) of all extensions L/K of degree n with F c K such that theGalois group of the Galois closure L of L/K viewed as a permutation group onthe set of embeddings of F into L is permutation isomorphic to G (i.e, n/m(G)times the number of extensions up to FJ-isomorphism, where m(G) is the numberof K-automorphisms of F). We writeN K , n (G,X) = \{L £ T n (G), \N[v(L/K))\ < X}\ ,where Q(L/K) denotes the relative ideal discriminant and TV the absolute norm.The aim of this paper is to give a survey of new methods, results, and conjectureson asymptotic and exact values of this quantity. It is usually easy to generalizethe results to the case where the behavior of a finite number of places of K in theextension L/K is specified, for example if K = Q when the signature (Fi, R 2 ) of Fis specified, with Fi + 2F 2 = n.Remarks.* Laboratoire A2X, Institut de Mathématiques, Université Bordeaux I, 351 Cours de laLibération, 33405 TALENCE Cedex, France. E-mail: cohen@math.u-bordeaux.fr

130 H. Cohen1. It is often possible to give additional main terms and rather good error termsinstead of asymptotic formulas. However, even in very simple cases such asG = S 3 , this is not at all easy.2. The methods which lead to exact values of NK,U(G,X) always lead to algorithmsfor computing the corresponding tables, evidently only whenNK,U(G,X) is not too large in comparison to computer memory, see for example[8] and [10].General conjectures on the subject have been made by several authors, forexample in [3]. The most precise are due to G. Malle (see [24], [25]). We need thefollowing definition.Definition 1.1. For any element g £ S n different from the identity, definethe index 'md(g) of g by the formula 'md(g) = n — |orbits of g\. We define the indexi(G) of a transitive subgroup G of S n by the formulaExamples.i(G) = min ind(p) .1. The index of a transposition is equal to 1, and this is the lowest possible indexfor a nonidentity element. Thus i(S n ) = 1.2. If G is an Abelian group, and if £ is the smallest prime divisor of \G\, theni(G) = \G\(l-l/t).Conjecture 1.2. For each number field K and transitive group G onn lettersas above, there exist a strictly positive integer 1>K(G) and a strictly positive constantCK(G) such thatN K , n (G,X) ~ c K (G)X 1 / i{G \\ogX) bK{G) - 1 .In [25], Malle gives a precise conjectural value for the constant bx(G) whichis too complicated to be given here.Remarks.1. This conjecture is completely out of reach since it implies the truth of theinverse Galois problem for number fields.2. If true, this conjecture implies that for any composite n, the proportion ofS^-extensions of K of degree n among all degree n extensions is strictly lessthan 1 (but strictly positive), contrary to the case of polynomials.The following results give support to the conjecture (see [2], [9], [18], [19], [20],[21], [22], [23], [28], [30]).Theorem 1.3. We will say that the above conjecture is true in the weak senseif there exists CK (G) > 0 such that for all e > 0 we haveCK (G)-X 1^G) < N K , n (G,X) < A-V'CGH* .1. (Moki, Wright). The conjecture is true for all Abelian groups G.2. (Davenport-Heilbronn, Datskovsky-Wright). The conjecture is true for n = 3and G = S 3 .

Constructing and Counting Number Fields 1313. (Cohen-Diaz-Olivier). The conjecture is true for n = 4 and G = F 4 .4- (Bhargava, Yukie). The conjecture is true for n = 4 and G = S 4 , in the weaksense if K ^ Q.5. (Klüners-Malle). The conjecture is true in the weak sense for all nilpotentgroups.6. (Kable-Yukie). The conjecture is true in the weak sense for n = 5 and G = S5.The methods used to prove these results are quite diverse. In the case ofAbelian groups G, one could think that class field theory gives all the answers sonothing much would need to be done. This is not at all the case, and in fact Kummertheory is usually more useful. In addition, Kummer theory allows us more generallyto study solvable groups. We will look at this method in detail.Apart from Kummer theory and class field theory, the other methods have adifferent origin and come from the classification of orders of degree n, interpretedthrough suitable classes of forms. This can be done at a very clever but still elementarylevel when the base field is Q, and includes the remarkable achievement ofM. Bhargava in 2001 for quartic orders. Over arbitrary K, one needs to use anddevelop the theory of prehomogeneous vector spaces, initiated at the end of the1960's by Sato and Shintani (see for example [26] and [27]), and used since withgreat success by Datskovsky-Wright, and more recently by Wright-Yukie (see [29]),Yukie and Kable-Yukie.2. Kummer theoryThis method applies only to Abelian, or more generally solvable extensions.2.1. Why not class field theory?It is first important to explain why class field theory, which is supposed tobe a complete theory of Abelian extensions, does not give an answer to countingquestions. Let us take the very simplest example of quadratic extensions, thus withG = C 2 . A trivial class-field theoretic argument gives the exact formulaN K , 2 (C 2 ,X) = -1+ J2 1 MCl+AK)) M K( Xwhere 0 runs over all integral ideals of K of norm less than or equal to X, Cl^(K)denotes the narrow ray class group modulo 0, rk(G) denotes the 2-rank of an Abeliangroup G, and MK(U) is the generalization to number fields of the summatory functionM(n) of the Möbius function.This formula is completely explicit, the quantities Cl^(K) and the functionMic(n) are algorithmically computable with reasonable efficiency, so we can computeiVif j2 (C 2 ,X) for reasonably small values of X in this way. Unfortunately, thisformula has two important drawbacks.The first and essential one is that, if we want to deduce from it asymptoticinformation on NK, 2 (C 2 ,X), we need to control rk(Cl+(K), which can be done,

132 H. Cohenalthough rather painfully, but we also need to control MK(U), which cannot bedone (recall for instance that the Riemann Hypothesis can be formulated in termsof this function).The second drawback is that, even for exact computation it is rather inefficient,compared to the formula that we obtain from Kummer theory. Thus, even thoughKummer theory is used in a crucial way for the constructions needed in the proofsof class field theory, it must not be discarded once this is done since the formulathat it gives are much more useful, at least in our context.2.2. Quadratic extensionsAs an example, let us see how to treat quadratic extensions using Kummertheory instead of class field theory. Of course in this case Kummer theory is trivialsince it tells us that quadratic extensions of K are parameterized by K*/K*" minusthe unit class. This is not explicit enough. By writing for any a £ K*, OLLK = nq 2with o an integral squarefree ideal, it is clear that K*/K*" is in one-to-one correspondencewith pairs (o, «), where o are integral squarefree ideals whose idealclass is a square, and u is an element of the so-called Selmer group of K, i.e., thegroup of elements u £ K* such that UZK = q 2 for some ideal q, divided by K**.We can then introduce the Dirichlet series $if, 2 (C7 2) «) = ^ L jV'(()(F/Kj)^s,wherethe sum is over quadratic extensions L/K in K. A number of not completely trivialcombinatorial and number-theoretic computations (see [9]) lead to the explicitformula

Constructing and Counting Number Fields 1332.3. General finite Abelian extensionsThe same method can in principle be applied to any finite Abelian groupG. I say "in principle", because in practice several problems arise. For the basefield K = Q, a complete and explicit solution was given by Maki in [23]. For ageneral base field, a solution has been given by Wright in [28], but the problemwith his solution is that the constant CK(G), although given as a product of localcontributions, cannot be computed explicitly without a considerable amount ofadditional work. It is always a finite linear combination of Euler products.In joint work with F. Diaz y Diaz and M. Olivier, using Kummer theoryin a manner analogous but much more sophisticated than the case of quadraticextensions, we have computed completely explicitly the constants CK (G) for G = Cithe cyclic group of prime order I, for G = C 4 and for G = V 4 = C 2 x C 2 . Althoughour papers are perhaps slightly too discursive, to give an idea the total number ofpages for these three results exceeds 100. We refer to [7], [13], [11], [15], [16] for thedetailed proofs, and to [12] and [14] for surveys and tables of results. We mentionhere the simplest one, for G = V 4 . We haveNK,A(VA, X) ~ c/f (V 4 ) X 1 ' 2 log 2 X with«=^« j n(i+i)(i-^n14 2 1 (1 - I/AVMP) + (i + WP) 2Np Np 2 Np 3 Np e{p)+1p|2Zj f 1 + —7-A/pOf course, the main difficulty is to compute correctly the local factor at 2.As usual, we can use our methods to compute very efficiently the N function.For example, we obtain (see [4]):NQ, 3 (C 3 , 10 37 ) = 501310370031289126,ÌVQ, 4 (C 4 , IO 32 ) = 1220521363354404,2.4. Dihedral £) 4 -extensionsÌVQ, 4 (V 4 , IO 36 ) = 22956815681347605884.We can also apply our method to solvable extensions. The case of quarticF 4 -extensions, where F 4 is the dihedral group of order 8, is especially simple andpretty. Such an extension is imprimitive, i.e., is a quadratic extension of a quadraticextension. Conversely, imprimitive quartic extensions are either F 4 -extensions, orAbelian with Galois group C 4 or V 4 . These can easily be counted as explained above,and in any case will not contribute to the main term of the asymptotic formula,so they can be neglected (or subtracted for exact computations). Since we havetreated completely the case of quadratic extensions, it is just a matter of showing3

134 H. Cohenthat we are allowed to sum over quadratic extensions of the base field to obtain thedesired asymptotic formula (for the exact formula nothing needs to be proved), andthis is not difficult. In this way, we obtain that NK,I(D±,X) ~ CK(D^) X for anexplicit constant CK(D±) (in fact we obtain an error term 0(X 3 / 4 +e)). This resultis new even for K = Q, although its proof not very difficult. In the case K = Q,we have for instance6 2- r2 WF((^),l)CQ(^)=Ê D2 L;^; 2 ; =0.1046520224...,where the sum is over fundamental discriminants D, r 2 (D) = r 2 (Q(VD)), andL((—),s) is the usual Dirichlet series for the character (—).Remark. In the Abelian case, it is possible to compute the Euler productswhich occur to hundreds of decimal places if desired using almost standard zetaproductexpansions, see for example [6]. Unfortunately, we do not know if it ispossible to express CQ(F 4 ) as a finite linear combination of Euler products (or atleast as a rapidly convergent infinite series of such), hence we have only been ableto compute 9 or 10 decimal places of this constant. We do not see any practicalway of computing 20 decimals, say.Our method also allows us to compute ÌVQ J4 (F 4 ,X) exactly. However, here amiracle occurs: when k is a quadratic field, in the formula that we have given abovefor $fc, 2 (C7 2) «) all the quadratic characters \ which we need are genus charactersin the sense of Gauss, in other words there is a decompositionL k (x,s) =L((^),.s)L((^),s)into a product of two suitable ordinary Dirichlet F-series. This gives a very fastmethod for computing ÌVQ J4 (F 4 , X), and in particular we have been able to computeiV Qi4 (F> 4 ,10 17 ) = 10465196820067560.We can also count the number of extensions with a given signature. Themethod is completely similar, but here not all characters are genus characters.In fact, it is only necessary to add a single nongenus character to obtain all thenecessary ones, but everything is completely explicit, and closely related to therational quartic reciprocity law. I refer to [5] for details.2.5. Other solvable extensionsWe can also prove some partial results in the case where G = A4 or G = S4 (ofcourse the results for S 4 are superseded by Bhargava's for K = Q, and by Yukie'sfor general K; still, the method is also useful for exact computations), see [17].In the case of quartic A4 and ^-extensions (or, for that matter, of cubic S 3 -extensions), we use the diagram involving the cubic resolvent (the quadratic onefor S 3 -extensions), also called the Hasse diagram. We then have a situation whichbears some analogies with the F 4 case. The differences are as follows. Instead ofhaving to sum over quadratic extensions of the base field K, we must sum over cubicextensions, cyclic for A4 and noncyclic for S4. As in the F 4 -case, we then have toconsider quadratic extensions of these cubic fields, but generated by an element of

Constructing and Counting Number Fields 135square norm. It is possible to go through the exact combinatorial and arithmeticcomputation of the corresponding Dirichlet series, the cubic field being fixed. Thisin particular uses some amusing local class field theory. As in the F 4 case, we thenobtain the Dirichlet generating series for discriminants of A4 (resp., S4) extensionsby summing the series over the corresponding cubic fields.Unfortunately, we cannot obtain from this any asymptotic formula. The reasonis different in the A4 and the S 4 case. In the A4 case, the rightmost singularity ofthe Dirichlet series is at s = 1/2. Unfortunately, this is simultaneously the mainsingularity of each individual Dirichlet series, and also that of the generating seriesfor cyclic cubic fields. Thus, although the latter is well understood, it seems difficult(but not totally out of reach) to paste things together. On the other hand, we cando two things rigorously in this case. First, we can prove an asymptotic formulafor .4 4 -extensions having a fixed cubic resolvent. Tables show that the formula isvery accurate. Second, we can use our formula to compute NK,ì(Sì,X) exactly.For instance, we have computed ÌVQ J4 (.4 4 , IO 16 ) = 218369252. This computationis much slower than in the F 4 -case, because we do not have the miracle of genuscharacters, and we must compute the class and unit group of all the cyclic cubicfields.In the S4 case, the situation is different. The main singularity of each individualDirichlet series is still at s = 1/2 (because of the square norm condition),and the rightmost singularity of the generating series for noncyclic cubic fields isat s = 1, so the situation looks better (and analogous to the F 4 situation withs replaced by s/2). Unfortunately, as already mentioned we know almost nothingabout the generating series for noncyclic cubic fields, a fortiori with coefficients. Sowe cannot go further in the asymptotic analysis. As in the A4 case, however, wecan compute exactly either the number of ^-extensions corresponding to a fixedcubic resolvent, or even NK,ì(Sì,X) itself. The problem is that here we must computeclass and unit groups of all noncyclic cubic fields of discriminant up to X,while cyclic cubic fields of discriminant up to X are much rarer, of the order ofX 1 / 2 instead. We have thus not been able to go very far and obtained for exampleÌVQ, 4 (S 4 , IO 7 ) = 6541232.3. Prehomogeneous vector spacesThe other methods for studying NK,U(G,X) are two closely related methods:one is the use of generalizations of the Delone-Fadeev map, which applies whenK = Q. The other, which can be considered as a generalization of the first, is theuse of the theory of prehomogeneous vector spaces, initiated by Sato and Shintaniin the 1960's.3.1. Orders of small degreeWe briefly give a sketch of the first method. We would first like to classifyquadratic orders. It is well known that, through their discriminant, such orders arein one-to-one correspondence with the subset of nonsquare elements of Z congruent

136 H. Cohento 0 or 1 modulo 4, on which SLi(Z) (the trivial group) acts. Thus, for fixeddiscriminant, the orbits are finite (in fact of cardinality 0 or 1). For maximalorders, we need to add local arithmetic conditions at each prime p, which are easyfor p > 2, and slightly more complicated for p = 2.We do the same for small higher degrees. For cubic orders, the classificationis due to Davenport-Heilbronn (see [19], [20]). These orders are in one-to-one correspondencewith a certain subset of Sym 3 (Z 2 ), i.e., binary cubic forms, on whichSL 2 (Z) acts. Since once again the difference in "dimensions" is 4 — 3 = 1, for fixeddiscriminant the orbits are finite, at least generically. For maximal orders, we againneed to add local arithmetic conditions at each prime p. These are easy to obtainfor p > 3, but are a little more complicated for p = 2 and p = 3. An alternate wayof explaining this is to say that a cubic order can be given by a nonmonic cubicequation, which is almost canonical if representatives are suitably chosen.For quartic orders, the classification is due to M. Bhargava in 2001, who showedin complete detail how to generalize the above. These orders are now in one-toonecorrespondence with a certain subset of Z 2 ® Sym^Z 3 ), i.e., pairs of ternaryquadraticforms, on which SL 2 (Z) x SL 3 (Z) acts. Once again the difference in"dimensions" is 2 x 6 — (3 + 8) = 1, so for fixed discriminant the orbits are finite,at least generically. For maximal orders, we again need to add local arithmeticconditions at each prime p, which Bhargava finds after some computation. Analternate way of explaining this is to say that a quartic order can be given by theintersection of two conies in the projective plane, the pencil of conies being almostcanonical if representatives are suitably chosen.For quintic orders, only part of the work has been done, by Bhargava andKable-Yukie in 2002. These are in one-to-one correspondence with a certain subsetof Z 4 ® A 2 (Z 5 ), i.e., quadruples of alternating forms in 5 variables, on which SL 4 (Z) xSL 5 (Z) acts. Once again the difference in "dimensions" is 4 x 10 — (15 + 24) = 1,so for fixed discriminant the orbits are finite, at least generically. The computationof the local arithmetic conditions, as well as the justification for the process ofpoint counting near the cusps of the fundamental domain has however not yet beencompleted.Since prehomogeneous vector spaces have been completely classified, this theorydoes not seem to be able to apply to higher degree orders, at least directly.3.2. ResultsUsing the above methods, and generalizations to arbitrary base fields, thefollowing results have been obtained on the function NK,U(G,X) (many other deepand important results have also been obtained, but we fix our attention to thisfunction). It is important to note that they seem out of reach using more classicalmethods such as Kummer theory or class field theory mentioned earlier.Theorem 3.1. Let K be a number field of signature (r\,r 2 ), and as abovewrite (K(1) for the residue of the Dedekind zeta function of K at s = 1.1. (Davenport-Heilbronn [19], [20]). We have NQ :3 (S 3 ,X) ~ cq(S 3 ) X withC Q( S 3)= TTjjy •

Constructing and Counting Number Fields 1372. (Datskovsky-Wright [18]). We have N Kß (S 3 ,X) ~ c K (S 3 )X withf. (a, (2Y 1 - 1 (IY 2 (K(1),3; \6j CK(3) '3. (Bhargava [1], [2]). We have ÌVQ J4 (S' 4 ,X) ~ CQ(S 4 )X with1 1,a-> 5TTA !6 Y V P -P P4- (Yukie [30]). There exist two strictly positive constants c\(K) and C2(K) suchthatci X < NK,4(S4,X) < c 2 Xlog" J (X) .Under some very plausible convergence assumptions we should have in factA r if j4 (S' 4 ,X) ~ CK(S4) X with5. (Kable-Yukie [21]). There exists a strictly positive constant c\ such that forall e > 0 we haveciX

138 H. Cohen[7] H. Cohen, F. Diaz y Diaz and M. Olivier, Densité des discriminants des extensionscycliques de degré premier, C. R. Acad. Sci. Paris 330 (2000), 61-66.[8] H. Cohen, F. Diaz y Diaz and M. Olivier, Construction of tables of quarticfields using Kummer theory, Proceedings ANTS IV, Leiden (2000), LectureNotes in Computer Science 1838, Springer-Verlag, 257-268.[9] H. Cohen, F. Diaz y Diaz and M. Olivier, Enumerating quartic dihedral extensions,Compositio Math., 28p., to appear.[10] H. Cohen, F. Diaz y Diaz and M. Olivier, Constructing complete tables ofquartic fields using Kummer theory, Math. Comp., lip., to appear.[11] H. Cohen, F. Diaz y Diaz and M. Olivier, On the density of discriminants ofcyclic extensions of prime degree, J. reine und angew. Math., 40p., to appear.[12] H. Cohen, F. Diaz y Diaz and M. Olivier, A Survey of Discriminant Counting,Proceedings ANTS V Conference, Sydney (2002), Lecture Notes in Comp. Sci.,15p., to appear.[13] H. Cohen, F. Diaz y Diaz and M. Olivier, Cyclotomic extensions of numberfields, 14p., submitted.[14] H. Cohen, F. Diaz y Diaz and M. Olivier, Counting discriminants of numberfields, 36p., submitted.[15] H. Cohen, F. Diaz y Diaz and M. Olivier, Counting cyclic quartic extensionsof a number field, 30p., submitted.[16] H. Cohen, F. Diaz y Diaz and M. Olivier, Counting biquadratic extensions ofa number field, 17p., submitted.[17] H. Cohen, F. Diaz y Diaz and M. Olivier, Counting A4 and S4 extensions ofnumber fields, 20p., in preparation.[18] B. Datskovsky and D. J. Wright, Density of discriminants of cubic extensions,J. reine und angew. Math. 386 (1988), 116—138.[19] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fieldsI, Bull. London Math. Soc. 1 (1969), 345-348.[20] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fieldsII, Proc. Royal. Soc. A 322 (1971), 405-420.[21] A. Kable and A. Yukie, On the number of quintic fields, preprint.[22] J. Kliiners and G. Malle, Counting Nilpotent Galois Extensions, submitted.[23] S. Maki, On the density of abelian number fields, Thesis, Helsinki, 1985.[24] G. Malle, On the distribution of Galois groups, J. Number Theory, to appear.[25] G. Malle, personal communication.[26] T. Shintani, On Dirichlet series whose coefficients are class numbers of integralbinary cubic forms, J. Math. Soc. Japan 24 (1972), 132-188.[27] T. Shintani, On zeta-functions associated with the vector space of quadraticforms, J. Fac. Sci. Univ. Tokyo, Sec. la 22 (1975), 25-66.[28] D. J. Wright, Distribution of discriminants of Abelian extensions, Proc. LondonMath. Soc. (3) 58 (1989), 17-50.[29] D. J. Wright and A. Yukie, Prehomogeneous vector spaces and field extensions,Invent. Math. 110 (1992), 283-314.[30] A. Yukie, Density theorems related to prehomogeneous vector spaces, preprint.

ICM 2002 • Vol. II • 139-148Analyse p-adique etReprésentations GaloisiennesJean-Marc Fontaine*AbstractThe notion of a p-adic de Rhani representation of the absolute Galoisgroup of a p-adic field was introduced about twenty years ago (see e.g. [Fo93]).Three important results for this theory have been obtained recently: The structuretheorem for the almost Cp-representations, the theorem weakly admissibleimplies admissible and the theorem de Rham implies potentially semi-stable.The proofs of the first two theorems are closely related to the study of a newkind of analytic groups, the Banaeh-Golmez spaces and the proof of the thirduses deep results on p-adic differential equations on the Robba ring.2000 Mathematics Subject Classification: 11F80, 11S25, 12H25, 14G22.Keywords and Phrases: Galois representations, de Rham representations,Semi-stable representations, p-adic Banach spaces, p-adic differential equations.1. Représentations p-adiques1.1. — Dans tout ce qui suit, K est un corps de caractéristique 0, complet pour unevaluation discrète, à corps résiduel k parfait de caractéristique p > 0. On choisitune clôture algébrique K de K, on note C son complété et | | p la valeur absolue deC normalisée par \p\ p = p^1.On pose GK = Gal(K/K).Une représentation banachique (de GK) est un espace de Banachp-adique munid'une action linéaire et continue de GK- Avec comme morphismes les applicationsQp -linéaires continues G A-équi variantes, ces représentations forment une catégorieadditive Q p -linéaire B(GK)-Une C-représentation (de GK) est un C-espace vectoriel de dimension finiemuni d'une action semi-linéaire et continue de GK- Lorsque k est fini, la catégorieRep c (Gif ) des C-représentations s'identifie à une sous-catégorie pleine de B(GK) '•* Institut Universitaire de France et UMR 8628 du CNRS, Mathématique, Université de Paris-Sud, Bâtiment 425, 91405 ORSAY Cedex, France. E-mail: fontaine@math.u-psud.fr

140 J.-M. FontainePROPOSITION [FoOO]. — Supposons k fini. Si W\ et W 2 sont des C-représentations,toute application Q p -linéaire continue GK-équivariante de W\ dans W 2 estC-linéaire.Disons que deux représentations banachiques Si et S 2 sont presque isomorphess'il existe un triplet (Vi,V 2 ,o;) où V» est un sous-Qp-espace vectoriel de dimensionfinie de Si, stable par GK, et où a : Si/Vi —¥ S 2 /V 2 est un isomorphisme (dansB(GK))- Une presque-C-représentation (de GK) est une représentation banachiquequi est presque isomorphe à une C-représentation. On note C(GK) la sous-catégoriepleine de B(GK) dont les objets sont les presque-C-représentations. Elle contientla catégorie Rep c (GK) et la catégorie Rep Q (GK) des représentations p-adiques dedimension finie (de GK) comme sous-catégories pleines.THéORèME A [Fo02]. — Supposons k fini. La catégorie C(GK) est abélienne. Ilexiste sur les objets de C(GK) une unique fonction additive dh :S H> (d(S),h(Sj) £N x Z telle que dh(W) = (dime W, 0) si W est une C-représentation et dh(V) =(0, diniQ p V) si V est de dimension finie sur Q p .Si S et T sont des objets de C(GK), les Q p -espaces vectoriels Ext l Ct GK^(S,T)sont de dimension finie et sont nuls pour i $ {0,1,2}. On aE? =0 (^l) i dim Qp Ex4 (Gjf) (S,F) = -[Ä- : Q p ]h(S)h(T).1.2. — Soit W FJi l'ensemble des suites (x^)neN d'éléments de C vérifiant^x(n+i)y _ x(n) p 0ur f 0U i n# Avec les lois(x + y)W = lïmm,ôo(x {n+m) + y( n + m )y m et (xy) {n) = x {n) y {n)c'est un corps algébriquement clos de caractéristique p, complet pour la valeurabsolue définie par |x| = |x^| p et on note F l'anneau de la valuation. Son corpsrésiduel s'identifie au corps résiduel k de K. L'anneau W(R) des vecteurs de Wittà coefficients dans F est intègre. Choisissons e,n £ R vérifiant e^ = 1, gC 1 ) ^ 1et n(°ï = p et, pour tout a £ R notons [a] = (a, 0,0,...) son représentant deTeichmüller dans W(R). L'application 6 : W(R) —¥ Oc qui envoie (ao,ai,...) surS»eN-P" a « est un homomorphisme d'anneaux surjectif dont le noyau est l'idéalprincipal engendré par £ = [n] —p. On note encore 9 : W(R)[l/p] —¥ C l'applicationdéduite en rendant p inversible. Rappelons que B^R = |im ^KW(R)[l/p]/(Ç n ) et-neNque le corps B^R des périodes p-adiques est le corps des fractions de Fj fl . Touteunité a de F s'écrit d'une manière unique sous la forme a = aoa + avec ao £ ket \a + — 1| < 1, la série J2 =A^^n+1 n ([ a+ ~\ ~ l)"/ n converge dans Fj~ fl vers unélément noté log [a] ; on pose t = log[e]. On a B^R = B^R[l/t]. On note A cris leséparé complété pour la topologie p-adique de la sous-W(F)-algèbre de W(R)[l/p]engendrée par les £ TO /m! pour m G N. Alors A cris s'identifie à un sous-anneau deFjjj contenant t et on pose Bf ris = A cris [l/p] et F cr j S = Bf ris [l/t] C B^R- La sérieJ2 n =A^^n+1^n/ n P n conv erge dans B^R vers un élément log[7r] = log([7r]/p) et on*- ' Voir [Fo88a] (resp. [Fo88b]) pour plus de détails sur la construction de B r i R , B CTis et B st(resp. sur les représentations p-adiques de de Rham et potentiellement semi-stables).

Analyse p-adique et Représentations Galoisiennes 141note B„t la sous-F cr j S -algèbre de B^R engendrée par log[7r]. Pour tout b £ R nonnul, il existe r,s £ Z, avec s > 1 et une unité a de F tels que 6 S = n r a et on poselog[6] = (r log[7r] + log[a])/s. On a B st = B cris [log[b]] dès que 6 n'est pas une unité.Soit £ l'ensemble des extensions finies de K contenues dans K. Pour toutF £ £, on pose GL = Gal(K/L) et on note F 0 le corps des fractions de l'anneau desvecteurs de Witt à coefficients dans le corps résiduel de F. Le corps K se plongede façon naturelle dans Fj~ fi et l'action de GK s'étend de façon naturelle à B^R,l'anneau B st est stable par GK- Pour tout F £ £, on a (B(ìR) GL = L tandis que(B st ) GL = L 0 et l'application naturelle F ®L 0 B st —¥ B^R est injective.Pour toute représentation p-adique V de GK de dimension finie h sur Q p , onpose D,IR(V) = (K ®q p V) GK , D st (V) = (B st ®q p V) GK et, si F est une extensionfinie de K contenue dans K, D st^(V) = (B st ®q p V) GL . On a dim^ D

142 J.-M. Fontainea) on a tn(D) = ìN(D),b) pour tout sous-F 0 -espace vectoriel D' de D, stable par ip,N et Gal(L/K),on a ìH(D') < ìN(D') (on a muni D' K c DK de la filtration induite).THéORèME C [CFOO]. — Soit L d K une extension finie galoisienne de K.i) Pour toute représentation L-semi-stable V, D st^(V) est admissible.ii) Le fondeur qui à, V associe D st^(V) induit une équivalence^ entre la souscatégoriepleine Rep st L(GK) de Rep Q (GK) dont les objets sont les représentationsL-semi-stables et la catégorie des (ip, N,Gal(L/K))-modules filtrés admissibles.Remarque. Il était jusqu'à présent d'usage [Fo88b] d'appeler faiblement admissiblece que nous appelons ici admissible. On savait (loc.cit., th.5.6.7) que D st: Linduit une équivalence entre la catégorie Rep st L(GK) et une sous-catégorie pleinede la catégorie des modules filtrés (faiblement) admissibles ; on conjecturait que cefoncteur est essentiellement surjectif et c'est ce qui est prouvé dans [CFOO].2. Espaces de Banach-Colmez^2.1. — Une C-algèbre de Banach est une C-algèbre normée complète A ; son spectremaximal est l'ensemble Spm c A des sections continues s : A —t C du morphismestructural. Si / £ A et s £ Spm c A, on pose f(s) = s(f). Une C-algèbre spectraleest une C-algèbre de Banach A telle que la norme est la norme spectrale, i.e. telleque, pour tout / £ A, ||/|| = sup se spm c A\f(s)\ p ; dans ce cas, Spm c A est unespace métrique complet (la distance étant définie par d(si,s 2 ) = sup||j|i

Analyse p-adique et Représentations Galoisiennes 143p r So C S C P S SQ- H revient au même de dire qu'il existe une norme équivalente àla norme donnée pour laquelle S est la boule unité.Une C-structure analytique sur S est la donnée d'un C-groupe spectral commutatifaffine S et d'un homomorphisme continu du groupe topologique sous-jacentà S dans S dont l'image est un réseau et le noyau un Z p -module de type fini. Ondit que deux C-structures analytiques S et T sur S sont équivalentes si S x s T estun sous-groupe spectral de S x T. Un (espace de) Banach analytique (sur C) estla donnée d'un espace de Banach muni d'une classe d'équivalence de C-structuresanalytiques (on les appelle les structures admissibles de S). On dit que S est effectifs'il existe une structure admissible S telle que l'application S —¥ S est injective.Un morphisme de Banach analytiques f : S —¥ T est une application Q p -linéaire continue telle qu'il existe des structures admissibles S de S et T de T etun morphisme S —¥ T qui induit /. Les Banach analytiques forment une catégorieadditive BAc-Si S est un Banach analytique et si V est un sous-Q p -espace vectoriel de dimensionfinie, le quotient S/V a une structure naturelle de Banach analytique. On ditque deux Banach analytiques Si et S 2 sont presque isomorphes s'il existe des sous-Qp -espaces vectoriels de dimension finie V\ de S\ et V 2 de S 2 et un isomorphismeSi/Vi —¥ S2/V2 (de Banach analytiques).Le groupe sous-jacent à Oc a une structure naturelle de groupe spectral commutatifaffine : on a oc = Spm c C{X} où C{X} est l'algèbre de Tate des sériesformelles à coefficients dans C en l'indéterminée X dont le terme général tend vers0. Ceci fait de C un espace de Banach analytique effectif. Un Banach analytiquevectoriel est un Banach analytique isomorphe à C d pour un entier d convenable.Un espace de Banach-Colmez est un Banach analytique presque vectoriel, i.e. unBanach analytique qui est presque isomorphe à un Banach analytique vectoriel.On note BCc la sous-catégorie pleine de BAc. dont les objets sont les espaces deBanach-Colmez.PROPOSITION (théorème de Colmez W). — La catégorie BCc

144 J.-M. Fontainetation est munie d'unee structure naturelle d'espace de Banach-Colmez ; touteapplication Q p -linéaire continue G/f-équivariante d'une presque C-représentationdans une autre est analytique. Le fait que C(GK) est abélienne et l'existence de lafonction dh résultent alors du théorème de Colmez.Le principe de la preuve du théorème C est le suivant : On se ramène facilementau cas semi-stable, i.e. au cas où F = K. Il s'agit de vérifier que, si D est un (ip, N)-module filtré (faiblement) admissible de dimension h, il existe une représentationp-adique V de dimension h telle que D st: K(V) soit isomorphe à D. Une torsionà la Tate permet de supposer que F'ù°DK = DK- Notons V^' (D) le Q p -espacevectoriel des applications FJ 0 -linéaires de D dans B st riB^R qui commutent à l'actionde ip et de N et V^' le quotient du FJ-espace vectoriel des applications FJ-linéairesde DK dans B^R par le sous-espace des applications qui sont compatibles avec lafiltration. On commence par vérifier que le noyau V* t (D) de l'application évidenteß : Vrf' (D) —^ Vgl"' (D) est un Q p -espace vectoriel de dimension finie < h et que,s'il est de dimension h, alors la représentation duale V s t(D) est semi-stable et Dest isomorphe à D st (V s t.(D)). La théorie des espaces de Banach analytiques permetde munir V^' (D) et V^' (D) d'une structure d'espace de Banach-Colmez et on adh(Vrf' (D)) = (tN(D),h) tandis que dh(V^' (D)) = (tn(D),0). Il suffit alors devérifier que l'application ß est analytique. Comme tn(D) = ìN(D), l'additivité dedh implique que ß est surjective et que dh(V* t (D)) = (0, h), ce qui signifie bien queàim Qp V s * t XD) = h.3. Equations différentielles3.1. — Soit A un anneau commutatif et d : A —t ÛA une dérivation de A dansun .4-module QA- Ici, un .4-module à connexion (sous-entendu relativement à d)est un .4-module libre de rang fini T> muni d'une application V : T> —t T> ® ÛAvérifiant la règle de Leibniz. On dit que ce module est trivial s'il est engendré parle sous-groupe ï>v=o des sections horizontales.Pour tout corps F de caractéristique 0, complet pour une valuation discrète,notons (cf. par exemple [Ts98], §2) 1Z X: L l'anneau de Robba de L (ou "anneau desfonctions analytiques sur une couronne d'épaisseur nulle"), c'est-à-dire l'anneau desséries X^nez a n,x n à coefficients dans F vérifiantV« < 1, |a n |s n H- 0 si n H- +oo et 3r < 1 tel que |a„|r" n 0 si n n ôo.Le sous-anneau £ s tx L de 1Z X: L des fonctions ^ a n x n telles que les a n sont bornésest un corps muni d'une valuation discrète (définie par \â n x n \ = sup|a„|) quin'est pas complet mais est hensélien. Son corps résiduel s'identifie au corps desséries formelles F = kL,((x)) où fc^ désigne le corps résiduel de F. Pour touteextension finie separable F de F, il existe une, unique à isomorphisme unique près,extension non ramifiée £' F de £^ L de corps résiduel F . Posons TZF = 1Z X ,L ® + £ F -£r..LSi kp désigne le corps résiduel de F, F' l'unique extension non ramifiée de F detcorps résiduel kp et si x' est un relèvement dans l'anneau des entiers de £ F d'une

Analyse p-adique et Représentations Galoisiennes 145uniformisante de F, l'anneau Tl.p s'identifie à l'anneau de Robba TZ X ' : L'-Notons Q^ le 7?. x ^-module libre de rang 1 de base dx, solution du problèmeuniversel pour les dérivations continues en un sens évident. Les modules à connexionsur l'anneau 1Z- X ,L forment une catégorie artinienne. Si T> est un objet de cettecatégorie, on dit qu'il est unipotent si son semi-simplifié est trivial. On dit qu'ilest quasi-unipotent s'il existe une extension finie separable F de k((Xj) telle que lemodule à connexion sur TZp déduit de T> par extension des scalaires soit unipotent.Pour tout z dans l'aneeau des entiers de £tx K , il existe un unique endomorphismecontinu ip de TZ X: K 0 qui prolonge le Frobenius absolu sur K 0 et vérifietp(x) = x p +pz ; on appelle Frobenius un tel endomorphisme. Pour un tel ip, on noteencore ip : 0^ —t Qi^ l'application induite. Soit T> un module à connexionsur 1Z X: K 0 . Une structure de Frobenius sur T> consiste en la donnée d'un Frobeniusip sur 1Z X: K 0 et d'une application tp-semi-linéaire ip-v :T> —¥T> commutant à V.THéORèME (André, Kedlaya, Mebkhout ^). — Tout module à, connexion surTZX,K 0 qui admet une structure de Frobenius est quasi-unipotent.Avant de montrer comment Berger [Be2] déduit le théorème B de cet énoncé,rappelons quelques résultats de [FoOO], [Fo90] et [CC98] (cf. aussi [Co98]). Danstout ce qui suit, V est une représentation p-adique de GK de dimension finie h.3.2. — Soit FJQO le sous-corps de K engendré sur K par les racines de l'unité d'ordreune puissance de p. Posons HK = Gal(K/K^,) et T'K = GK/HK- En utilisantla théorie de Sen [Se80], on montre [FoOO] que l'union A d R(V) des sous-FJ^[[£]]-modules de type fini de (B d R ®Q P V) HK stables par F^ est un FJ 0O ((t))-espacevectoriel de dimension h et qu'il existe une unique connexionV:A dR (V)Â dR (V)®dt/tqui a la propriété que, pour tout sous-F^ff^-module de type fini Y stable par GK,tout entier r > 0 et tout y £ Y, il existe un sous-groupe ouvert T r^y de F tel que, siV(y) = Vo(y) ® dt/t, alors7(y) = exp(logx(7)-Vo)(y) (mod t r Y), pour tout 7 £ T r^y.Cette connexion est régulière : le F^ff^-module A^R(V) = (B^R ® V) nAdR(V) est un réseau de A d R(V) vérifiant V(Aj fl (F)) C A^R(V) ® dt/t. Ona D(ìR(V) = (A d R(Vj) TK . L'action de F^ est discrète sur A d R(V)y = o ; on endéduit que A d R(V)y = o = K^, ®K D d R(V) donc que V est de de Rham si et seulementsi le module à connexion A d R(V) est trivial. Ceci se produit si et seulements'il existe un réseau (nécessairement unique) A® R (V) de A d R(V) vérifiantV(A 0 dR(V))cA 0 dR(V)®dt.W Crew [Cr98] a suggéré que ce théorème pouvait être vrai ; il a été prouvé indépendammentpar André [An02], Mebkhout [Me02] et Kedlaya [KeOl]. Pour André comme pour Mebkhout, c'estun cas particulier d'un résultat plus général dont la preuve repose sur la théorie de Christol-Mebkhout [CM]. La preuve de Kedlaya est plus directe : elle utilise une classification à laDieudonné-Manin des modules munis d'un Frobenius pour se ramener à un résultat de Tsuzuki[Ts98]. Voir [CoOl] pour une étude comparative plus détaillée.

146 J.-M. Fontaine3.3. — Rappelons brièvement la théorie des (ip, F)-modules [F08O]. Soit Og 0 l'adhérencedans W(FR) de la sous-W(fc)-algèbre engendrée par [e] et l/([e] — 1). C'estun anneau de valuation discrète complet dont l'idéal maximal est engendré par p etdont le corps résiduel F 0 est le corps des séries formelles k((e — 1)) vu comme souscorpsfermé de FR. Notons 0~ nr le séparé complété pour la topologie p-adique del'union de toutes les sous-€>£ 0 -algèbres finies étales de Og 0 contenues dans W(FR).C'est un anneau de valuation discrète complet dont le corps résiduel est une clôtureseparable F s de F 0 . Son corps des fractions £ nr s'identifie à un sous-corps fermé ducorps F = W(FR)[l/p], stable par l'action de GK et par le Frobenius tp. Le corps£K = (£ nr ) Hli est une extension finie non ramifiée du corps des fractions de Og 0 .Son corps résiduel EK est une extension finie separable de F 0 ; le corps résiduel k'de EK est celui de K^.Alors, D(V) = (£ nr ®q p V) HK est un (ip,Y K)-module sur £K, i.e. un EKespacevectoriel de dimension finie D muni d'un Frobenius tp-semi-linéaire (que l'onnote encore ip) et d'une action semi-linéaire continue de F^ commutant à l'actionde ip ; ce ( de D tel que T>est le Og K -module engendré par tp(T>). La correspondance V H> D(V) définit uneéquivalence entre Rep Q (GK) et la catégorie des ( n °t° ns B^R le sous-anneau de Bformé des séries de ce type qui convergent dans Fj~ fi . L'application B^R —t B^R estinjective et permet d'identifier B^R à une sous-W(F)[l/p]-algèbre de B^R. Pourtout r e N, posons E' = £ nrn tp r (B^R) et, pour tout 6 £ E' , notons tp r (b)l'unique c £ Fj~ fi c Fj~ fi tel que tp r (c) = b. On & £ r r '' c £ r +i ; soit £" r >îl'union des £ rr. Alors £ K = (E nr >\) HK est un sous-corps dense de EK stable parip. On pose D'(V) = (£ nr * ®q p V) HK . On peut le calculer à partir de D(V) :tc'est l'union des sous-£^--espaces vectoriels de dimension finie de D(V) stables partp. Le résultat principal de [CC98] est que V est surconvergente, c'est-à-dire quel'application naturelle EK ® D* (V) —^ D(V) est un isomorphisme.Pour tout r G N, soit £ Kr = (£? r^) HK . Alors D\(V) = (£? r 't ® Qp V) HKt 'est aussi le plus grand sous-5^ r-module M de type fini de D(V) tel que tp(M) c4 jr+1 M. Pour r assez grand, l'application naturelle 4® f DÎ(V) -+ rt(V) estun isomorphisme. Lorsqu'il en est ainsi, on a tp r (£ K r ) c ^00[M] etA dR (V) = lfoo((*)) ® | Dl(V) et donc D dR (V) = (K^t)) ® | Dl(V)Y K .3.5. — Wach [Wa96] a montré comment calculer D st (V) à partir de D*(V) lorsquet

Analyse p-adique et Représentations Galoisiennes 147V est de hauteur finie. C'est Berger [Be02] qui a compris comment traiter le casgénéral : Choisissons un relèvement x dans l'anneau des entiers de £tK d'une unitformisante de EK- Si K' Q désigne le corps des fractions de W(k'), £ K s'identifieprécisément au sous-anneau £!tK, de l'anneau de Robba TZ X K 1 • Ce dernier nedépend pas du choix de x et nous le notons £^s; il contient t = log[e]. Si F estune extension finie de K contenue dans K, le corps F = (E s ) GaX ( K / LK^ est uneextension finie de EK et l'anneau £ r F a s'identifie à l'anneau noté TZp au §3.1.Rappelons (§1.2) que si u = log[e — 1], on a B st = B cris [u]. Les actions de tpet de Y'K s'étendent de façon évidente à £ T 'KK S , , à £]f£ T K\S/^\S [l/£] et à l'anneau £jf £^6 S [l/£][u]3des polynômes en « à coefficients dans £^s[l/t].Berger montre queKD r ,(V) = (££»[!/*] ® f DÌ(V)f« et D st (V) = (£ r^[l/t][u] ® f DÌ(V)f«(l'action de N sur D st (V) est la restriction de —d/du ® id D ±> V) ).3.6. - Posons D = D^s(V) = £ r K S [l/t] ® | Ft (F). En utilisant l'action de F*-comme au §3.4, on définit une connexion V : D —ï D ® dt/t qui commute à l'actionde tp. Cette connexion est régulière au sens qu'il existe un sous-£]^s-moduleD +de D, libre de rang h, stable par tp et vérifiant V(F+) C D + ® dt/t (prendreD + = £ r^9 ® J. D'(V)). On vérifie que le £]^s-modulelibre fl 1 ig admet d[e]4bli ,comme base. Mais dt/t = [e] -1 /td[é] et t n'est pas inversible dans £^s.On déduitalors facilement du théorème d'André-Kedlaya-Mebkhout que V est potentiellementsemi-stable si et seulement s'il existe un sous-£]^s-modulelibre D° de D, libre derang h, stable par tp et vérifiant V(F°) C D° ® dt.Il ne reste plus qu'à construire un tel D° lorsque V est de de Rham. Fixons unentier r Q > 1 suffisamment grand pour que DÎ 0 (V) contienne une base de D^(V)sur £ K et pour que x £ £r 0 . Pour tout r > r 0 le sous-anneau £^grde £^g= H XZ K'formé des X^»ez a n,x n vérifiantV« < 1, |a n |s n H- 0 si n H- +oo et a n (e r — 1)" H- 0 si n H- ôof r' r' fest stable par F^ et contient £ Kr - Si D r = D r^9r(V) = £^gr® ± Dl(V), alorsD est la réunion croissante des D r et tp(D r ) c D r +i- L'application tp r induit unhomomorphisme de £^srdans FJ 0O ((t)) et un isomorphisme de FJ 0O ((t)) ® £r -i g D rsur A d R(V). L'application $ r : D r —t A d R(V) qui envoie a sur 1 ® a est injective.Soit A® R (V) le sous-FJ 00 ((t))-module de A d R(V) engendré par les sectionshorizontales. Pour tout r > r 0 , soit D® = {a £ D r \ $ s (a) £ A® R (V) pour tout s >r}. On afljc F° +1 et F 0 = U r > ro F° est un sous-£]^s-modulede D, stable par tpet vérifiant V(F°) C D° ® dt. Si V est de de Rham on déduit du fait que A® R (V)est un réseau de A d R(V) que F 0 est libre de rang h sur £^s.D'où le théorème B.

148 J.-M. FontaineBibliographie[An02[Be02[CM;[Co98[CoOl[Co02[CC98[CFOO;[Cr98[Fo83[Fo88a[Fo88b[Fo90;[FoOO;[Fo02[FP02[Ke02[Me02[Sen80[Ts98[Wa96Y. André, Filtration de type Hasse-Arf et monodromie p-adique, Inv. Math.148 (2002), 285-317.L. Berger, Représentations p-adiques et équations différentielles, Inv. Math.148 (2002), 219-284.G. Christol et Z. Mebkhout, Sur le théorème de l'indice des équations différentiellesp-adiques I, Ann. Inst. Fourier 43 (1993), 1545-1574 ; II, Ann. ofMaths. 146 (1997), 345-410 ; III, Ann. of Maths. 151 (2000), 385-457 ;IV, Inv. Math. 143 (2001), 629-671.P. Colmez Représentations p-adiques d'un corps local in Proceedings of theI.CM. Berlin, vol. II, Documenta Mathematica (1998), 153-162.P. Colmez, Les conjectures de monodromie p-adique, Sém. Bourbaki, exp.897, novembre 2001.P. Colmez, Espaces de Banach de dimension finie, J. Inst. Math. Jussieu, àparaître.F. Cherbonnier et P. Colmez, Représentations p-adiques surconvergentes,Inv. Math. 133 (1998), 581-611.P. Colmez et J.-M. Fontaine, Construction des représentations semi-stables,Inv. Math. 140 (2000), 1-43.R. Crew, Finiteness theorems for the cohomology of an over convergent isocrystalon a curve, Ann. scient. E.N.S. 31 (1998), 717-763.J.-M. Fontaine, Représentations p-adiques, in Proceedings of the I.CM.,Warszawa, vol. I, Elsevier, Amsterdam (1984), 475-486.J.-M. Fontaine, Le corps des périodes p-adiques, avec un appendice par PierreColmez, in Périodes p-adiques, Astérisque 223, S.M.F., Paris (1994), 59-111.J.-M. Fontaine, Représentations p-adiques semi-stables, in Périodes p-adiques,Astérisque 223, S.M.F., Paris (1994), 113-184.J.-M. Fontaine, Représentations p-adiques des corps locaux, in the GrothendieckFestschrift, vol II, Birkhàuser, Boston (1990), 249-309).J.-M. Fontaine, Arithmétique des représentations galoisiennes p-adiques, prépublication,Orsay 2000-24. A paraître dans Astérisque.J.-M. Fontaine, Presque-C p -représentations, prépublication, Orsay 2002-12.J.-M. Fontaine et Jérôme Plût, Espaces de Banach-Colmez, en préparation.K. Kedlaya, A p-adic local monodromy theorem, preprint, Berkeley (2001).Z. Mebkhout, Analogue p-adique du théorème de Turrittin et le théorème dela monodromie p-adique, Inv. Math. 148 (2002), 319-351.S.Sen, Continuous Cohomology and p-adic Galois Representations, Inv. Math.62 (1980), 89-116.N. Tsuzuki, Slope filtration of quasi-unipotent overconvergent F-isocrystals,Ann. Inst. Fourier 48 (1998), 379-412.N. Wach, Représentations p-adiques potentiellement cristallines, Bull. S.M.F.124 (1996), 375-400.

ICM 2002 • Vol. II • 149-162Equivariant Bloch-Kato Conjecture andNon-abelian Iwasawa Main ConjectureA. Huber* G. Kings 1 'AbstractIn this talk we explain the relation between the (equivariant) Bloch-Katoconjecture for special values of L-functions and the Main Conjecture of (nonabelian)Iwasawa theory. On the way we will discuss briefly the case of Dirichletcharacters in the abelian case. We will also discuss how "twisting" in thenon-abelian case would allow to reduce the general conjecture to the caseof number fields. This is one the main motivations for a non-abelian MainConjecture.2000 Mathematics Subject Classification: 11G40, 11R23, 19B28.Keywords and Phrases: Iwasawa theory, L-function, Motive.1. IntroductionThe class number formula expresses the leading coefficient of a Dedekind-(-function of a number field F in terms of arithmetic invariants of F:CF(0)* =JALWF(h the class number, Rp the regulator, wp the number of roots of unity in F). Byworkof Lichtenbaum, Bloch, Beilinson, and Kato among others, the class numberformula has been generalized to other F-functions of varieties (or even motives)culminating in the Tamagawa number conjecture by Bloch and Kato.Iwasawa, on the other hand, initiated the study of the growth of the classnumbers in towers of number fields. His decisive idea was to consider the classgroup of the tower as a module under the completed group ring of the Galois groupof the tower. From his work evolved the "Main Conjecture" describing this growthin terms of the p-adic F-function.*Math. Institut, Universität Leipzig, Augustusplatz 10/11, 04109 Leipzig, Germany. E-mail:huber@mathematik.uni-leipzig.detNWF I-Mathematik, Universität Regensburg, 93040 Regensburg, Germany. E-mail:guido.kings@mathematik.uni-regensburg.de

150 A. Huber G. KingsIt is a surprising insight of Kato that an equivariant version of the Tamagawanumber conjecture can be viewed as a version of the Main Conjecture of Iwasawatheory. Perrin-Riou, in her efforts to develop a theory of p-adic F-functions, arrivedat a similar conclusion.The purpose of this paper is to make the connection between the equivariantTamagawa number conjecture and the Iwasawa Main Conjecture precise. In thespirit of Kato, we formulate an Iwasawa Main Conjecture (3.2.1) for arbitrary motivesand towers of number fields whose Galois group is a p-adic Lie group. Thisformulation does not involve p-adic F-functions. We show that it is implied by theequivariant Tamagawa number conjecture formulated by Burns and Flach. For easeof exposition, we restrict to the case of F-values at very negative integers, wherethe Bloch-Kato exponential does not play a role. The study of non-abelian Iwasawatheory was initiated by Coates. Recently, there have been systematic results byCoates, Howson, Ochi, Schneider, Sujatha and Venjakob.Our interest in allowing general towers of number fields is motivated by thepossibility of reducing the Tamagawa number conjecture to an equivariant classnumber formula (modulo hard conjectures, see 3.).Important special cases of the Main Conjecture were considered by (alphabeticalorder) Coates, Greenberg, Iwasawa, Kato, Mazur, Perrin-Riou, Rubin, Schneider,Wiles and more recently by Ritter and Weiss.It is a pleasure to thank C. Deninger, S. Howson, B. Perrin-Riou, A. Schmidt,P. Schneider for helpful comments and discussions.2. Non-abelian equivariant Tamagawa number conjecture2.1. NotationFix p^ 2 and let M be a motive over Q with coefficients in Q, for exampleM = h r (X), X a smooth projective variety over Q. It has Betti-realization MRand p-adic realization M p . Let M v be the dual motive. In the p-adic realization itcorresponds to the dual Galois module. We denote by Hj^(Z,M(k)) the "integral"motivic cohomology of the motive M in the sense of Beilinson [1].For any finite Galois extension K/Q with Galois group G, let Q[G] be thegroup ring of G. It is a non-commutative ring with center denoted Z(Q[G]).We consider the deformation Q[G] ® M := h°(K) ® M. If M = ¥'(X) andK/Q is a number field, then h°(K) ® M = h r (X x K) considered as a motive overQ-We consider a finite set of primes S satisfying:(*) Q[G] ® M and K have good reduction at all primes not dividing S, and p £ S.2.2. Equivariant i-functions

Equiv. Bloch-Kato Conjecture and Non-abelian Iwasawa Main Conjecture 151We assume the usual conjectures about the F-functions of motives, like meromorphiccontinuation and functional equation etc., to be satisfied.In order to define the equivariant F-function for G (without the Euler factorsat the primes dividing S), consider a Galois extension E/Q such that E[G] =(J) End E (V(pj), where V(p) are absolutely irreducible representations of G. Thenthe center of E[G] is Z(E[G]) = ® F and the motives V(p) ® M have coefficientsin F. We defineL S (G, M, k)* := (L S (V(P) ® M, k)*) £ Z(E ® Q C[G])*to be the element with p-component the leading coefficient at s = k of the F ®Q devaluedF-functions Ls(V(p)®M, s) without the Euler factors at S. Then Lg(G, M, k)*has actually values in Z(R[G])* (see [4] Lemma 7) and is independent of the choiceof F. We will always consider Lg(G, M, k)* as an element in Z(R[G]) c R[G].Remark In [22] Kato uses a different description of this equivariant F-function.2.3. Non-commutative determinantsWe follow the point of view of Burns and Flach. Let A be a (possibly noncommutative)ring and V(A) the category of virtual objects in the sense of Deligne[12]. V(A) is a monoidal tensor category and has a unit object 1A- Moreover it isa groupoid, i.e., all morphisms are isomorphisms. There is a functordet^ : {perfect complexes of .4-modules and isomorphisms} —¥ V(A)which is multiplicative on short exact sequences. The group of isomorphism classesof objects of V(A) is K 0 (A) andAut(l^) = Ki(A) =Gl 00 (A)/E(A)(E(A) the elementary matrices). In general Honiy^^det^X, det^F) is eitherempty or a K\ (-A)-torsor. If A —t B is a ring homomorphism, we get a functorB® : V(A) —t V(B) such that tensor product and det^ commute.Convention By abuse of notation we are going to write z £ detA X for z : 1A —^det^ X and call this a generator of det^ X.If A is commutative and local, then the category of virtual objects is equivalentto the category of pairs (L,r) where F is an invertible .4-module and r £ Z. Onerecovers the theory of determinants of Knudson and Mumford. The unit objectis 1A = (A,0) and one has Aut(l^) = Ki(A) = A*. Thus K\(A) is used asgeneralization of A* to the non-commutative case. Generators of det^ X = (L, 0)in the above sense correspond to .4-generators of F.2.4. Formulation of the conjectureThe original conjecture dates back to Beilinson [1] and Bloch-Kato [3]. Theidea of an equivariant formulation is due to Kato [23] and [22]. Fontaine and Perrin-Riou gave a uniform formulation for mixed motives and all values of F-functions at

152 A. Huber G. Kingsall integer values [14], [15]. The generalization to non-abelian coefficients is due toBurns and Flach [4].For simplicity of exposition, we restrict to values at very negative integers.In the absolute case this coincides with the formulation given by Kato in [23].We consider a motive M and values at 1 — k where k is big enough. In the caseM = h r (X), k big enough means that• k > inf{r,dim(X)}, (r,k) ^ (1,0); (2dim(X), dim(X) + 1) and 2k ^ r + 1.• for all £ £ S the local Euler factor Li(M p , s)^1at £ does not vanish at 1 — k.Consider the (injective) reduced norm map rn : Ki(R[G]) —¥ Z(R[G])* andrecall that L$(G, M v , 1 — fc)* £ Z(M[G])*. By strong approximation (see [4] Lemma8) there is A G Z(Q[G])* such that AF S (G, M v (l -k))* is in the image of FJi(R[G])under rn. LetAF s (G,M v (l^fc))*el R[Gn]be the corresponding generator. For k big enough, we define the fundamental linein V(Q[G]) asA f (G, M v (l - kj) = det^j F^(Z,QG] ® M (kj) ® det Q[G] (Q[G] ® M B (k - 1))+ .Here + denotes the fixed part under complex conjugation.Conjecture 2.4.1 Let M be as in 2., p ^ 2 a prime and k big enough.1. The Beilinson regulator r-p induces an isomorphismA / (G,M v (l^fc))®R-l R[G] .2. Under this isomorphism the generator (XLg(G,M v (I — k))*)^1a (unique) generatoris induced by(\- 1 ö(G,M,k)) £ A f (G,M v (l^k)).The reduced norm is an isomorphism Ki(Q p [G]) — Z(Q P [G])*. Using the operationof Ki(Q p [G]) on generators in A f (G, M v (l - kj) ® Q p , we putö p (G,M,k) := (\- 1 ó(G,M,k,))\£ Af(G,M v (l^k))®Q p .Note that this generator is independent of the choice of A.3. The p-adic regulator r p induces an isomorphismA / (G,M v (l^fc))®Q p -det^[G] H^Zll/Sl^lG] ® M p (k)) ® det Qp[G] (Q p [G] ® M B (k - 1))+.4- Let TR C MR be a lattice such that T p = TR ® Z p c M p is Galois stable.Under the last isomorphism ö p (G,M,k) is induced by a generatorì p (G,M,k) £ det Zp[G] RY^l/S],Z p [G]®T p (k))®det Zp[G] (Z p [G]®T B )(k^l))+.

Equiv. Bloch-Kato Conjecture and Non-abelian Iwasawa Main Conjecture 153Remark a) The conjecture is compatible with change of group G. If G —¥ G' isa surjection, then the equivariant conjecture for G tensored with Q[G'] over Q[G]gives the conjecture for G'.b) The element 8 P (G, M, k) is determined up to an element in the kernel of the mapKi(Zp[G]) —¥ Ki(Qp[G]). In the commutative case, this map is always injective.c) The conjecture is independent of T. It is also independent of S. This computationshows that the definition of the equivariant F-function forces the use of the reducednorm in the formulation of the conjecture.3. Non-abelian Main Conjecture3.1. Iwasawa algebra and modulesLet K n be a tower of finite Galois extensions of Q with Galois groups G n suchthat GQO : = fimG„ is a p-adic Lie group of dimension at least 1. Moreover, weassume that only finitely many primes ramify in K œ := (J n K n .The classical example is the cyclotomic tower K n := Q(C P » ) with £ p » a p"-throot of unity. A non-abelian example is the tower K n := Q(F[p"]), where E[p n ] arethe p n -torsion points of an elliptic curve F without CM defined over Q.The Iwasawa algebraA:=Z p [[Goo]] = ]jmZ p [G n ]is the ring of Z p -valued distributions on GQO- It is a possibly non-commutativeNoetherian semi-local ring. If GQO is in addition a pro-p-group without p-torsion itis even a regular and local ring.For the cyclotomic tower, A = Z p [Gi][[r]] is the classical Iwasawa algebra. Forthe tower of p n -torsion points of F, the Iwasawa algebra was studied by Coates andHowson [8], [9]. Modules over such algebras are studied recently by Venjakob [36]and by Coates-Schneider-Sujatha [10].We are concerned with the complex of A-modules RY(Z[1/S], A®z p T p (k)) and(A ®z Tß(k — 1)) + . They are perfect complexes. Note thatFF(Z[1/S], A ® Zp T p (k)) = lim RY(0 Knwhere OK„ is the ring of integers of K n .[1/S], T p (k))3.2. Formulation of the non-abelian Main ConjectureThe Main Conjecture can be viewed as a Bloch-Kato conjecture for the deformed"motive" A ® M with coefficients in A.Recall from 2.4.1 that the generators ö p (G n ,M,k) are compatible under thetransition maps Q P [G„] —¥ Q p [G„_i]. They defineô p (G 00 ,M,k) = ^mô p (G n ,M,k) £lta[det Qp[Gn] FF(Z[l/S],%[G„]®M p (fc))®det Qp[Gn] (Q p [G„]®M B (fc^l)r

154 A. Huber G. Kingsmore precisely an element of fimHom l /(Q p [ Gn ])(lQ p [ Gn ], • ).The map A —t Q p [G„] induces an isomorphismQ P [G n ] ®A RY(Z[l/S],A®T p (kj) -> FF(Z[1/S],Q,[G„] ® Qp M p (kj) .Conjecture 3.2.1 (Non-abelian Main Conjecture) Let M and S be as in 2.,GQO as in 3., p ^ 2, TR C MR a lattice such that T p := TR ® Z p is Galois stableand k big enough (cf. section 2.). Then ö p (G 00 ,M,k) is induced by a generatorì p (G 00 ,M,k)£ [det A FF(Z[l/S],A®r p (fc)) ®det A (A®F B (fc - 1))+] .The conjecture translates into the Iwasawa Main Conjecture in the case of Dirichletcharacters or CM-elliptic curves. See section 5. for more details.Remark a) The conjecture is independent of the choice of lattice TR. The correctionfactor (A ®T B (k — 1)) + compensates different choices of lattice.b) Perrin-Riou [31] has defined a p-adic F-function and stated a Main Conjecturefor motives in the abelian case. She starts at the other side of the functional equation,where the exponential map of Bloch-Kato comes into play. Her main tool isthe "logarithme élargi", which maps Galois cohomology over K^ to a module ofp-adic analytic nature. It would be interesting to compare her approach with theabove.c) A Main Conjecture for motives and the cyclotomic tower was formulated byGreenberg [16], [17]. Ritter and Weiss consider the case of the cyclotomic towerover a finite non-abelian extension [32].Proposition 3.2.2 (see section 6.) The equivariant Bloch-Kato conjecture forM, k and all G n is equivalent to the Main Conjecture for M, k and G^.3.3. TwistingAssume that T p becomes trivial over K^,, for example let GQO be the imageof Gal(Q/Q) in Aut(F p ). Let T nvp be the Z p -module underlying T p with trivialoperation of the Galois group. The map g ® t >-¥ g ® g^1tinduces an equivariantisomorphism A ®z p T p = A ®z p T nv p . Hence there is an isomorphismdet A FF(Z[1/S], A® T p (kj) ® det A (A ®T B (k^ 1))+ ~det A FF(Z[1/S], A®T* riv (k)) ® det A (A ® T B riv (k - 1))+.Note that T B " V can be viewed as a lattice in the Betti-realization of the trivialmotive h°(Q) ® M triv = Q(0) ® M triv where M triv is M B considered as Q-vectorspace.Corollary 3.3.1 // the Main Conjecture is true for M and Q(0) ® M tnv and k,then _ _(5 p (G 0O ,M,fc) = ^(G 0O ,M triv ,fc)up to an element in FJi(A) under the above isomorphism.

Equiv. Bloch-Kato Conjecture and Non-abelian Iwasawa Main Conjecture 155Remark Even if T p is not trivial over K^,, the same method allows to twist with amotive whose p-adic realization is trivial over K^. A particular interesting case isthe motive Q(l) if K^ contains the cyclotomic tower. It allows to pass from valuesof the F-function at k to values at k + 1.Strategy This observation allows the following strategy for proving the Main Conjectureand the Bloch-Kato conjecture for all motives:• first prove the equivariant Bloch-Kato conjecture for the motive h°(Q) = Q(0),one fixed k and all finite groups G n . For k = 1 this is an equivariant classnumber formula.• by proposition 3.2.2 this implies the Main Conjecture for the motives Q(k) ®M tnv and all p-adic Lie groups GQO .• for any motive M there is a K^ such that T p becomes trivial. Using corollary3.3.1 it remains to show that ö p (G 00 ,M triv , k) induces 8 p (G n , M, k) for all n.This is a compatibility conjecture for elements in motivic cohomology andallows to reduce to the case of number fields.• the equivariant Bloch-Kato conjecture follows by 3.2.2.4. Relation to classical Iwasawa theory in the criticalcase4.1. Characteristic idealsWe restrict to the case GQO a pro-p-group without p-torsion. In this case theIwasawa algebra is local and Auslander regular ([36]). Its total ring of quotients isa skew field D. Then K 0 (A) ^K 0 (D)^Z, K ± (A) = (A*) ab , and Ki(D) = (F*) abwhere - ab denotes the abelianization of the multiplicative group.Let T be the category of finitely generated A-torsion modules. The localizationsequence for FJ-groups implies an exact sequence(A*) ab -• (F*) ab -• K Q (T) -• 0.If X is a A-torsion module, then we call its class in K 0 (T) the characteristic ideal.By the above sequence it is an element of D* up to [D*,D*] Im A*. If GQO is abelian,K 0 (T) is nothing but the group of fractional ideals that appears in classical Iwasawatheory.The characteristic ideal can also be computed from the theory of determinants.The class of X in K 0 (A) is necessarily 0, hence there exists a generator x £ detA(X).Its image in F®det A (X) = det£>(0) = ID is an element of K\(D). This constructionyields a well-defined element of Ki(D)/lmKi(A) = K 0 (T), in fact the inverse ofthe characteristic ideal of X.Note that a complex is perfect if and only if it is a bounded complex withfinitely generated cohomology. Such complexes also have characteristic ideals if

156 A. Huber G. Kingstheir cohomology is A-torsion.Remark Coates, Schneider and Sujatha study the category of A-torsion modulesin [10]. In particular, they also define a notion of characteristic ideal as object ofK 0 (T b /T 1 ) where T b '/T 1 denotes the quotient category of bounded finitely generatedA-torsion modules by the sub-category of pseudo-null modules. They constructa mapK 0 (T) -+ Focr/r 1 ) -+ Kotro/T 1 )which maps the class of a module to the characteristic ideal in their sense. If GQO isabelian, then the two maps are isomorphisms and all notions of characteristic idealsagree. In the general case, we do not know whether the map is injective. However,it seems to us that the problem is not so much in passing to the quotient categorymodulo pseudo-null modules but rather in projecting to the bounded part.4.2. Zeta distributionsLet M, k, S and GQO as before. AssumeHj^(Z,qG„] ® M(k)) = 0 for all G„.For k big enough, this implies that M B (k — 1) + = 0 and K n totally real. Themotives Q[G n ] ® M(k) are critical in the sense of Deligne. Note that the onlymotivesexpected to be critical and to satisfy our condition k big enough (see 2.)are Artin motives (with k > 1).In this case, the Beilinson conjecture asserts that Ls(G n , M v , l—k)€ Z(Q[G n ])*(no leading coefficients has to be taken). We call£ 5 (Goo,M v ,l -k) = HmL s (G n ,M v , 1 -k) £ Hm^(Q p [G„])*the zeta distribution.Let /,g £ A such that the images f n ,9nthe reduced norm, they define a distribution€ Z p [G n ] are units in Q P [G„]. Via(Tn(f n g- 1 )) n £lfiaiZ(Q p [G n ]r.Remark It is not clear to us if the class of f/g £ K\(D) = (F*) ab is uniquelydeterminedby the sequence fnQn 1 - In the abelian case this is true and f/g is ageneralization of Serre's pseudo measure (cf. [35]).In this case the complexes RY(ÖK n [l/S],T p (k)) are torsion. Hence the complexRY(Z[l/S],A®Tp(k)) = lim n FF(O ifn [l/S],F p (fc)) is bounded and its cohomologyis A-torsion (see [18]). The main conjecture 3.2.1 takes the following form:Conjecture 4.2.1 Let M be an Artin motive, k > 1, S, G^ as before (in particularGQO pro-p and without p-torsion) and Q[G n ] ® M(k) critical for all n. There exist

Equiv. Bloch-Kato Conjecture and Non-abelian Iwasawa Main Conjecture 157f,g £ A such that the induced distribution (m(f n g n 1 j) n £ limZ(Q p [G„])* is thezeta distribution £s(Goo, M v , 1 — k) and the characteristic idealcoincides with the image of fg^1[RY(Z[l/S],A®T p (k))[l]]£K 0 (T)£ (F*) ab .Remark a) The conjecture is isogeny invariant, i.e., independent of the choice oflattice T p . The correction term (A®T B (kj) + vanishes.b) In the abelian case this means that the zeta distribution is a pseudo measureand generates the characteristic ideal.c) In the case of the cyclotomic tower, a similar conjecture is formulated by Greenberg,[16], [17].d) If GQO is abelian, the above conjecture is easily seen to be implied by conjecture3.2.1. The argument also works in the non-abelian case if the set of all elements ofA which, for all n, are units in Q p [G„] is an Ore set.5. Examples5.1. Dirichlet charactersLet x be a Dirichlet character, V(x) its associated motive with coefficients inF. Let Qoo = U» Q» be the cyclotomic Z p -extension of Q and GQO = Gal(Qoo/Q) =lim G n . In this case the equivariant F-function is Ls(G n ,V(x),s) = (I j s(px, s ))p,where p runs through all characters of G n and Ls(px, s) is the Dirichlet F-functionassociated to px- Yet k be big enough, i.e., k > 1.Critical case x( — 1) = ( — !)*•Here Hj i4 (Z,E[G„] ® V(x)(kj) = 0 for all n. As in section 4., the equivariantF-values give rise to the zeta distribution £s(Goo,V(x) v , l—k)€ Um E[G„]. It isa classical calculation (Stickelberger elements) that this is in fact a pseudo measure,which gives rise to the Kubota-Leopoldt p-adic F-function. Let Ö C F be the ringof integers, A = ö p [[Goo]] the Iwasawa algebra and T p (x) C V p (x) a Galois stablelattice. The Iwasawa Main Conjecture 4.2.1 amounts to the following theorem:Theorem 5.1.1 The zeta distribution £s(G 0O ,I / (x) v , 1 — k) generatesdet^1 H^Z^/S], A® T p (x)(k)) ® det A H 2 (Z[1/S], A® T p ( X )(k)).Remark This is a reformulation of the main theorem of Mazur and Wiles in [29].There is an extension to the case of totally real fields by Wiles [37] and an equivariantversion by Burns and Greither [6].Non-critical case x( — 1) = (^l)* -1 -Here Hj i4 (Z,E[G„] ® V(x)(kj) has F[G„]-rank 1. It is a theorem of Borei(resp. Soulé) that r-p ® R (resp. r p ® Q p ) is an isomorphism. By a theorem ofBeilinson-Deligne (see [21] or [19]), the image of ö p (G n ,V(x),k) under r p is givenbyc^Gn-Mx))' 1 ®t p (x)(k ^ I),

158 A. Huber G. KingswhereCk(Gn,t P (xï) eis a twist of a cyclotomic unit and t p (x)(kCk(Goo,t p (x)) •= ^nCk(G n ,t p (x))-H 1 (Z[l/S],O p [G n ]®T p (x)(k))— 1) is a generator of T p (x)(k — 1). LetTheorem 5.1.2 There is a canonical isomorphism of A-determinantsdetA (H 1 (Z[l/S],A®T p (x)(kj)/c k (G 00 ,t p (x))) =detA H 2 (Z[1/S], A® T p (x)(k)).Remark For p { ord(x) this is a consequence of theorem 5.1.1 and was showndirectly by Rubin [33] with Euler system methods. The restriction at the order ofX is removed in Burns-Greither [5] and Huber-Kings [20] by different methods.The Tamagawa number conjecture for V(x)(r) (and hence for h°(F)(r) withF an abelian number field) can be deduced from theorems 5.1.1 and 5.1.2, seeBurns-Greither [5] or Huber-Kings [20]. Previous partial results were proved inMazur-Wiles [29], Wiles [37], Kato [22], [23], Kolster-Nguyen Quang Do-Fleckinger[26] and Benois-Nguyen Quang Do[2].We would like to stress that the strategy 3. is used in Huber-Kings [20] toprove theorems 5.1.1, 5.1.2 and the Tamagawa number conjecture from the classnumber formula.5.2. Elliptic curvesLet F be an elliptic curve over an imaginary quadratic field K with CM by OK-The motive ft 1 (F) considered with coefficients in K decomposes into V(ip) ® V('ip),where ip is the Grössencharacter associated to F. The F-function of V(ip) is theHecke F-function of ip, which has a zero of order 1 at 2 — k, where k > 2. YetS = Np, where N is the conductor of ip and let K n := K(E[p n ]).It is not known if Hj^(OK,K[G„] ® V(ip)(k)) has FJ[G„]-rank 1 but Deninger[13] shows that r-p ® R is surjective and that the Beilinson conjecture holds. Itis a result of Kings [25] that the image in étale cohomology of the zeta elementöp(G n ,V(ip),2 — k) given by Beilinson's Eisenstein symbol is given byefe(G„,£ p (i/0) _1 ®t p (ip) ,where ek(G n ,t p (ipj) £ H 1 (Z[l/S],O p [G„]®T p (ip)(kj) is the twist of an elliptic unit.Let A := O p [[Goo]j and e k (G oo, t p (ij))) = ljm n e k (G n ,t p (ipj).Theorem 5.2.1 There is a canonical isomorphism of determinantsdetA (H 1 (Z[l/S],A®T p (

Equiv. Bloch-Kato Conjecture and Non-abelian Iwasawa Main Conjecture 159the condition that H 2 (Z[l/S],T p (ip)(k)) is finite (fulfilled for almost all k for fixedP).Kato [24] has investigated the case of elliptic curves over Q and the cyclotomictower. His approach to the Birch-Swinnerton-Dyer conjecture uses the ideaof twisting cup-products of Eisenstein symbols to the value of the F-function at 1.As a consequence he can prove one inclusion of the Iwasawa main conjecture in thiscase. The result supports our general philosophy of twisting to the case of numberfields.6. Proof of proposition 3.2.2We want to give the proof of proposition 3.2.2. The implication from the MainConjecture to the equivariant Bloch-Kato conjecture is trivial. Conversely, we haveto show the following abstract statement:Lemma 6.1 Let V £ V(A) and ö(n) £ Z P [G„] ® V generators such that theirimages ö(n) £ Q p [G„] ® V are compatible under transition maps. Then there is agenerator Ö' (oo) £ V inducing all ö(n).The proposition follows with ö(n) = 8 p (G n , M, k) andV = detA RY(Z[l/pS],A®T p (kj) ® det A (A ® T B (k - 1))+.We now prove the lemma. We first reduce to a statement about elements ofKi. By assumption, Z p [G n ] ® V has a generator, in particular, its isomorphismclass is zero in K 0 (Z P [G„]). As K 0 (A) —t ljmK 0 (Z p [G„]) is an isomorphism, thisimplies that the class of V is zero in K 0 (A). Without loss of generality we canassume V = 1 A . Recall that by our convention, a generator of 1A is nothing butan element of the abelian group K\(A) for all rings A.Let B n = ImFJi(Z p [G„]) —¥ Ki(Q p [G„]). By assumption ö(n) £ B n . There isa system of short exact sequences0 -+ SKi(Z p [G n ]) -+ Ki(Z p [G n ]) -+ B n -+ 0.By [11] 45.22 the groups SKi(Z p [G n ]) are finite. The system of these groups isautomatically Mittag-Leffler. Hence we get a surjective maplimFJ 1 (Z p [G„])^fimF„.The system (ö(n)) n has a preimage (ö'(nj) n £ HmF'i(Z p [G„]).All Z p [G n ] are semi-local, hence by [11] 40.44Ki(Z p [G n ]) - Gl 2 (Z p [G n ])/E 2 (Z p [G n ])where F 2 is the subgroup of elementary matrices. We represent ö'(n) by an elementof Gl2(Z p [G n ]). By assumption the image of ö'(n) in Ki(Z p [G n -i]) differs from5'(n — 1) by some elementary matrix in F 2 (Z p [G„_i]). Elementary matrices canbe lifted to elementary matrices in Gi2(Z p [G n ]). Hence we can assume that the

160 A. Huber G. Kingselements ö'(nj) £ G1 2 (Z P [G„]) form a projective system. The system defines anelementô' p (n) £ G1 2 (A)whose class in K\ (A) has the necessary properties.References[I] A. Beilinson, Higher regulators and values of L-functions, Jour. Soviet. Math.,30 (1985), 2036-2070.[2] D. Benois, Thong Nguyen Quang Do, La conjecture de Bloch et Kato pour lesmotifs Q(m) sur un corps abélien, Preprint 2000.[3] S. Bloch, K. Kato, F-functions and Tamagawa numbers of motives, TheGrothendieck Festschrift, Vol. I, 333-400, Progr. Math., 86, Birkhäuser Boston,Boston, MA, 1990.[4] D. Burns, M. Flach, Tamagawa numbers for motives with (non-commutative)coefficients, Doc. Math., 6 (2001), 501-570 (electronic).[5] D. Burns, C. Greither, On the equivariant Tamagawa conjecture for Tate motives,Preprint 2001.[6] D. Burns, C. Greither, Equivariant Weierstrass preparation and values of F-functions at negative integers, preprint 2002.[7] J. Coates, Fragments of the GL 2 Iwasawa theory of elliptic curves withoutcomplex multiplication. Arithmetic theory of elliptic curves (Cetraro, 1997),1-50, Lecture Notes in Math., 1716.[8] J. Coates, S. Howson, Euler characteristics and elliptic curves, Elliptic curvesand modular forms (Washington, DC, 1996), Proc. Nat. Acad. Sci. U.S.A. 94(1997), no. 21, 11115-11117.[9] J. Coates, S. Howson, Euler characteristics and elliptic curves. II, J. Math. Soc.Japan, 53 (2001), no. 1, 175-235.[10] J. Coates, P. Schneider, R. Sujatha, Modules over Iwasawa algebras, Preprint2001.[II] C.W. Curtis, I. Reiner, Methods of representation theory, Vol. I. and Vol. II,John Wiley & Sons, Inc., New York, 1981 and 1987.[12] P. Deligne, Le déterminant de la cohomologie, Current trends in arithmeticalalgebraic geometry (Areata, Calif, 1985), 93-177, Contemp. Math., 67, Amer.Math. Soc, Providence, RI, 1987.[13] C. Deninger, Higher regulators and Hecke F-series of imaginary quadratic fieldsI, Invent. Math., 96 (1989), no. 1, 1-69.[14] J.-M. Fontaine, Valeurs spéciales des fonctions F des motifs, Séminaire Bourbaki,Vol. 1991/92. Astérisque No. 206, (1992), Exp. No. 751, 4, 205-249.[15] J.-M. Fontaine, B. Perrin-Riou, Autour des conjectures de Bloch et Kato: cohomologiegaloisienne et valeurs de fonctions F, Motives (Seattle, WA, 1991),599-706, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc, Providence,RI, 1994.[16] R. Greenberg, Iwasawa theory for motives, L-functions and arithmetic, Pro-

[17;[is;[19;[20;[21[22[23;[24;[25;[26;[27[28;[29;[30;[31[32;[33;[34;[35;[36;Equiv. Bloch-Kato Conjecture and Non-abelian Iwasawa Main Conjecture 161ceedings of the Durham Symposium 1989, LMS Lecture Notes Series Vol. 153,Cambridge University press, 211-234.R. Greenberg, Iwasawa Theory and p-adic Deformation of Motives, Motives(Seattle, WA, 1991), 193-223, Proc. Sympos. Pure Math., 55, Part 2, Amer.Math. Soc, Providence, RI, 1994.M. Harris, p-adic representations arising from descent on Abelian Varieties,Harvard PhD Thesis 1977, also Comp. Math., 39 (1979), 177-245.A. Huber, G. Kings, Degeneration of l-adic Eisenstein classes and of the ellipticpolylog, Invent, math., 135 (1999), 545-594.A. Huber, G. Kings, Bloch-Kato Conjecture and Main Conjecture of IwasawaTheory for Dirichlet Characters, Preprint 2001, revised 2002. To appear: DukeMath. Journal.A. Huber, J. Wildeshaus, Classical motivic polylogarithm according to Beilinsonand Deligne, Doc. Math. J. DMV, 3 (1998), 27-133 and 297-299.K. Kato, Iwasawa theory and p-adic Hodge theory, Kodai Math. J., 16 (1993),no. 1, 1-31.K. Kato, Lectures on the approach to Iwasawa theory for Hasse-Weil F-functions via FdR. I, Arithmetic algebraic geometry (Trento, 1991), 50-163,Lecture Notes in Math., 1553, Springer, Berlin, 1993.K. Kato, p-adic Hodge theory and values of zeta functions of modular forms,preprint 2000.G. Kings, The Tamagawa number conjecture for CM elliptic curves, Invent.Math., 143 (2001), no. 3, 571-627.M. Kolster, Th. Nguyen Quang Do, V. Fleckinger, Twisted S-units, p-adicclass number formulas, and the Lichtenbaum conjectures, Duke Math. J., 84(1996), no. 3, and Correction, Duke Math. J., 90 (1997), no. 3, 641-643.M. Kolster, Th. Nguyen Quang Do, Universal distribution lattices for abeliannumber fields, Preprint 2000.T.Y. Lam, A first course in noneommutative rings, Second edition, GraduateTexts in Mathematics, 131.B. Mazur, A. Wiles, Class fields of abelian extensions of Q, Invent. Math., 76(1984), no. 2, 179-330.Y. Ochi, O. Venjakob, On the structure of Selmer groups of p-adic Lie extensions,to appear: Journal of Alg. Geom.B. Perrin-Riou, p-adic L-functions and p-adic representations, SMF/AMSTexts and Monographs, 3., Paris, 2000.J. Ritter, A. Weiss, Toward equivariant Iwasawa theory, to appear:Manuscripta Mathematica.K. Rubin, Euler Systems, Annals of Mathematics Studies, 147, Princeton, NJ,2000.K. Rubin, The "main conjectures" of Iwasawa theory for imaginary quadraticfields, Invent. Math., 103 (1991), no. 1, 25-68.J-P. Serre: Sur le résidu de la fonction zêta p-adique d'un corps de nombres,CR. Acad. Sci. Paris, Série A, 287 (1978), A183-A188.O. Venjakob, On the structure theory of the Iwasawa algebra of a p-adic Lie

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ICM 2002 • Vol. II • 163-171Tamagawa NumberConjecture for zeta ValuesKazuya Kato*AbstractSpencer Bloch and the author formulated a general conjecture (Tamagawanumber conjecture) on the relation between values of zeta functions ofmotives and arithmetic groups associated to motives. We discuss this conjecture,and describe some application of the philosophy of the conjecture to thestudy of elliptic curves.2000 Mathematics Subject Classification: 11G40.Keywords and Phrases: zeta function, Etale cohomology, Birch Swinnerton-Dyer conjecture.Mysterious relations between zeta functions and various arithmetic groupshave been important subjects in number theory.(0.0) zeta functions arithmetic groups.A classical result on such relation is the class number formula discovered in19th century, which relates zeta functions of number field to ideal class groups andunit groups. As indicated in (0.1)-(0.3) below, the formula of Grothendieck expressingthe zeta functions of varieties over finite fields by etale cohomology groups,Iwasawa main conjecture proved by Mazur-Wiles, and Birch and Swinnerton-Dyerconjectures for abelian varieties over number fields, considered in 20th century, alsohave the form (0.0).(0.1) Formula of Grothendieck.zeta functions etale cohomology groups.(0.2) Iwasawa main conjecture.zeta functions, zeta elements ideal class groups, unit groups.(0.3) Birch Swinnerton-Dyer conjectures (see 4).zeta functions groups of rational points, Tate-Shafarevich groups.Here in (0.2), "zeta elements" mean cyclotomic units which are units in cyclotomicfields and closely related to zeta functions. Roughly speaking, the relations*Department of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Meguro, Tokyo,Japan. E-mail: kkato@ms.u-tokyo.ac.jp

164 Kazuya Kato(often conjectural) say that the order of zero or pole of the zeta function at an integerpoint is equal to the rank of the related finitely generated arithmetic abeliangroup (Tate, the conjecture (0.3), Beilinson, Bloch, ...) and the value of the zetafunction at an integer point is related to the order of the related arithmetic finitegroup.In [BK], Bloch and the author formulated a general conjecture on (0.0) (Tamagawanumber conjecture for motives). Further generalizations of Tamagawa numberconjecture by Fontaine, Perrin-Riou, and the author [FP], [Pei] [Kai], [Ka 2 ] havethe form(0.4) zeta functions (= Euler products, analytic)•try zeta elements (= Euler systems, arithmetic)•try arithmetic groups.Here the first -try means that zeta functions enter the arithmetic world transformingthemselves into zeta elements, and the second -try means that zeta elements generate"determinants" of certain etale cohomology groups.The aim of this paper is to discuss (0.4) in an expository style. We review(0.1) in §1, and then in §2, we describe the generalized Tamagawa number conjecture(0.4), the relation with (0.2), and an application of the philosophy (0.4) to (0.3).In this paper, we fix a prime number p. For a commutative ring F, let Q(R)be the total quotient ring of F obtained from F by inverting all non-zerodivisors.1. Grothendieck formula and zeta elementsLet X be a scheme of finite type over a finite field F g . We assume p is differentfrom char(F g ).In this §1, we first review the formula (1.1.2) of Grothendieck representing zetafunctions of p-adic sheaves on X by etale cohomology. We then show that thosezeta functions are recovered from p-adic zeta elements (1.3.5).1.1. Zeta functions and etale cohomology groups in positive characteristiccase. The Hasse zeta function ((X,s) = Yl xe i x \ (1 — tM 1 ) -8 ) -1 ) where|X| denotes the set of all closed points of x and K(X) denotes the residue field of x,has the form ((X,s) = ((X/F q ,q^s)whereC(X/F g ,«)= JJ (l-u^W)- 1 , deg(x) = [K(x):F q ]. (1.1.1)x€|X|A part of Weil conjectures was that ((X/F q ,u) is a rational function in u,and it was proved by Dwork and then slighly later by Grothendieck. The proof ofGrothendieck gives a presentation of ((X/F q , u) by using etale cohomologyy. Moregenerally, for a finite extension F of Q p and for a constructible F-sheaf J 7 on X,Grothendieck proved that the L-function L(X/F q ,J 7 ,u) has the presentationL(X/F q ,F,u) = l[det L (l -

Tamagawa Number Conjecture for zeta Values 165where F c is the etale cohomology with compact supports and tp q is the action ofthe q-th power morphism on X.In the case F = Q p = T, C,(X/F q ,u) = L(X/F q ,J 7 ,u).1.2. p-adic zeta elements in positive characteristic case. Determinantsappear in the theory of zeta functions as above, rather often. The regulator of anumber field, which appears in the class number formula, is a determinant. Suchrelation with determinant is well expressed by the notion of "determinant module".If F is a field, for an F-module V of dimension r, detfl(F) means the 1 dimensionalF-module A r R(V). For a bounded complex C of F-modules whose cohomologiesH m (C) are finite dimensional, detfl(C) means ® TOG z {detfl(F TO (G))}®^1^"*.This definition is generalized to the definition of an invertible F-module detp(C)associated to a perfect complex C of F-modules for a commutative ring F (see[KM]). det^j 1 (G) means the inverse of the invertible module detfl(G).By a pro-p ring, we mean a topological ring which is an inverse limit of finiterings whose orders are powers of p. Let A be a commutative pro-p ring. By a ctfA-complex on X, we mean a complex of A-sheaves on X for the etale topology withconstructible cohomology sheaves and with perfect stalks. For a ctf A-complex Ton X, RY et:C (X, T) ( c means with compact supports) is a perfect complex over A.For a commutative pro-p ring A and for a ctf A-complex J 7 on X, we define thep-adic zeta element ((X, T, A) which is a A-basis of det7 1 v FF etjC (X, T). Considerthe distinguished triangleFF et;C (X,^) -+ RY et , c (X® Fq F q ,F) ^ RY et , c (X ® Fq F q ,F). (1.2.1)Since det is multiplicative for distinguished triangles, (1.2.1) induces an isomorphismdet i - 1 i FF etiC (X,^) ~ det^FFrf^X ® Fq F q ,F) ® A det A RY et , c (X ® Fq F q ,F) s* A.(1.2.2)We define ((X, T, A) to be the image of 1 G A in det 1 A FF etjC (X,T) under (1.2.2).It is a A-basis of the invertible A-module det7 1 v FF etjC (X, T).1.3. Zeta functions and p-adic zeta elements in positive characteristiccase. Let F be a finite extension of Q p , let Op be the valuation ring ofF, and let J 7 be a constructible O^-sheaf on X. We show that the zeta functionL(X/F q ,J 7 p,u) of the F-sheaf Tp = T ®o L L is recovered from a certain p-adiczeta element as in (1.3.5) below. LetA = 0 L [[Gal(F g /F g )]] = hjnO L [Gal(F r /F g )]. (1.3.1)nYet s(A) be the A-module A which is regarded as a sheaf on the etale site of X viathe natural action of Gal(F g /F g ). ThenH C (X,T® 0L s(Aj) - fimF c (X ® Fq F qn ,T) (1.3.2)

166 Kazuya Katowhere the transition maps of the inverse system are the trace maps. From this, wecan deduce that H c (X,J 7 ®o L s(Aj) is a finitely generated O^-module for any rn.Hence we have Q(N) ® A FF etjC (X, J 7 ®o L s(Aj) = 0 and this gives an identificationcanonical isomorphismNoteQ(A) ® A detX 1 RY et , c (X,T®o L t(A)) = Q(A). (1.3.3)Q(A) = Q(hjnO L [«]/(«" - 1)) D Q(0 L [u]) = L(u). (1.3.4)nBy a formal argument, we can prove the following (1.3.5) (1.3.6) which showrespectively.zeta function = zeta element,zeta value = zeta element,L(X/F q ,T L ,u) = aX,T®o L s(A),A) in Q(A). (1.3.5)If F C (X,Tp) = 0 for any rn, L(X/F q ,J 7 p,u)has no zero or pole at u = 1, andL(X/F q ,T L ,l) = aX,T,0 L ) in F. (1.3.6)2. Tamagawa number conjectureIn 2.1, we describe the generalized version of Tamagawa number conjecture.In 2.2 (resp. 2.3), we consider p-adic zeta elements associated to 1 (resp. 2) dimensionalp-adic representations of Gal(Q/Q), and their relations to (0.2) (resp.(0.3)).2.1. The conjecture. Let X be a scheme of finite type over Z[-]. For acomplex of sheaves J 7 on X for the etale topology, we define the compact supportversion FF etjC (X, J 7 ) of FF et (X, J 7 ) as the mapping fiber ofFF et (Z[-], RfsF) -+ FF et (R, RfiF) ® FF et (Q p ,PRfiF).where / : X —t Spec(Z[-]).It can be shown that for a commutative pro-p ring A and for a ctf A-complexT on X, RY et:C (X, J 7 ) is perfect.The following is a generalized version of the Tamagawa number conjecture[BK] (see [FP], [Pei], [Kai], [Ka 2 ]). In [BK], the idea of Tamagawa number ofmotives was important, but it does not appear explicitly in this version.Conjecture. To any triple (X, A, J 7 ) consisting of a scheme X of finite type overZ[ì], a commutative pro-p ring A, and a ctf A-complex on X, we can associate aA-basis ((X, T, A) ofA(X,T,A) = det^RTet^X, J 7 ),

Tamagawa Number Conjecture for zeta Values 167which we call the p-adic zeta element associated to T, satisfying the following conditions(2.1.1)-(2.1.5).(2.1.1) If X is a scheme over a finite field F q , ((X, T, A) coincides with the elementdefined in §3.2.(2.1.2) (rough form) If T is the p-adic realization of a motive M, ((X, T, A)recovers the complex value lim s _s. 0 s~ e L(M, s) where L(M, s) is the zeta function ofM and e is the order of L(M, s) at s = 0.(2.1.3) If A' is a pro-p ring and A —t A' is a continuous homomorphism, ((X, T® AA', A') coincides with the image ofC,(X,T,A) under A(X,J 7 ®j(A',A') =ë A(X,F)® AA'.(2.1-4) For a distinguished triangle T' —¥ T —¥ J 7 " with common X and A, we have((X,T,A) = C(X,^',A)®C(X,^",A) in A(X,^,A) = A(X,F,A)® A A(X,T V ,A).(2.1.5) If Y is a scheme of finite type over Z[-] and f : X —¥ Y is a separatedmorphism,C(F, RfiF, A) = C(X, T, A) in A(Y, RfiF, A) = A(X,T, A).By this (2.1.5), the constructions of p-adic zeta elements are reduced to thecase X = Spec(Z[^]). How to formulate the part (4.1.2) of this conjecture is reducedto the case of motives over Q by (2.1.5) and L(M, s) = L(Rf\(M), s) (by philosophyof motives), where / : X —t Spec(Z[-]).The conditions (2.1.3)-(2.1.5) are formal properties which are analogous toformal properties of zeta functions. The conditions (2.1.1) and (2.1.3)-(2.1.5) canbe interpreted as(2.1.6) The system (X, A, J 7 ) H> ((X,T, A) is an "Euler system".In fact, let F be a finite extension of Q p , S a finite set of prime numberscontaining p, and let F be a free O^-module of finite rank endowed with a continuous0£-linear action of Gal(Q/Q) which is unramified outside S. For m > 1, letR m = 0 L [Gal(Q(C ro )/Q)] and letz m = (R m (Z[^],j m ,(T®o L s(R m )),R m ) £ det^FF etiC (Z[C ro ,^s],F).(j m : Spec(Z[^]) -+ Spec(Z[i])).Then the conditions (4.1.1) and (4.1.3)-(4.1.5) tell that when m varies, thep-adic zeta elements z m form a system satisfying the conditions of Euler systemsformulated by Kolyvagin [Ko].We illustrate the relation (2.1.2) with zeta functions.Let M be a motive over Q, that is, a direct summand of the motive H m (X)(r)for a proper smooth scheme X over Q and for r £ Z, and assume that M is endowedwith an action of a number field K. Then the zeta function L(M, s) lives in C, andthe p-adic zeta element lives in the world of p-adic etale cohomology. Since thesetwo worlds are too much different in nature, L(M, s) and the p-adic zeta elementare not simply related.

168 Kazuya KatoHowever in the middle of C and the p-adic world,(a) there is a 1 dimensional FT-vector space AK (M) constructed by the Bettirealization and the de Rham realization of M, and F'-groups (or motivic cohomologygroups) associated to M.Yet oo be an Archimedean place of K. Then(b) there is an isomorphismA K (M)® K K 00^Kvoconstructed by Hodge theory and F'-theory.Let w be a place of K lying over p, let M w be the representation of Gal(Q/Q)over K w associated to M, and let F be a Gal(Q/Q)-stable OK W -lattice in M w .Then(c) there is an isomorphismA K (M) ®K K„ -+ det^FF etjC (Z[-], j,M w )— p=detö Kw RY et , c (Z[-],;j*T) ® 0liw K WPwhere j : Spec(Q) —¥ Spec(Z[|]), constructed by p-adic Hodge theory and F'-theory.See [FP] how to construct (a)-(c) (constructions require some conjectures).The part (2.1.2) of the conjecture is:(d) there exists a F'-basis ((M) of AK(M) (called the rational zeta elementassociated to M), which is sent to lim s _s. 0 s~ e L(M, s) under the isomorphism(b) where e is the order of L(M,s) at s = 0, and to £(Z[^], j»T,OK W )in det^FF etjC (Z[ì], j*M w ) under the isomorphism (c).The existence of ((M) having the relation with lim s _s. 0 s~ e L(M, s) was conjecturedby Beilinson [Be].How zeta functions and p-adic zeta elements are related is illustrated in thefollowing diagram.zeta functions side (Betti) < — (de Rham)regulatorp-adic Hodge theory(F'-theory) • (etale) p-adic zeta elements side.Chern classWe have the following picture.?automorphic rep

Tamagawa Number Conjecture for zeta Values 169The left upper arrow with a question mark shows the conjecture that the map{motives} —¥ {zeta functions} factor through automorphic representations, whichis a subject of non-abelian class field theory (Langlands correspondences). As theother question marks indicate, we do not know how to construct zeta elements ingeneral, at present.2.2. p-adic zeta elements for 1 dimensional galois representations.Let A be a commutative pro-p ring, and assume we are given a continuous homomorphismp:Gal(Q/Q)^GF„(A)which is unramified outside a finite set S of prime numbers S containing p. Letj7 — j\_(Sn on which Gal(Q/Q) acts via p, regarded as a sheaf on Spec(Z[^]) for theetale topology. We consider how to construct the p-adic zeta element £(Z[^|], T, A).In the case n = 1, we can use the "universal objects" as follows. Such p comesfrom the canonical homomorphismPu„iv : Gal(Q/Q)-*-GLi(A univ ) where A univ = Z p [[Gal(Q(0v p °°)/Q)]]for some N > 1 whose set of prime divisors coincide with S and for some continuousring homomorphism A un ; v —t A. We have T — .Funiv ®A univ A. Hence £(Z[^|], T, A)should be defined to be the image of £(Z[^|],-F un j v , A un ; v ). As is explained in [Ka 2 ]Ch. I, 3.3, £(Z[^|], .Funivj A u „iv) is the pair of the p-adic Riemann zeta function anda system of cyclotomic units. Iwasawa main conjecture is regarded as the statemnetthat this pair is a A un ; v -basis of A(Z[^], -F un j v , A un ; v ).2.3. p-adic zeta elements for 2 dimensional Galois representations.Now consider the case n = 2. The works of Hida, Wiles, and other people suggestthat the universal objects A un ; v and -F un iv for 2 dimensional Galois representationsin which the determinant of the action of the complex conjugation is -1, are givenbyA U niv = ^Tû p-adic Hecke algebras of weight 2 and of level Np n ,nFumy = hm F 1 of modular curves of level Np n .nBeilinson [Be] discovered ratinai zeta elements in K 2 of modular curves, and the imagesof these elements in the etale cohomology under the Chern class maps becomep-adic zeta elements, and the inverse limit of these p-adic zeta elements should beC(Z[iy],.Funivj A u „iv) at least conjecturally. By using this plan, the author obtainedp-adic zeta elements for motives associated to eigen cusp forms of weight > 2, fromBeilinson elements. Here it is not yet proved that these p-adic zeta elements areactually basis of A, but it can be proved that they have the desired relations withvalues L(E,x, 1) and L(f,x,r) (1 < r < k — 1) for elliptic curves over Q (whichare modular by [Wi], [BCDT]) and for eigen cusp forms of weight k > 2, and forDirichlet charcaters x- Beilinson elements are related in the Archimedean world

170 Kazuya Katoto lim s _s.o s^1L(E,x,s)for elliptic curves F over Q, but not related to L(E,x, 1).However since they become universal (at least conjecturally) in the inverse limit inthe p-adic world, we can obtain from them p-adic zeta elements related to L(E, x, 1).Using these elements and applying the method of Euler systems [Ko], [Pe 2 ], [Ru 2 ],[Ka3], we can obtain the following results ([Raj).Theorem. Let E be an elliptic curve over Q, let N > 1, and let x '• Ga/(Q((iv)/Q)= (Z/NZ) X -t C be a homomorphism. If L(E,x,l) # 0, the XrP ar t °f E(Q((N))and the x P^rt of the Tate-shafarevich group of E over Q((JV) are finite.The p-adic L-function L P (E) of F is constructed from the values L(E,x, 1).Theorem. Let E be an elliptic curve over Q which is of good reduction at p.(1) rank(E(Q) < ord 8=1 L p (E).(2) Assume E is ordinary at p. Let A = Z p [[Gal(Q(( p^/Q)]]. Then thep-primary Seltner group of E over Q((p~) is A-cotorsion and its characteristicpolynomial divides p n L p (E) for some n.This result was proved by Rubin in the case of elliptic curves with complexmultiplication ([Rui]).As described above, we can obtain p-adic zeta elements of motives associatedto eigen cusp forms of weight > 2. For such modular forms, we can prove theanalogous statement as the above (2).Mazur and Greenberg conjectured that the charcteristic polynomial of theabove p-primary Selmer group and the p-adic L-function divide each other.References[Be] Beilinson, A., Higher regulators and values of F-functions, J. Soviet Math.,30 (1985), 2036-2070.[BK] Bloch, S. and Kato, K., Tamagawa numbers of motives and L-functions,in The Grothendieck Festschrift, 1, Progress in Math., 86, Burkhauser(1990), 333-400.[BCDT] Breuil, C, Conrad, B, Diamond, F., Taylor, R., On the modularity ofelliptic curves over Q: wild 3-adic exercises, J. Amer. Math. Soc, 14(2001), 834-939.[FP] Fontaine, J. -M., and Perrin-Riou, B., Autour des conjectures de Blochet Kato, cohomologie Galoisienne et valeurs de fonctions L, Proc. Symp.Pure Math. 55, Amer. Math. Soc., (1994), 599-706.[Kai] Kato, K., Iwasawa theory and p-adic Hodge theory, Rodai Math. J., 16(1993), 1-31.[Ka 2 ] Kato, K., Lectures on the approach to Iwasawa theory for Hasse-WeilF-functions via B d p. I, Arithmetic algebraic geometry (Trento, 1991),50-163, Lecture Notes in Math., 1553, Springer, Berlin (1993).[Ka3] Kato, K., Euler systems, Iwasawa theory, and Selmer groups, Kodai Math.J., 22 (1999), 313-372.

Tamagawa Number Conjecture for zeta Values 171[Kai][KM][Ko][Pei][Pe 2 ][Rui][Ru 2 ][Wi]Kato, K., p-adic Hodge theory and values of zeta functions of modularforms, preprint.Knudsen, F., and Mumford, D., The projectivity of the moduli space ofstable curves I, Math. Scand., 39, 1 (1976), 19-55.Kolyvagin, V. A., Euler systems, in The Grothendieck Festchrift, 2, Birkhouser(1990), 435-483.Perrin-Riou, B., Fonction L p-adiques des représentations p-adiques,Astérisque 229 (1995).Perrin-Riou, B., Systèmes d'Euler p-adiques et théorie d'lwaswa, Ann.Inst. Fourier, 48 (1998), 1231-1307.Rubin, K., The "main conjecture" of Iwasawa theory for imaginary quadraticfields, Inventiones math., 103 (1991), 25-68.Rubin, K., Euler systems, Hermann Weyl Lectures, Annals of Math. Studies,147, Princeton Univ. Press (2000).Wiles, A., Modular elliptic curves and Fermât's last theorem, Ann. ofMath., 141 (1995), 443-551.

ICM 2002 • Vol. II • 173-183Derivatives of Eisenstein Seriesand Arithmetic Geometry*Stephen S. Kudla tAbstractWe describe connections between the Fourier coefficients of derivatives ofEisenstein series and invariants from the arithmetic geometry of the Shimuravarieties M associated to rational quadratic forms (V,Q) of signature (n,2). Inthe case n = 1, we define generating series 4>i{r) for 1-cycles (resp. «feM for0-cycles) on the arithmetic surface M associated to a Shimura curve over Q.These series are related to the second term in the Laurent expansion of anEisenstein series of weight § and genus 1 (resp. genus 2) at the Siegel-Weilpoint, and these relations can be seen as examples of an 'arithmetic' Siegel^Weil formula. Some partial results and conjectures for higher dimensionalcases are also discussed.2000 Mathematics Subject Classification: 14G40, 14G35, 11F30.Keywords and Phrases: Heights, Derivatives of Eisenstein series, Modularforms.1. IntroductionIn this report, we will survey results about generating functions for arithmeticcycles on Shimura varieties defined by rational quadratic forms of signature(n, 2). For small values of n, these Shimura varieties are of PEL type, i.e., canbe identified with moduli spaces for abelian varieties equipped with polarization,endomorphisms, and level structure. By analogy with CM or Heegner points onmodular curves, cycles are defined by imposing additional endomorphisms. Relationsbetween the heights or arithmetic degrees of such cycles and the Fouriercoefficients of derivatives of Siegel Eisenstein series are proved in [10] and in subsequentjoint work with Rapoport, [14], [15], [16], and with Rapoport and Yang [17],[18]. These relations may be viewed as an arithmetic version of the classical Siegel-Weil formula, which identifies the Fourier coefficients of values of Siegel Eisenstein*Partially supported by NSF grant DMS-9970506 and by a Max-Planck Research Prize fromthe Max-Planck Society and Alexander von Humboldt Stiftung.t Mathematics Department, University of Maryland, College Park, MD 20742, USA. E-mail:ssk@math.umd.edu

174 S. S. Kudlaseries with representation numbers of quadratic forms. The most complete exampleis that of anisotropic ternary quadratic forms (n = 1), so that the cycles are curvesand 0-cycles on the arithmetic surfaces associated to Shimura curves. Other surveysof the material discussed here can be found in [11] and [12].2. Shimura curvesLet F be an indefinite quaternion algebra over Q, and let D(B) be the productof the primes p for which B p = B ®Q Q P is a division algebra. The rational vectorspaceV = { x £ B | tr(x) = 0 }with quadratic form given by Q(x) = —x 2 = v(x), where tr(x) (resp. v(x)) is thereduced trace (resp. norm) of x, has signature (1,2). The action of F x on V byconjugationgives an isomorphism G = GSpin(V) ~ F x . LetD = { w £ V(C) | (w,w) = 0, (w,w) < 0 }/C x~ P^C) \P 1 (R)be the associated symmetric space. Let 0 B be a maximal order in F and let F = Oßbe its unit group. The quotient M(C) = Y\D is the set of complex points of theShimura curve M (resp. modular curve, if D(B) = 1) determined by F. This spaceshould be viewed as an orbifold [F\F]. For a more careful discussion of this andof the stack aspect, which we handle loosely here, see [18]. The curve M has acanonical model over Q. From now on, we assume that D(B) > 1, so that M isprojective. Drinfeld's model M for M over Spec (Z) is obtained as the moduli stackfor abelian schemes (.4, i) with an action i : 0 B < L -¥ End(.4) satisfying the 'special'condition, [3]. It is proper of relative dimension 1 over Spec(Z), with semi-stablereduction at all primes and is smooth at all primes p at which F splits, i.e., forp\D(B). We view M. as an arithmetic surface in the sense of Arakelov theory andconsider its arithmetic Chow groups with real coefficients CH (A4) = CH R (A4),as defined in [2]. Recall that these groups are generated by pairs (Z,g), where Zis an R-linear combination of divisors on A4 and g is a Green function for Z, withrelations given by R-linear combinations of elements div (/) = (div(/), ^log|/| 2 )where / £ Q(M) X is a nonzero rational function on A4. These real vector spacescome equipped with a geometric degree map deg Q : CH (A4) —^ CH 1 (MQ) —^ R,where MQ is the generic fiber of A4, an arithmetic degree map deg : CH (A4) —¥ R,and the Gillet-Soulé height pairing, [2],( , ) : CF^M) x CF^M) —• R.Let A be the universal abelian scheme over M. Then the Hodge line bundle u =e*(0^,_ M ) determined by A has a natural metric, normalized as in [18], section3, and defines an element û £ Pic(^M), the group of metrized line bundles on

Eisenstein Series and Arithmetic Geometry 175M. We also write û for the image of this class in CH (M) under the naturalmap, which sends a metrized line bundle £ = (£,\\ ||) £ Pic(M) to the class of(div(s), — log ||«|| 2 ), for any nonzero section s of £.Arithmetic cycles in M are defined by imposing additional endomorphisms ofthe following type.Definition 1. ([10]) The space of special endomorphisms V(A,i)scheme (A,i), as above, isof an abelianV(A, c) = { x £ End(.4) | x o t(fe) = t(fe) ox, V6 G 0 B , and tr(x) = 0 },with Z-valued quadratic form given by —x 2 = Q(x) id^.2.1. DivisorsTo obtain divisors on A4, we impose a single special endomorphism. For apositive integer t, let Z(t) be the divisor on M determined by the moduli stackof triples (A,L,X) where (A,i) is as before and where x £ V(A,i) is a specialendomorphism with Q(x) = t. Note that, for example, the complex points Z(t)(C)of Z(t) correspond to abelian surfaces (.4, i) over C with an 'extra' action of the orderZ[y^t] in the imaginary quadratic field Q(\/—t), i.e., to CM points on the Shimuracurve M(C). On the other hand, the cycles Z(i) can have vertical components inthe fibers of bad reduction M p for p | D(B). More precisely, in joint work with M.Rapoport we show:Proposition 1. ([15]) For p | D(B), Z(t) contains components of the fiber of badreduction M p if and only if ord p (t) > 2 and no prime £ | D(B), £ ^ p, splits inh •= Q(V~t).The precise structure of the vertical part of Z(t) is determined in [15] usingthe Drinfeld-Cherednik p-adic uniformization of M p . For example, for p | D(B),the multiplicities of the vertical components in the fiber M p of the cycle Z(p 2r t)grow with r, while the horizontal part of this cycle remains unchanged.To obtain classes in CH (M), we construct Green functions by the procedureintroduced in [10]. Let F = 0 B n V. For t £ Z >0 and v £ R>o, define a functionE(t,v) on M(C) byE(t,v)(z)= J2 ßi(^vR(x,z)),x£L(t)where L(t) = {x £ L \ Q(x) = t}, and, for z £ D with preimage w £ V(C),R(x,z) = |(a;,w)| 2 |(w,w)| _1 . Here/•OOßi(r) = / e- ru u- x du = Êi(^r)

t.176 S. S. Kudlais the exponential integral. Recall that this function has a log singularity as r goesto zero and decays exponentially as r goes to infinity. In fact, as shown in [10],section 11, for any x £ V(M.) with Q(x) ^ 0, the function(,(x,z) :=ßi(2nR(x,z))can be viewed as a Green function on D for the divisor D x := {z £ D | (x, z) = 0}.A simple calculation, [10], shows that, for t > 0, E(t,v) is a Green function oflogarithmic type for the cycle Z(t), while, for t < 0, E(t,v) is a smooth function onM(C).Definition 2. (i) For t £ Z and v > 0, the class Z(t,v)£ CH (M) is defined by:Z(t,v)( (Z(t),E(t,v)) ift>0,^0! + (0,c^log(w)) ift = 0,{ (0,E(t,v)) ift p\D(B)where C,D(B)(S) = C,( s ) U P \D(B)(^ ^P S )anI(T) is a (nonholomorphic) modular„ —~~iform of weight |, valued in CH (A4), for a subgroup Y' c SL 2 (Z).The proof of Theorem 1 depends on Borcherd's result [1] and on the modularityof various complex valued g-expansions obtained by taking height pairings of

Eisenstein Series and Arithmetic Geometry 177with other classes in CH (A4). We now describe some of these in terms of valuesand derivatives of a certain Eisenstein series, [18], of weight §£I(T,S,D(B))= Y, (cT + d)-i\cT + d\- {s -i ) v^s-i )î(sr/,D(B)),associated to F and the lattice F, and normalized so that it is invariant unders H> —s. The main result of joint work with M. Rapoport and T. Yang is thefollowing:Theorem 2. ([18]) (i)Î(T,—,D(B)) = deg(i(T)) =J2deg Q (Z(t,v))q t .(ii)£[(T,\;D(B))2'= (MT),û) = 5^(f (t, w ),w>g*.tNote that this result expresses the Fourier coefficients of the first two terms inthe Laurent expansion at the point s = \ of the Eisenstein series £I(T, S; D(Bj) interms of the geometry and the arithmetic geometry of cycles on A4.Next consider the image of-MrWi(r, i; F(F)) • degM- 1•cDin CH 1 (MQ), the usual Chow group of the generic fiber. By (i) of Theorem 2, itlies in the Mordell-Weil space CH 1 (Mq)° ® C ~ Jac(M)(Q) ® z C. In fact, it isessentially the generating function defined by Borcherds, [1], for the Shimura curveM, and hence is a holomorphic modular of weight §. For the case of modularcurves, such a modular generating function, whose coefficients are Heegner points,was introduced by Zagier, [25]. By the Hodge index theorem for CH (A4), [2], theproof of Theorem 1 is completed by showing that the pairing of

178 S. S. Kudlawhere Z = J2ì n ìPì f° r dosed points Pi of M with residue field fc(Fj), is an isomorphism.Let r = u + iv £ SJ2, the Siegel space of genus 2, and for T £ Sym 2 (Z), letq T = e 27rrtr ( TT ). To define the generating series4> 2 (T)= Y. 2(2» g T ,T€Sym 2 (Z)we want to define classes Z(T,v) £ CH (M) for each T £ Sym 2 (Z) and v £Sym 2 (R) >0 .We begin by considering cycles on M which are defined by imposing pairs ofendomorphisms. For T £ Sym 2 (Z) >0 a positive definite integral symmetric matrix,let Z(T) be the moduli stack over M consisting of triples (.4, i, x) where (.4, i) is asbefore, and x = [ari,a: 2 ] £ Y(A,i) 2 is a pair of special endomorphisms with matrixof inner products Q(x) = |((x»,Xj)) = T. We call T the fundamental matrix of thetriple (A,i,x). The following result of joint work with M. Rapoport describes thecases in which Z(T) is, in fact, a 0-cycle on M.Proposition 2. ([15]) Suppose thatT £ Sym 2 (Z) >0 . (i) The cycle Z(T) is eitherempty or is supported in the set of supersingular points in a fiber M p for a uniqueprime p determined by T. In particular, Z(T)Q = 0. The prime p is determinedby the condition that T is represented by the ternary quadratic space V^ = { x £F^ | tr(ar) = 0 }, with Q^(x) = —x 2 , where F^ is the definite quaternion algebraover Q with Bf ~ B( for all primes £ ^ p. If there is no such prime, then Z(T)is empty.(ii) (T regular) Let p be as in (i). Then, if p \ D(B) or if p | F(F) but p 2 \ T,then Z(T) is a 0-cycle in M p .(in) (T irregular) Letp be as in (i). If p | F(F) andp 2 | T, then Z(T) is a union,with multiplicities, of components of A4 P , cf. [15], 176.For T £ Sym 2 (Z) >0 regular, as in (ii) of Proposition 2, we letFor T = f1Z(T,v) := Z(T) = (Z(T),0) £ Gif (M).J £ Sym 2 (Z) >0 irregular, we use the results of [15], section8 (where the quadratic form is taken with the opposite sign). We must thereforeassume that p^2, although the results of the appendix to section 11 of [18] suggestthat it should be possible to eliminate this restriction. In this case, the vertical cycleZ(T) in the fiber M p is the union of those connected components of the intersectionZ(h) XM Z(t2) where the 'fundamental matrix', [15], is equal to T. Here Z(ti)and Z(t2) are the codimension 1 cycles defined earlier. Note that, by Proposition1, they can share some vertical components. We base change to Z p and setZ(T,v) := x(Z(T), O z(tl) ® h O z(t2) ) • log(p) G R ~ CH 2 (M),

Eisenstein Series and Arithmetic Geometry 179where x ' 1S the Euler-Poincaré characteristic of the derived tensor product of thestructure sheaves Oz(t{) an d Oz(t 2 ),CI> - [15]) section 4. Note that the same definitioncould have been used in the regular case.Next we consider nonsingular T £ Sym 2 (Z) of signature (1,1) or (0,2). Inthis case, Z(T) is empty, since the quadratic form on V(A,i) is positive definite,and our 'cycle' should be viewed as 'vertical at infinity'. For a pair of vectorsx = [#i,#2] € V'(Q) 2 with nonsingular matrix of inner products Q(x) = \ ((xi,Xj)),the quantityA(x) := / £(xi)*£(x 2 ),JDwhere £(#i) * Ç(x 2 ) is the *-product of the Green functions £(#i) and £(12), [6],is well defined and depends only on Q(x). In addition, A(x) has the followingremarkable invariance property.Theorem 3. ([10, Theorem 11.6]) For k £ 0(2), A(x • k) = A(x).For T £ Sym 2 (Z) of signature (1,1) or (0,2) and for v £ Sym 2 (R) >0 , choosea £ GF 2 (R) such that v = a 1 a, and defineZ(T,v):=YlxGL 2 , Q(x) = T, mod FA ( xa )eR~CF 2 (,V().Here F = 0 B n V and F = Oß, as before. Note that the invariance property ofTheorem 3 is required to make the right side independent of the choice of a.We omit the definition of the terms for singular F's, cf. [11].By analogy with Theorem 1, we conjecture that, with this definition, thegenerating series

180 S. S. Kudla(ii) T £ Sym 2 (Z)> 0 is regular a,ndp\2D(B), [10].(iii) T G Sym 2 (Z) >0 is irregular with p ^ 2, or regular with p \ D(B) and p ^ 2,[15].(iv) T G Sym 2 (Z) is nonsingular of signature (1,1) or (0,2), [10].Theorem 4 is proved by a direct computation of both sides of (Cip). In case(ii), the computation of the Fourier coefficient £' 2 T (T,0;D(B)) depends on the formulaof Kitaoka, [8], for the local representation densities a p (S,T) for the given Tand a variable unimodular S. The computation of Z(T,v) = deg ((Z(T),Oj) dependson a special case of a result of Gross and Keating, [7], about the deformationsof a triple of isogenies between a pair of p-divisible formal groups of dimension 1and height 2 over F p . Their result is also valid for p = 2, so it should be possible toextend (ii) to the case p = 2 by extending the result of Kitaoka.In case (iii), an explicit formula for the quantity x(Z(T),Oz(t 1 ) ® L Oz(t 2 )) ' 1Sobtained in [15] using p-adic uniformization. The analogue of Kitaoka's result is adetermination of a p (S,T) for arbitrary S due to T. Yang, [22]. In both of theseresults, the case p = 2 remains to be done.Case (iv) is proved by directly relating the function A, defined via the *-product to the derivative at s = 0 of the confluent hypergeometric function of amatrix argument defined by Shimura, [21]. The invariance property of Theorem 3plays an essential role. The case of signature (1,1) is done in [10]; the argument forsignature (0,2) is the same.A more detailed sketch of the proofs can be found in [11].As part of ongoing joint work with M. Rapoport and T. Yang, the verificationof (Cip) for singular T of rank 1 is nearly complete.3. Higher dimensional examplesSo far, we have discussed the generating functions 4>I(T) £ CH (M) and4>2(T) £ CH (M) attached to the arithmetic surface M, and the connections ofthese series to derivatives of Eisenstein series. There should be analogous series definedas generating functions for arithmetic cycles for the Shimura varieties attachedto rational quadratic spaces (V,Q) of signature (n,2). At present there are severaladditional examples, all based on the accidental isomorphisms for small values ofn, which allow us to identify the Shimura varieties in question with moduli spacesof abelian varieties with specified polarization and endomorphisms. Here we brieflysketch what one hopes to obtain and indicate what is known so far. The resultshere are joint work with M. Rapoport.Hilbert-Blumenthal varieties (n = 2), [14]. When the rational quadraticspace (V,Q) has signature (2,2), the associated Shimura variety M is a quasiprojectivesurface with a canonical model over Q. There is a model M of Mover Spec(Z[iV -1 ]) defined as the moduli scheme for collections (A,X,i,f]) whereA is an abelian scheme of relative dimension 8 dimension with polarization A, level

Eisenstein Series and Arithmetic Geometry 181structure fj, and an action of Oc ® Ok, where Oc is a maximal order in the Cliffordalgebra C(V) of V and Ok is the ring of integers in the quadratic field k = Q(Vd) ford = discr(V), the discriminant field of V, [14]. Again, a space V(A, i) = V(A, X, i, fj)of special endomorphisms is defined; it is a Z-module of finite rank equipped witha positive definite quadratic form Q. For T £ Sym r (Z), we let Z(T) be the locusof (A,\,i,fj,x)'s where x = [xi,... ,x r ], xi £ V(A,i) is a collection of r specialendomorphisms with matrix of inner products Q(x) = |((XJ,£J)) = ^ •One would like to define a family of generating functions according to the followingconjectural chart. Again there is a metrized Hodge line bundle û £ CH(A4).r = l, Z(t)q = HZ - curved(r) = w+? + ]T t#0 Jj(£, v)

182 S. S. KudlaTheorem 6. ([13], [9], [11]) Suppose thatV is anisotropic (i) C1(çV(T)) is a Siegelmodular form of genus r and weight 2 valued in H 2r (M,C).(ii) For the cup product pairing, ( cl((f> r (TJ),cl(û) ) = £ r (T,so), where so = |(3 —r).Part (ii) here generalizes (i) of Theorem 2 above, so that, again, the value at soof the Eisenstein series £ r (r, s) involves the complex geometry, while, conjecturally,the second term involves the height pairing.Siegel modular varieties (n = 3), [16]. Here, an integral model M of theShimura variety M attached to a rational quadratic space of signature (3,2) canbe obtained as a moduli space of polarized abelian varieties of dimension 16 withan action of a maximal order Oc in the Clifford algebra of V. We just give therelevant conjectural chart:1, Z(t)q --Humbert0i (T)=W+? + £ M O %«)?*> (Mr),û 3 )= £{(T 3 -surface '' 2 i-,Si7 , V^ 9/rp „,\ „Til l^\ ,-,2\2, Z(t)q •- curve 2 (r) = ûr+? + X ^ Z(F, v)

Eisenstein Series and Arithmetic Geometry 183[9] S. Kudla, Algebraic cycles on Shimura varieties of orthogonal type, Duke Math.J. 86 (1997), 39-78.[10] , Central derivatives of Eisenstein series and height pairings, Ann. ofMath. 146 (1997), 545-646.[11] , Derivatives of Eisenstein series and generating functions for arithmeticcycles, Sém. Bourbaki n° 876, Astérisque, vol. 276, 2002, pp. 341-368.[12] , Special cycles and derivatives of Eisenstein series, Proc. of MSRIWorkshop on Heegner points (to appear).[13] S. Kudla and J. Millson, Intersection numbers of cycles on locally symmetricspaces and Fourier coefficients of holomorphic modular forms in severalcomplex variables, Pubi. Math. IHES 71 (1990), 121-172.[14] S. Kudla and M. Rapoport, Arithmetic Hirzebruch-Zagier cycles, J. reineangew. Math. 515 (1999), 155-244.[15] , Height pairings on Shimura curves and p-adic unformization, Invent.math. 142 (2000), 153-223.[16] , Cycles on Siegel threefolds and derivatives of Eisenstein series, Ann.Scient. Éc Norm. Sup. 33 (2000), 695-756.[17] S. Kudla, M. Rapoport and T. Yang, On the derivative of an Eisenstein seriesof weight 1, Int. Math. Res. Notices, No.7 (1999), 347-385.[18] , Derivatives of Eisenstein series and Faltings heights, preprint (2001).[19] U. Kühn, Generalized arithmetic intersection numbers, J. reine angew. Math.534 (2001), 209-236.[20] W. J. McGraw, On the rationality of vector-valued modular forms, preprint(2001).[21] G. Shimura, Confluent hypergeometric functions on tube domains, Math. Annalen260 (1982), 269-302.[22] T. Yang, An explicit formula for local densities of quadratic forms, J. NumberTheory 72 (1998), 309-356.[23] , The second term of an Eisenstein series, Proc. of the ICCM, (toappear).[24] , Faltings heights and the derivative of Zagier's Eisenstein series, Proc.of MSRI workshop on Heegner points, preprint (2002).[25] D. Zagier, Modular points, modular curves, modular surfaces and modularforms, Lecture Notes in Math. 1111, Springer, Berlin, 1985, 225-248.

ICM 2002 • Vol. II • 185-195Elliptic Curves and Class Field TheoryBarry Mazur*Karl RubinÂbstractSuppose E is an elliptic curve defined over Q. At the 1983 ICM the firstauthor formulated some conjectures that propose a close relationship betweenthe explicit class field theory construction of certain abelian extensions ofimaginary quadratic fields and an explicit construction that (conjecturally)produces almost all of the rational points on E over those fields.Those conjectures are to a large extent settled by recent work of Vatsaland of Cornut, building on work of Kolyvagin and others. In this paperwe describe a collection of interrelated conjectures still open regarding thevariation of Mordell-Weil groups of E over abelian extensions of imaginaryquadratic fields, and suggest a possible algebraic framework to organize them.2000 Mathematics Subject Classification: 11G05, 11R23.Keywords and Phrases: Elliptic curves, Iwasawa theory, Heegner points.1. IntroductionEighty years have passed since Mordell proved that the (Mordell-Weil) groupof rational points on an elliptic curve F is finitely generated, yet so limited isour knowledge that we still have no algorithm guaranteed to compute the rankof this group. If we want to ask even more ambitious questions about how therank of the Mordell-Weil group E(F) varies as F varies, it makes sense to restrictattention only to those fields for which we have an explicit construction, such asfinite abelian extensions of a given imaginary quadratic field K. Taking our leadfrom the profound discovery of Iwasawa that the variational properties of certainarithmetic invariants are well-behaved if one restricts to subfields of Z^-extensionsof number fields, we will focus on the following Mordell-Weil variation problem:Fixing an elliptic curve E defìned over Q, an imaginary quadratic fìeldK, and a prime number p, study the variation of the Mordell-Weil groupof E over fìnite subfìelds of the (unique) Z 2 -extension of K in K.* Department of Mathematics, Harvard University, Cambridge, MA 02138, USA. E-mail:mazur@math.harvard.edu^Department of Mathematics, Stanford University, Stanford, CA 94305, USA. E-mail:rubin@math.stanford.edu

186 B. Mazur K. RubinThis problem was the subject of some conjectures formulated by the firstauthor at the 1983 ICM [8], conjectures which have recently been largely settled byworkof Vatsal [15] and Cornut [1] building on work of Kolyvagin and others.Example. Let F be the elliptic curve y 2 + y = x 3 —x,p= 5, and let K = Q(-\/—7).If F is a finite extension of K, contained in the Z\ extension of K, then rank E(F) =[F n Kû: K] where Kûis the anticyclotomic Z 5 -extension of K (see §2 for thedefinition). One only has an answer like this in the very simplest cases.Now with the same F and p, take K = Q(V^26). A guess here would be thatrank E(F) = [F n K%£ u : K] + 2, but this seems to be beyond present technology.The object of this article is to sketch a package of still-outstanding conjecturesin hopes that it offers an even more precise picture of this piece of arithmetic. Theseconjectures are in some cases due to, and in other cases build on ideas of, Bertolini& Darmon, Greenberg, Gross & Zagier, Haran, Hida, Iwasawa, Kolyvagin, Nekovâr,Perrin-Riou, and the authors, among others.In sections 3 through 5 we describe the three parts of our picture: the arithmetictheory (the study of the Selmer modules over Iwasawa rings that contain theinformation we seek), the analytic theory (the construction and study of the relevantF-functions, both classical and p-adic), and the universal norm theory which arisesfrom purely arithmetic considerations, but provides analytic invariants.In the final section we suggest the beginnings of a new algebraic structure toorganize these conjectures. This structure should not be viewed as a conjecture,but rather as a mnemonic to collect our conjectures and perhaps predict new ones.More details and proofs will appear in a forthcoming paper.2. Running hypotheses and notationFix a triple (E,K,p) where F is an elliptic curve of conductor N over Q, Kis an imaginary quadratic field of discriminant D < —4, and p is a prime number.To keep our discussion focused and as succinct as possible, we make the followinghypotheses and conventions.Assume that p is odd, that N, p and D are pairwise relatively prime, and thatif F has complex multiplication, then K is not its field of complex multiplication.Let OK C K denote the ring of integers of K. Assume further that there exists anideal Af C OK such that OK/A" is cyclic of order N (this is sometimes called theHeegner Hypothesis), and that p is a prime of ordinary reduction for F. Forsimplicity we will assume throughout this article that the p-primary subgroups ofthe Shafarevich-Tate groups of F over the number fields we consider are all finite.Proposition 1. Under the assumptions above, rank E(K) is odd.Proof. This follows from the Parity Conjecture recently proved by Nekovâr [11].Let KQO denote the (unique) Z 2 -extension of K and T := Gal(K 0O /Ä'), sor = Z 2 . We define the Iwasawa ringA:=Z p [[r]]® Zp Q p .

Elliptic Curves and Class Field Theory 187(To simplify notation and to avoid some complications, we will often work withQp-vector spaces instead of natural Z p -modules; in particular we have tensored theusual Iwasawa ring with Q p .) For every (finite or infinite) extension F of K in K œwe also defineA F := Z p [[Gal(F/K)]] ® Zp Q p , I F := ker{A -» A F }.Then IK is the augmentation ideal of A, and if [F : K] is finite then Ap is just thegroup ring Q p [Gal(F/K)]. If Gal(F/K) is Z p or Z 2 , and M is a finitely generatedtorsion Af-module, then charA F (M) will denote the characteristic ideal of M. Inparticular charA F (M) is a principal ideal of Ap.There is a Q p -projective line of Z p -extensions of K, all contained in K^.Among these are two distinguished Z p -extensions:• the cyclotomic Z p -extension K^ci , the compositum of K with the unique(cyclotomic) Z p -extension of Q (write F cyc i = Gal(K^d/K),A cyc i = A^-cyd),• the anticyclotomic Z p -extension K%£ u , the unique Z p -extension of K thatis Galois over Q with non-abelian, and in fact dihedral, Galois group (writeFanti = Gal(F^J 1 /K), A a „ti = A K^)-Then T = Y cyci ® F anti and A = A cyd ® Zp A anti .Complex conjugation r : K —¥ K acts on F, acting as +1 on F cyc i and ~1 onF ant i. This induces nontrivial involutions of A and A ant i, which we also denote by r.If M is a module over A (or similarly over A ant i), let M^ denote the module whoseunderlying abelian group is M but where the new action of 7 £ T on TO £ M^ isgiven by the old action of 7 T on rn.Our A-modules will usually come with a natural action of Gal(K 0O /Q). Theseactions are continuous and Z p -linear, and satisfy the formula f (7-771) = 7 T -f(ro) forevery lift f of r to Gal(K 0O /Q). Thus the action of any lift f induces an isomorphismAf ^ M( T \ We will refer to such A or A a „ti-modules as semi-linear r-modules.If M is a semi-linear r-module and is free of rank one over A ant j, we define the signof M to be the sign ±1 of the action of r on the one-dimensional Q p -vector spaceM ®A„ti A if. Such an M is completely determined (up to isomorphism preservingits structure) by its sign.Definition 2. If M and A are semi-linear T-modules, then a (A-bilinear) A-valuedr-Hermitian pairing IT is a A-module homomorphism ix : M ® A M^ —^ A suchthat for every lift f of r to Gal(K 0O /Q)n(m ® n) = n(n ® m) T= n(fn ® fro).3. Universal normsDefinition 3. If K C F C Koo, the universal norm module U(F) is the projectivelimitU(F):=Q p ® Hm (E(L)®Z p )KCLCF

188 B. Mazur K. Rubin(projective limit with respect to traces, over finite extensions L of K in F) withits natural Ap-structure. If F is a finite extension of K, then U(F) is simplyE(F)®Q P .If F is a Z p -extension of K, then U(F) is a free A^-module of finite rank, andis zero if and only if the Mordell-Weil ranks of F over subfields of F are bounded(cf. [8] §18 or [12] §2.2). The first author conjectured some time ago [8] that forZp-extensions F/K, and under our running hypotheses, U(F) = 0 if F ^ K%g u andU(K%£ U ) is free of rank one over A an ti- The following theorem follows from recentwork of Kato [6] for K^ci and Vatsal [15] and Cornut [1] for K%g u .Theorem 4. U(K^cl ) = 0 and U(K%£ U ) is free of rank one over A ant i.For the rest of this paper we will write U for the anticyclotomic universal normmodule U(KÛ).Complex conjugation gives U a natural semi-linear r-modulestructure. Since U is free of rank one over A ant i, we conclude that U is completelydetermined(up to isomorphism preserving its r-structure) by its sign.Let i^ be the rank of the ±1 eigenspace of r acting on E(K), so rank F(Q) =r+ and rank E(K) = r + + r^. By Proposition 1, rank E(K) is odd so r+ ^ r^.Conjecture 5 (Sign Conjecture). The sign of the semi-linear r-module U is +1if r + > r - , and is — 1 if r - > r + .Remark. Equivalently, the Sign Conjecture asserts that the sign of U is +1 if twicerankF(Q) is greater than rank E(K), and —1 otherwise.As we discuss below in §4, the Sign Conjecture is related to the nondegeneracyof the p-adic height pairing (see the remark after Conjecture 11).The A ant i-module U comes with a canonical Hermitian structure. That is, thecanonical (cyclotomic) p-adic height pairing (see [10] and [12] §2.3)h '• U ®A„ti U (T) > r cyc i ® Zp A a „tiis a T-Hermitian pairing in the sense of Definition 2.Conjecture 6 (Height Conjecture). The homomorphism h is an isomorphismof free A wt \-modules of rank oneh '• U ®A„ti U (T) ^ r cyc i ® Zp Aanti-The Aanti-module U has an important submodule, the Heegner submodule% C U. Fix a modular parameterization X 0 (N) —t E. The Heegner submodule% is the cyclic A a „ti-module generated by a trace-compatible sequence c = {cp} ofHeegner points cp £ E(L)®Z P for finite extensions F of K in Kû.See for example[8] §19 or [12] §3. Call such a c £ % a Heegner generator. The Heegner generatorsof % are well-defined up to multiplication by an element of ±F c (A ant i) x . TheAanti-submodule W C Wis stable under the semi-linear r-structure of U, so theaction of r gives an isomorphism U/H, ^t (UfiH)^ — U^/H^.Let c^ denote the element c viewed in the A ant ;-module 'HS T \ Since(±7C) ®A„ ti (±7C) (T) = C®A„ ti C (T)

Elliptic Curves and Class Field Theory 189for every ±7 £ ±Y, the element c®c^ £ 'H®^Dii 'H^T' > is independent of the choiceof Heegner generator, and is therefore a totally canonical generator of the free, rankone A anti -module % ®A„ ti %^ •Definition 7. The Heegner F-function (for the triple (E,K,p)running hypotheses) is the element£ := h(c® C (T) ) G F cyc i ® Zp A anti .Conjecture 8. F cyc i ® cha,r(U/H) 2 = A anti £ inside F cyc i ® A anti .satisfying ourOne sees easily that F cyc i ® char(U/H) 2 D A ant j£, and that Conjecture 8 isequivalent to the Height Conjecture (Conjecture 6).4. The analytic theoryThe ( "two-variable" ) p-adic F-function for F over K is an element L £ Aconstructed by Haran [3] and by a different, more general, method by Hida [4] (seealso the papers of Perrin-Riou [13, 14]). The F-function L is characterized by thefact that it interpolates special values of the classical Hasse-Weil F-function of twistsof F over K. More precisely, embedding Q both in C and Q p , if x '• T —¥ Z xis a character of finite order thenx(L) = C( X ) Lda 7 a :; i !^X' 1) (4.1)where L c i aiSS j cai i(E/ K , X, s ) ' 1S the Hasse-Weil F-function of the twist of E/ K by x,c(x) is an explicit algebraic number (cf. [13] Théorème 1.1), fp is the modular formon Y'o(N) corresponding to F, and ||/#|| is its Petersson norm.Projecting L £ A to the cyclotomic or the anticyclotomic line via the naturalprojections A —t A cyc i and A —t A ant ;, we get "one-variable" p-adic F-functionsL 1-^ F cyc i £ A cyc i and L 1-^ F ant ; £ A ant ;.It follows from the functional equation satisfied by L ([13] Théorème 1.1) and theHeegner Hypothesis that F ant i = 0. In other words, viewing A = A ant ;[[r cyc i]] as thecompleted group ring of F cyc i with coefficients in A ant ;, we have that the "constantterm" of L £ A ant ; [[F cyc i]] vanishes. We now consider its "linear term."There is a canonical isomorphism of (free, rank one) A ant ;-modulescZ pxIcycl ®Z P A ant i — I if >nti /IK%which sends 7 1 £ F cyc i ® Zp A anti to 7 - 1 £I K^/12ifantiConjecture 9 (A-adic Gross-Zagier Conjecture). Let L' denote the image ofL under the map Isanti/I^„ ti ^y F cyc i ®z p A ant i. ThenL' = d- x £where d is the degree of the modular parametrization X 0 (N) —t E.

190 B. Mazur K. RubinRemark. Perrin-Riou [13] proved that if p splits in K and the discriminant D ofK is odd, then F' and d^1£have the same image under the projection A an ti —*A A- = Qp.Let I := IK, the augmentation ideal of A. For every integer r > 0 we havejjyjr+i ^ Syniz (r) ® Qp. Using the direct sum decomposition F = F cyc i ® F ant i weget a canonical direct sum decompositionSymL (r) = ê Y r -J'J where Y^ := (1^,)®' ® Zp (Y wti f j . (4.2)P3=0Consider the canonical (two-variable) p-adic height pairing( , ) : E(K) x E(K) —• T ® Q p . (4.3)Set r = rankE(K), which is odd by Proposition 1. Define the two-variable p-adicregulator R p (E,K) to be the discriminant of this pairing:R P (E, K) := r 2 det(F i , P 5 ) £ Sym^ (r) ® Q p ~ F/F+ 1 ,where {Pi,..., F r } generates a subgroup of E(K) of finite index t. For each integerj = 0,..., r let R P (E, K) r^j' j be the projection of R P (E, K) into F r_ -" ® Q p under(4.2), so thatR P (E,K)= ê R p (E,Ky-".3=0Recall that r 1 * 1 is the rank of the ±l-eigenspace Efö) 1^ of r acting on E(K).Proposition 10. R p (E,K) r^J>J = 0 unless j is even and j < 2min(r+,r _ ).Proof. This follows from the fact that the height pairing (4.3) is r-Hermitian, so(rx,Ty) = (x,y) T , and therefore the induced height pairingsvanish.E(K)± x E(K)± -+ F anti ® Qp, E(K)+ x E(K)~ -+ F cycl ® Q pConjecture 11 (Maximal nondegeneracy of the height pairing). If j is evenand 0 < j < 2min(r+,r-) then R p (E,K) r '-^ # 0.Remark. Conjecture 11, or more specifically the nonvanishing of R p (E,K) r^J>Jwhen j = 2min(r+,r _ ), implies the Sign Conjecture (Conjecture 5). This is provedin the same way as Proposition 10, using the additional fact that the anticyclotomicuniversal norms in E(K) ® Z p are in the kernel of the anticyclotomic p-adic heightpairing (E(K) ® Z p ) x (E(K) ® Z p ) -• F anti ® Q p .

Elliptic Curves and Class Field Theory 1915. The arithmetic theoryFor every algebraic extension F of K, let Sel p (E/ F ) denote the p-power Selmergroup of F over F, the subgroup of ^(Gp, E[p°°]) that sits in an exact sequence0 —• E(F) ® Qp/Zp —• Selp(F /F ) —• HI(F /F )[p°°] —• 0where IH(E/ F )is the Shafarevich-Tate group of F over F. Also writeS p (E/p)= Hom(Selp(F /F ), Qp/Z p ) ® Q pfor the tensor product of Q p with the Pontrjagin dual of the Selmer group.The following theorem is proved using techniques which go back to [7]; see [2]and [12] Lemme 5, §2.2.Theorem 12 (Control Theorem). Suppose K C F C K œ .(i) The natural restriction map ^(FjEfy 00 ]) —t F 1 (K 0O ,F[p°°]) induces an isomorphismSp(E/jâo ) ®A Ap ^y S p (E/p).(ii) There is a canonical isomorphism U(F) ^y Ylom Af ,(S p (E/p),Ap).Conjecture 13 (Two-variable main conjecture [8, 12]). The two-variable p-adic L-function L generates the ideal chax\(S p (E/ Kao )) of A.Restricting the two-variable main conjecture to the cyclotomic and anticyclotomiclines leads to the following "one-variable" conjectures originally formulatedin [9] and [12], respectively. Let F' denote the image of L in F cyc i ® Zp A ant ; as inConjecture 9, and S P (E/Kîtors the A ant i-torsion submodule of S P (E/ K^; ti j)-Conjecture 14 (Cyclotomic and anticyclotomic main conjectures).(i) F cyc i generates the ideal char A

192 B. Mazur K. Rubin6. Orthogonal A-modulesIn this final section we introduce a purely algebraic template which, when it"fits", gives rise to many of the properties conjectured in the previous sections.Keep the notation of the previous sections. In particular r : A —t A is theinvolution of A induced by complex conjugation on K, and if V is a A-module,then V^ denotes V with A-module structure obtained by composition with r. LetV* = HOIIIA(V, A). If V is a free A-module of rank r, then detA(V' T ) will denotethe r-th exterior power of V and a r-gauge on V is a A-isomorphism between thefree A-modules of rank onet v : det A (F*) ^det A (V'( T) )or equivalently an isomorphism detA(V) ® detA(V r(T^) -^ A.By an orthogonal A-module we mean a free A-module V with semi-linearr-structure endowed with a r-gauge tv and a A-bilinear r-Hermitian pairing (Definition2)IT : V ®A V (T) —y A.Viewing n as a A-linear map V^ —^ V*, the compositiont v o det A (7r) : det A (V'( T) ) —y det A (F*) —y det A (V'( T) )must be multiplication by an element disc(V) £ A that we call the discriminantof the orthogonal A-module V. We further assume that disc(V) ^ 0, and we defineM = M(V, ix) to be the cokernel of the (injective) map ix : V^ —^ V*, so we have0 —• V {T) —• V* —• M —• 0. (6.1)If K C F c KQO, recall that IF = ker{A -» Ap} and defineV(F) := {x £ V : TT(X,V {T) ) C I F }/IFF = ker{F ® A A F ^ (V {T) )* ® A A F }and similarly V^ (F) := ker{V r M ®\ F _+ V*®A F }. Any lift f of r to Ga^K^/Q)induces an isomorphism V(F) —^ V^(F). From (6.1) we obtain0 —• V {T) (F) —• V {T) ® A A F —• V* ®A Ap —• M ®A Ap —• 0 (6.2)and (applying Hom( • , Ap) and using the Hermitian property of n)We have an induced pairingV(F) ~ Hom AF (M ®A Ap, A F ). (6.3)np : V^(F) ® AF V(F) —• I F /F F ,which we call the F-derived pairing. If F is stable under complex conjugation then\/( T i(F) is canonically isomorphic to V(F)^T' > and -Kp is r-Hermitian.

Elliptic Curves and Class Field Theory 193Now suppose F = K%g u . By (6.3), V(Ä'^tl ) is free over A an ti- Applying thedeterminant functor to (6.2), the r-gauge tv induces an isomorphismdet A „ ti V(K^tì )^ = det A „ ti V^(K^tì ) ^ Hom(det A „ ti (M ®A A anti ), A anti ).If V(K%£ U ) has rank one over A ant i, then V(Ä'^tl ) contains a unique maximalr-stable submodule H such that the mapV{Kanti )( r) ^ Hom(det A „ ti (M ® A anti ), Aanti)D Hom(M ® Aanti, Aanti) ~ F(FT^nti )sends H^ into F. (Namely, F = JV(ÜT^ltl ) where J is the largest ideal of A an tisuch that J T = J and J 2 C charA„ ti (M ® A an ti)tors-)Recall that Sel p (E/ F ) denotes the p-power Selmer group of F over F andS P (E /F ) = Hom(Selp(F /F ), Qp/Z p ) ® Q p .Proposition 16. With notation as above, suppose that V is an orthogonal A-module and tpv '• M ^y S p (E/ Kao ) is an isomorphism. Then for every extensionF of K in KQO, ifiv induces an isomorphismV(F) ^ U(F)where U(F) is the universal norm module defined in §3.proof. This follows directly from Theorem 12 and (6.3).Definition 17. We say that the orthogonal A-module V organizes the anticyclotomicarithmetic of (E,K,p) if the following three properties hold.(a) disc(V) = L, the two-variable p-adic L-function of E.(b) There is an isomorphism tpv '• M ^y S p (E/jâo ).(c) The isomorphism V(K%£ U ) = U of Proposition 16 identifies H c V(K%£ U )with the Heegner submodule % C U, and identifies the Kû-derived pairingwith the canonical p-adic height pairing into I K »•>« /I 2 K^Bti — F cyc i ® A an ti-Question. Given F, K, and p satisfying our running hypotheses, is there an orthogonalA-module V that organizes the anticyclotomic arithmetic of (E,K,p)lIf one is not quite so (resp., much more) optimistic one could formulate ananalogous question with the ring A replaced by the localization of A at I (resp.,with A replaced by Z p [[r]]).Question. If V is an orthogonal A-module V which organizes the anticyclotomicarithmetic of (E,K,p), then for every finite extension F of FT in Koo, we have anisomorphism E(F) ® Q p = U(F) = V(F) as in Proposition 16, a p-adic heightpairing on E(F) ®Q P , and the F-derived pairing on V(F). How are these pairingsrelated?When F = Kûcondition (c) says that the two pairings are the same, but itseems that in general they cannot be the same for finite extensions F/K.

194 B. Mazur K. RubinTheorem 18. Suppose that there is an orthogonal A-module V that organizes theanticyclotomic arithmetic of (E,K,p). Then Conjectures 13 (the 2-variable mainconjecture), and H(i) (the cyclotomic main conjecture) hold.If further the induced pairing V^F^J* 1 ) ® Vfó^* 1 )^ —^ F cyc i ® A an ti is surjective,then Conjectures 6 (the Height Conjecture), 8, 9 (the A-adic Gross-Zagierconjecture), and 14(H) (the anticyclotomic main conjecture) also hold.Brief outline of the proof of Theorem 18. Since disc(F) is a generator of char A (M),the two-variable main conjecture follows immediately from (a) and (b) of Definition17. The cyclotomic main conjecture follows from the two-variable main conjecture.Now suppose that the induced pairing V(K^tl ) ® V(K^tl )^ —^ F cyc i ® A an tiis surjective. By (c) of Definition 17 this is equivalent to the Height Conjecture,which in turn is equivalent to Conjecture 8.Howard proved in [5] that S P (E/ K^ti ) is pseudo-isomorphic to A an ti © B 2where F is a r-stable torsion A an tr m odule. By Theorem 12(1) the same is trueof M ® Aanti) and so the remark at the end of the definition of F shows thatF = chai(B)\ / (K^tl ). Using (6.2), (6.3), and our assumption that the inducedpairing is surjective, one can show that the image of L in I K »•>« /I 2 K^Bti generateschar(F) 2 Ii f!1 nti/I 2 ?[inti. The A-adic Gross-Zagier conjecture and the anticyclotomicmain conjecture follow from these facts and (c).DReferences[1] C. Cornut, Mazur's conjecture on higher Heegner points, Invent, math. 148(2002), 495-523.[2] R. Greenberg, Galois theory for the Selmer group of an abelian varietypreprint).[3] S. Haran, p-adic F-functions for elliptic curves over CM fields, thesis, MIT1983.[4] H. Hida, A p-adic measure attached to the zeta functions associated with twoelliptic modular forms. I, Invent. Math. 79 (1985), 159-195.[5] B. Howard, The Heegner point Kolyvagin system, thesis, Stanford University2002.[6] K. Kato, p-adic Hodge theory and values of zeta functions of modular forms(preprint).[7] B. Mazur, Rational points of abelian varieties with values in towers of numberfields, Invent. Math. 18 (1972), 183-266.[8] , Modular curves and arithmetic. In: Proceedings of the InternationalCongress of Mathematicians (Warsaw, 1983), PWN, Warsaw (1984), 185—211.[9] B. Mazur, P. Swinnerton-Dyer, Arithmetic of Weil curves, Invent. Math. 25(1974), 1-61.[10] B. Mazur, J. Tate, Canonical height pairings via biextensions. In: Arithmeticand Geometry, Progr. Math. 35, Birkhaiiser, Boston (1983), 195-237.[11] J. Nekovâr, On the parity of ranks of Selmer groups. II, G R. Acad. Sci. ParisSér. I Math. 332 (2001), 99-104.

Elliptic Curves and Class Field Theory 195[12] B. Perrin-Riou, Fonctions F p-adiques, théorie d'Iwasawa et points de Heegner,Bull. Soc. Math. France 115 (1987), 399-456.[13] , Points de Heegner et dérivées de fonctions F p-adiques, Invent. Math.89 (1987), 455-510.[14] , Fonctions F p-adiques associées à une forme modulaire et à un corpsquadratique imaginaire, J. London Math. Soc. 38 (1988), 1-32.[15] V. Vatsal, Special values of anticyclotomic F-functions (preprint).

ICM 2002 • Vol. II • 197-206Théorie Ergodique et GéométrieArithmétiqueEmmanuel Ullmo*AbstractWe will present several examples in which ideas from ergodic theory canbe useful to study some problems in arithmetic and algebraic geometry.2000 Mathematics Subject Classification: 11F32, 11G10, 11G15, 11G40,22D40, 22E40.Keywords and Phrases: Equidistribution, Variétés abeliennes, Variétés deShimura.1. IntroductionLe but de ce rapport est d'expliquer différentes techniques permettant de montrerl'équidistribution de certains ensembles de points de nature arithmétique surdes variétés algébriques définies sur des corps de nombres et de donner des applicationsarithmétiques et géométriques de ces résultats.Si X est une variété algébrique sur C et F une ensemble fini de X(C) on note\E\ son cardinal et Ap la mesure de Dirac normalisée1' x£ESi E n est une suite d'ensembles finis de X(C) et ß une mesure de probabilité surX(C), on dit que les E n sont équidistribués pour ß si pour toute fonction continuebornée / sur X(C) on aA E M) = T^n E /(*)—• / f"-l^nl xeE p„ n JX(C)"Universit Paris-Sud Orsay Bât 425, 91405 Orsay Cedex France. E-mail: ullmo@math.u-psud.fr

198 E. UllmoSoit X une variété algébrique, une suite de points x n de X est dite "générique"si pour toute sous-variété Y de X, Y ^ X, {n £ N ,x n £ Y} est un ensemble fini.(Il revient au même de dire que x n converge vers le point générique pour la topologiede Zariski).André et Oort ont formulé un analogue de la conjecture de Manin-Mumforddémontrée par Raynaud [18] [19] dans le cadre des variétés de Shimura. Dans cesdeux conjectures, on dispose de points spéciaux et de variétés spéciales. Pour laconjecture de Manin-Mumford l'espace ambiant est une variété abélienne, les pointsspéciaux sont les points de torsion et les variétés spéciales sont les "sous-variétésde torsion" (translatés, par un point de torsion, d'une sous-variété abélienne).Pour la conjecture d'André-Oort l'espace ambiant est une variété de Shimura,les points spéciaux sont les points à multiplication complexe (ou points CM) etles sous-variétés spéciales sont les "sous-variétés de type de Hodge" (des composantesirréductibles de translatés par un opérateur de Hecke de sous-variétés deShimura). Nous préciserons ces définitions plus bas. Dans les deux cas ces conjecturess'énoncent sous la forme: une composante irréductible de l'adhérence deZariski d'un ensemble de points spéciaux est une sous-variété spéciale.Dans ce cadre une suite de points x n de X (X variété abélienne ou X variétéde Shimura) est dite "stricte" si pour toute sous-variété spéciale Y de X, Y ^X, {n £ N ,x n £ Y} est un ensemble fini. On remarque qu'avec ces définitionsles conjectures d'André-Oort et de Manin-Mumford se retraduisent de la manièresuivante: Toute suite stricte de points spéciaux est générique.Une conséquence géométrique (conjecturale pour les variétés de Shimura) quel'on obtient en considérant l'adhérence de Zariski de l'ensemble des points spéciauxd'une sous-variété M de X est l'existence d'un ensemble fini {Si,... ,S r } de sousvariétésspéciales avec S, C M telle que toute sous-variété spéciale S C M estcontenue dans l'un des Si.Dans la première partie nous décrivons des résultats d'équidistribution pourdes suites de points de petite hauteur sur des variétés algébriques utilisant lagéométrie d'Arakelov. Le résultat le plus marquant est la résolution de la conjecturede Bogomolov (qui généralise la conjecture de Manin-Mumford et en donneune nouvelle démonstration) pour les variétés abéliennes due à Zhang [24] et àl'auteur du rapport [22].Dans la deuxième partie nous expliquons des résutats d'équidistribution depoints de Hecke sur des variétés de la forme X = F\G(R) pour un groupe algébriquesemi-simple et simplement connexe G et un réseau F. Les méthodes combinentthéorie spectrale et théorie des représentations.Dans la troisième partie nous présentons des énoncés largement conjecturauxpour l'équidistribution des points à multiplication complexe des variétés de Shimura.La théorie analytique des nombres via les familles de fonctions F et la théorie desformes automorphes y jouent un rôle central.Dans une dernière partie nous expliquons comment la théorie de Ratner etMargulis permet de démontrer des résultats d'équidistribution pour des suites desous-variétés "fortement spéciales" (appartenant à une classe assez large de sousvariétésspéciales de dimension positive) des variétés de Shimura. Nous expli-

Théorie Ergodique et Géométrie Arithmétique. 199querons la relation avec la conséquence géométrique de la conjecture d'André-Oortprécédemment décrite.2. Equidistribution des points de petite hauteurExemple 2.1 On prend X = G TO , E n l'ensemble des racines n-ième de l'unité,E n est équidistribué pour la mesure uniforme sur le cercle unité |^. En utilisantl'irréductibilité du polynôme cyclotomique on voit que l'orbite sous Galois d'uneracine n-ième primitive de l'unité est aussi équidistribuée pour |^.Exemple 2.2 On prend X = F une courbe elliptique sur C et E n l'ensemble despoints de n torsion, alors E n est équidistribué pour la mesure de Harr normaliséesur F(C). Si F est défini sur un coprs de nombres K et E n'a pas de multiplicationcomplexe, par le théorème de l'image ouverte de Serre, pour tout nombre premier passez grand le groupe de Galois agit transitivement sur les points d'ordre p. On endéduit encore que les orbites sous Galois des points d'ordre p sont équidistribuéespour la mesure de Haar normalisée.La théorie d'Arakelov a permis de comprendre ces énoncés d'une manière bienplus générale. On montre [21] pour une variété arithmétique un théorème générald'équidistribution des orbites sous Galois de suite génériques de points dont lahauteur (à la Arakelov) tend vers 0. Les exemples précédents correspondent à dessuites de points de hauteurs nulles. Pour les variétés abéliennes on obtient avecSzpiro et Zhang le résultat suivant (qui donne des informations nouvelles mêmepour les points de torsion des courbes elliptiques à multiplication complexe):Théorème 2.3 [21] Soit A une variété abélienne sur un corps de nombres K. Onnote fiMP la hauteur de Néron-Tate sur les points algébriques de A (associée àun fibre inversible ample symétrique sur X). Soit x n une suite générique de pointsalgébriques de A telle que hNp(x n ) tend vers 0. Pour toute place à l'infini a l'orbitesous Galois de x n est équidistribuée pour la mesure de Haar normalisée dß a deA AC).L'analogue de cet énoncé pour G^ a été montré par Bilu [2] sans théoried'Arakelov. Une extension pour certaines variétés semi-abéliennes de ces résultatsa été obtenue par Chambert-Loir [6] par des méthodes Arakeloviennes. On peutaussi comprendre grâce aux travaux de Autissier [1] l'exemple 2.1 comme un casparticulier de théorème d'équidistribution vers la mesure d'équilibre d'un compactde capacité 1 de l'orbite sous Galois d'une suite de points entiers algébriques.On trouvera dans [25] comment on obtient la conjecture de Bogomolov enproduisant une contradiction sur les mesures limites de suites de mesures associéesà des orbites sous Galois de points de petite hauteur. Retenons l'énoncé suivant dûà l'auteur [22] pour les courbes de genre g > 2 dans leur jacobienne et étendu endimension arbitraire par Zhang [24]:

200 E. UllmoThéorème 2.4 Soit X une sous-variété d'une variété abélienne A définie sur uncorps de nombres K. Grâce à la conjecture de Manin-Mumford démontrée parRaynaud [19], on sait qu'il existe des sous-variétés de torsion (éventuellement dedimension 0) {Ti,...,T r }, Ti c X tels que si T c X est une sous-variété detorsion alors T c T t pour un certain i. Il existe alors c > 0 tel que si F est unpoint algébrique de X et F ^ U^=1 Fj alors îIMP(P) > c.3. Equidistribution des points de HeckeSoient G un groupe algébrique linéaire presque simple et simplement connexesur Q, F C G(Q) un réseau de congruence et X = F\G(R). Soit ßo la mesureinvariante normalisée sur X. Pour tout a £ G(Q) on a une décompositionFaF =öâ) Yaiavec deg(a) = |F\FaF| £ N. Pour tout a; G X, on note T a .x l'ensemble des a,a;compté avec multiplicité. L'opérateur de Hecke T a ainsi défini est une correspondancede degré deg (a) sur X; il induit une opération sur les espaces de fonctionsL 2 (X,ß 0 ) (fonctions de carrés intégrables sur X) et G°(X) (fonctions continuesbornées sur X) pardeg(a)Avec Clozel et Oh nous obtenons [3]:Théorème 3.1 On suppose que le Q-rang de G est différent de 0. Soit a n £ G(Q)une suite telle que deg(a n ) —¥ oo. Pour tout x £ X les T an .x sont équidistribuéspour ß 0 - De plus pour tout f £ L 2 (X,ß Q ) on a la convergence L 2\T a J- [ /-A*olU=« —• 0.JxOn a en fait des résultats aussi dans le cas ou le Q-rang de G vaut 0. Laméthode de démonstration fournit des estimations très précises pour la vitesse deconvergence dans le théorème F 2 . Si on dispose de plus de régularité sur / (parexemple / C°° à support compact), cete vitesse est obtenue aussi pour la convergencesimple (ou uniforme sur les compacts). Pour G = SL n (n > 3) ou G = Sp2 n(n > 2) ces estimations sont essentiellement optimales.On montre par des méthodes classiques que l'énoncé de convergence simpledu théorème se déduit de l'énoncé F 2 . Pour montrer le théorème F 2 on écrit ladécomposition spectrale de L 2 (X,ß Q ) sous la forme adélique. Une fonction intervenantdans la décomposition spectrale est alors propre pour les opérateurs deHecke et les valeurs propres s'interprètent comme des coefficients matriciaux dereprésentations locales associées à . Pour montrer le théorème sous la forme F 2 ,on doit montrer que T an —¥ 0 quand n-*ooau sens F 2 . On se ramène ainsi àcontrôler la décroissance de ces coefficients matriciaux. En Q-rang r > 2 on dispose

Théorie Ergodique et Géométrie Arithmétique. 201d'assez d'informations sur le dual unitaire pour conclure grâce aux travaux de Oh([17], théorème 5.7). En Q-rang 1 on utilise un principe de restriction à la Burger-Sarnak en une place finie démontré dans [4] et une approximation de la conjecturede Ramanujan pour SL 2 .4. Equidistribution des points CM des variétés deShimuraNous devons préciser un peu les définitions relatives aux variétés de Shimuraafin d'expliquer ce que l'on entend par l'équidistribution des points CM.Soit (G, X) une donnée de Shimura; G est un groupe algébrique réductif surQ et X est une G(R) classe de conjuguaison de morphismesh : S —y GR(S = Res C/RG TO est le tore de Deligne) vérifiant les 3 propriétés de Deligne [10] [11].Les composantes irréductibles de X sont alors des domaines symétriques hermitiens.Soient Â/ l'anneau des adèles finies de Q et F' un sous-groupe compact ouvertde G (A), on définit sur le corps C la variété de ShimuraSh K (G,X) = G(Q)\X x G(A f )/K.On vérifie que SîIK(G,X) est une réunion finie de quotients de composantesirréductibles de X par des sous-groupes de congruences de G(Q). Par ailleursSîIK(G, X) a un "modèle canonique" sur un corps de nombres E(G, X) ne dépendantque de la donnée de Shimura (G,X).Soit (Gi,Xi) une sous-donnée de Shimura de (G,X), on dispose alors d'uneapplication canonique/: ShKnGi(Kf) —y Sh,K(G,X).Une sous-variété de type de Hodge est une composante irréductible d'un translatéde l'image d'un tel morphisme par une correspondance de Hecke. (Moonen [15]caractérise ces sous-variétés en termes de variations de structures de Hodge, d'où lenom.)Pour /i:§-> GR, h £ X, on définit le groupe de Mumford-Tate MT(h) de hcomme le plus petit Q-sous-groupe H de G tel que h se factorise par FR. Si MT(h)est un tore, on dit que h est spécial. Les points spéciaux de SîIK(G,X) sont lespoints de la forme [h,gK] avec g £ G(A/) et h spécial.Fixons ho £ X un élément spécial et T 0 = MT(h Q ). L'ensembleS(ho) = {[ho,gK], g£G(A f j\est appelé ensemble des points spéciaux de "type ho" de X. On a une action deT 0 (Af) sur S(ho) donnée par t.[ho,gK] = [h Q ,tgK]. Pour tout g £ G(A/), l'orbitesous T 0 (Af) de [ho,gK] est finie, on appelle "orbite torique" de [ho,gK] cette orbite.La première question naturelle est

202 E. UllmoQuestion 4.1 Soit x n = [h n ,g n K] une suite générique de points spéciaux de S =SìIK(G,X). Est-il vrai que l'orbite torique de x n est équidistribuée pour la mesureinvariante normalisée de SîIK(G,X).Notons qu'il n'est déjà pas à priori évident de prévoir la proportion des pointsde l'orbite torique dans les composantes de S. Il est peut-être plus réaliste detravailler dans chaque composante connexe de S (comme dans la dernière partie dece texte). Nous tairons dans la suite ces problèmes de non connexité.Les premiers résultats pour ces questions sont dus à Duke [12] pour la courbemodulaire Y (Y) = SL(2,Z)\M. Il montre l'équidistribution des points à multiplicationcomplexe par l'anneau des entiers OK quand le discriminant tend vers l'infini.Nous expliquons dans [4], en utilisant en plus des résultats sur l'équidistributiondes points de Hecke comment obtenir l'équidistribution des points à multiplicationcomplexe par un ordre arbitraire de OK quand le discriminant tend versl'infini. Nous pensons plus généralement que la question 4.1 est liée aux problèmesd'équidistribution des points de Hecke décrits précédemment.Des résultats pour l'équidistribution des orbites toriques de points CM sontannoncés par S. Zhang [26] pour les courbes de Shimura et plus généralement desvariétés de Shimura de type quaternionique via un avatar de la formule de Gross-Zagier. Pour les variétés modulaires de Hilbert des résultats de ce type sont annoncésindépendamment par P. Cohen [7] (par la méthode originale de Duke) etpar S. Zhang.Les méthodes pour prouver ces énoncés comportent trois étapes que l'on vadécrire de manière imprécise pour la concision de ce rapport. Soit S une variétéde Shimura, soit / une fonction non constante intervenant dans la décompositionspectrale de S, soit x n £ S une suite de points CM et E n son orbite torique. Ondoit montrer quelim ïh E f(y) = / f d *>- (!)La fonction / est alors une forme automorphe. La première étape est de montrer une"formule de classe" reliant -rg-r Y^y€En f(y) à la valeur de la fonction F de /, torduepar une forme automorphe que l'on définit à partir de E n , au point critique. Ce typede formule est obtenu par Waldspurger [23] pour des algèbres de quaternions surun corps de nombres F et revisité par Zhang [26]dans le but d'obtenir les résultatsd'équidistribution.Une fois la formule de classe établie, on dispose d'une famille de fonctionsF indexée par les entiers. On définit à partir de l'équation fonctionnelle de cesfonctions une notion de "conducteur analytique" q n . L'hypothèse de Riemann (oude Lindelôf) prévoit une borne en 0(q ( n) pour la valeur critique de la fonction Fconsidérée. Dans tous les exemples considérés, il est remarquable que pour montrerl'équidistribution il faut améliorer la borne triviale (donnée par le principe deconvexité de Phragmen-Lindelôf). Ce genre de questions a reçu une attention considérableen théorie analytique des nombres et a été résolue dans de nombreux cas.On pourra consulter la série de papiers [13] et [14] pour une présentation des principauxrésultats et applications de ce cercle d'idées. Notons que la démonstration

Théorie Ergodique et Géométrie Arithmétique. 203de l'équidistribution des orbites toriques de points CM sur les variétés modulairesde Hilbert utilise les résultats spectaculaires récents [8].Pour les applications éventuelles à des énoncés arithmétiques, il paraît importantde remplacer les orbites toriques par les orbites sous Galois. De manièregénérale si [h,gK] est un point CM d'une variété de Shimura, T = MT(h) est letore associé et F = E(T, h) est le corps réflexe de la variété de Shimura associé àla donnée de Shimura (T,h), l'action de Galois (cf [10], [11] )se factorise à traversl'action de F(A/) via un morphisme de réciprocité (et la théorie du corps de classe).r : Resp/qG m , : E —y Tqui induit un morphisme non surjectif en généralr : Res iï/Q G roi i ï (A / ) —y T(A f ).On s'attend néanmoins à une réponse positive à la question suivante:Question 4.2 Soit x n une suite générique de points CM sur une variété de ShimuraS, est-il vrai que les orbites sous Galois 0(x n ) sont équidistribuées dans S pour lamesure invariante?De manière encore plus optimiste, on espère (par analogie avec le cas desvariétés abéliennes) que le même résultat est encore vrai pour des suites strictesde points CM. Ce serait une conséquence de la conjecture d'André-Oort et de laquestion précédente. Notons que nous espérons que des résultats d'équidistributionpour les points CM soient en fait une étape pour montrer la conjecture en question. (C'est au moins ce qui se passe dans le cas des variétés abéliennes).5. Equidistribution de sous-variétés spécialesCette partie décrit un travail [5] en cours de préparation en commun avecL. Clozel. Soit S une composante irréductible d'une variété de Shimura. Uneconséquence géométrique frappante de la conjecture d'André et Oort est la suivante:Soit Y une sous-variété de S, il existe un ensemble fini {Si,..., S r } de sous-variétésspéciales avec S, C Y pour tout i tel que toute variété spéciale Z de S contenuedans Y est en fait contenue dans un des Si.Supposons que S est une composante irréductible de SHK(G,X) pour ungroupe G que l'on suppose adjoint (pour simplifier). On a vu qu'une sous-variétéspéciale M est associée à une sous-donnée de Shimura (G\,X\). Si G\ est semisimpleet Xi contient un point spécial xi tel que le tore associé T = MT(x\) C G\est tel que FR est un tore maximal compact de G, on dit que M est fortementspéciale. Par exemple les variétés modulaires de Hilbert (associées à des corps totalementréels de degré n sur Q) sont fortement spéciales dans l'espace de module A ndes variétés abéliennes principalement polarisées de dimension n. On peut montrer:Théorème 5.1 Soit Y une sous-variété d'une variété de Shimura S. Il existe unensemble fini {Si,... ,Sk} de sous-variétés fortement spéciales de dimension positiveSi C Y tel que si Z est une sous-variété fortement spéciale de dimensionpositive avec Z c Y alors Z c Si pour un certain i £ {!,... ,k}.

204 E. UllmoNotons que cet énoncé ne dit rien sur les sous-variétés spéciales de dimension0 (les points spéciaux), notons cependant le corollaire suivant:Corollaire 5.2 Soit Y une sous-variété stricte de A n , il existe au plus un nombrefini de sous-variétés modulaires de Hilbert contenu dans Y.Le théorème 5.1 se déduit d'un énoncé ergodique. Toute sous-variété spécialeZ de S est muni d'une manière canonique d'une mesure de probabilité ßz-Théorème 5.3 Soit S n une suite de sous-variétés fortement spéciales Soit ß n lamesure de probabilité associée à S n . Il existe une sous-variété fortement spécialeZ et une sous-suite ß nk qui converge faiblement vers ßz- De plus Z contient S nkpour tout k assez grand.On obtient la preuve du théorème 5.1 en considérant une suite de sous-variétésfortement spéciales maximales S n parmi les sous-variétés fortement spéciales contenuesdans Y. En passant à une sous-suite on peut supposer que ß n convergefaiblement vers ßz- Comme le support de ßz est contenu dans Y, on en déduit queZ c Y. Par la maximalité des S n et le fait que S n C Z pour tout n assez grand,on en déduit que la suite S n est stationaire.On peut aussi réécrire cet énoncé avec la terminologie de [21]. On dit qu'unesuite S n de sous-variétés fortement spéciales est stricte si pour toute sous-variétéfortement spéciale M de S,{n G N, S n C M}est fini. On peut d'ailleurs prendre dans cette définition M spéciale car une sousvariétéspéciale contenant une sous-variété fortement spéciale est automatiquementfortement spéciale. Dans ce language le théorème 5.3 admet comme corollaireimmédiat:Corollaire 5.4 Soit S n une suite stricte de sous-variétés fortement spéciales de S.Soit ß n et ß les mesures de probabilités associées sur S n et S. La suite ß n convergefaiblement vers ß.On peut appliquer cet énoncé à des suites de sous-variétés fortement spécialesmaximales. La condition d'être stricte signifie alors de ne pas avoir de sous-suitesconstantes. C'est par exemple le cas pour les variétés modulaires de Hilbert dansle modules des variétés abéliennes principalement polarisées A n .La preuve du théorème 5.3 repose sur des résulats de Mozes et Shah [16]qui précisent la conjecture de Raghunathan démontrée par Ratner [20]. Si S =F\G(R)/F' 0O pour un sous-groupe compact maximal K œ et un réseau de congruenceF, on note F+ = G(R) + fiF et S = F+\G(R) + . Si F est un sous-groupe semi-simplede G(R) + tel que F+ n F est un réseau de F alors MH = F+ n H\H est fermé dansS et est muni canoniquement d'une mesure de probabilité F-invariante ßn-Si MH„ est une suite de telles sous-variétés de S, le théorème de Mozes Shah[16] permet sous certaines hypothèses; au besoin en passant à une sous-suite; demontrer la convergence faible de ßn n vers une mesure ßn canoniquement associée

Théorie Ergodique et Géométrie Arithmétique. 205à un MH = F+ n H\H. En général les sous-groupes H n n'induisent pas de sousvariétésspéciales sur S car H n n'est pas toujours réductif et même si H n est réductifl'espace symétrique associé à H n n'a aucune raison d'être hermitien. Un des pointsclefs de la démonstration est de vérifier que si les H n induisent des sous-variétésfortement spéciales il en est de même pour F. Pour passer de résultats sur S à desrésultats sur S on utilise aussi des résultats de Dani et Margulis ([9]thm. 2) quidonnent des critères de retour vers des compacts pour des flots unipotents sur S.References[s;[7;[8[9[io;[12:[is;[w:[15[16P. Autissier Points entiers et Théorèmes de Bertini arithmétiques. J. ReineAngew. Math.531, (2001), 201^235.Y. Bilu Limit distribution of small points on algebraic tori. Duke Math. J. 89(1997), n.o 3, 465-176.L. Clozel, H. Oh, E. Ullmo. Hecke operators and equidistribution of Heckepoints. Invent. Math., 144, (2001), 327-351.L. Clozel, E. Ullmo. Equidistribution des points de Hecke, à paraître dans"Contributions to Automorphic Forms, Geometry and Arithmetic" volume enl'honneur de Shalika, Johns Hopkins University Press, éditeurs: Hida, Ramakrishnanet Shaidi.L. Clozel, E. Ullmo. Equidistribution de sous-variétés spéciales. En préparation.A. Chambert-Loir Points de petite hauteur sur les variétés semi-abéliennes.Ann. Ecole Norm. Sup. 33, (2000) no.6, 789-821.P. Cohen. Travail en prépararation.J. Cogdel, LI Piateskii-Shapiro, P. Sarnak. En préparartion.S.G Dani, G. A Margulis. Limit distribution of orbits of unipotent flows andvalues of quadratic forms. Adv. Sov. Math. 16, (1993), 9H37.P. Deligne. Travaux de Shimura. Séminaire Bourbaki, Exposé 389, Février 1971,Lecture Notes in Maths. 244, Springer-Verlag, Berlin 1971, 123^165.P. Deligne. Variétés de Shimura: interprétation modulaire et techniques de constructionde modèles canoniques, dans Automorphic Forms, Representations,and L-functions part. 2; Editeurs: A. Borei et W Casselman; Proc. of Symp.in Pure Math. 33, American Mathematical Society, 1979, 247^290.W. Duke. Hyperbolic distribution problems and half-integral weight Maassforms. Invent, math. 92, (1988), 73^90.W. Duke, J. Friedlander, H. Iwaniec. Bounds for automorphic L-functions I,II, III. Invent. Math 112 (1993) No. 1, 1-8; Invent. Math, bf 115, No 2 (1994),219^239; Invent. Math. 143 (2001) No.2, 221-248.J. Friedlander. Bounds for L-functions. Proceedings of the InternationalCongress of Mathematicians, (Zürich 1994), Birkhäuser (1995), Basel, 363^373.B. Moonen. Linearity properties of Shimura varieties I. Journal of AlgebraicGeometry 7 (1998), 539^567.S. Mozes, N. Shah On the space of ergodic invariant measures of unipotentflows. Ergod. Th. and Dynam. Sys. 15, (1995), 149^159.

206 E. Ullmo[17] H. Oh. Uniform Pointwise bounds for matrix coefficients of Unitary representationsand applications to Kasdhan constants. To appear in Duke Math. Journal.[18] M. Raynaud. Courbe sur une variété abélienne et points de torsion. Invent.Math. 71, (1983), 207^223.[19] M. Raynaud. Sous-variété d'une variété abélienne et points de torsion. Arithmeticand Geometry, Paper dedicated to I. R. Shafarevich on the ocasion ofhis sixties birthday, vol 1, J. Coates, S. Helgason editors. (1983) Birkhäuser.[20] M. Ratner. On Raghunathan's measure conjecture, Ann. Math. 134, (1991),545^607.[21] L. Szpiro, E. Ullmo, S. Zhang Equirépartition des petits points. Invent. Math127, 337^347 (1997).[22] E. Ullmo. Positivité et dicrétion des points algébriques des courbes. Ann. ofMaths, 147 (1998), 167^179.[23] J.-L. Waldspurger. Sur les valeurs de certaines fonction F automorphes en leurcentre de symétrie. Compositio Math. 54 (1985), 173^242.[24] S. Zhang. Equidistribution of small points on abelian varieties. Ann. of Maths,147, (1998), 159^165.[25] S. Zhang. Small points and Arakelov theory. Proceedings of the InternationalCongress of Mathematicains, Vol II (Berlin 1998); Doc. Math. (1998) ExtraVol II, 217^225.[26] S. Zhang. Gross-Zagier formula for GL 2 . Asian J. Math. 5, (2001), 183^290.

ICM 2002 • Vol. II • 207^217Diophantine Methods forExponential Sums, and ExponentialSums for Diophantine ProblemsTrevor D. Wooley*AbstractRecent developments in the theory and application of the Hardy-Littlewood method are discussed, concentrating on aspects associated withdiagonal diophantine problems. Recent efficient differencing methods for estimatingmean values of exponential sums are described first, concentrating ondevelopments involving smooth Weyl sums. Next, arithmetic variants of classicalinequalities of Bessel and Cauchy-Schwarz are discussed. Finally, someemerging connections between the circle method and arithmetic geometry arementioned.2000 Mathematics Subject Classification: 11P55, 11L07, 11P05, 11D72,14G05.Keywords and Phrases: The Hardy-Littlewood method, Exponential sums,Waring's problem, Equations in many variables, Rational points, Representationproblems.1. IntroductionOver the past fifteen years or so, the Hardy-Littlewood method has experienceda renaissance that has left virtually no facet untouched in its application todiophantine problems. Our purpose in this paper is to sketch what might be termedthe past, present, and future of these developments, concentrating on aspects associatedwith diagonal diophantine problems, and stressing modern developmentsthat make increasing use of less traditional diophantine input within ambient analyticmethods. We avoid discussion of the Kloosterman method and its importantrecent variants (see [5] and [8]), because the underlying ideas seem inherently constrainedto quadratic, and occasionally cubic, diophantine problems. Our accountbegins with a brief introduction to the Hardy-Littlewood (circle) method, using*Department of Mathematics, University of Michigan, East Hall, 525 East University Avenue,Ann Arbor, MI 48109-1109, USA. E-mail: wooley@umich.edu

208 T. D. WooleyWaring's problem as the basic example. The discussion here illustrates well theissues involved in the analysis of systems of diagonal equations over arbitrary algebraicextensions of Q, and motivates that associated with more general systems ofhomogeneous equations (see [1] and [14]).Let s and k be natural numbers with s > k > 2, and consider an integer nsufficiently large in terms of s and k. The circle method employs Fourier analysisin order to obtain asymptotic information concerning the number, R(n) = R s^(n),of integral solutions of the equation x k + • • • + x k = n. Write F = n x l k and definethe exponential sum f(a) = f(a; P) byf(a) = Y, • 0. Finally, we use the convention that whenever e occurs in a formula, then it isasserted that the statement holds for each fixed positive number e.

Exponential Sums and Diophantine Problems 209An asymptotic formula for R(n), with leading term determined by the majorarc contribution (1.1), now follows provided that the corresponding contributionarising from the minor arcs m = [0,1) \ 9Jt is asymptotically smaller. Although suchis conjectured to hold as soon as s > max{4,fc + 1}, this is currently known onlyfor larger values of s. It is here that energy is focused in current research. Onetypically estimates the minor arc contribution via an inequality of the typef(a) s e(—na)da < ( sup \f(a)\ ) / |/(a)| 2 *do;. (1.2)\aGm / JoFor suitable choices of t and Q, one now seeks bounds of the shapesup \f(a)\ -C F!- T+e and f \f(a)\ 2t da -C p?t-k+s+^ âGmJOwith r > 0 and Ö small enough that (s — 2t)r > Ö. The right hand side of (1.2) isthen o(n s / k ~ r ), which is smaller than the main term of (1.1) whenever & s ,k(n) ^ 1-The latter is assured provided that non-singular p-adic solutions can be found foreach prime p, and in any case when s > 4k. Classically, one has two apparentlyincompatibleapproaches toward establishing the estimates (1.3). On one side isthe differencing approach introduced by Weyl [23], and pursued by Hua [9], thatyields an asymptotic formula for R(n) whenever s > 2 k + 1. The ideas introducedby Vinogradov [21], meanwhile, provide the desired asymptotic formula when s >Ck 2 log k, for a suitable positive constant C.2. Efficient differencing and smooth Weyl sumsSince the seminal work of Vaughan [15], progress on diagonal diophantineproblems has been based, almost exclusively, on the use of smooth numbers, bywhich we mean integers free of large prime factors. In brief, one seeks serviceablesubstitutes for the estimates (1.3) with the underlying summands restricted to besmooth, the hope being that this restriction might lead to sharper bounds. Beforedescribing the kind of conclusions now available, we must introduce some notation.Let A(P, R) denote the set of natural numbers not exceeding F, all of whose primedivisors are at most F, and define the associated exponential sum h(a) = h(a; P, R)byh(a;P,R)= J^ e(ax k ).xeA(P,R)When £ is a positive integer, we consider the mean value St(P, R) = J 0 \h(a)\ 2t da,which, by orthogonality, is equal to the number of solutions of the diophantineequation x\ + • • • + x k = y\ + • • • + y k , with x»,y, £ A(P, R) (1 < i < t). Wetake F x P v in the ensuing discussion, with n a small positive number 2 . In these2 We adopt the convention that whenever r} appears in a statement, implicitly or explicitly,then it is asserted that the statement holds whenever r\ > 0 is sufficiently small in terms of e.

210 T. D. Wooleycircumstances one has card(„4(F, Rj) ~ c(n)P, where the positive number c(n) isgiven by the Dickman function, and it follows that St(P,R) >• F* + p 2t — k . It isconjectured that in fact St(P, R) -C P e (P t + p 2 *-*). We refer to the exponent Xt asbeing permissible when, for each e > 0, there exists a positive number n = n(t, k,e)with the property that whenever F < P v , one has St(P,R) -C P Xt+e . One expectsthat the exponent Xt = max{£, 2t — k} should be permissible, and with this in mindwe say that 5t is an associated exponent when Xt = t + 5t is permissible, and thatA* is an admissible exponent when Xt = 2t — k + A t is permissible.The computations required to determine sharp permissible exponents for aspecific value of k are substantial (see [20]), but for larger k one may summarise somegeneral features of these exponents. First, for 0 < t < 2 and k > 2, it is essentiallyclassicalthat the exponent 5t = 0 is associated, and recent work of Heath-Brown[6] provides the same conclusion also when t = 3 and k > 238,607,918. Whent = o(Vk), one finds that associated exponents exhibit quasi-diagonal behaviour,and satisfy the property that 5t —t 0 as k —¥ oo. To be precise, Theorem 1.3 of [28]shows that whenever k > 3 and 2 < t < 2e~ 1 k 1 / 2 , then the exponentA4fcV2(A k \ (OUis associated. For larger t, methods based on repeated efficient differencing yield thesharpest estimates. Thus, the corollary to Theorem 2.1 of [26] establishes that fork > 4, an admissible exponent A t is given by the positive solution of the equationA t e At^k = fee 1-2 */*. The exponent Xt = 2t — k + fee 1-2 */* is therefore always permissible.Previous to repeated efficient differencing, analogues of these permissibleexponents had a term of size fee - */* in place of fee 1-2 */* (see [15]), so that in asense, the modern theory is twice as powerful as that available hitherto.The above discussion provides a useable analogue of the mean-value estimatein (1.3). We turn next to localised minor arc estimates. Take Q = P, and define mas in the introduction. Suppose that s, t and w are parameters with 2« > k + 1 forwhich A s , A* and A w are admissible exponents, and definemkÂ tÂ s A wa(k) 2(s(k + A W^ A t ) + tw(l + A s j) 'Then Corollary 1 to Theorem 4.2 of [27] shows that sup QGm \h(a)\ -C p 1^cr (* ! )+' ! ) andfor large k this estimate holds with a(k)^1= k(logk + O(loglogfc)). Applying ananalogue of (1.2) with h in place of/, and taking 3 t = [|fc(logfc + loglogfc+ 1)] ands = 2t+k+ [Ak log log k/ log k], for a suitable A > 0, we deduce from our discussionof permissible exponents that J m h(a) s e(—na)da = o(n s / k^r). By considering therepresentations of a given integer n with all of the fcth powers F-smooth, it is now3 We write [z] to denote max{n e Z : n < z}.

Exponential Sums and Diophantine Problems 211apparent that a modification of the argument sketched in the introduction showsthat R(n) >• 6 Sj fc(n)n s/ '* !_1 as soon as one confirms thath(a)'e(-na)da ~ c^) 8^^ ]i^S&s ,u(n)n s ' k - 1 . (2.2)Sharp asymptotic information concerning h(a) is available throughout HfJl(Q) onlywhen Q is a small power of log F, and so the proof of (2.2) involves pruning technology.Such machinery, in this case designed to estimate the contribution from a setof the shape HfJl(P) \ 9Jt((log P) s ), has evolved into a powerful tool. Such issues canbe handled these days with a number of variables barely exceeding max{4, k + 1}.This approach leads to the best known upper bounds on the function G(k) inWaring's problem, defined to be the least integer r for which all sufficiently largenatural numbers are the sum of at most r positive integral fcth powers.Theorem 2.1. One has G(k) < k(logk + log log k + 2 + O(log log A;/log A;)).This upper bound (Theorem 1.4 of [27]) refines an earlier one of asymptoticallysimilarstrength (Corollary 1.2.1 of [24]) that gave the first sizeable improvement ofVinogradov's celebrated bound G(k) < (2 + o(l))fclogfc, dating from 1959 (see [22]).Aside from Linnik's bound C7(3) < 7 (see [11]), all of the sharpest known bounds onG(k) for smaller k are established using variants of these methods. Thus one hasG # (4) < 12 (see [15], and here the # denotes that there are congruence conditionsmodulo 16), C7(5) < 17, C7(6) < 24, G(7) < 33, G(8) < 42, G(9) < 50, G(10) < 59,G(ll) < 67, G(12) < 76, G(13) < 84, G(U) < 92, G(15) < 100, G(16) < 109,G(17) < 117, G(18) < 125, G(19) < 134, G(20) < 142 (see [17], [18], [19], [20]).Unfortunately, shortage of space obstructs any but the crudest account ofthe ideas underlying the proof of the mean value estimates that supply the abovepermissible exponents. The use of exponential sums over smooth numbers occursalready in work of Linnik and Karatsuba (see [10]), but only with Vaughan's newiterative method [15] is a flexible homogeneous approach established. An alternativeformulation suitable for repeated efficient differencing is introduced by the authorin [24]. Suppose that the exponent A s is permissible, and consider a polynomialip £ Z[t] of degree d > 2. Given positive numbers M and T with M < T, and anelement x £ A(T, R) with x > M, there exists an integer m with m £ [M, MR]for which m\x. Consequently, by applying a fundamental lemma of combinatorialflavour, one may bound the number of integral solutions of the equation'ip(z) — '(p('W) = ^(xï - VÏi=l(2.3)with 1 < z,w

212 T. D. Wooleywith 1 < z,w < F, M < m < MR, (ip'(z)ip'(w), m) = 1 and m,Vi £ A(T/M,R)(1 < i < s). The implicit congruence condition ip(z) = ip(w) (mod m k ) maybeanalytically refined to the stronger one z = w (mod m k ), and in this wayoneis led to replace the expression ip(z) — ip(w) by the difference polynomialip\(z;h;m) = mr k (tp(z + hm k ) — fp(zj). Notice that when M > P x l k , one isforced to conclude that z = w, and then the number of solutions of (2.4) is boundedabove by PMRS S (T/M,R) of Schwarz's inequality to the associated mean value of exponential sums, onemay recover an equation of the shape (2.3) in which ip(z) is replaced by ip\(z), andT is replaced by T/M, and repeat the process once again. This gives a repeateddifferencing process that hybridises that of Weyl with the ideas of Vinogradov.It is now possible to describe a strategy for bounding a permissible exponentA s+ i in terms of a known permissible exponent A s . We initially take T = Pand ip(z) = z k , and observe that S s+ i(P,R) is bounded above by the number ofsolutions of (2.3). We apply the above efficient differencing process successivelywith appropriate choices for M at each stage, say M = P^1,with 0 <

Exponential Sums and Diophantine Problems 213integers not exceeding X that are represented as the sum of three positive integralcubes. One has N(X) > X 1 -^3-",where £ = (^2833 - 43)/41 = 0.24941301...arises from the permissible exponent A3 = 3 + £ for k = 3. Earlier, Vaughan [15]obtained an estimate of the latter type with 13/4 in place of 3 + £.3. Arithmetic variants of Bessel's inequalityAlready in our opening paragraph we alluded to some of the applicationsaccessible to the methods of §2. We now turn to less obvious applications thathave experienced recent progress. We illustrate ideas once again with a simpleexample, and consider the set Z(N) of integers n, with N/2 < n < N, that are notrepresented as the sum of s positive integral fcth powers. The standard approach toestimating Z(N) = ca,rd(Z(Nj) is via Bessel's inequality. We now take F = N 1^.When 03 Ç [0,1), write F*(n;Q3) = J m h(a) s e(—na)da, and write also R*(n) =R*(n; [0,1)). The theory of §2 ensures that when Q is a sufficiently small powerof logF, and s > 4k, then F*(n;9Jt) x n s l k^x. Under such circumstances, anapplication of Bessel's inequality reveals that Z(N) is bounded above byEN/2

214 T. D. Wooleyand also byl f \K(a)\ A da) ( f \h(a)\ As f 3 da) . (3.4)In either case, the diophantine equations underlying the integrals on the left handsides of (3.3) and (3.4) contain arithmetic information that can be effectively exploitedwhenever the set Z(N) is reasonably thin.The strategy sketched above has been exploited by Brüdern, Kawada and Wooleyin a series of papers devoted to additive representation of polynomial sequences.Typical of the kind of results now available is the conclusion [3] that almost all valuesof a given integral cubic polynomial are the sum of six positive integral cubes.Also, Wooley [30], [31], has derived improved (slim) exceptional set estimates inWaring's problem when excess variables are available. For example, write E(N) forthe number of integers n, with 1 < n < N, for which the anticipated asymptoticformula fails to hold for the number of representations of an integer as the sum of asquare and five cubes of natural numbers. Then in [31] it is shown that E(N) -C N e .As a final illustration of such ideas, we highlight an application to the solubilityof pairs of diagonal cubic equations. Fix k = 3, define h(a) as in §2, and putc(n) = J Q \h(a)\ 5 e(—na)da for each n £ N. Brüdern and Wooley [4] have appliedthe ideas sketched above to estimate the frequency with which large values of \c(n)\occur, and thereby have shown that, with £ defined as in the previous section,Ec(x z - y z )\ 2 = \h(afh(ßfh(a + ß) 2 \dadß -C F 6+ç+e .J0 J0x,yeA(P,R)On noting that 6 + £ < 6.25, cognoscenti will recognise that this twelfth momentof smooth Weyl sums, in combination with a classical exponential sum equippedwith Weyl's inequality, permits the discussion of pairs of diagonal cubic equationsin 13 variables via the circle method. The exponent 6 + £ improves an exponent6 + 2£ previously available for a (different) twelfth moment. Brüdern and Wooley[4] establish the following conclusion.Theorem 3.1. Suppose that s > 13, and that a,, 6, (1 of equationsaixf + h a s x 3 s = bixf + h b s x 3 s = 0.The condition s > 13 improves on the previous bound s > 14 due to Brüdern[2], and achieves the theoretical limit of the circle method for this problem.4. Arithmetic geometry via descentLet F(x) £ Z[xi,...,x s ] be a homogeneous polynomial of degree d, andconsider the number, N(B), of integral zeros of the equation F(x) = 0, with

Exponential Sums and Diophantine Problems 215x e [—B,B] S . When s is sufficiently large in terms of d, the circle method shows undermodest geometric conditions that N(B) is asymptotic to the expected productof local densities. For fairly general polynomials, the condition on s is as severe ass > (d— l)2 rf , though for diagonal equations the methods of §2 relax this conditionto s > (1 + o(l))dlogd. However, there is a class of varieties with small dimensionrelative to degree, for which the circle method supplies non-trivial informationconcerning the density of rational points. The idea is to apply a descent processin order to interpret points on the original variety in terms of corresponding pointson a new variety, with higher dimension relative to degree, more amenable to thecircle method.To illustrate this principle, consider a field extension K of Q of degree n withassociated norm form N(x) £ Q[#i,... ,x„]. Also, let I and k be natural numberswith (k,l) = 1, and let a be a non-zero rational number. Then Heath-Brown andSkorobogatov [7] descend from the variety t l (l — t) k = aN(x) to the associatedvariety aN(u) + bN(v) = z n , for suitable integers a and 6. The circle methodestablishes weak approximation for the latter variety, and thereby it is shown thatthe Brauer-Manin obstruction is the only possible obstruction to the Hasse principleand weak approximation on any smooth projective model of the former variety.One can artificially construct further examples amenable to the circle method. Forexample, if we take linearly independent linear forms Fj(x) £ Q[xi,... ,x„] (1

216 T. D. Wooleytions involving norms, Imperial College preprint (June 2001).[8] C. Hooley, On nonary cubic forms, J. Reine Angew. Math. 386 (1988), 32-98.[9] L.-K. Hua, On Waring's problem, Quart. J. Math. Oxford 9 (1938), 199-202.[10] A. A. Karatsuba, Some arithmetical problems with numbers having smallprime divisors, Acta Arith. 27 (1975), 489-492.[11] Ju. V. Linnik, On the representation of large numbers as sums of seven cubes,Mat. Sb. 12 (1943), 218-224.[12] S. T. Parseli, Multiple exponential sums over smooth numbers, J. Reine Angew.Math. 532 (2001), 47-104.[13] E. Peyre, Torseurs universels et méthode du cercle, Rational points on algebraicvarieties, Progr. Math. 199, Birkhâuser, 2001, 221-274.[14] W. M. Schmidt, The density of integer points on homogeneous varieties, ActaMath. 154 (1985), 243-296.[15] R. C. Vaughan, A new iterative method in Waring's problem, Acta Math. 162(1989), 1-71.[16] R. C. Vaughan, The Hardy-Littlewood Method, Cambridge University Press,1997.[17] R. C. Vaughan & T. D. Wooley, Further improvements in Waring's problem,III: eighth powers, Philos. Trans. Roy. Soc. London Ser. A 345 (1993),385-396.[18] R. C. Vaughan & T. D. Wooley, Further improvements in Waring's problem,II: sixth powers, Duke Math. J. 76 (1994), 683-710.[19] R. C. Vaughan & T. D. Wooley, Further improvements in Waring's problem,Acta Math. 174 (1995), 147-240.[20] R. C. Vaughan & T. D. Wooley, Further improvements in Waring's problem,IV: higher powers, Acta Arith. 94 (2000), 203-285.[21] I. M. Vinogradov, The method of trigonometric sums in the theory of numbers,Trav. Inst. Math. Stekloff 23 (1947), 109.[22] I. M. Vinogradov, On an upper bound for G(n), Izv. Akad. Nauk SSSR Ser.Mat. 23 (1959), 637-642.[23] H. Weyl, Über die Gleichverteilung von Zahlen mod Eins, Math. Ann. 77(1916), 313-352.[24] T. D. Wooley, Large improvements in Waring's problem, Ann. of Math. (2)135 (1992), 131-164.[25] T. D. Wooley, On Vinogradov's mean value theorem, Mathematika 39 (1992),379-399.[26] T. D. Wooley, The application of a new mean value theorem to the fractionalparts of polynomials, Acta Arith. 65 (1993), 163-179.[27] T. D. Wooley, New estimates for smooth Weyl sums, J. London Math. Soc.(2) 51 (1995), 1-13.[28] T. D. Wooley, Breaking classical convexity in Waring's problem: sums of cubesand quasi-diagonal behaviour, Invent. Math. 122 (1995), 421-451.

Exponential Sums and Diophantine Problems 217[29] T. D. Wooley, On exponential sums over smooth numbers, J. Reine Angew.Math. 488 (1997), 79-140.[30] T. D. Wooley, Slim exceptional sets for sums of cubes, Canad. J. Math. 54(2002), 417-448.[31] T. D. Wooley, Slim exceptional sets in Waring's problem: one square and fivecubes, Quart. J. Math. 53 (2002), 111-118.[32] T. D. Wooley, Sums of three cubes, Mathematika (to appear).

Section 4. Differential GeometryB. Andrews: Positively Curved Surfaces in the Three-sphere 221Robert Bartnik: Mass and 3-metrics of Non-negative Scalar Curvature 231P. Biran: Geometry of Symplectic Intersections 241Hubert L. Bray: Black Holes and the Penrose Inequality in GeneralRelativity 257Xiuxiong Chen: Recent Progress in Kahler Geometry 273Weiyue Ding: On the Schrödinger Flows 283P. Li: Differential Geometry via Harmonic Functions 293Yiming Long: Index Iteration Theory for Symplectic Paths withApplications to Nonlinear Hamiltonian Systems 303Anton Petrunin: Some Applications of Collapsing with BoundedCurvature 315Xiaochun Rong: Collapsed Riemannian Manifolds with Bounded SectionalCurvature 323Richard Evan Schwartz: Complex Hyperbolic Triangle Groups 339Paul Seidel: Fukaya Categories and Deformations 351Weiping Zhang: Heat Kernels and the Index Theorems on Even and OddDimensional Manifolds 361

ICM 2002 • Vol. II • 221-230Positively Curved Surfacesin the Three-sphereB. Andrews*AbstractIn this talk I will discuss an example of the use of fully nonlinear parabolicflows to prove geometric results. I will emphasise the fact that there is a widevariety of geometric parabolic equations to choose from, and to get the bestresults it can be very important to choose the best flow. I will illustrate thisin the setting of surfaces in a three-dimensional sphere.There are quite a few relevant results for surfaces in the sphere satisfyingvarious kinds of curvature equations, including totally umbillic surfaces,minimal surfaces and constant mean curvature surfaces, and intrinsically flatsurfaces. Parabolic flows can strengthen such results by allowing classes ofsurfaces satisfying curvature inequalities rather than equalities: This was firstdone by Huisken, who used mean curvature flow to deform certain classes ofsurfaces to totally umbillic surfaces. This motivates the question "What is theoptimal result of this kind?" — that is, what is the weakest pointwise curvaturecondition which defines a class of surfaces which retracts to the space ofgreat spheres?The answer to this question can be guessed in view of the examples. Toprove it requires a surprising choice of evolution equation, forced by the requirementthat the pointwise curvature condition be preserved.I will conclude by mentioning some other geometric situations in whichstrong results can be proved by choosing the best possible evolution equation.2000 Mathematics Subject Classification: 53C44, 53C40.Keywords and Phrases: Surfaces, Curvature, Parabolic equations.1. IntroductionMy aim in this talk is to demonstrate the use of fully nonlinear parabolic evolutionequations as tools for proving results in differential geometry. I will emphasisethe fact that there is a wide variety of flows which are geometrically defined and* Centre for Mathematics and its Applications, Australian National University, ACT 0200,Australia. E-mail: andrews@maths.anu.edu.au

222 B. Andrewspotentially applicable to geometric problems, and that there is great benefit to behad by choosing the flow carefully. I will focus on a particular application, relatingto surfaces in the 3-sphere, but the method has much wider applicability.There are some well-known examples of geometric evolution equations of thekind I want to consider: Eells and Sampson [8] used a heat flow to prove existenceof harmonic maps into non-positively curved targets; Hamilton considered the flowof Riemannian metrics in the direction of their Ricci tensor, and proved that itdeforms metrics of positive Ricci curvature on three-manifolds [12] and metrics ofpositive curvature operator on four-manifolds [13] to constant curvature metrics.The Ricci flow also gives results in higher dimensions, proved by Huisken [14],Nishikawa [24] and Margerin [19]—[21], if the curvature tensor is suitably pinched.The mean curvature flow of submanifolds of Euclidean space is also well-known asthe gradient descent flow of the area functional, and because it arises in modelsof interfaces such as in annealing metals. The examples I will concentrate on areclosest to the last example, as they are evolution equations describing submanifoldsmoving with curvature-dependent velocity. There are many parabolic flows of thiskind, particularly for the codimension one (hypersurface) case: William Firey [11]introduced the motion by Gauss curvature as a model for pebbles wearing away asthey tumble, and other flows which have been considered include motion by powersof Gauss curvature [28], [6], the square root of the scalar curvature [7], the harmonicmean of the principal curvatures [2]-[3], and the reciprocal of the mean curvature[17]. More generally, one can take the velocity to be a function of the principalcurvatures which is monotone increasing in each argument.This gives a huge variety of flows to choose from, so it makes sense to choosethe flow carefully to suit the problem. I will illustrate a strategy for choosing theflow by asking that some desired curvature inequality be preserved under the flow.I will begin, in the next two sections, by discussing some old results concerningsurfaces in the three-sphere. This motivates the results of the later sections.2. Constant mean curvature surfacesThere is a well-known result of Simons [27] which says that a minimal hypersurfacein a S n+1 with the squared norm of the second fundamental form \A\ 2less than n is in fact totally geodesic (hence a great n-sphere). This result comesfrom an application of Simons' identity which relates the second derivatives of meancurvature to the Laplacian of the second fundamental form:VjVjF = Ahij + \A\ 2 hij - Hh\hpj + F^y - n%.From this we can deduce if the hypersurface is minimal (so F = 0)0 = A|,4| 2 - 2|V-4| 2 + 2|,4| 2 (|,4| 2 - n).If |.4| 2 < n at a maximum, then the maximum principle implies |.4| 2 is identicallyzero,and the result follows. Also, if the maximum of |.4| 2 is equal to n, then Mmust be a product S k (a) x S n^k(b) in R k+1 x R n+1^k, with radii a and 6 determinedby the fact that M lies in S n+1 C R n+2 and is minimal.

Positively Curved Surfaces in the 3-sphere 223Simons' argument was taken up by other authors ([25], [5], [1]) in the slightlymoregeneral setting of constant mean curvature hypersurfaces. The results aresimilar: If the hypersurface has constant mean curvature H, and |.4| is boundedby a constant depending on n and H, then the hypersurface is totally umbillic,hence a geodesic sphere in S n+1 ; if the inequality is not strict then the only extrapossibilities are products of spheres. The argument is similar to that above, butcomplicated by the non-vanishing of the mean curvature.Let me look closer at the situation for surfaces in the three-sphere: The intrinsiccurvature of the surface is given by I + K1K2 = 1 + |F 2^||.4| 2 . If M is minimal,then F = 0, so \A\ 2 < 2 is equivalent to positivity of the intrinsic curvature. Thisis also true for constant mean curvature surfaces: In two dimensions, the curvaturecondition from [25] and [5] is equivalent to positivity of the intrinsic curvature.3. Flat toriThe condition of positive intrinsic curvature seems natural in view of the resultson constant mean curvature surfaces. For surfaces in space, positive curvature isa rather restrictive condition — a compact surface satisfying this condition is theboundary of a convex region. In the 3-sphere it seems somewhat less restrictive,as we can see by considering the 'boundary' case of flat surfaces, where there arethe beautiful results of Weiner [32] and Enomoto [9] which classify flat tori in the3-sphere by their Gauss maps. It was known for some time that there are manyexamplesof these (see [26]), since the inverse image of any smooth curve in S 2under the Hopf projection is a flat torus in S 3 . These examples are all invariantunder the action of U(Y) on C 2 ~ F 4 , but Weiner and Enomoto showed that thereare many examples which are not symmetric.The Gauss map of a surface in S 3 can be thought of in several ways: Onecan consider the tangent plane of the surface as a subspace of F 4 , which gives amap from the surface to the Grassmannian G 2 ,4 of 2-planes in F 4 . The latter is ametric product S 2 x S 2 , and the projections onto each factor are called the self-dualand anti-self-dual Gauss maps. Alternatively, since S 3 is a group, one can mapthe unit normal of the surface by either left or right translations to the Lie algebra— this again gives two maps to S 2 , and of course these are the same as before:The self-dual Gauss map is the same as the left-translation Gauss map, and theanti-self-dual Gauss map is the same as the right-translation Gauss map.Enomoto [9] observed that if M 2 is intrinsically flat in S 3 , then both Gaussmaps are degenerate (their images are just curves in S 2 ). Weiner gave the completeclassification result: The image curves 71 and 72 necessarily have zero totalcurvature, and if fi and I2 are subintervals of 71 and 72 respectively, thenI fj nds\ + \ fj nds\ < IT. Conversely, if 71 and 72 are any curves satisfying theseconditions, then there is a flat torus with these curves as the images of the twoGauss maps, and the torus is unique up to motion by unit speed in the normaldirection.This gives a very large family of flat tori in the 3-sphere, and from these we seethat surfaces with positive intrinsic curvature in S 3 can look quite complicated: The

224 B. Andrewssurface can look metrically like a long thin cylinder with caps on the ends, placedin S 3 by 'winding around' a flat torus many times before closing off the ends.4. Curvature flowCurvature flow can give powerful generalisations of results like those from[27], [25] and [5]: Huisken [16] extended techniques developed earlier for convexhypersurfaces in Euclidean space [14] to prove the following result:Theorem: Let MQ = XQ(M) be a hypersurface in S n+1 which satisfiesif n > 2, and\A\ 2 < -^—H 2 + 2n — 1W- < -^ + -if n = 2. Then there exists a smooth family of hypersurfaces {M t = xt(M)}o

Positively Curved Surfaces in the 3-sphere 225so that in the case we are interested in, (p(x) = \/4 + x 2 . We can also writeThen the evolution equation for G is as follows:F = f(Ki+K 2 ,G). (5.2)BC— = F ij ViVjG + Q(h)(Vh,Vh) + Z(h), (5.3)where F is the matrix of derivatives of F with respect to the components of thesecond fundamental form, which is positive definite as long as F is an increasingfunction of each of the principal curvatures. The second term is a quadratic functionof the components of the derivative of the second fundamental form, with coefficientsdepending on curvature h, explicitly given byQ = (G i JF kl ' mn - p ij G kl ' mn^ VihaVjhr,where F is the second derivative of F with respect to the components of h. Thelast term Z depends on the curvature alone, and has the formZ = G ij (F(h%j + gij) + F kl (hijhli - h kl h^ + gijh k i -

226 B. Andrewsboundary of the set {G = 0} in the curvature plane, so we can consider F as definedby (5.2) with G = 0. Then the conditions can be written explicitly as follows:1 — tp' tp fi 1 + tp' tpIn the case of interest, we have ip = \/4 + H 2 , and the first and last quantities areboth equal to —2H/(4 + H 2 ). The only possibilities for F are the following:F = C\ + C2 arctan f —This applies only along the curve {G = 0}, so we are reasonably free to choose Fin the region where G > 0, as long as it is monotone in both principal curvatures.5.2. The extreme caseThe remarkably restricted form of the evolution equation is illuminated somewhatby considering the extreme case of flat surfaces: If the flow preserves positiveintrinsic curvature, then it must also preserve zero curvature. As outlined above,the structure of surfaces with zero curvature is very well understood, and in particularthe Gauss map G : M 2 —t S 2 x S 2 has the remarkable property that theprojection onto each factor is one-dimensional. This must be preserved under theflow.The flow we have ended up with is characterised by the fact that the Gaussmap evolves according to the mean curvature flow (now for codimension 2 surfacesin S 2 xS 2 , which means that each of the two curves coming from the two projectionsof the Gauss map evolves according to the curve-shortening flow in S 2 . Since eachof the curves divides the area of the sphere into two equal parts, the image ofthe Gauss map never develops singularities (at least in the case where the twocurves are homotopic to great circles traversed once), but in fact the flat tori willin general develop singularities — this is analogous to the motion of a curve inthe plane with constant normal speed, which develops singularities even though thenormal direction stays constant at each point. Incidentally, there has been somevery impressive recent progress on mean curvature flow in higher codimension, dueto Mu-Tao Wang [29]-[31], who has used it to prove several very interesting resultsregarding maps between manifolds.The examples of flat tori can be used to prove that there is no other curvaturedrivenflow of surfaces which preserves the condition of positive curvature, by givingexamples for any other flow of flat tori which do not stay flat.5.3. RegularityA technical issue which arises is the following: The speed we ended up with isnot concave or convex as a function of the second fundamental form. The regularityestimatesdue to Krylov [18] and Evans [10] for fully nonlinear equations (neededto prove that we get classical solutions of the flow) require concavity, so we cannot

Positively Curved Surfaces in the 3-sphere 227use these. Instead it is possible to adapt the estimates for elliptic equations in twovariables (due to Morrey [22] and Nirenberg [23]) to give good C 2,a estimates forsolutions of fully nonlinear parabolic equations in two space variables.5.4. Curvature pinchingNow we come to the problem of choosing a good way to extend the speed fromthe boundary {G = 0} to the interior of the region {G > 0}. The idea is to do thisin such a way that any compact surface with strictly positive curvature necessarilyhas very strongly controlled curvature in the future — that is, we want the region{G > 0} to be exhausted by a nested family of regions which stay away from theboundary, and only approach infinity near the 'umbillic' line Ki = K 2 . This meansthat any singularity which occurs will have to be totally umbillic, so occurs onlywhen the surface shrinks to a point while becoming spherical in shape.This can be done in many ways. One which is relatively simple to describe,but results in solutions which are only C 2 ' a , is as follows: TakeJ arctanKi +arctanK2, K1K2 < 1;lf(«l«2 + l), KiK 2 > 1.This is then a Lipschitz, monotone increasing function of the curvatures, and onecan check that the following regions of the curvature plane are preserved:„ f, 1 1 + K1K2 Ì . ,, f, , 21ii e = < 1 «i — K 2 \ n {K1K2 < i) u < |KI — K 2 \ < - > n {K1K2 > l).This means that the difference between the principal curvatures stays bounded evenif the curvature becomes large, which implies very strong control on singularities.This is similar to the estimate used in [4] to prove that worn stones (i.e. convexsurfaces moving by their Gauss curvature) become round as they shrink to points.With a little more work we can choose the speed to be a smooth function ofthe principal curvatures, and then solutions are also smooth.In the choice above, we also have the nice feature that minimal surfaces donot move. We can with slight modifications arrive at a speed for which constantmean curvature surfaces do not move, for any particular choice of the mean curvature,as long as we are willing to work in the category of oriented surfaces. Moregenerally, we can contrive that for a given monotone increasing function of theprincipal curvatures, surfaces satisfying = 0 do not move. Here F (and ) mustbe symmetric. We can also choose if desired a speed which is always positive, sothat there are no stationary solutions.5.5. The resultsThe main result for the above speed is the following:Theorem 1. Let XQ be an immersion of S 2 in S 3 , with non-negative intrinsiccurvature in the induced metric. Then the flow constructed above deforms M 0 =XQ(S 2 ) through a family M t = xt(S 2 ), with intrinsic curvature strictly positive for

228 B. Andrewseach t > 0, to either a great sphere (in infinite time) or to a point, with sphericallimiting shape (in finite time). If M 0 is embedded, then so is M t for each t > 0.This includes in particular Simons' result on mimimal surfaces. If we modifythe speed somewhat, then we get the following result, which gives in particular anew result for Weingarten surfaces in the 3-sphere:Theorem 2. Let 0}. Then there exists a function F which is smoothly defined on{K1K2 + 1 > 0}, and strictly monotone increasing in each argument, with sgnF =sgncf) everywhere, such that the following holds: If M 0 = XQ(S 2 ) is a smooth compactsurface in S 3 with non-negative intrinsic curvature, then the motion with speed Fdeforms M 0 through a smooth family {M t } 0

Positively Curved Surfaces in the 3-sphere 229The methods also give good results for hypersurfaces in higher-dimensionalspheres: Hypersurfaces with positive sectional curvatures can be deformed in sucha way as to preserve that condition, and similar results can be deduced. Thecondition of positive sectional curvature can probably be relaxed: Positive sectionalcurvature is implied by the condition of Okumura [25] for constant mean curvaturehypersurfaces, but not by the sharper condition of Cheng and Nakagawa [5] andAlencar and do Carmo [1].References[i[2:[3;[4;[*.[6;[9[io;[n[12:[is;[w;[15[16[17;[18H. Alencar and M. do Carmo, Hypersurfaces with constant mean curvature inspheres, Proc. Amer. Math. Soc. 120 (1994), 1223-1229.B. Andrews, Contraction of convex hypersurfaces in Euclidean space, CaleVar. P.D.E. 2 (1994), 151-171.B. Andrews, Contraction of convex hypersurfaces in Riemannian spaces, J.Differential Geometry 39 (1994), 407-431.B. Andrews, Gauss Curvature Flow: The Fate of the Rolling Stones, Invent.Math. 138 (1999), 151-161.Q.-M. Cheng and H. Nakagawa, Totally umbillic hypersurfaces, HiroshimaMath. J. 20 (1990), 1-10.B. Chow, Deforming convex hypersurfaces by the nth root of the Gaussiancurvature, J. Differential Geom. 22 (1985), 117-138.B. Chow, Deforming convex hypersurfaces by the square root of the scalarcurvature, Invent. Math. 87 (1987), 63-82.J. Eells and J. Sampson, Harmonic mappings of Riemannian manifolds, Amer.J. Math. 86 (1964), 109-160.K. Enomoto, The Gauss image of flat surfaces in F 4 , Kodai Math. J. 9 (1986),19-32.L. C. Evans, Classical solutions of fully nonlinear, convex, second order ellipticequations, Comm. Pure Appi. Math. 24 (1982), 333-363.W. J. Firey, Shapes of worn stones. Mathematika 21 (1974), 1-11.R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. DifferentialGeometry, 17 (1982), 255-306.Four-manifolds with positive curvature operator, J. Differential Geometry 24(1986), 153-179.G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. DifferentialGeometry 20 (1984), 237-266.G. Huisken, Ricci deformation of the metric on a Riemannian manifold, J.Differential Geometry 21 (1985), 47-62.G. Huisken, Deforming hypersurfaces of the sphere by their mean curvature,Math. Z. 195 (1987), 205-219.G. Huisken and T. Ilmanen, The Riemannian Penrose Inequality, Internat.Math. Res. Notices 1997, no. 20, 1045-1058.N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations,Izvestia Akad. Nauk. SSSR 46 (1982), 487-523. English translation in Math.

230 B. AndrewsUSSR Izv. 20 (1983).[19] C. Margerin, Pointwise pinched manifolds are space forms, Proc. Symp. PureMath. 44, 1986.[20] C. Margerin, Une caractrisation optimale de la structure diffrentielle standardde la sphre en terme de courbure pour (presque) toutes les dimensions, C. R.Acad. Sci. Paris Sér I Math. 319 (1994) 713-716 and 605-607.[21] C. Margerin, A sharp characterization of the smooth 4-sphere in curvatureterms, Comm. Anal. Geom. 6 (1998), 21-65.[22] C.B. Morrey, Jr., On the solutions of quasi-linear elliptic partial differentialequations, Trans. Amer. Math. Soc. 43, (1938), 126-166.[23] L. Nirenberg, On nonlinear elliptic partial differential equations and Holdercontinuity, Comm. Pure Appi. Math. 6 (1953), 103-156.[24] S. Nishikawa, Deformation of Riemannian metrics and manifolds with boundedcurvature ratios, Proc. Sympos. Pure Math. 44, 1986.[25] M. Okumura, Hypersurfaces and a pinching problem on the second fundamentaltensor, Amer. J. Math. 96 (1974), 207-213.[26] U. Pinkall, Hopf Tori in S 3 , Invent. Math. 81 (1985), 379-386.[27] J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88(1968), 62-105.[28] Raising Tso, Deforming a hypersurface by its Gauss-Kronecker curvature,Comm. Pure Appi. Math. 38 (1985), 867-882.[29] M.-T. Wang, Mean curvature flow of surfaces in Einstein four-manifolds, J.Differential Geom. 57 (2001), 301-338.[30] M.-T. Wang, Deforming area preserving diffeomorphism of surfaces by meancurvature flow, Math. Res. Lett. 8 (2001), 651-661.[31] M.-T. Wang, Subsets of Grassmannians preserved by mean curvature flow,preprint, 2002.[32] J. Weiner, Flat tori in S 3 and their Gauss maps, Proc. London Math. Soc. 62(1991), 54-76.

ICM 2002 • Vol. II • 231-240Mass and 3-metries of Non-negativeScalar CurvatureRobert Bartnik*AbstractPhysicists believe, with some justification, that there should be a correspondencebetween familiar properties of Newtonian gravity and properties ofsolutions of the Einstein equations. The Positive Mass Theorem (PMT), firstproved over twenty years ago [45, 53], is a remarkable testament to this faith.However, fundamental mathematical questions concerning mass in generalrelativity remain, associated with the definition and properties of quasi-localmass. Central themes are the structure of metrics with non-negative scalarcurvature, and the role played by minimal area 2-spheres (black holes).2000 Mathematics Subject Classification: 53C99, 83C57.Keywords and Phrases: Quasi-local mass, Einstein equations, Scalar curvature.1. Positive Mass TheoremThe Positive Mass Theorem provides a good example of "the unreasonableeffectiveness of physics in mathematics 1 ". The need to define mass in general relativityis motivated directly by the physics imperative to establish a correspondencebetween general relativity and classical Newtonian gravity. Already difficulties arise:although the vacuum Einstein equations Ric a ß — \Rg a ß = 0 for the Lorentz metricg a ß suggest (by analogy with the wave equation, for example) that a mass (energy)which includes contributions from the gravitational field, should be built from thefirst derivatives of the field g a ß, it is clear that this is incompatible with coordinateinvariance.The Schwarzschild vacuum spacetime metric, for r > max(0, 2M),ds 2 = - (1 - 2M/r) dt 2 + *' 2 + r 2 (dd 2 + sin 2 ê dtp 2 ), (1.1)* School of Mathematics and Statistics, University of Canberra, ACT 2601, Australia. E-mail:robert.bartnik@canberra.edu.au1 with apologies to Eugene Wigner [52].

232 Robert Bartnikprovides an important clue, since the parameter M £ R governs the behaviour oftimelike geodesies and may be regarded as the total mass. Note that M > 0 ensuresthe boundary r = 2M is smooth and totally geodesic in the hypersurfaces t = const.A Riemannian 3-manifold (M, g) is said to be asymptotically flat if M\K ~R 3 \Fi(0) for some compact K, and M admits a metric g which is flat outside K,and the metric components #y in the induced rectangular coordinates satisfy\ 9ij - gij\ = O^- 1 ), \d k9ij \ = 0(r- 2 ), \d k d mj \ = 0(r- 3 ). (1.2)The total mass of (M,g) is defined informally by [1]m ADM = -T7T- f (di9ij -djgu)dSj. (1.3)lt)7rJ.S 2 (oo)If the scalar curvature R(g) £ L 1 (M) then UIADM is well-defined, independent ofthe choices of rectangular coordinates and of exhaustion of M used to define §32/^— see [3, 15, 37] for weaker decay and smoothness assumptions.For simplicity, the discussion here is restricted to C°° Riemannian 3-dimensionalgeometry. This corresponds to the case of time-symmetric initial data: (M, g) is atotally geodesic spacelike hypersurface in a Lorentzian manifold, and we can identifythe local matter (equivalently, energy) density with the scalar curvature R(g) > 0.This simplification entails a small loss of generality: most, but not all, of the resultswe describe have been extended to general asymptotically flat space-time initialdata (M,g,K), where FTy is the second fundamental form of a spacelike hypersurfaceM. Some results also generalize to the closely related Bondi mass, whichmeasures mass and gravitational radiation flux near null infinity, and to mass onasymptotically hyperbolic and anti-deSitter spaces cf. [51, 16], but these involveadditional complications which we will not discuss here.The Positive Mass Theorem (PMT) in its simplest form isTheorem 1 Suppose (M, g) is a complete asymptotically flat 3-manifold with nonnegativescalar curvature R(g) > 0. Then UIADM > 0, and UIADM = 0 iff (M,g) =(R 3 ,ô).The rigidity conclusion in the case UIADM = 0 shows that UIADM > 0 for(M,g) scalar flat ("matter-free") but non-flat, so UIADM does provide a measure ofthe gravitational field.Three distinct approaches have been successfully used to prove the PMT:with stable minimal surfaces [45, 46]; with spinors [53, 36] and the Schrödinger-Lichnerowicz identity [48, 35]; and using the Geroch foliation condition [23, 30].A number of other appproaches have produced partial results: using spacetimegeodesies [42]; a nonlinear elliptic system for a distinguished orthonormal frame[39, 18]; and alternative foliation conditions [32, 33, 6]. The connection betweenthese approachs remains mysterious; the only discernable common thread is meancurvature, and this is quite tenuous.The application of the positive mass theorem to resolve the Yamabe conjecture[44, 34] is well known. Less well known is the proof of the uniqueness of the

Mass and Scalar Curvature 233Schwarzschild spacetime amongst static metrics with smooth black hole boundary[13], which we briefly outline.A static spacetime is a Lorentzian 4-manifold with a hypersurface-orthogonaltimelike Killing vector. With V denoting the length of the Killing vector, the metricg on the spacelike hypersurface satisfies the static equationsRiCg = V~ l V 2 V, ( .{A g V = 0.'Smoothness implies the boundary set S = {V = 0} is totally geodesic; analyticityof g, V can be used to show the asymptotic expansions9ij = (1 + 2m/r)% + 0(r- 2 ),V = 1 — m/r + 0(r~ 2 ),as r —t oo for some constant m £ R. The metrics g ± = 4 g where = ± on the two copies of M, gives a complete AF manifold with R(g) = 0 andvanishing mass. The PMT shows (M, g) is flat and it follows without difficulty that(M,g) is Schwarzschild. This extends previous results [31, 43] which required theboundary to be connected.2. Penrose conjectureA boundary component S with mean curvature F = 0 is called a black hole orhorizon, since if (M, g) is a totally geodesic hypersurface then S is a trapped surfaceand hence, by the Penrose singularity theorem [26], lies within an event horizon andis destined to encounter geodesic incompleteness in the predictable future.The spatial Schwarzschild metric g = 1_' 2 M/ r + r 2 (d'à 2 + sin édtp 2 ) with M < 0shows that the completeness condition in the PMT is important, but it can beweakened to allow horizon boundary components of M. This follows immediatelyfrom the minimal surface argument [45]; or by an extension to the Witten argument[22], imposing one of the boundary conditionsip = ±eip on S = dM, (2.1)on the spinor field tp, where e = 7"7° satisfies e 2 = 1. An interesting extension isobtained by imposing the spectral boundary conditionP+'ip = 0 on S (2.2)where F + is the projection onto the subspace of positive eigenspinors of the inducedDirac operator T>-£. Using the remarkable Hijazi-Bär estimate [28, 2]|A| > V47r/|S|, (2.3)for the eigenvalues of VY. when S ~ S 2 , Herzlich showed [27]

234 Robert BartnikTheorem 2 If (M, g) is asymptotically flat with R(g) > 0 and boundary S ~ S 2with mean curvature satisfyingHn < 2/r (2.4)where r = ^/|£|/47r, then UIADM > 0, with equality iff (M,g) = (R 3 \F(r),5).The proof starts with the Riemannian form of the Schrödinger-Lichnerowicz-Witten identity [48, 35, 53]M(\V'tp\ 2 + \R(g)\'tp\ 2 - \Vip\ 2 )dv M = 4Tx\'tp 00 \ 2 m A DM + f p(ip), (2.5)where ß('ip) is the Nester-Witten form [38]ß(iP) = (^(^ + 1^*5:. (2.6)The boundary condition P + ip\j: = 0 is elliptic and it can be shown [8] there isa spinor on M satisfying Vip = 0 with boundary conditions ip —¥ rpoo ^ 0 asr ^ oo and (2.2) on S. It follows from (2.3) and (2.2) that {tp, (£> s + |F E )i/>) \/|S|/167T, (2.7)A closed minimal surface is said to be an outermost horizon or outer-minimizinghorizon if M contains no least area surfaces homologous to S in the asymptoticregion exterior to S. The outermost condition is essential, since examples of nonnegativescalar curvature manifolds can be constructed by forming the connectedsum of M and large spheres by arbitrarily small and large necks.The Penrose conjecture has been established by Huisken and Ilmanen [29, 30]using a variational level set formulation of the inverse mean curvature flow [23], andby Bray [12] by a very interesting conformai deformation argument. Bray's proofis more general since it takes into account contributions from all the connectedcomponents of the boundary.3. Quasi-local massThus it is natural to consider y/\S\/16w as the mass of a black hole (minimalsurface) S. More generally, the correspondence with Newtonian gravity suggeststhat any bounded region (Q, g) should have a quasi-local mass, which measures bothJT.

Mass and Scalar Curvature 235the matter density (represented in this case by the scalar curvature R(g) > 0), andsome contribution from the gravitational field. The rather satisfactory positivityproperties of the total mass, as established by the PMT, motivate the propertieswe might expect such a geometric mass to possess [20, 14, 7].1. (non-negativity) ntQL^l) > 0;2. (rigidity/strict positivity) ntQL^l) = 0 if and only if (ii,g) is flat;3. (monotonicity) rriQL^li) < TOQL(0 2 ) whenever Qi c 0 2 , where it is understoodthat the inclusion is a metric isometry;4. (spherical mass) UIQL should agree with the spherical mass, for sphericallysymmetricregions;5. (ADM limit) UIQL should be asymptotic to the ADM mass;6. (black hole limit) UIQL should agree with the black hole mass (2.7).Many candidates have been proposed for quasi-local mass (see for example [10] fora comparison of some definitions), the most significant being that of Hawking [25],'""V^î'-àl" 1 )where S = 90. This equals M for standard spheres in Schwarschild. AlthoughrriH < 0 for surfaces in R 3 , it was shown in [14] that mij(S) > 0 for a stable constantmean curvature 2-sphere S in a 3-manifold of non-negative scalar curvature. Thusfor such "round" spheres, rriR is nonegative, and the black hole limit conditionis trivially satisfied. However the remaining properties, in particular rigidity andmonotonicity, are rather problematic. Although the twistorially-defined Penrosequasi-local mass [41] is well-behaved in special cases [50], it is defined unambiguouslyonlyfor surfaces arising from embedding into a conformally flat spacetime, and eventhen numerical experiments [11] strongly suggest that monotonicity is violated.In fact, of the various proposals for rriQL, only the definitions of [14, 5, 19]are known to satisfy positivity. Dougan and Mason [19] show that the integral§^ ß('ip) of the Nester-Witten 2-form (2.6) is positive for spinor fields ip on S whichsatisfy a certain elliptic system on S. However, Bergqvist [9] shows that positivityholds under much weaker conditions on ip, and there are many variant definitionswith similar properties. It would be useful to understand these DM-style definitionsbetter, and in particular whether any satisfy monotonicity.Monotonicity and ADM-compatibility imply niQL(ii) < mADM(M,g) for anyregion Q embedded isometrically in an (M,g) satisfying (as always) the PMT conditions.This motivates the following definition [4, 30]Definition 4 Let VM denote the set of all asymptotically flat 3-manifolds (M,g)of non-negative scalar curvature, with boundary which if non-empty, consists ofcompact outermost horizons, and such that (M, g) has no other horizons. For anybounded open connected region (ii,g), let VM(ii) be the set of (M,g) £ VM suchthat 0 embeds isometrically into M, and definem QL (ii) = inî{m A DM(M,g) : (M,g) £ VM(iì)}. (3.2)i3A >We say that M satisfying these conditions is an admissible extensionofii.

236 Robert BartnikThe horizon condition serves to exclude examples which hide 0 inside an arbitrarilysmall neck, which would force the infimum to zero. This is a refinement[30] of the original definition [4], which prohibited horizons altogether.Clearly irtQL^l) is well-defined and finite, once the region Q admits just oneadmissible extension. The PMT with horizon boundary implies non-negativity, andmonotonicity follows directly. Strict positivity of UIQL was established in [30], withthe slightly weaker rigidity conclusion that if niQL(ii) = 0 then Q is locally flat.Agreement with the spherical mass, and the ADM limit condition, follows also from[30]. Bray's results imply that niQL(ii) agrees with the black hole mass in the limitas 0 shrinks down to a black hole. In addition, niQL(ii) < niADM(M) for anyadmissibleextension M, so UIQL is the optimal quasi-local mass definition withrespect to this condition.The optimal form of the horizon condition remains conjectural. Bray hassuggested an alternative condition, that 0 be a "strictly minimizing hull" [30] inM, so S = 90 has the least area amongst all enclosing surfaces in the exterior.In this case we say S is outer minimizing, and denote by ìTIQL(ìì) the quasilocalmass function defined by restricting admissible extensions to those M in which Sis outer minimizing. For this modified definition the Penrose inequality [30, 12]applies to show that if 90 embeds into the Schwarzschild 3-manifold with the sameinduced metric and mean curvature (cf. (4.1), (4.2)) and encloses the horizon, thenniQL(ii) = M. It is not clear how to establish this natural result for the unmodifieddefinition niQL(ii).4. Static metricsAlthough in many respects the definition of UIQL is quite satisfactory, it is notconstructive, and thus it is important to determine computational methods. Thekey is the following [4]Conjecture 5 The infimum in UIQL is realised by a 3-metric agreeing with Q inthe interior, static (1-4) in the exterior region, and such that the metric is Lipschitzcontinuousacross the matching surface S, and the mean curvatures of the two sidesagree along S.A similar conjecture for the space-time generalisation of the quasi-local mass,asserts that the exterior metric is stationary, ie. admits a timelike Killing field [4, 7].As motivation for this conjecture, note first that if F (g) > 0 in some region,then a conformai factor

Mass and Scalar Curvature 237Theorem 6 If (M,g) realizes the infimum in Definition 4, then there is a V £C°°(M\ii) such that g,V satisfy the static metric equations (1-4) in M\Q.This suggests a computational algorithm for determining roQL(O): find anasymptotically flat static metric with boundary geometry matching that of 90. Todetermine the appropriate boundary conditions, recall the second variation formulafor the area of the leaves of a foliation labelled by r:R(g) = 2D n H - \II\ 2 - H 2 + 2K - 2A" 1 A r A (4.1)where II, H, K are respectively the second fundamental form, mean curvature andGauss curvature of the leaves, À is the lapse function, n = X^1dr is the normal vectorand A r is the Laplacian on the leaves. Our conventions give F = ^F„(log y/detg r )where g r is the volume element of the leaves. This shows that R(g) will be defineddistributionally across a matching surface as a bounded function ifglran = g\r-£, , .F 0fi = F E .('jConjecture 7 (ii,g) determines a unique static asymptotically flat manifold (S,g)with boundary S ~ 90 satisfying (4-2).If true, this would give a prime candidate for the minimal mass extension. Itis known (Pengzi Miao, private communication) that the boundary conditions (4.2)are elliptic for (1.4).It is tempting to conjecture that mass-minimizing sequences for UIQL shouldconverge to a static metric. For example, [3, Theorem 5.2] shows that a sequenceof metrics g k , close in the weighted Sobolev space Wl'?, q > 3,r > 1/2, to theflat metric Ö on R 3 and such that mADAf(gk) —* 0, converges strongly to ö inW 1 ' 2 . Similar results, under rather different size conditions, are given in [21], anda discussion of the general "weak compactness" conjecture may be found in [30].5. Estimating quasi-local massTo estimate UIQL from above, it suffices to construct admissible extensions —metrics with non-negative scalar curvature and satisfying (4.2). These boundaryconditionsexclude the usual conformai method. Instead, metrics in quasi-sphericalform [6]g = u 2 dr 2 + (rdâ + ß 1 dr) 2 + (r sine dtp + ß 2 dr) 2 (5.1)satisfy a parabolic equation for u on S 2 evolving in the radial direction, when R(g) =0, with ß 1 ,ß 2 freely specifiable. Since the metric 2-spheres S 2 have mean curvatureH r = (2 — div$2ß)/ur > 0, (5.1) provides admissible extensions for 90 = S 2 withmean curvature F > 0. The underlying parabolic equation derives from (4.1), andhas been generalized to non-spherical foliations in [49]. As an application, choosingß = 0 we can show

238 Robert BartnikTheorem 8 Suppose 90 = S 2metrically, with H > 0. Thenm QL (iì) < |r(l - |r 2 minF 2 ). (5.2)This bound is sharp when 0 is a flat ball or a Schwarzschild horizon.Finding lower bounds for roQL(O) is more difficult. Bray's definition of innermass [12, p243] gives a lower bound, but for ?BQL(0). The difficulty here as abovelies in showing that a horizon inside 0 remains outermost when the inner region isglued to a general exterior region M ext C M £ VM(iì). This follows easily whenS = 90 is outer-minimizing in M ext , as guaranteed by the definition for ?BQL(0).On physical grounds one expects that if "too much" matter is compressed intoregion which is "too small", then a black hole must be present. The geometricchallenge lies in making this heuristic statement precise, and the only result inthis direction has been [47], which gives quantitative measures which guarantee theexistence of a black hole. An observation by Walter Simon (private communication)is thus very interesting: if roQL(O) = 1 (say) and 0 embeds isometrically into acomplete asymptotically flat manifold M without boundary and with non-negativescalar curvature, and such that niADM(M) < 1, then M must have a horizon. Thisreinforces the importance of finding good lower bounds for UIQL , since the existenceof a horizon in a similar situation with ûIQL does not follow.References[i[2[3;[4;[6;[9[io;[n[12:R. Arnowitt, S. Deser, and C. Misner. Coordinate invariance and energy expressionsin general relativity. Phys. Rev., 122:997^1006, 1961.C. Bär. Lower eigenvalue estimates for Dirac operators. Math. Ann., 293:39^46,1992.R. Bartnik. The mass of an asymptotically flat manifold. Comm. Pure Appi.Math., 39:661^693, 1986.R. Bartnik. New definition of quasilocal mass. Phys. Rev. Lett., 62(20):2346^2348, May 1989.R. Bartnik. The regularity of variational maximal surfaces. Acta Math., 1989.R. Bartnik. Quasi-spherical metrics and prescribed scalar curvature. J. Diff.Geom., 37:31-71, 1993.R. Bartnik. Energy in general relativity. In Shing-Tung Yau, editor, Tsing HuaLectures on Analysis and Geometry, pages 5^28. International Press, 1997.R. Bartnik and P. Chrusciel. On spectral boundary conditions for Dirac-typeequations, preprint, 2002.G. Bergqvist. Quasilocal mass for event horizons. Class. Quant. Grav., 9:1753-1768, 1992.G. Bergqvist. Positivity and definitions of mass. Class. Quant. Gravity, 9:1917-1922, 1992.D. H. Bernstein and K. P. Tod. Penrose's quasilocal mass in a numericallycomputedspace-time. Phys. Rev., D49:2808^2819, 1994.H. Bray. Proof of the Riemannian Penrose inequality using the positive masstheorem. J. Diff. Geom., 59:177^267, 2001.

Mass and Scalar Curvature 239[13] G. Bunting and A. K. M. Masood ul alam. Non-existence of multiple blackholes in asymptotically Euclidean static vacuum space-time. Gen. Rei. Grav.,19:147, 1987.[14] D. Christodoulou and S.-T. Yau. Some remarks on the quasi-local mass. InJ. Isenberg, editor, Mathematics and General Relativity, Contemporary Mathematics.American Math. Society, 1988.[15] P. T. Chrusciel. Boundary conditions at spatial infinity from a Hamiltonianpoint of view. In P. G. Bergmann and V. de Sabbata, editors, Topologicalproperties and global structure of space-time. Plenum, New York, 1986.[16] P. T. Chrusciel and G. Nagy. The mass of spacelike hypersurfaces in asymptoticallyandi-de Sitter spacetimes. Adv. Theor. Math. Phys., 5, 2001.[17] J. Corvino. Scalar curvature deformation and a gluing construction for theEinstein constraint equations. Commun. Math. Phys., 214:137-189, 2000.[18] A. Dimakis and F. Müller-Hoissen. Spinor fields and the positive enegy theorem.Class. Quantum Grav., 7:283-295, 1990.[19] A. J. Dougan and L. J. Mason. Quasi-local mass constructions with positivegravitational energy. Phys. Rev. Lett., 67:2119-2123, 1991.[20] D. M. Eardley. Global problems in numerical relativity. In L. Smarr, editor,Sources of Gravitational Radiation, pages 127-138. Cambridge UP, 1979.[21] F. Finster and I. Kath. Curvature estimates in asymptotically flat manifoldsof positive scalar curvature. MPI Leipzig preprint, 2001.[22] G. T. Horowitz G. W. Gibbons, S. W. Hawking and M. J. Perry. Positive masstheorems for black holes. Commun. Math. Phys., 88:295-308, 1983.[23] R. Geroch. Energy extraction. Ann. NY. Acd. Sci., 224:108-117, 1973.[24] G. W. Gibbons. The isoperimetric and Bogomolny inequalities for black holes.In T. Willmore and N. Hitchin, editors, Global Riemannian Geometry, chapter6, pages 194-202. Ellis Harwood, Chichester, April 1984.[25] S. W. Hawking. Gravitational radiation in an expanding universe. J. Math.Phys., 9:598-604, 1968.[26] S. W. Hawking and G. R. Ellis. The large-scale structure of spacetime. CambridgeUP, Cambridge, 1973.[27] M. Herzlich. A Penrose-like inequality for the mass on Riemannian asymptoticallyflat manifolds. Commun. Math. Phys., 188:121-133, 1997.[28] O. Hijazi. A conformai lower bound for the smallest eigenvalue of the Diracoperator and Killing spinors. Commun. Math. Phys., 104:151-162, 1986.[29] G. Huisken and T. Ilmanen. The Riemannian Penrose inequality. Int. Math.Res. Not, 20:1045-1058, 1997.[30] G. Huisken and T. Ilmanen. The inverse mean curvature flow and the RiemannianPenrose inequality. J. Diff. Geom., 59:353-438, 2001.[31] W. Israel. Event horizons in static electrovac space-times. Commun. Math.Phys., 8:245-260, 1968.[32] P. S. Jang. On positivity of mass for black hole space-times. Commun. Math.Phys., 69:257-266, 1979.[33] J. Kijowski. Unconstrained degrees of freedom of gravitational field and thepositivity of gravitational energy. In Gravitation, Geometry and Relativistic

240 Robert BartnikPhysics, LNP 212. Springer, 1984.[34] J. Lee and T. Parker. The Yamabe problem. Bull. AMS, 17:37-81, 1987.[35] A. Lichnerowicz. Spineurs harmonique. C.R. Acad. Sci. Paris Sér. A-B, 257:7-9, 1963.[36] J. Lohkamp. Scalar curvature and hammocks. Math. Ann., 313:385-407, 1999.[37] N. O Murchadha. Total energy momentum in general relativity. J. Math.Phys., 27:2111-2128, 1986.[38] J. M. Nester. The gravitational Hamiltonian. In F. J. Flaherty, editor, Asymptoticbehaviour of mass and space-time geometry (Oregon 1983), Lecture Notesin Physics 212, pages 155-163. Springer Verlag, 1984.[39] J. M. Nester. A gauge condition for orthonormal three-frames. J. Math. Phys.,30:624-626, 1988.[40] R. Penrose. Naked singularities. Ann. N. Y. Acad. Sci., 224:125-134, 1973.[41] R. Penrose. Quasi-local mass and angular momentum in general relativity.Proc. Roy. Soc. Lond. A, 381:53-63, 1982.[42] E. Woolgar R. Penrose, R.D. Sorkin. A positive mass theorem based on thefocusing and retardation of null geodesies, grqc/9301015, 1993.[43] D.C.Robinson. A simple proof of the generalisation of Israel's theorem. Gen.Rei. Grav., 8:695-698, 1977.[44] R. Schoen. Conformai deformation of a Riemannian metric to constant scalarcurvature. J. Diff. Geom., 20:479-495, 1984.[45] R. Schoen and S.-T. Yau. Proof of the positive mass theorem. Comm. Math.Phys., 65:45-76, 1979.[46] R. Schoen and S.-T. Yau. Proof of the positive mass theorem II. Comm. Math.Phys., 79:231-260, 1981.[47] R. Schoen and S.-T. Yau. The existence of a black hole due to the condensationof matter. Comm. Math. Phys., 90:575-579, 1983.[48] E. Schrödinger. Diracsches Elektron im Schwerfeld. Preuss. Akad. Wiss. Phys.-Math., 11:436-460, 1932.[49] B. Smith and G. Weinstein. On the connectedness of the space of initial datafor the Einstein equations. Electron. Res. Announc Amer. Math. Soc, 6:52-63,2000.[50] K. P. Tod. Some examples of Penrose's quasi-local mass construction. Proc.Roy. Soc. Lond. A, 388:457-477, 1983.[51] X. Wang. The mass of asymptotically hyperbolic manifolds. J. Diff. Geom.,57:273-300, 2001.[52] E. Wigner. The unreasonable effectiveness of mathematics in the natural sciences.Comm. Pure and Appi. Math., 13, 1960.[53] E. Witten. A simple proof of the positive energy theorem. Comm. Math. Phys.,80:381-402, 1981.

ICM 2002 • Vol. II • 241-255Geometry of Symplectic IntersectionsP. Biran*AbstractIn this paper we survey several intersection and non-intersection phenomenaappearing in the realm of symplectic topology. We discuss their implicationsand finally outline some new relations of the subject to algebraicgeometry.2000 Mathematics Subject Classification: 53D35, 53D40, 14D06, 14E25.Keywords and Phrases: Symplectic, Lagrangian, Algebraic variety.1. IntroductionSymplectic geometry exhibits a range of intersection phenomena that cannotbe predicted nor explained on the level of pure topology or differential geometry.The main players in this game are certain pairs of subspaces (e.g. Lagrangiansubmanifolds, domains, or a mixture of both) whose mutual intersections cannotbe removed (or reduced) via the group of Hamiltonian or symplectic diffeomorphisms.The very first examples of such phenomena were conjectures by Arnold inthe 1960's, and eventually established and further explored by Gromov, Floer andothers starting from the mid 1980s.The first part of the paper will survey several intersection phenomena and themathematical tools leading to their discovery. We shall not attempt to present themost general results and since the literature is vast the exposition will be far fromcomplete. Rather we shall concentrate on various intersection phenomena trying tounderstand their nature and whether there is any relations between them.The second part is dedicated to "non-intersections", namely to situationswhere the principles of symplectic intersections break down. In the case of Lagrangiansubmanifolds this absence of intersections is reflected in the vanishing ofa symplectic invariant called Floer homology. This vanishing when interpreted algebraicallyleads to restrictions on the topology of Lagrangian submanifolds. Asa byproduct we shall explain how these restrictions can be used to study someproblems in algebraic geometry concerning hyperplane sections and degenerations.* School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel. Email:biran@math.tau.ac.il

242 P. Biran2. Various intersection phenomenaIn this section we shall make a brief tour through the zoo of symplectic intersections,encountering three different species.Before we start let us recall two important notions from symplectic geometry.Let (M,(jj) be a symplectic manifold. A submanifold F c M is called Lagrangian ifdim F = | dim M and OJ vanishes on T(L). From now on we assume all Lagrangiansubmanifolds to be closed. The second notion is of Hamiltonian isotopies. Anisotopy of diffeomorphisms {ht : M —t M}o

2.2. Balls intersect ballsGeometry of Symplectic Intersections 243Denote by B 2n (R) the closed Euclidean ball of radius F, endowed with thestandard symplectic structure induced from R 2 ". Denote by CF" the complexprojective space, endowed with its standard Kahler form a, normalized so that/cpi f = 7T. The following obstruction for symplectic packing was discovered byGromov [22]:Theorem C. Let M be either B 2n (l) or 1then B Vl n B V2 ^ 0.Since symplectic embeddings are also volume preserving there is an obviousvolume obstruction for having B lfil fl B V2 = 0. However, volume considerationspredict an intersection only if R 2n + Ff" > 1 (moreover for volume preservingembeddings the latter inequality is sharp).When one considers embeddings of several balls things become more complicatedand interesting. Here results are currently available only in dimension 4.Theorem D. Let M be either F 4 (l) or CP 2 , and let B Vï ,...,B VN c M be theimages of symplectic embeddings tp k : B 4 (R) —t M, k = 1,...,N, of N balls ofthe same radius R. Then there exist i ^ j such that B ipi n B lfii ^ $ in each of thefollowing cases:1. N = 2 or 3 and R 2 > 1/2.2. N = 5 or 6 and R 2 > 2/5.3. N = 7 and R 2 > 3/8.4. N = 8 and R 2 > 6/17.Moreover all the above inequalities are sharp in the sense that in each case if theinequality on R is not satisfied then there exist symplectic embeddings tpi,..., tpMas above with disjoint images B iPl ,..., F^ c M.Statement 2 for N = 5 was proved by Gromov [22]. The rest was establishedby McDuff and Polterovich [27]. Let us mention that for N = 4 and any N > 9this intersection phenomenon completely disappears in the sense that an arbitrarilylargeportion of the volume of M can be filled by a disjoint union of N equal balls(see [27] for N = 4 and N = k 2 , and [5, 6] for the remaining cases).2.3. Balls intersect LagrangiansIt turns out that there exist (symplectically) irremovable intersections alsobetween contractible domains (e.g. balls) and Lagrangian submanifolds.Denote by RF" C CF" the Lagrangian n-dimensional real projective space(embedded as the fixed point set of the standard conjugation of CP n ). The followingwas proved in [7]:Theorem E. Let B v c CF" be the image of a symplectic embedding tp : B 2n (R) —tCP n . If R 2 > 1/2 then B v n RF" ^ 0. Moreover the inequality is sharp, namelyfor every R 2 < 1/2 there exists a symplectic embedding tp : B 2n (R) —t CP n whoseimage avoids RF".In fact this pattern of intersections occurs in a wide class of examples (see [7]):

244 P. BiranTheorem E'. Let (M,oj) be a closed Kahler manifold with [OJ] £ H 2 (M;Q) andn 2 (M) = 0. Then for every e > 0 there exists a Lagrangian CW-complex A f _ c(M,(jj) with the following property: every symplectic embedding tp : B 2n (e) —¥ (M,oj)must satisfy Image (ip) n A f _ ^ 0.By a Lagrangian CW-complex we mean a subspace A f _ c M which topologicallyis a CW-complex and the interior of each of its cells is a smoothly embeddeddisc of M on which u vanishes.2.4. Methods for studying intersectionsLagrangian intersections. The first systematic study of Lagrangian intersectionswas based on the theory of generating function [12, 26] (an equivalent theory wasindependently developed in contact geometry [13]). Gromov's theory of pseudoholomorphiccurves [22] gave rise to an alternative approach which culminated inwhat is now called Floer theory. Each of these theories has its own advantage. Floertheory works in larger generality and seems to have a richer algebraic structure, onthe other hand the theory of generating functions leads in some cases to sharperresults (see [20]).Since Floer theory will appear in the sequel, let us outline a few facts aboutit (the reader is referred to the works of Floer [16] and of Oh [29, 30] for details).Let (M,(jj) be a symplectic manifold and F 0 ,Fi c (M,oj) two Lagrangiansubmanifolds. In "ideal" situations Floer theory assigns to this data an invariantHF(L 0 ,Li). This is a Z2-vector space obtained through an infinite dimensionalversion of Morse-Novikov homology performed on the space of paths connectingF 0 to L\. The result of this theory is a chain complex CF(L 0 ,Li) whose underlyingvector space is generated by the intersection points F 0 fl L\ (one perturbsF 0 ,Fi so their intersection becomes transverse). The homology of this complexHF(L 0 ,Li) is called the Floer homology of the pair (F 0 ,Fi). The most importantfeature of FF(F 0 ,Fi) is its invariance under Hamiltonian isotopies: if L' 0 ,L'iare Hamiltonianly isotopie to F 0 ,Fi respectively, then HF(L' 0 ,L'i) = FF(F 0 ,Fi).From this point of view HF(L 0 , L\) can be regarded as a quantitative obstructionfor Hamiltonianly separating F 0 from L\. Indeed, the rank of HF(L 0 , L\) is a lowerbound on the number of intersection points of any pair of transversally intersectingLagrangians L' 0 , L\ in the Hamiltonian deformation classes of F 0 , L\ respectively.Let us explain the "ideal situations" in which Floer homology is defined. Firstof all there are restrictions on M : due to analytic difficulties manifolds are requiredto be either closed or to have symplectically convex ends (e.g. C", cotangent bundlesor any Stein manifold). More serious restrictions are posed on the Lagrangians. Forsimplicity we describe them only for the case when L\ is Hamiltonianly isotopieto F 0 . From now on we shall write L = L 0 and F' = L\. In Floer's originalsetting [16] the theory was defined under the assumption that the homomorphismA u : n 2 (M,L) —¥ R, defined by D >-¥ f D uj, vanishes. The reason for this comesfrom the construction of the differential of the Floer complex: the main obstructionfor defining a meaningful differential turns out to be existence of holomorphic discswith boundary on F or F'. These discs appear as a source of non-compactness ofthe space of solutions of the PDEs involved in the construction. Since holomorphic

Geometry of Symplectic Intersections 245discs must have positive symplectic area the assumption A u = 0 rules out theirexistence. Under this assumption Floer defined HF(L, L') and proved its invarianceunder Hamiltonian isotopies. Moreover he showed that HF(L, L) is isomorphic tothe singular cohomology F*(F;Z 2 ) of F. This together with the invariance give:Theorem F. Let (M,oj) be a symplectic manifold, either compact or with symplecticallyconvex ends. Let L c (M,oj) be a Lagrangian submanifold with A u = 0.Then for every Lagrangian L' which is Hamiltonianly isotopie to L and intersects Ltransversally we have: jpL

246 P. Biransymplectic packings in dimension higher than 4. The only packing obstructionsknown in these dimensions are described in Theorem C. Note that CF" admits fullpacking by N = k n equal balls [27], but it is unclear what happens for other valuesof N. In view of this and Theorem C, the first unknown case (for n > 3) is ofN = 2 n + 1 equal balls.The situation in dimension 4 is only slightly better. Except of CF 2 and a fewother rational surfaces no packing obstructions are known. It is known that for everysymplectic 4-manifold (M,oj) with [OJ] £ H 2 (M;Q) packing obstruction (for equalballs) disappear once the number of balls is large enough (see [6]), but nothing isknown when the number of balls is small. In fact even the case of one ball is poorlyunderstood(namely, what is the maximal radius of a ball that can be symplecticallyembedded in M). The reason here is that the methods yielding packing obstructionsstrongly rely on the geometry of algebraic and pseudo-holomorphic curves in themanifold. The problem is that most symplectic manifolds have very few (or noneat all) J-holomorphic curves for a generic choice of the almost complex structure.Thus, even in dimension 4 it is unknown whether or not packing obstructions is aphenomenon particular to a sporadic class of manifolds such as CF 2 .Is everything Lagrangian? Weinstein's famous saying could be relevant for theintersection described in Theorems C,D and E. In other words, it could be thatthese intersections are in fact Lagrangian intersections under disguise. To be moreconcrete, let | < R 2 < ^r and consider a Lagrangian LR lying on the boundarydB 2n (R). Is it possible to Hamiltonianly separate LR from itself inside F 2 "(l) ?If we can find a Lagrangian LR for which the answer is negative then thiswould strongly indicate that Theorem C is in fact a Lagrangian intersections result.Namely it would imply Theorem C for Fi = F2 under the additional assumptionthat tpi,tp2 are symplectically isotopie. A good candidate for LR seems to be thesplit torus dB 2 (x/R/n) x • • • x dB 2 (x/R/n) C dB 2n (R), but one could try otherLagrangians as well.Attempts to approach this question with traditional Floer homology fail. Thereason is that Floer homology is blind to sizes: both to the "size" of the LagrangianLR as well as to the "size" of the domain in which we work F 2 "(l). Indeed itis easy to see that HF(LR,LR) whether computed inside F 2 "(l) or in R 2 " is thesame, hence vanishes. The meaning of "sizes" can be made precise: the size of LRis encoded in its Liouville class, and the size of F 2 "(l) could be encoded here bythe action spectrum of its boundary.It would be interesting to try a mixture of symplectic field theory [19] withFloer homology. This would require a sophisticated counting of holomorphic discswith k punctures (for all k > 0), where the boundary of the discs goe to LR andthe punctures to periodic orbits on 9F 2 "(1).It is interesting to note that when the radii of the balls are not equal thingsbecome more complicated. Indeed suppose that R\ + F| > 1 and consider twoLagrangian submanifolds LR X , LR 2 lying on the boundaries of the balls B iPl , B V2 .Then clearly LR X and LR 2 can be disjoint even though the balls B iPl , B V2 do intersect(e.g. two concentric balls B iPl c B iP2 , where Fi < F2). It would be interesting

Geometry of Symplectic Intersections 247to see to which extent this mutual position can be detected on the level of theLagrangians LR X and LR 2 alone. Or, in more pictorial (but less mathematical)terms, do the Lagrangians LR X and LR 2 know that they lie one "inside" the other ?Returning to the case of equal balls, if the above plan is feasible, it wouldbe interesting to try similar approaches for more than two balls as described inTheorem D. A similar approach could be tried in the situation of Theorem E. Hereone could expect an irremovable intersection between a Lagrangian submanifoldLR C dB v and RF".Quantitative intersections. In contrast to the quantitative version of Lagrangianintersections given by Theorems B and F, Theorems CÊ provide only existence ofintersections. Is it possible to measure the size of these intersections ?More concretely, consider two balls B iPl ,B iP2 c F 2 "(l) with F 2 + Ff > 1 butwith R 2n + Ff" < 1 (so that Vo^F^) + Vo^F^) < 1). Is it possible to boundfrom below the size of B iPl n B V2 ?It is not hard to see that volume is a wrong candidate for the size since for everye > 0 there exist two such balls with Vo^F^ C\B V2 ) < e. Symplectic capacities seemalso to be inappropriate for this task. It could be that "size" should be replacedhere by a kind of "complexity" or a trade-off between capacity and complexity:namely if the intersection has large capacity (e.g. when B iPl c B V2 ) the complexityislow, and vice-versa. Note that in dimension 2 a possible notion of complexity ofa set is the number of connected components of its interior.A related problem is the following. Consider two symplectic balls B iPl , B V2 cCF" of radii R\, R 2 , where Ff+Ff. = 1. Assume further that Int (F^Jnlnt (B V2 ) =0. Theorem C implies that the balls must intersect hence the intersection occurson the boundaries: dB iPl n dB iP2 ^ 0. What can be said about the intersectiondB iPl n dB iP2 ^ 0, in terms of size, dynamical properties etc. ?It is easy to see that this intersection cannot be discrete. Moreover, an argumentbased on the work of Sullivan [37] shows that the intersection must contain atleast one entire (closed) orbit of the characteristic foliation of the boundaries of theballs (see [33] for a discussion on this point). Looking at examples however suggeststhat the number of orbits in the intersection should be much larger.The same problem can be considered also for (some of) the extremal casesdescribed in Theorem D. Similarly one can study the intersection dB v n RF" whereB v c CF" is a symplectic ball of radius R 2 = 1/2 whose interior is disjoint fromRF". It is likely that methods of symplectic field theory [19] could shed some lighton this circle of problems.Stable intersections. The problems described here come from Polterovich [32].Let (M, OJ) be a symplectic manifold and A c M a subset. We say that A has theHamiltonian intersection property if for every Hamiltonian diffeomorphism / wehave f(A) n A ^ 0. We say that A has the stable Hamiltonian intersection propertyif Osi x A c T* (S 1 ) x M has the Hamiltonian intersection property. Polterovich discoveredin [32] that if there exists a subset A c M with open non-empty complementand with the stable Hamiltonian property then the universal cover Ham(M, OJ) of the

248 P. Birangroup of Hamiltonian diffeomorphisms has infinite diameter with respect to Hofer'smetric. Note that when 7Ti(Ham(M,u;)) is finite the same holds also for Ham(M,w)itself. (See [32] for the details and references for other results on the diameter ofHam). This is applicable when (M,OJ) contains a Lagrangian submanifold A withHF(A,A) # 0, since then HF(0 S i x A, 0 S i x A) = (Z 2 ®Z 2 ) ® H F(A, A) # 0. Forexample, taking A = RF" c CF" Polterovich proved that diamHam(CF") = oo(for n= 1,2 the same holds for diaroHam(CF")).In view of the above the following question seems natural: does every closedsymplectic manifold contain a subset A with open non-empty complement and withthe stable Hamiltonian intersection property ? Note that besides Lagrangian submanifolds(with H F ^ 0) no other stable Hamiltonian intersection phenomena areknown. It would also be interesting to find out whether the intersections describedin Theorems C,D,E and especially E' continue to hold after stabilization.4. Intersections versus non-intersectionsIn contrast to cotangent bundles there are manifolds in which every compactsubset can be separated from itself by a Hamiltonian isotopy. The simplest exampleis C" : indeed linear translations are Hamiltonian, and any compact subset can betranslated away from itself. Clearly the same also holds for every symplectic manifoldof the type M x C by applying translations on the C factor. Note that manifoldsof the type M x C sometime appear in "disguised" forms (e.g. as subcriticai Steinmanifolds, see Cieliebak [14]).The "non-intersections" property has quite strong consequences on the topologyof Lagrangian submanifolds already in C". Denote by u; s td the standard symplecticstructure of C" and let A be any primitive of uj s td- Note that the restrictionA|T(L) °f A to any Lagrangian submanifold F c C" is closed. The following wasproved by Gromov in [22]:Theorem G. Let F c C" be a Lagrangian submanifold. Then the restriction of Xto L is not exact. In particular H 1^; R) ^ 0.Indeed if A were exact on F then A u : iT2(C n ,L) —t R must vanish, hence by-Theorem F it is impossible to separate F from itself by a Hamiltonian isotopy. Onthe other hand, as discussed above, in C" this is always possible. We thus get acontradiction. (Gromov's original proof is somewhat different, however a carefulinspection shows it uses the failure of Lagrangian intersections in an indirect way).Arguments exploiting non-intersections were further used in clever ways by Lalondeand Sikorav [25] to obtain information on the topology of exact Lagrangians incotangent bundles (see also Viterbo [42] for further results).An important property of symplectic manifolds W having the "non-intersections"property is the following vanishing principle: for every Lagrangian submanifoldL c W with well defined Floer homology we have HF(L, L) = 0. Applying thisvanishing to C" yields restrictions on the possible Maslov class of Lagrangian submanifoldsof C". (Conjectures about the Maslov class due to Audin appear alreadyin [1]. First results in this directions are due to Polterovich [31] and to Viterbo [41].The interpretation in Floer-homological terms is due to Oh [29]. Generalizations

Geometry of Symplectic Intersections 249to other manifolds appear in [2] and [11]. Finally, consult [21] for recent resultsanswering old questions on the Maslov class).4.1. Lagrangian embeddings in closed manifoldsThe ideas described above can be applied to obtain information on the topologyof Lagrangian submanifolds of some closed manifolds. Note that in comparisonto closed manifolds the case of C" can be regarded as local (Darboux Theorem). Ofcourse, "local" should by no means be interpreted as easy. On the contrary, characterizationof manifolds that admit Lagrangian embeddings into C" is completelyout of reach with the currently available tools.Below we shall deal with the "global" case, namely with Lagrangians in closedmanifolds. One (coarse) way to "mod out" local Lagrangians is to restrict to LagrangiansF with H\(L;Z) zero or torsion (so that by Theorem G they cannot liein a Darboux chart). The pattern arising in the theorems below is that under suchassumptions in some closed symplectic manifolds we have homological uniquenessof Lagrangian submanifolds. Let us view some examples.We start with CF". It is known that a Lagrangian submanifold F c CF"cannot have H\(L;Z) = 0 (see Seidel [39], see also [10] for an alternative proof).However, F c CF" may have torsion H\(L;Z) as the example RF" c CF" shows.Theorem H. Let L c CF" be a Lagrangian submanifold with H\(L; Z) a 2-torsiongroup (namely, 2H\(L;Z) = 0). Then:1. H*(L;Z 2 ) = F*(RF";Z 2 ) as graded vector spaces.2. Let a £ F 2 (CF";Z 2 ) be the generator. Then O\L £ H 2 (L;Z 2 ) generates thesubalgebra F even (F;Z 2 ). Moreover if n is even the isomorphism in 1 is ofgraded algebras.Statement 1 of the theorem was first proved by Seidel [39]. An alternativeproof based on "non-intersections" can be found in [8]. Let us outline the mainideas from [8]. Consider CF" as a hypersurface of CF" +1 . Let U be a smalltubular neighbourhood of CF" inside CF" +1 . The boundary dU looks like a circlebundle over CF" (in this case it is just the Hopf fibration). Denote by F^ —t Lthe restriction of this circle bundle to F c CF". A local computation shows thatU can be chosen so that F^ c CF" +1 \ CF" becomes a Lagrangian submanifold.(This procedure works whenever we have a symplectic manifold S embedded as ahyperplane section in some other symplectic manifolds M). The next observationis that Y'L C CF" +1 \ CF" is monotone and moreover its minimal Maslov numberNr L is the same as the one of F. Due to our assumptions on H\(L;Z) this numberturns out to satisfy Nr L > n + 1. The crucial point now is that HF(Y'L,Y'L) = 0.Indeed, the symplectic manifold CF" +1 \ CF" can be completed to be C" +1 whereFloer homology vanishes.Having this vanishing we turn to an alternative computation of HF(Y'L,Y'L).This computation is based on the theory developed by Oh [29] for monotone Lagrangiansubmanifolds. According to [29] Floer homology can be computed via aspectral sequence whose first stage is the singular cohomology of the Lagrangian.The minimal Maslov number has an influence both on the grading as well as on the

250 P. Birannumber of steps it takes the sequence to converge to HF. In our case we have a spectralsequence starting with H*(Y'L; Z 2 ) and converging to HF(Y'L,Y'L) = 0. A computationthrough this process together with the information that Nr L > n+1 makesit possible to completely recover F*(FL;Z 2 ). It turns out that H % (TL;Z 2 ) = Z 2for i = 0,1, n and n + 1, while H 1 (YL; Z 2 ) = 0 for all 1 of the circlebundle Y'L —ï L and noting that the second Stiefel-Whitney class of this bundle isnothing but the restriction O\L of the generator a £ F 2 (CF"; Z 2 ).Summarizing the proof, there are three main ingredients:1. Transforming the Lagrangian F into a related Lagrangian F^ living in a differentmanifold such that F^ can be Hamiltonianly separated from itself. Consequentlywe obtain HF(Y'L,Y'L) = 0.2. Relating HF(Y'L,Y'L) to H*(Y'L) via the theory of Floer homology (e.g. aspectral sequence).3. Passing back from H*(Y L ) to H*(L).Similar ideas work in various other cases (see [8]). For example, considerCF" x CF". This manifold has Lagrangians with H\(L;Z) = 0, e.g. CF" whichcan be embedded as the "anti-diagonal" {(z,w) £ CF" x CF"|w = z}.Theorem I. Let L c CF" x CF" be a Lagrangian with H X (L;Z) = 0. ThenH*(L;Z 2 ) — F*(CF";Z 2 ), the isomorphism being of graded algebras.Another application of this circle of ideas is for Lagrangian spheres. RecentlyLagrangian spheres have attracted special attention due to their relations to interestingsymplectic automorphisms [38, 39] and to symplectic Lefschetz pencils [15].Theorem J. 1) Let M be a closed symplectic manifold with n 2 (M) = 0, anddenote by m = dime M its complex dimension. If M x CF" (where m,n > 1) hasa Lagrangian sphere then m = n + 1 (mod 2n + 2).2) Let M = CF" x CF, m + n > 3, be endowed with the split symplectic form(n + 1)

Geometry of Symplectic Intersections 251example, Sommese proved that Abelian varieties of (complex) dimension > 2 havethis property (see [35] for more examples).Let us outline an alternative approach to this problem using symplectic geometry.Let X c CF^ be a smooth variety. Denote by X v c (CF^)* the dual variety(namely, the variety of all hyperplanes F £ (CP N )* that are non-transverse to X).Theorem K. Suppose that £ = X n F 0 C X is a smooth hyperplane section of Xobtained from a projective embedding X c CF^. Then either S has a Lagrangiansphere (for the symplectic structure induced from CP N ), or codimc(X v ) > 1.Here is an outline of the proof. Suppose that codimc(X v ) = 1. Choose ageneric line £ C (CF^)* intersecting X v transversely (and only at smooth points ofX v ). Consider the pencil {Xr\H}net parametrized by £. Passing to the blow-up Xof X along the base locus of the pencil we obtain a holomorphic map n : X —t £ asCF 1 . The critical values of n are in 1-1 correspondence with the point of £ n X v .Moreover, the fact that £ intersects X v transversely implies that n is a so calledLefschetz fibration, namely each critical point of n has non-degenerate (complex)Hessian (in other words, locally n looks like a holomorphic Morse function). Thecondition codimc(X v ) = 1 ensures that £ n X v ^ 0 hence at least one of the fibresof n is singular. Let X 0 be such a fibre and p £ X 0 a critical point of n. Theimportant point now is that the vanishing cycle (corresponding to p) that lies inthe nearby smooth fibre X f _ can be represented by a (smooth) Lagrangian sphere.By Moser argument all the smooth divisors in the linear system {X fl H} H€^CP Nyaresymplectomorphic. In particular S has a Lagrangian sphere too.The existence of Lagrangian vanishing cycles was known folklorically for longtime. Its importance to symplectic geometry was realized by Arnold [4], Donaldson[15] and by Seidel [38].Theorem K can be applied as follows: given a smooth variety S, use methodsof symplectic geometry to prove that S contains no Lagrangian spheres, say forany symplectic structure compatible with the complex structure of S. Then by-Theorem K the only chance for S to be a hyperplane section is inside a variety Xwith "small dual", namely codimc(X v ) > 1. Let us remark that smooth varietiesX c CF^ with codimc(X v ) > 1 are quite rare, and have very restricted geometry(see e.g. Zak [43] and Ein [17, 18]). Using the theory of "small dual varieties" wecan either rule out this case or get strong restrictions on the pair (X, S).Let us illustrate this on the example mentioned at the beginning of the section.Let S be an Abelian variety of complex dimension n>2. Note that S cannot havea Lagrangian sphere for any Kahler form. Indeed, if S had such a sphere then thesame would hold also for the universal cover of S which is symplectomorphic to C".But this is impossible in view of Theorem G. Thus if S is a hyperplane section ofX c CF^ then codimc(X v ) > 1. It is well known [24] that in this case X musthave rational curves (in fact lots of them). In particular n 2 (X) ^ 0. By Lefschetz'stheorem we get ^(S) ^ 0. But this is impossible since S is an Abelian variety. Wetherefore conclude that S cannot be a hyperplane section in any smooth variety X.An analogous (though symplectically more involved) argument should applyalsoto any algebraic variety S with c\ = 0 and 6i(S) ^ 0 (see [9]). An applicationof more refined symplectic tools (e.g. methods described in Section 4.1 above) can

252 P. Biranbe used to obtain many more examples.Here is another typical application: let C be a projective curve of genus > 0.It was observed by Silva [34] that C x CF" can be realized as a hyperplane sectionin various smooth varieties. Note that by Theorem J, C x CF" cannot have anyLagrangian spheres. It immediately follows that the only smooth varieties X thatsupport C x CF" as their hyperplane section must have small dual. For n < 5results of Ein [17, 18] make it even possible to list all such X's.We conclude with a remark on the methods. The symplectic approach outlinedabove gives coarser results. Indeed Sommese [35] provides examples of varieties thatcannot be ample divisors whereas the methods above only rule out the possibilityof being very ample. On the other hand the symplectic approach has an advantagein its robustness with respect to small deformations (see [9], cf. [36]).5.2. Degenerations of algebraic varietiesThe methods of the previous section can also be used to study degenerationsof algebraic varieties. Let Y be a smooth projective variety. We say that Y admitsa Kahler degeneration with isolated singularities if there exists a Kahler manifoldX and a proper holomorphic map n : X —t D to the unit disc D C C with thefollowing properties:1. Every 0 ^ t £ D is a regular value of n (hence, all the fibres X t = 7r -1 (i),t ^ 0, are smooth Kahler manifolds).2. 0 is a critical value of n and all the critical points of n are isolated.3. Y is isomorphic (as a complex manifold) to one of the smooth fibres of n, sayXt 0 ,to^0.As in the previous section this situation is related to symplectic geometrythroughthe Lagrangian vanishing cycle construction. As pointed out by Seidel [39]one can locally morsify each of the critical points in X 0 = 7r _1 (0) and then byapplyingMoser's argument obtain for each critical point of IT at least one Lagrangiansphere in the nearby fibre X f _. Since all the smooth fibres are symplectomorphic weobtain Lagrangian spheres also in Y.Applying results from Section 4 to this situation we obtain examples of projectivevarieties that do not admit any degeneration with isolated singularities. Forexample, let Y be any of the following:• CF", n > 2. Or more generally CF" x M, where M is a smooth variety withn 2 (M) = 0 and dime M ^ n + 1 (mod n + 1).• Any variety whose universal cover is C", (n > 2), or a domain in C".Then by the results in Section 4, Y has no Lagrangian spheres, hence does notadmit any degeneration as above. More examples can be found in [9].This point of view seems non-trivial especially when H n (Y;Z) = 0, wheren = dime Y. In these cases the vanishing cycles are zero in homology and it seemsthat there are no obvious topological obstructions for degenerating Y as above.^From the list above, the first non-trivial example should be CF" with n = odd > 3.It would be interesting to figure out to which extent the above statement could be

Geometry of Symplectic Intersections 253proved within the tools of pure algebraic geometry. Note that Lagrangian spheresare a non-algebraic object and it seems that their existence/non-existence cannotbe formalized in purely algebro-geometric terms.Another direction of applications should be to find an upper bound on thenumber of singular points of an algebraic variety X 0 that can be obtained from adegeneration of Y. Note that the vanishing cycles of different singular points ofX 0 are disjoint. Thus the idea here is to obtain an upper bound on the numberof possible disjoint Lagrangian spheres that can be embedded in Y. The simplesttest case here should be the quadric Q = {z 2 + • • • + z 2 +1 = 0} C CF" +1 , wheren > 2. Clearly Q can be degenerated to a variety X 0 with isolated singularities(e.g. to a cone over a smaller dimensional quadric). It seems reasonable to expectthat in every such degeneration the singular fibre X 0 will have only one singularpoint. Note that for n = even this easily follows from topological reason but it maynot be so when n = odd > 3 because H n (Q;Z) = 0. From a symplectic point ofview the above statement would follow if we could prove that every two Lagrangianspheres in Q must intersect. This is currently still unknown but there are evidencessupporting this conjecture [8]. It is likely that a refinement of the methods from [40]would be useful for this purpose. More generally, one could try to bound the numberof singular fibres in a degeneration of other hypersurfaces S c CF" +1 (in terms ofdeg(S) and n). See [8, 9] for the conjectured bounds.Acknowledgments. I am indebted to Leonid Polterovich for sharing with me hisinsight into symplectic intersections in general and especially regarding the problemspresented in Section 3. Numerous discussions with him have influenced myconceptions on the subject. I wish to thank him also for valuable comments onearlier drafts of the paper. I would like to thank also Jonathan Wahl who toldme about Sommese's work [35], Olivier Debarre for the reference to the works ofZak [43] and Ein [17,18], and Paul Seidel for useful discussions on symplectic aspectsof singularity theory and the material presented in section 5.References[1] M. Audin, Fibres normaux d'immersions en dimension double, points doublesd'immersions lagragiennes et plongements totalement réels. Comment. Math.Helv. 63 (1988), 593^623.[2] M. Audin, F. Lalonde & L. Polterovich, Symplectic rigidity: Lagrangian submanifolds.Holomorphic curves in symplectic geometry. Edited by M. Audin &J. Lafontaine. Progr. Math., 117. Birkhäuser 1994.[3] V. Arnold, The first steps of symplectic topology. Uspekhi Mat. Nauk 41 (1986),no. 6 (252), 3-18, 229.[4] V. Arnold, Some remarks on symplectic monodromy of Milnor fibrations. TheFloer memorial volume, 99^103, Progr. Math., 133, Birkhäuser 1995.[5] P. Biran, Symplectic packing in dimension 4, Geom. Funct. Anal. 7 (1997),420-437.

254 P. Biran[6] P. Biran, A stability property of symplectic packing, Invent. Math. 136 (1999),123-155.[7] P. Biran Lagrangian barriers and symplectic embeddings. Geom. Funct. Anal.11 (2001), 407-164.[8] P. Biran, Lagrangian non-intersections., in preparation.[9] P. Biran, Symplectic obstructions in algebraic geometry, in preparation.[10] P. Biran & K. Cieliebak, Symplectic topology on subcriticai manifolds. Comment.Math. Helv. 76 (2001), 712-753.[11] P. Biran & K. Cieliebak, Lagrangian embeddings into subcriticai Stein manifolds.Israel J. Math. 127 (2002), 221-244.[12] M. Chaperon, Quelques questions de géométrie symplectique. Bourbaki seminar,Astérisque, 105-6, 231-249, Soc. Math. France, Paris, 1983.[13] Y. Chekanov, Critical points of quasifunctions, and generating families of Legendrianmanifolds. Funct. Anal. Appi. 30 (1996), 118—128.[14] K. Cieliebak, Subcriticai Stein manifolds are split. Preprint.[15] S. Donaldson, Polynomials, vanishing cycles and Floer homology. Mathematics:frontiers and perspectives, 55-64, Amer. Math. Soc, 2000.[16] A. Floer, Morse theory for Lagrangian intersections. J. Differential Geom. 28(1988), 513-547.[17] L. Ein, Varieties with small dual varieties. I. Invent. Math. 86 (1986), 63-74.[18] L. Ein, Varieties with small dual varieties. II. Duke Math. J. 52 (1985), 895-907.[19] Y. Eliashberg, A. Givental & H. Hofer Introduction to symplectic field theory.G AFA 2000 (Tel Aviv, 1999). Geom. Funct. Anal. 2000, Special Volume, PartII, 560-673.[20] Y. Eliashberg & M. Gromov, Lagrangian intersection theory: finitedimensionalapproach. Geometry of differential equations, 27-118, Amer. Math.Soc. Transi. Ser. 2, 186, Amer. Math. Soc. 1998.[21] K. Fukaya, Y.-G. Oh, H. Ohta & K. Ono, Lagrangian intersection Floer theory- anomaly and obstruction. Preprint.[22] M. Gromov, Pseudoholomorphic curves in symplectic manifolds. Invent. Math.82 (1985), 307-347.[23] H. Hofer, Lagrangian embeddings and critical point theory. Ann. Inst. H.Poincaré Anal. Non Linéaire 2 (1985), 407-462.[24] S. Kleiman, About the conormal scheme. Complete intersections (Acireale,1983), 161-197, Lecture Notes in Math., 1092, Springer 1984.[25] F. Lalonde & J.-C Sikorav, Sous-variétés lagrangiennes et lagrangiennes exactesdes fibres cotangents. Comment. Math. Helv. 66 (1991), 18-33.[26] F. Laudenbach & J.-C Sikorav, Persistance d'intersection avec la section nulleau cours d'une isotopie hamiltonienne dans un fibre cotangent. Invent. Math.82 (1985), 349-357.[27] D. McDuff & L. Polterovich, Symplectic packings and algebraic geometry. Invent.Math. 115 (1994), 405-434.[28] D. McDuff and D. Salamon, Introduction to symplectic topology. Oxford MathematicalMonographs. Oxford University Press, New York, 1998.

Geometry of Symplectic Intersections 255[29] Y.-G. Oh, Floer cohomology, spectral sequences, and the Maslov class of Lagrangianembeddings. Internat. Math. Res. Notices 1996, 305-346.[30] Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudoholomorphicdisks. I. Comm. Pure Appi.[31] L. Polterovich, Monotone Lagrange submanifolds of linear spaces and theMaslov class in cotangent bundles. Math. Z. 207 (1991), 217-222.[32] L. Polterovich, Hofer's diameter and Lagrangian intersections. Internat. Math.Res. Notices 1998, 217-223.[33] L. Polterovich & K. F. Siburg, Lagrangian submanifolds in convex domains ofcotangent bundles. Preprint.[34] A. Silva, Relative vanishing theorems. I. Applications to ample divisors. Comment.Math. Helv. 52 (1977), 483-489.[35] A. Sommese, On manifolds that cannot be ample divisors. Math. Ann. 221(1976), 55-72.[36] A. Sommese, Nonsmoothable varieties. Comment. Math. Helv. 54 (1979), no.1, 140-146.[37] D. Sullivan, Cycles for the dynamical study of foliated manifolds and complexmanifolds. Invent. Math. 36 (1976), 225-255.[38] P. Seidel, Floer homology and the symplectic isotopy problem, PhD thesis, OxfordUniversity 1997.[39] P. Seidel, Graded Lagrangian submanifolds. Bull. Soc. Math. France 128 (2000),103-149.[40] P. Seidel, A long exact sequence for symplectic Floer cohomology. Preprint.[41] C. Viterbo, A new obstruction to embedding Lagrangian tori. Invent. Math.100 (1990), 301-320.[42] C. Viterbo, Exact Lagrange submanifolds, periodic orbits and the cohomologyof free loop spaces. J. Differential Geom. 47 (1997), 420-468.[43] F. Zak, Tangents and secants of algebraic varieties. Mathematical Monographs,127. American Mathematical Society, 1993.

ICM 2002 • Vol. II • 257-271Black Holes and the Penrose Inequalityin General RelativityHubert L. BravAbstractIn a paper [23] in 1973, R. Penrose made a physical argument that thetotal mass of a spacetime which contains black holes with event horizons oftotal area A should be at least -\/A/167r. An important special case of thisphysical statement translates into a very beautiful mathematical inequalityin Riemannian geometry known as the Riemannian Penrose inequality. Oneparticularly geometric aspect of this problem is the fact that apparent horizonsof black holes in this setting correspond to minimal surfaces in Riemannian 3-manifolds. The Riemannian Penrose inequality was first proved by G. Huiskenand T. Ilmanen in 1997 for a single black hole [17] and then by the author in1999 for any number of black holes [6]. The two approaches use two differentgeometric flow techniques. The most general version of the Penrose inequalityis still open.In this talk we will sketch the author's proof by flowing Riemannian manifoldsinside the class of asymptotically flat 3-manifolds (asymptotic to R 3at infinity) which have nonnegative scalar curvature and contain minimalspheres. This new flow of metrics has very special properties and simulatesan initial physical situation in which all of the matter falls into the blackholes which merge into a single, spherically symmetric black hole given bythe Schwarzschild metric. Since the Schwarzschild metric gives equality in thePenrose inequality and the flow decreases the total mass while preserving thearea of the horizons of the black holes, the Penrose inequality follows. We willalso discuss how these techniques can be generalized in higher dimensions.2000 Mathematics Subject Classification: 53, 83.Keywords and Phrases: Black holes, Penrose inequality, Positive masstheorem, Quasi-local mass, General relativity.1. IntroductionA natural interpretation of the Penrose inequality is that the mass contributedby a collection of black holes is (at least) y/A/16w, where A is the total area of theevent horizons of the black holes. More generally, the question "How much matter* Mathematics Department, 2-179, Massachusetts Institute of Technology, 77 MassachusettsAvenue, Cambridge, MA 02139, USA. E-mail: bray@math.mit.edu

258 Hubert L. Brayisin a given region of a spacetime?" is still very much an open problem [12]. In thispaper, we will discuss some of the qualitative aspects of mass in general relativity,look at examples which are informative, and sketch a proof of the RiemannianPenrose inequality.1.1. Total mass in general relativityTwo notions of mass which are well understood in general relativity are localenergy density at a point and the total mass of an asymptotically flat spacetime.However, defining the mass of a region larger than a point but smaller than theentire universe is not very well understood at all.Suppose (M 3 ,g) is a Riemannian 3-manifold isometrically embedded in a(3+1) dimensional Lorentzian spacetime. Suppose that M 3 has zero second fundamentalform in the spacetime. This is a simplifying assumption which allows usto think of (M 3 ,g) as a "t = 0" slice of the spacetime. The Penrose inequality(which allows for M 3 to have general second fundamental form) is known as theRiemannian Penrose inequality when the second fundamental form is set to zero.We also want to only consider (M 3 ,g) that are asymptotically flat at infinity,which means that for some compact set K, the "end" M 3 \K is diffeomorphic toR 3 \Bi(0), where the metric g is asymptotically approaching (with certain decayconditions)the standard flat metric % on R 3 at infinity. The simplest example ofan asymptotically flat manifold is (R 3 , #y) itself. Other good examples are the conformalmetrics (R 3 , u(a:) 4 (%), where u(x) approaches a constant sufficiently rapidlyat infinity. (Also, sometimes it is convenient to allow (M 3 ,g) to have multipleasymptotically flat ends, in which case each connected component of M 3 \K musthave the property described above.)The purpose of these assumptions on the asymptotic behavior of (M 3 ,g) atinfinity is that they imply the existence of the limitm=—lim / y2(gij,iVj - giijvj) dp, (1)where S a is the coordinate sphere of radius a, vis the unit normal to S a , and dß isthe area element of S a in the coordinate chart. The quantity m is called the totalmass (or ADM mass) of (M 3 ,g) (see [1], [2], [24], and [27]).Instead of thinking of total mass as given by equation 1, it is better to considerthe following example. Going back to the example (R 3 , u(x) 4 öij), if we suppose thatu(x) > 0 has the asymptotics at infinityu(a:) = a+6/|a:| + 0(l/|a:| 2 ) (2)(and derivatives of the C(l/|ar| 2 ) term are ö(l/|a:| 3 )), then the total mass of (M 3 ,g)ism = 2ab. (3)Furthermore, suppose (M 3 ,g) is any metric whose "end" is isometric to (R 3 \K,u(a:) 4 (5y), where u(x) is harmonic in the coordinate chart of the end (R 3 \K, öij)

Black Holes and the Penrose Inequality in General Relativity 259and goes to a constant at infinity. Then expanding u(x) in terms of spherical harmonicsdemonstrates that u(x) satisfies condition 2. We will call these Riemannianmanifolds (M 3 ,g) harmonically flat at infinity, and we note that the total massof these manifolds is also given by equation 3.A very nice lemma by Schoen and Yau is that, given any e > 0, it is alwayspossible to perturb an asymptotically flat manifold to become harmonically flatat infinity such that the total mass changes less than e and the metric changesless than e pointwise, all while maintaining nonnegative scalar curvature (discussedin a moment). Hence, it happens that to prove the theorems in this paper, weonly need to consider harmonically flat manifolds! Thus, we can use equation 3as our definition of total mass. As an example, note that (R 3 ,%) has zero totalmass. Also, note that, qualitatively, the total mass of an asymptotically flat orharmonically flat manifold is the 1/v rate at which the metric becomes flat atinfinity.1.2. Local energy densityAnother quantification of mass which is well understood is local energy density.In fact, in this setting, the local energy density at each point isM = ^ (4)where R is the scalar curvature of the 3-manifold (which has zero second fundamentalform in the spacetime) at each point. Thus, we note that (R 3 ,%) has zeroenergy density at each point as well as zero total mass. This is appropriate since(R 3 ,öij) is in fact a "t = 0" slice of Minkowski spacetime, which represents a vacuum.Classically, physicists consider p > 0 to be a physical assumption. Hence,from this point on, we will not only assume that (M 3 ,g) is asymptotically flat, butalso that it has nonnegative scalar curvature,R > 0. (5)This notion of energy density also helps us understand total mass better. Afterall, we can take any asymptotically flat manifold and then change the metric tobe perfectly flat outside a large compact set, thereby giving the new metric zerototal mass. However, if we introduce the physical condition that both metrics havenonnegative scalar curvature, then it is a beautiful theorem that this is in factnot possible, unless the original metric was already (R 3 ,%)! (This theorem isactually a corollary to the positive mass theorem discussed in a moment.) Thus,the curvature obstruction of having nonnegative scalar curvature at each point is avery interesting condition.Also, notice the indirect connection between the total mass and local energydensity.At this point, there does not seem to be much of a connection at all. Totalmass is the 1/v rate at which the metric becomes flat at infinity, and local energydensityis the scalar curvature at each point. Furthermore, if a metric is changedin a compact set, local energy density is changed, but the total mass is unaffected.

260 Hubert L. BrayThe reason for this is that the total mass is not the integral of the localenergy density over the manifold. In fact, this integral fails to take potential energyinto account (which would be expected to contribute a negative energy) as well asgravitational energy (discussed in a moment). Hence, it is not initially clear whatwe should expect the relationship between total mass and local energy density tobe, so let us begin with an example.1.3. Example using super harmonic functions in R 3Once again, let us return to the (R 3 ,u(a:) 4 (%) example. The formula for thescalar curvature isR = ^8u(x)- 5 Au(x). (6)Hence, since the physical assumption of nonnegative energy density implies nonnegativescalar curvature, we see that u(x) > 0 must be superharmonic (Au < 0).For simplicity, let's also assume that u(x) is harmonic outside a bounded set sothat we can expand u(x) at infinity using spherical harmonics. Hence, u(x) has theasymptotics of equation 2. By the maximum principle, it follows that the minimumvalue for u(x) must be a, referring to equation 2. Hence, 6 > 0, which impliesthat m > 0! Thus we see that the assumption of nonnegative energy density ateach point of (R 3 ,u(a:) 4 (%) implies that the total mass is also nonnegative, whichis what one would hope.1.4. The positive mass theoremMore generally, suppose we have any asymptotically flat manifold with nonnegativescalar curvature, is it true that the total mass is also nonnegative? Theanswer is yes, and this fact is know as the positive mass theorem, first proved bySchoen and Yau [25] in 1979 using minimal surface techniques and then by Witten[30] in 1981 using spinors.Theorem 1 (Schoen-Yau) Let (M 3 ,g) be any asymptotically flat, complete Riemannianmanifold with nonnegative scalar curvature. Then the total mass m > 0,with equality if and only if (M 3 ,g) is isometric to (R 3 ,ö).1.5. Black holesAnother very interesting and natural phenomenon in general relativity is theexistence of black holes. Instead of thinking of black holes as singularities in aspacetime, we will think of black holes in terms of their horizons. Given a surfacein a spacetime, suppose that it admits an outward shell of light. If the surface areaof this shell of light is decreasing everywhere on the surface, then this is called atrapped surface. The outermost boundary of these trapped surfaces is called theapparent horizon of the black hole. Apparent horizons can be computed based ontheir local geometry, and an apparent horizon always implies the existence of anevent horizon outside of it [15].Now let us return to the case we are considering in this paper where (M 3 ,g)is a "t = 0" slice of a spacetime with zero second fundamental form. Then it is a

Black Holes and the Penrose Inequality in General Relativity 261very nice geometric fact that apparent horizons of black holes intersected with M 3correspond to the connected components of the outermost minimal surface S 0 of(M 3 ,g).All of the surfaces we are considering in this paper will be required to besmooth boundaries of open bounded regions, so that outermost is well-defined withrespect to a chosen end of the manifold [6]. A minimal surface in (M 3 ,g) is a surfacewhich is a critical point of the area function with respect to any smooth variationof the surface. The first variational calculation implies that minimal surfaces havezero mean curvature. The surface S 0 of (M 3 ,g) is defined as the boundary of theunion of the open regions bounded by all of the minimal surfaces in (M 3 ,g). Itturns out that S 0 also has to be a minimal surface, so we call S 0 the outermostminimal surface.We will also define a surface to be (strictly) outer minimizing if everysurface which encloses it has (strictly) greater area. Note that outermost minimalsurfaces are strictly outer minimizing. Also, we define a horizon in our context tobe any minimal surface which is the boundary of a bounded open region.It also follows from a stability argument (using the Gauss-Bonnet theoreminterestingly) that each component of a stable minimal surface (in a 3-manifold withnonnegative scalar curvature) must have the topology of a sphere. Furthermore,there is a physical argument, based on [23], which suggests that the mass contributedby the black holes (thought of as the connected components of S 0 ) should be definedto be y/Ao/16n, where A Q is the area of S 0 . Hence, the physical argument that thetotal mass should be greater than or equal to the mass contributed by the blackholes yields that following geometric statement.The Riemannian Penrose InequalityLet (M 3 ,g) be a complete, smooth, 3-manifold with nonnegative scalar curvaturewhich is harmonically flat at infinity with total mass m and which has an outermostminimal surface S 0 of area A 0 . Then"^VïS-with equality if and only if(M 3 ,g) is isometric to the Schwarzschild metric (R 3 \{0},(1 + 5rn) 4

262 Hubert L. Braywe will require every surface (which could have multiple connected components) inthis paper to enclose all of the ends of the manifold except the chosen end.Other contributions on the Penrose Conjecture have also been made by Herzlich[16] using the Dirac operator which Witten [30] used to prove the positivemass theorem, by Gibbons [14] in the special case of collapsing shells, by Tod [29],by Bartnik [4] for quasi-spherical metrics, and by the author [7] using isoperimetricsurfaces. There is also some interesting work of Ludvigsen and Vickers [21] usingspinors and Bergqvist [5], both concerning the Penrose inequality for null slices ofa space-time.1.6. The Schwarzschild metricThe Schwarzschild metric (R 3 \{0}, (1 + ^T) 4 %)J referred to in the abovestatement of the Riemannian Penrose Inequality, is a particularly important exampleto consider, and corresponds to a zero-second fundamental form, space-like sliceof the usual (3+l)-dimensional Schwarzschild metric (which represents a sphericallysymmetricstatic black hole in vacuum). The 3-dimensional Schwarzschild metricshave total mass m > 0 and are characterized by being the only spherically symmetric,geodesically complete, zero scalar curvature 3-metrics, other than (R 3 ,#y).They can also be embedded in 4-dimensional Euclidean space (x,y,z,w) as the setof points satisfying \(x,y,z)\ = f^ + 2m, which is a parabola rotated around anS 2 . This last picture allows us to see that the Schwarzschild metric, which has twoends, has a Z 2 symmetry which fixes the sphere with w = 0 and \(x,y,z)\ = 2m,which is clearly minimal. Furthermore, the area of this sphere is 47r(2ro) 2 , givingequality in the Riemannian Penrose Inequality.2. The conformai flow of metricsGiven any initial Riemannian manifold (M 3 ,g 0 ) which has nonnegative scalarcurvature and which is harmonically flat at infinity, we will define a continuous,one parameter family of metrics (M 3 ,g t ), 0 < t < oo. This family of metricswill converge to a 3-dimensional Schwarzschild metric and will have other specialproperties which will allow us to prove the Riemannian Penrose Inequality for theoriginal metric (M 3 ,g 0 ).In particular, let S 0 be the outermost minimal surface of (M 3 ,g 0 ) with areaA 0 . Then we will also define a family of surfaces £(£) with S(0) = S 0 such that£(£) is minimal in (M 3 ,g t ). This is natural since as the metric gt changes, weexpect that the location of the horizon £(£) will also change. Then the interestingquantities to keep track of in this flow are A(t), the total area of the horizon £(£)in (M 3 ,g t ), and m(t), the total mass of (M 3 ,g t ) in the chosen end.In addition to all of the metrics gt having nonnegative scalar curvature, wewill also have the very nice properties thatA'(t) = 0, (8)m'(t) < 0 (9)

Black Holes and the Penrose Inequality in General Relativity 263for all t > 0. Then since (M 3 ,g t ) converges to a Schwarzschild metric (in anappropriate sense) which gives equality in the Riemannian Penrose Inequality asdescribed in the introduction,which proves the Riemannian Penrose Inequality for the original metric (M 3 ,g 0 ).The hard part, then, is to find a flow of metrics which preserves nonnegative scalarcurvature and the area of the horizon, decreases total mass, and converges to aSchwarzschild metric as t goes to infinity.2.1. The definition of the flowIn fact, the metrics gt will all be conformai to go- This conformai flow ofmetrics can be thought of as the solution to a first order o.d.e. in t defined byequations11, 12, 13, and 14. Letand UQ(X) = 1. Given the metric gt, defineg t = utixfgo (11)Ti(t) = the outermost minimal area enclosure of S 0 in (M 3 ,g t ) (12)where So is the original outer minimizing horizon in (M 3 ,go). In the cases in whichwe are interested, £(£) will not touch S 0 , from which it follows that £(£) is actuallya strictly outer minimizing horizon of (M 3 ,g t ). Then given the horizon £(£), definevt(x) such thatA go vt(x) = 0 outside £(£)vt(x) = 0 on £(f) (13)limôot^x) = -e - *and vt(x) = 0 inside £(£). Finally, given vt(x), defineut(x) = 1 + / v s (x)ds (14)Joso that ut(x) is continuous in t and has uo(x) = 1-Note that equation 14 implies that the first order rate of change of ut(x) isgiven by Vt(x). Hence, the first order rate of change of gt is a function of itself, ga,and vt(x) which is a function of go, t, and £(£) which is in turn a function of gt andS 0 . Thus, the first order rate of change of gt is a function of t, gt, go, and S 0 .Theorem 2 Taken together, equations 11, 12, 13, and 14 define a first ordero.d.e. in t for ut(x) which has a solution which is Lipschitz in the t variable, C 1 inthe x variable everywhere, and smooth in the x variable outside S(£). Furthermore,£(£) is a smooth, strictly outer minimizing horizon in (M 3 ,g t ) for all t > 0, and£(i 2 ) encloses but does not touch S(£i) for all t2 > h > 0.

264 Hubert L. Bray-Since Vt(x) is a superharmonic function in (M 3 ,g 0 ) (harmonic everywhereexcept on £(£), where it is weakly superharmonic), it follows that Ut(x) is superharmonicas well. Thus, from equation 14 we see that lim x _ >0O ut(x) = e - * andconsequently that Ut(x) > 0 for all t by the maximum principle. Then sinceR(g t ) = ut(x)- 5 (^8A go + R(go))ut(x), (15)it follows that (M 3 ,g t ) is an asymptotically flat manifold with nonnegative scalarcurvature.Even so, it still may not seem like gt is particularly naturally defined since therate of change of gt appears to depend on t and the original metric go in equation13. We would prefer a flow where the rate of change of gt can be defined purely asa function of gt (and So perhaps), and interestingly enough this actually does turnout to be the case. In section 2.4. we prove this very important fact and define anew equivalence class of metrics called the harmonic conformai class. Then oncewe decide to find a flow of metrics which stays inside the harmonic conformai classof the original metric (outside the horizon) and keeps the area of the horizon £(£)constant, then we are basically forced to choose the particular conformai flow ofmetrics defined above.Theorem 3 The function A(t) is constant in t and m(t) is non-increasing in t, forall t > 0.The fact that A'(t) = 0 follows from the fact that to first order the metricis not changing on £(£) (since Vt(x) = 0 there) and from the fact that to firstorder the area of £(£) does not change as it moves outward since £(£) is a criticalpoint for area in (M 3 ,g t ). Hence, the interesting part of theorem 3 is proving thatm'(t) < 0. Curiously, this follows from a nice trick using the Riemannian positivemass theorem, which we describe in section 2.3..Another important aspect of this conformai flow of the metric is that outsidethe horizon £(£), the manifold (M 3 ,g t ) becomes more and more spherically symmetricand "approaches" a Schwarzschild manifold (R 3 \{0}, s) in the limit as t goesto oo. More precisely,Theorem 4 For sufficiently large t, there exists a diffeomorphism 0, there exists a T such that for allt > T, the metrics gt and t(s) (when determining the lengths of unit vectors of(M 3 ,g t )) are within e of each other and the total masses of the two manifolds arewithin e of each other. Hence,mUtl = J_Theorem 4 is not that surprising really although a careful proof is reasonablylong.However, if one is willing to believe that the flow of metrics converges to aspherically symmetric metric outside the horizon, then theorem 4 follows from two

Black Holes and the Penrose Inequality in General Relativity 265facts. The first fact is that the scalar curvature of (M 3 ,g t ) eventually becomesidentically zero outside the horizon £(£) (assuming (M 3 ,g 0 ) is harmonically flat).This follows from the facts that £(£) encloses any compact set in a finite amount oftime, that harmonically flat manifolds have zero scalar curvature outside a compactset, that ut(x) is harmonic outside £(£), and equation 15. The second fact is thatthe Schwarzschild metrics are the only complete, spherically symmetric 3-manifoldswith zero scalar curvature (except for the flat metric on R 3 ).The Riemannian Penrose inequality, inequality 7, then follows from equation10 using theorems 2, 3 and 4, for harmonically flat manifolds [6]. Since asymptoticallyflat manifolds can be approximated arbitrarily well by harmonically flatmanifolds while changing the relevant quantities arbitrarily little, the asymptoticallyflat case also follows. Finally, the case of equality of the Penrose inequalityfollowsfrom a more careful analysis of these same arguments.2.2. Qualitative discussionThe diagrams above and below are meant to help illustrate some of the propertiesof the conformai flow of the metric. The above picture is the original metricwhich has a strictly outer minimizing horizon S 0 . As t increases, £(£) movesoutwards, but never inwards. In the diagram below, we can observe one of theconsequences of the fact that A(t) = A 0 is constant in t. Since the metric is notchanging inside £(£), all of the horizons £(s), 0 < s < t have area A 0 in (M 3 ,g t ).Hence, inside £(£), the manifold (M 3 ,g t ) becomes cylinder-like in the sense that itis laminated (meaning foliated but with some gaps allowed) by all of the previoushorizons which all have the same area A 0 with respect to the metric gt-

266 Hubert L. Bray(M 3 ,öo)Now let us suppose that the original horizon £ 0 of (M 3 ,g) had two components,for example. Then each of the components of the horizon will move outwardsas t increases, and at some point before they touch they will suddenly jump outwardsto form a horizon with a single component enclosing the previous horizon withtwo components. Even horizons with only one component will sometimes jump outwards,but no more than a countable number of times. It is interesting that thisphenomenon of surfaces jumping is also found in the Huisken-Ilmanen approach tothe Penrose conjecture using their generalized 1/H flow.2.3. Proof that m'(t) < 0The most surprising aspect of the flow defined in section 2.1. is that m'(t) < 0.As mentioned in that section, this important fact follows from a nice trick using theRiemannian positive mass theorem.The first step is to realize that while the rate of change of gt appears to dependon t and go, this is in fact an illusion. As is described in detail in section 2.4., therate of change of gt can be described purely in terms of gt (and So). It is also truethat the rate of change of gt depends only on g t and £(£). Hence, there is no specialvalue of t, so proving m'(t) < 0 is equivalent to proving ro'(O) < 0. Thus, withoutloss of generality, we take t = 0 for convenience.Now expand the harmonic function vo(x), defined in equation 13, using sphericalharmonics at infinity, to getv 0 (x) = -l + ^- i +o( T^) (17)Fl \\ x \ Jfor some constant c. Since the rate of change of the metric gt at t = 0 is given byvo(x) and since the total mass m(t) depends on the 1/r rate at which the metric gtbecomes flat at infinity (see equation 3), it is not surprising that direct calculationgives us thatm'(0) = 2(c-m(0)). (18)Hence, to show that ro'(0) < 0, we need to show thatc

Black Holes and the Penrose Inequality in General Relativity 267surface or that (M 3 ,g 0 ) have nonnegative scalar curvature. Hence, we quickly seethat equation 19 is a fairly deep conjecture which says something quite interestingabout manifold with nonnegative scalar curvature. Well, the Riemannian positivemass theorem is also a deep conjecture which says something quite interesting aboutmanifolds with nonnegative scalar curvature. Hence, it is natural to try to use theRiemannian positive mass theorem to prove equation 19.Thus, we want to create a manifold whose total mass depends on c fromequation 17. The idea is to use a reflection trick similar to one used by Bunting andMasood-ul-Alani for another purpose in [11]. First, remove the region of M 3 insideS(0) and then reflect the remainder of (M 3 ,g 0 ) through S(0). Define the resultingRiemannian manifold to be (M 3 ,g 0 ) which has two asymptotically flat ends since(M 3 ,g 0 ) has exactly one asymptotically flat end not contained by S(0). Note that(M 3 ,g 0 ) has nonnegative scalar curvature everywhere except on S(0) where themetric has corners. In fact, the fact that S(0) has zero mean curvature (since itis a minimal surface) implies that (M 3 ,g 0 ) has distributional nonnegative scalarcurvature everywhere, even on S(0). This notion is made rigorous in [6]. Thus wehave used the fact that S(0) is minimal in a critical way.Recall from equation 13 that vo(x) was defined to be the harmonic functionequal to zero on S(0) which goes to — 1 at infinity. We want to reflect vo(x) to bedefined on all of (M 3 ,g 0 ). The trick here is to define vo(x) on (M 3 ,g 0 ) to be theharmonic function which goes to —1 at infinity in the original end and goes to 1 atinfinity in the reflect end. By symmetry, VQ(X) equals 0 on S(0) and so agrees withits original definition on (M 3 ,g 0 ).The next step is to compactify one end of (M 3 , g 0 ). By the maximum principle,we know that VQ(X) > —1 and c > 0, so the new Riemannian manifold (M 3 , (vo(x) +l) 4 9o) does the job quite nicely and compactifies the original end to a point. Infact, the compactified point at infinity and the metric there can be filled in smoothly(using the fact that (M 3 ,g 0 ) is harmonically flat). It then follows from equation 15that this new compactified manifold has nonnegative scalar curvature since vo(x) +1is harmonic.The last step is simply to apply the Riemannian positive mass theorem to(M 3 , (vo(x) + l) 4 go)- R is not surprising that the total mass rô(0) of this manifoldinvolves c, but it is quite lucky that direct calculation yieldsm(0) =-4(c-m(0)), (20)which must be positive by the Riemannian positive mass theorem. Thus, we havethatro'(0) = 2(c-m(0)) = -^m(O) < 0. (21)2.4. The harmonic conformai class of a metricAs a final topic which is also of independent interest, we define a new equivalenceclass and partial ordering of conformai metrics. These new objects providea natural motivation for studying conformai flows of metrics to try to prove the

268 Hubert L. BrayRiemannian Penrose inequality. Let92 = u(x)^gi, (22)where #2 and gi are metrics on an n-dimensional manifold M n , n > 3. Then weget the surprisingly simple identity thatA gi (u 0 and u(x) > 0.Then from equation 23 we get the following lemma.Lemma 2 The relation > is reflexive and transitive, and hence is a partial ordering.Since y is defined in terms of superharmonic functions, we will call it the superharmonicpartial ordering of metrics on M n . Then it is natural to define the followingset of metrics.Definition 4 Define\d]s = {S I9h9}-This set of metrics has the property that if g £ [g]s, then [g]s C [g]sAlso, the scalar curvature transforms nicely under a conformai change of themetric. In fact, assuming equation 22 again,R(g 2 ) = u(x)~ ( -^)(^c n A gi + R(gij) u(x) (24)where c n =_ 9 '. This gives us the following lemma.

Black Holes and the Penrose Inequality in General Relativity 269Lemma 3 The sign of the scalar curvature is preserved pointwise by ~. That is, if92 ~ gi, then sgn(R(g 2 )(x)) = sgn(R(gi)(xj) for all x £ M n . Also, if g 2 > gi, andpi has non-negative scalar curvature, then g2 has non-negative scalar curvature.Hence, the harmonic conformai equivalence relation ~ and the superharmonicpartial ordering > are useful for studying questions about scalar curvature. Inparticular, these notions are useful for studying the Riemannian Penrose inequalitywhich concerns asymptotically flat 3-manifolds (M 3 ,g) with non-negative scalarcurvature. Given such a manifold, define m(g) to be the total mass of (M 3 ,g) andA(g) to be the area of the outermost horizon (which could have multiple components)of (M 3 ,g). Define P(g) = "p£L to be the Penrose quotient of (M 3 ,g).y'Ma)Then an interesting question is to ask which metric in [g]$ minimizes P(g).Section 2. of this paper can be viewed as an answer to the above question. Weshowed that there exists a conformai flow of metrics (starting with g 0 ) for which thePenrose quotient was non-increasing, and in fact this conformai flow stays inside[go]s- Furthermore, gt 2 £ [gt t ]s for all t 2 > t\ > 0. We showed that no matterwhich metric we start with, the metric converges to a Schwarzschild metric outsideits horizon. Hence, the minimum value of P(g) in [g]$ is achieved in the limit bymetricsconverging to a Schwarzschild metric (outside their respective horizons).In the case that g is harmonically flat at infinity, a Schwarzschild metric (outsidethe horizon) is contained in [g]s- More generally, given any asymptotically flatmanifold (M 3 ,g), we can use R 3 \B r (0) as a coordinate chart for the asymptoticallyflatend of (M 3 ,g) which we are interested in, where the metric #y approaches #yat infinity in this coordinate chart. Then we can consider the conformai metricCx4gc ={ 1 + W\) 9 (25)in this end. In the limit as C goes to infinity, the horizon will approach the coordinatesphere of radius C. Then outside this horizon in the limit as C goes to infinity,the function (1 + ßr) will be close to a superharmonic function on (M 3 ,g) and themetric gc will approach a Schwarzschild metric (since the metric g is approachingthe standard metric on R 3 ). Hence, the Penrose quotient of gc will approach(ltm)^1/ 2 , which is the Penrose quotient of a Schwarzschild metric.As a final note, we prove that the first order o.d.e. for {g t } defined in equations11, 12, 13, and 14 is naturally defined in the sense that the rate of change of gt is afunction only of gt and not of go or t. To see this, given any solution g t = Ut(x) 4 goto equations 11, 12, 13, and 14, choose any s > 0 and define üt(x) = ut(x)/u s (x)so thatg t = u t (xfg s (26)and u s (x) = 1. Then define vt(x) such thatAg B vt(x) = 0 outside £(£)vt(x) = 0 on £(f) (27)fmiaôoWâ:) = _ e -(*-»)

270 Hubert L. Brayand vt(x) = 0 inside £(£). Then what we want to show isü t (x) = 1+ / v r (x)dr (28)J sTo prove the above equation, we observe that from equations 23, 27, and 13 itfollows thatvt(x) = vt(x)u s (x) (29),u s (x) = e _s . Hence, sinceut(x) = u s (x) + / v r (x)dr (30)Jaby equation 14, dividing through by u s (x) yields equation 28 as desired. Thus, wesee that the rate of change of gt (x) at t = s is a function of v s (x) which in turn isjust a function of g s (x) and the horizon £(s). Hence, to understand properties ofthe flow we need only analyze the behavior of the flow for t close to zero, since anymetricin the flow may be chosen to be the base metric.References[1] R. Arnowitt, S. Deser and C. Misner, Coordinate Invariance and Energy Expressionsin General Relativity, Phys. Rev. 122 (1961), 997^1006.[2] R. Bartnik, The Mass of an Asymptotically Flat Manifold, Comm. Pure Appi.Math. 39 (1986), 661^693.[3] R. Bartnik, New Definition of Quasi-Local Mass, Phys. Rev. Lett. 62 (1989),2346.[4] R. Bartnik, Quasi-Spherical Metrics and Prescribed Scalar Curvature, J. Diff.Geom. 37 (1993), 31-71.[5] G. Bergquist, On the Penrose Inequality and the Role of Auxiliary SpinorFields, Class. Quantum Grav. 14 (1997), 2577^2583.[6] H. L. Bray, Proof of the Riemannian Penrose Inequality Using the PositiveMass Theorem, Jour. Diff. Geom. (to appear).[7] H. L. Bray, The Penrose Inequality in General Relativity and Volume ComparisonTheorems Involving Scalar Curvature, thesis, Stanford University, 1997.[8] H. L. Bray, F. Finster, Curvature Estimates and the Positive Mass Theorem,Communications in Analysis and Geometry (to appear).[9] H. L. Bray, K. Iga, A Nonlinear Property of Superharmonic Functions in R"and the Penrose Inequality in General Relativity, Communications in Analysisand Geometry (to appear).[10] H. L. Bray, R. M. Schoen, Recent Proofs of the Riemannian Penrose Conjecture,Current Developments in Mathematics 1999, edited by S.-T. Yau.[11] Bunting, Masood-ul-Alani, Non-Existence of Multiple Black Holes in AsymptoticallyEuclidean Static Vacuum Space-Time, Gen. Rei. and Grav., Vol. 19,No. 2, 1987.

[23;[24;[25;[26;[27[28;[29'[30^Black Holes and the Penrose Inequality in General Relativity 271[12] D. Christodoulou and S.-T. Yau, Some Remarks on the Quasi-Local Mass,Contemporary Mathematics 71 (1988), 9-14.[is; R. Geroch, Energy Extraction, Ann. New York Acad. Sci. 224 (1973), 108-17.G. Gibbons, Collapsing Shells and the Isoperimetric Inequality for Black Holes,Class. Quant. Grav. 14 (1997), 2905.[is; S. W. Hawking and G. F. R. Ellis, The Large-ScaleCambridge University Press, Cambridge, 1973.Structure of Space-Time,[ie; M. Herzlich, A Penrose-like Inequality for the Mass of Riemannian AsymptoticallyFlat Manifolds, Comm. Math. Phys. 188 (1997), 121.[ir G. Huisken and T. Ilmanen, The Inverse Mean Curvature Flow and the RiemannianPenrose Inequality, J. Diff. Geom (to appear).[is; G. Huisken and T. Ilmanen, The Riemannian Penrose Inequality, Int. Math.Res. Not. 20 (1997), 1045-1058.[19; G. Huisken and T. Ilmanen, A Note on Inverse Mean Curvature Flow, Proceedingsof the Workshop on Nonlinear Partial Differential Equations (SaitamaUniversity, Sept. 1997), available from Saitama University.[20; P. S. Jang and R. M. Wald, The Positive Energy Conjecture and the CosmicCensor Hypothesis, J. Math. Phys. 18 (1977), 41-44.[21 M. Ludvigsen and J. Vickers, 1983, J. Phys. A: Math. Gen. 16, 3349.[22; T. H. Parker and C. H. Taubes, On Witten's Proof of the Positive Energy-Theorem, Commun. Math. Phys. 84 (1982), 223-238.R. Penrose, Naked Singularities, Ann. New York Acad. Sci. 224 (1973), 125-134.R. Schoen, Variational Theory for the Total Scalar Curvature Functional forRiemannian Metrics and Related Topics, Topics in Calculus of Variations (M.Giaquinta, ed.) Lecture Notes in Math., 1365, 120-1-54, Springer, Berlin, 1987.R. Schoen and S.-T. Yau, On the Proof of the Positive Mass Conjecture inGeneral Relativity, Comm. Math. Phys. 65 (1979), 45-76.R. Schoen and S.-T. Yau, Positivity of the Total Mass of a General Space-Time,Phys. Rev. Lett. 43 (1979), 1457-1459.R. Schoen and S.-T. Yau, Proof of the Positive Mass Theorem II, Comm. Math.Phys. 79 (1981), 231-260.R. Schoen and S.-T. Yau, The Energy and the Linear Momentum of Space-Times in General Relativity, Comm. Math. Phys. 79 (1981), 47-51.K. P. Tod. Class. Quant. Grav. 9 (1992), 1581-1591.E. Witten, A New Proof of the Positive Energy Theorem, Comm. Math. Phys.80 (1981), 381-402.

ICM 2002 • Vol. II • 273-282Recent Progress in Kahler Geometry*Xiuxiong ChenÂbstractIn recent years, there are many progress made in Kahler geometry. Inparticular, the topics related to the problems of the existence and uniquenessof extremal Kahler metrics, as well as obstructions to the existence of suchmetrics in general Kahler manifold. In this talk, we will report some recentdevelopments in this direction. In particular, we will discuss the progressrecently obtained in understanding the metric structure of the infinite dimensionalspace of Kaehler potentials, and their applications to the problemsmentioned above. We also will discuss some recent on Kaehler Ricci flow.2000 Mathematics Subject Classification: 53, 35.Keywords and Phrases: Extremal Kahler metrics, Kähler-Einstein metrics,Holomorphic vector field, Holomorphic invariant, Kahler Ricci flow.In the last few years, we have witnessed a rapid progress in Kahler geometry.In particular, the topic related to the existence, to the uniqueness of extremal Kahlermetrics, and to obstructions to the existence of such metrics. In this talk, we willgive a brief survey of these exciting progress made in this direction.0.1. Some backgroundLet (M,OJ) be a polarized n-dimensional compact Kahler manifold, where OJ isa Kahler form on M. In local coordinates z\, • • • ,z n , we havenOJ = sfî ^2 9fjdz % Adz j > 0,where {gn} is a positive definite Hermitian matrix function. The Kahler conditionrequires that a; is a closed positive (l,l)-form. The Kahler metric corresponding toOJ is given byn9u = Yl 9aß dza ®dz$.a,/3=l*Paritally supported by NSF research grant DMS-0110321 (2001-2004).'Department of Mathematics, Princeton University. Department of Mathematics, Universityof Wisconsin at Madison, USA. E-mail: xiu@math.princeton.edu

274 Xiuxiong ChenFor simplicity, in the following, we will often denote by OJ the corresponding Kahlermetric. The Kahler class of w is its cohomology class [OJ] in H 2 (M, R). It followsfrom the Hodge-Dolbeault theorem that any other Kahler metric in the same Kahlerclass is of the formï£>0. . -, dz i dzi*J=Ifor some real valued function tp on M.Given a Kahler metric OJ, its volume form is—^ = (>/=!)" det (W) dz 1 A dz T A • • • A dz n Adz".Its Ricci (curvature) form is:Ric(oj) = y/^ÏRfj dwi dwj = —y^ïdd log det oj n .Note also that R(OJ) = g^Rfj corresponds to one half times the scalar curvatureas it is usually defines in Riemannian geometry. We say that the first Chern classof M is positive of negative definite, if there exists a real valued function ip onfi' 2 M such that Rß + dw,ß W . is, respectively, positive of negative definite. A Kahlermetric is Kähler-Einstein, if the Ricci form is proportional to the Kahler form bya constant factor. A Kahler metric is called extremal in the sense of E. Calabi [3],if it is a critical point of the functional / \Ric(oj)\ 2 oj n , or, equivalently, if theJ Mcomplex gradient vector field of the scalar curvature function g a/3 (OJ) d^' ^f- is aholomorphic vector field.0.2. Existence of extremal Kahler metricsIt is well known that a Kähler-Einstein metric satisfies a Monge-Ampere equationwhere [Ric(ojj] = X [OJ] andOJ^log det —- = —A tp+ h u0J nRic(oj) — À OJ = idd h w .In Calabi's work in the 1950s, he made conjectures about the existence of Kähler-Einstein metrics on compact Kahler manifolds with definite first Chern class. In1976, Aubin and Yau independently obtained existence when the first Chern classis negative. Around the same time, Yau proved also the existence of a Kähler-Einstein metric when the first Chern class vanishes. This is a celebrated work; andany Kahler manifold admit such a metric is called "Calabi-Yau" manifold. Thepositive case remains open, but significant progress has been made in the last twodecades. G. Tian proved in [29] the existence of Kähler-Einstein metrics on anycomplexsurface with positive first Chern class and reductive automorphism group.

Recent Progress in Kahler Geometry 275In 1997, Tian [30] proved that existence of Kähler-Einstein metrics with positivescalar curvature is equivalent to an analytic stability. It remains open how this analyticstability follows from certain algebraic stability in geometric invariant theory.The construction of complete non-compact Calabi-Yau manifolds has also enjoyeda good deal of success through the work of Calabi, Tian and Yau, Anderson,Kronheimer, LeBrun, Joyce and many others. These non-compact metrics are relatedto manifolds with G 2 and Spin(7) holonomy, which are important in M-theory.A lot of effort has also gone into constructing special or explicit examples ofKähler-Einstein metrics and extremal Kahler metrics. The same is true for hyperkaehlermetrics as well. Counter examples to the existence of extremal metrics havegiven by Levine, Burns-De Bartolomeis, and LeBrun.There has not been much progress made on the existence of extremal metricsin general. One of the possible reasons is the lack of maximum principle for nonlinearequations of 4th order. A general existence result, even in complex surfaces,will be highly interesting.0.3. ObstructionsIn 1983, A. Futaki [19] introduced a complex character T(X, [OJ]) on the complexLie algebra of all holomorphic vector fields X in M, depending only on theKahler class [OJ], and show that its vanishing is a necessary condition for the existenceof a Kähler-Einstein metric on the manifold. In 1985, E. Calabi[4] generalizedFutaki's result to cover the more general case of any extremal Kahler metric: thegeneralized Futaki invariant of a given Kahler class is zero or not, according towhether any extremal metric in that class has constant scalar curvature or not. S.Bando also obtained some generalizations of the Futaki invariant. More recently,a finite family of obstructions was introduced in [14]. For any holomorphic vectorfield X inducing the trivial translation on the Albanese torus there exists a complexvalued potential function 9x,u, uniquely determined up to additive constants, definedby the equation: LXOJ = \fi^ïdd9x( Here Lx denote the Lie derivative alongvector field X.). Now, for each k = 0,1, • • -n, define the functional c^s k (X,oj) by 1%(X,oj) = (n^k) [ e x oj nJM+ ((k+ l)Aöx RicM* A w""* -(n-k) B x Ric(u;) fc+1 A a;"-*- 1 ) .J M ^'Here and elsewhere, A w denotes the one half times the Laplacian-Beltrami operatorof the induced Riemannian structure OJ.The next theorem assures that the above integral gives rise to a holomorphicinvariant.1 This is a formula for canonical Kahler class. For general Kahler class, see [14].

276 Xiuxiong ChenTheorem 0.1. [14] The integral c^s k (X,oj) is independent of choices of Kahler metricsin the Kahler class [OJ], that is, c^s k (X,oj) = c^s k (X,oj') so long as the Kahlerforms OJ and OJ' represent the same Kahler class. Hence, the integral c^s k (X,oj) isa holomorphic invariant, which will be denoted by c^k(X, [OJ]). Note that % is theusual Futaki invariant.0.4. Uniqueness of extremal Kahler metricsWe now turn to the uniqueness of extremal metrics. In the 1950s, Calabi usedthe maximum principle to prove the uniqueness of Kähler-Einstein metrics whenthe first Chern class is non-positive. In 1987, Mabuchi introduced the "K-energy",which is essentially a potential function for the constant scalar curvature metricequation. Using the K-energy, he and Bando [2] proved that the uniqueness ofKähler-Einstein metric up to holomorphic transformations when the first Chernclass is positive. Recently, Tian and X. H. Zhu proved that the uniqueness ofKähler-Ricci Soliton on any Kahler manifolds with positive first Chern class.Theorem 0.2. [31], [32] The Kahler Ricci solitoti of a Kahler manifold M isunique modulo the automorphism subgroup Aut r (M); more precisely, if UJI,UJ 2 aretwo Kahler Ricci solitons with respect to a holomorphic vector field X, i.e., theysatisfiesRic(oJi) — oji = £x(oJi), where i = 1,2. (0.1)Then there are automorphism a in Aut°(M) and r in Aut r (M) such that a^X £r) r (M) and a*0J2 = T*a*oJi, where m(M) denotes the Lie algebra of Aut r (M). Infact, a^X lies in the center ofn r (M). Moreover, this vector field X is unique upto conjugations.Following a program of Donaldson (which will be explained in Subsection 0.7),we proved in 1998 [10] that the uniqueness for constant scalar curvature metric inany Kahler class when Ci < 0 along with some other interesting results:Theorem 0.3. [10] // the first Chern class is strictly negative, then the extremalKahler metric is unique in each Kahler class. Moreover, the K energy must have auniform lower bound if there exists an extremal Kahler metric in that Kahler class.Very recently, Donaldson proved a beautiful theorem which statesTheorem 0.4. [18] For algebraic Kahler class with no non-trivial holomorphicvector field, the constant scalar curvature metric is unique.The two theorems overlaps in a lot cases, but mutually non-inclusive.0.5. Lower bound of the K energyAccording to T. Mabuchi and S. Bando[2], the existence of a lower bound ofthe K energy is a necessary condition for the existence of Kähler-Einstein metricsin the first Chern class. Tian [30] showed that in a Kahler manifold with positivefirst Chern class and no non-trivial holomorphic fields, the Kähler-Einstein metric

Recent Progress in Kahler Geometry 277exists if and only if the Mabuchi functional is proper. When the first Chern class isnegative, making use of Tian's explicit formulation [30], a simple idea in [9] reducesa lower bound of the K energy to the existence of critical point for the followingconvex functional:n-lJ(tp) = - £ (p+1)i(n_ j> _ 1)! J v V Ricci(ojo) A a,?"*" 1 (ddtpf,where Ricci(oJo) < 0. In complex surfaces, we solves this existence problem completely,which leads to the following interesting result:Theorem 0.5. [9] Suppose dimV = 2 and Ci(V) < 0. For any Kahler class [OJQ],if 2 \ u p [wo] + [Ci(V)] > 0, then the K energy has a lower bound in thisKahler class.It will be very interesting to generalize this result to higher dimensional Kahlermanifold.0.6. Donaldson's programMabuchi defined in [25] a Weil-Petersson type metric on the space of Kahlerpotentials in a fixed Kahler class. Consider the space of Kahler potentials% = {tp | OJ V = OJ + Bdtp > 0, on M}.A tangent vector in % is just a real valued function in M.ip £ T v %, we define the length of this vector as:For any vectorW'PWl = Ifi'P 2d ßv-It is easy to see that the geodesic equation for this metric is* (t) 9 * dw a dw ß ~ 'pi 2where g a ß = go a ß + dw ß w^ > 0. It is first observed (cf. Semmes S. [27] )that onecan complexified the t variable, denoted it by w„+i. Then, the geodesic equationbecomes a homogenous complex Monge-Ampere equation:det/ f)2 \% i] + 7 T = =0, on S x M. (0.2)V dwißwj J {n+1){n+1)Here S = [0,1] x S 1 . It turns out that we don't need to restrict to this special case.For any Riemann surface S with boundary, and for any C°° map tpo from 9S to %,one can always ask the following existence problem:

278 Xiuxiong ChenQuestion 0.6. (Donaldson[16])For any smooth map tpo : 9S —t %, does thereexists a smooth map tp : S —t % which satisfies the Homogenous Monge-Ampereequation 0.2 such that tp = tpo in 9S ?Theorem 0.7. (X. Chen [10]) For any smooth map tpo : 9S —t %, there alwaysexists a C 1 ' 1 map tp : S —t'H which solves the Homogenous Monge-Ampere equation0.2 such that tp = tpo in 9S.An important conjecture by Donaldson in [16] was that the space of Kahlerpotentials is a metric space which is path-connected with respect to this Weil-Petersson metric. This conjecture was complete verified here.Theorem 0.8. [10] The space % is a genuine metric space: the minimal distancebetween any two Kahler metrics is realized by the unique C 1 ' 1 geodesic; and thelength of this geodesic is positive.Collaborating with E. Calabi, we proved the followingTheorem 0.9. [5] The space H in a fixed Kahler class is a non-positively curvedspace in the sense of Alexandrov: Suppose A, B, C are three smooth points in %and P\ is a geodesic interpolation point for 0 < X < 1 : the distance from P to Band C are respectively Xd(B, C) and (1 — X)d(B, c) 2 . Then the following inequalityholds:d(A,P x ) 2 < (1 - X)d(A,B) 2 + Xd(A,C) 2 - A • (1 - X)d(B,C) 2 .Theorem 0.10. [5] Given any two Kahler potentials tpi andtp2 in % and a smoothcurve tp(t),0

Recent Progress in Kahler Geometry 2790.7.1. The Calabi flow on Riemann surfacesThe Calabi flow is the gradient flow of the K energy and it is a 4th orderparabolic equation, proposed by E. Calabi in 1982. Namely, for a given Kahlermanifold (M, [OJ]), the Calabi flow was defined by^ ß = R(OJ,„) -£— / R(OJ) oj n .dt v v ;^ vol(M)J MThe local existence for this flow is known, while very little is known for its longterm existence since this is a 4th order flow. The only known result is in Riemannsurface where Chrusciel proved that the flow converges exponentially fast to a uniqueconstant scalar curvature metric. In 1998 [11], we gave a new proof based on somegeometrical integral estimate and concentration compactness principle. Now thechallenging question is:Question 0.11. Does the Calabi flow exists globally for any smooth initial metric?0.7.2. The Kahler Ricci flowA Kahler Ricci flowis defined byd_Ft OJ V = OJ V — Ric(oJ v ).This flow was first studied by H. D. Cao , following the work of R. Hamilton onthe Ricci flow 3 . Cao[6] proved that the flow always exists for all the time alongwith some other interesting results. It was proved by S. Bando [1] for 3-dimensionalKahler manifolds and by N. Mok [26] for higher dimensional Kahler manifolds thatthe positivity of bisectional curvature is preserved under the Kahler Ricci flow.The main issue here is the global convergence on manifold with positive bisectionalcurvature. In the work with Tian, we found a set of new functionals {E k }^=Qon curvature tensors such that the Ricci flow is the gradient like flow of thesefunctionals. On Kähler-Einstein manifold with positive scalar curvature, if theinitial metric has positive bisectional curvature, we can prove that these functionalshave a uniform lower bound, via the effective use of Tian's inequality. Consequently,we are able to prove the following theorem:Theorem 0.12. [14],[12] Let M be a Kähler-Einstein manifold with positive scalarcurvature. If the initial metric has nonnegative bisectional curvature and positiveat least at one point, then the Kahler Ricci flow will converge exponentially fast toa Kähler-Einstein metric with constant bisectional curvature.The above theorem in complex dimension 1 was proved first by Hamilton [21].B. Chow [15] later showed that the assumption that the initial metric has positive3 The Ricci flow was introduced by R. Hamilton [20] in 1982. There are extensive study in thissubject (cf. [22]) since his famous work in 3-dimensonal manifold with positive Ricci curvarure (cf.[22] for further references). Another important geometric flow is the so called "mean curvatureflow. " The codimension 1 case was studied extensively by G. Huisken and many others. Recently,there are some interesting progress made in codimension 2 case (cf. [7] [24] for further references).

280 Xiuxiong Chencurvature in S 2 can be removed since the scalar curvature will become positive afterfinite time anyway.Corollary 0.13. The space of Kahler metrics with non-negative bisectional curvatureis path-connected.Moreover, we can carry over the proof of Theorem 0.12 to a more general caseof Kahler orbifolds, for which we will not go into details here. Now the definitionof these functionals E k = E® — J k (k = 0,1, • • • ,n):Definition 0.14. For any k = 0,1, • • • ,n, we define a functional E® on % bywherec 1k = T71T7V / hu, I ^2 Ric(w)* Ao)' * ) A OJvoï(M) J MandRic(oj) - OJ = y/=îddh u , and [ (e h - - l)oj n = 0.J MDefinition 0.15. For each k = 0,1,2, • • • ,n — l, we will define J k:l0 as follows: Lettp(t) (t £ [0,1]) be a path from 0 to tp in %, we definei=0Put J n = 0 for convenience in notations.Note that E 0 is the well known K energy function introduced by T. Mabuchiin 1987. Direct computations lead toTheorem 0.16. For any k = 0,1, • • • ,n, we haveAJtWicK/Ac^-dEk k+1 f-vdt vol(M) J M\dtHere {tp(i){ is any path in %.Note that under the Kahler Ricci flow, these functionals essentially decreases!We then prove the derivative of these functionals along a curve of holomorphicautomorphisms give rise to a set of holomorphic invariants c^s k (k = 0,1, • • • , n) (cf.Theorem 0.1). In case of Kähler-Einstein manifolds, all these invariants vanishes.This give us freedom to re-adjust the flow so that the evolving Kahler potentialsare perpendicular to the first eigenspace of a fixed Kähler-Einstein metric. Thenwe will be able to show that the evolved volume form has a uniform lower bound.From this point on, the boot-strapping process will give us necessary estimates toobtain global convergence.

Recent Progress in Kahler Geometry 2810.8. Some new result with G. TianIn 2001, Donaldson proved the followingTheorem 0.17. [17] (Openness) For any smooth solution to the geodesic equationwith a disc domain, there are always exists a smooth solution to the geodesic equationif we perturb the boundary data in a small open set (of the given boundary data).This is somewhat surprising result since it is very hard to deform any solutionof a homogenous Monge-Ampere equation even locally. However, Donaldson wasable to make clever use of the Fredholm theory of holomorphic discs with totally realboundary in his proof. Then the problem of closed-ness becomes very important inlight of this theorem. Tian and I are able to establish the closed-ness in this case.Theorem 0.18. [13] (Closure property) The defamation of geodesic solution in thepreceding theorem is indeed closed, provided we allow solution to be smooth almosteverywhere.This is a deep theorem and we will not go into detail here due to the expositorynatureof this talk. However, this theorem, along with the ideas of proof, shall haveimplication in both geometry and other Monge-Ampere type equation in the future.References[i[2[3;[4;[5;[6;[7;[8"[9[10;[HS. Bando. On the three dimensional compact Kahler manifolds of nonnegativebisectional curvature. J. D. G., 19:283^297, 1984.S. Bando and T. Mabuchi. Uniquness of Einstein Kahler metrics modulo connectedgroup actions. In Algebraic Geometry, Advanced Studies in Pure Math.,1987.E. Calabi. Extremal Kahler metrics. In Seminar on Differential Geometry,volume 16 of 102, 259^290. Ann. of Math. Studies, University Press, 1982.E. Calabi. Extremal Kahler metrics, II. In Differential geometry and Complexanalysis, 96^114. Springer, 1985.E. Calabi and X. X. Chen. Space of Kahler metrics (II), 1999. to appear inJ.D.G.H. D. Cao. Deformation of Kahler metrics to Kähler-Einstein metrics on compactKahler manifolds. Invent. Math., 81:359^372, 1985.J.Y. Chen and J.Y. Li. mean curvature flow of surfaces in 4-manifolds, 2000.Adv. in Math, (to appear).J.Y. Chen and J.Y. Li. quaternionic maps between hyperkähler manifolds, 2000.J. D. G.X. X. Chen. On lower bound of the Mabuchi energy and its application. InternationalMathematics Research Notices, 12, 2000.X. X. Chen. Space of Kahler metrics. Journal of Differential Geometry,,56:189^234, 2000.X. X. Chen. Calabi flow in Riemann surface revisited: a new point of views.(6):276^297, 2001. "International Mathematics Research Notices".

282 Xiuxiong Chen[12] X. X. Chen and G. Tian. Ricci flow on Kähler-Einstein manifolds, 2000. Submittedto Annals of Mathematics.[13] X. X. Chen and G. Tian. Space of Kahler metrics (III), 2000. preprint.[14] X. X. Chen and G. Tian. Ricci flow on complex surfaces, 2002. Inventionesmathemticae.[15] B. Chow. The Ricci flow on the 2-sphere. J. Diff. Geom., 33:325^334, 1991.[16] S.K. Donaldson. Symmetric spaces, kahler geometry and Hamiltonian dynamics.Amer. Math. Soc. Transi. Ser. 2, 196, 13^33, 1999. Northern CaliforniaSymplectic Geometry Seminar.[17] S.K. Donaldson. Holomorphic Discs and the complex Monge-Ampere equation,2001. to appear in Journal of Sympletic Geometry.[18] S.K. Donaldson. Scalar curvature and projective embeddings, I, 2001. to appearin Journal of Differential Geometry.[19] A. Futaki. An obstruction to the existence of Einstein Kahler metrics. Inv.Math. Fase, 73(3):437^443, 1983.[20] R. Hamilton. Three-manifolds with positive Ricci curvature. J. Diff. Geom.,17:255^306, 1982.[21] R. Hamilton. The Ricci flow on surfaces. Contemporary Mathematics, 71:237^261, 1988.[22] R. Hamilton. The formation of singularities in the Ricci flow, volume IL Internat.Press, 1993.[23] J.Y. Li J.Y. Chen and G. Tian. two dimensional graphs moving by meancurvature flow, 2000. preprint.[24] Wang M. T. Mean curvature flow of surfaces in einstein four-manifolds. J. D.Geometry, 57(2):301^338, 2001.[25] T. Mabuchi. Some Sympletic geometry on compact kahler manifolds I. Osaka,J. Math., 24:227^252, 1987.[26] N. Mok. The uniformization theorem for compact Kahler manifolds of nonnegativeholomorphic bisectional curvature. J. Differential Geom., 27:179^214,1988.[27] S. Semmes. Complex monge-ampere and sympletic manifolds. Amer. J. Math.,114:495^550, 1992.[28] M. S. Struwe. Curvature flows on surfaces, August 2000. priprint.[29] G. Tian. On Calabi's conjecture for complex surfaces with positive first chernclass. Invent. Math., 101(1):101-172, 1990.[30] G. Tian. Kähler-Einstein metrics with positive scalar curvature. Invent. Math.,130:1^39, 1997.[31] G. Tian and X. H. Zhu. Uniqueness of kähler-Ricci Soliton, 1998. priprint.[32] G. Tian and X. H. Zhu. A new holomorphic invariant and uniqueness of Kähler-Ricci Soliton, 2000. priprint.

ICM 2002 • Vol. II • 283^291On the Sehrödinger FlowsWeiyue Ding*AbstractWe present some recent results on the existence of solutions of the Sehrödingerflows, and pose some problems for further research.2000 Mathematics Subject Classification: 53C44, 35Q55.Keywords and Phrases: Sehrödinger equation, Hamiltonnian flow, Kahlermanifold.1. IntroductionRecently the research on so-called Sehrödinger flow (or Sehrödinger map [l]-[4])has been carried out by several authors. This is an infinite-dimensional Hamiltonianflow defined on the space of mappings from a Riemannian manifold (M,g) into aKahler manifold (N,J,h), where g is the Riemannian metric on M, and h is theKahler metric on N, with J being the complex structure on N. This flow is definedby the following equationut = J(U)T(U), (1.1)where T(U) is the so-called tension field well-known in the theory of harmonic maps.In local coordinates, T(U) is given by-, • f)ii^ ßuT(U) x1 = A M u' tf - g aß Y) k }kX (u)" '''dx a dxßR.Here AM is the Laplace-Beltrami operator on M and Y % - k are the Christoffel symbolsof the Riemannian connection on N. Obviously, the Sehrödinger flows preserves theenergy E(u) of mapping u, i.e. E(u(tj) = E(u(0j), where'Peking Univ. and AMSS, CAS, China. E-mail: dingwy@math.pku.edu.cn

284 Weiyue DingSehrödinger flows are related to various theories in mechanics and physics. Awell-known and important example is the so-called Heisenberg spin chain system(also called ferromagnetic spin chain system [7]). This is just the Sehrödinger flowinto S 2 . Consider S 2 as the unit sphere in R 3 , then the equation for the system isgiven byut = u x Au.Note that, for a mapping u from M into S 2 ,J(u) = ux : T U S" —• T U S"is the standard complex structure on S 2 , and the tension field of the map u intoS 2 is given by T(U) = Au + |V«| 2 «. So, we have u x Au = J(U)T(U). Anotherinteresting example of the Sehrödinger flow is the anisotropic Heisenberg spin chainsystem, i.e. the Sehrödinger flow into Poincaré disk H( — l).Comparing to other geometric nonlinear evolutionary systems, such as the heatflow of harmonic maps (parabolic system) and wave maps (hyperbolic system), thestudy of Sehrödinger flows is still at the beginning stage. There are some remarkableresults on the existence of solutions for certain specific cases. E.g. for the Heisenbergspin chain system (N = S 2 ), Zhou et. al. [9] proved the global existence for M = S 1 ,and Sulem et. al. [10] proved the local existence for M = R m . There are somemore recent works, see [1], [3] and [11]. For the general case, however, it turns outthat even local existence is hard to prove. In this respect, a recent result obtainedby Youde Wang and this author ([4]) statesTheorem Let (M, g) be a closed Riemannian manifold of dimension m, andlet (N, J, h) is a closed Kahler manifold. If mo is the smallest integer greater thanro/2 (i.e. mo = [m/2] + 1), and «o £ W k,2 (M,N) for any k > mo + 3, thenthe initial value problem for (1.1) with initial value «o has a unique local solution.Moreover, if uo £ C°°(M,N), the local solution is C°° smooth.We remark that, the maximal existence time of the local solution in the aboveresult, depends only on the W mo+1 -norm of the initial map «o for any k. This iswhy we can get local existence in the C°° case. Also, for the existence part, theregularity of «o can be lowered to W k ' 2 with k > mo + 1, however we do not knowhow to get the uniqueness if k < mo + 3.In the following, we give a description of the proof of the above Theorem inSection 2 and 3. Then, in Section 4, we pose some important problems for futureresearch of the Sehrödinger flows.2. Some inequalities for Sobolev section norms ofmapsLet n : E —y M be a Riemannian vector bundle over M. Then we have thebundle A P T*M ® E —y M over M which is the tenser product of the bundle Eand the induced p-form bundle over M, where p = 1,2,• • • ,dim(Af). We defineF(A*>T*M ® E) as the set of all smooth sections of hPT*M ®E —y M. There

On the Sehrödinger Flows 285exists a induced metric on SPT*M®Ethat for any si, s 2 £ Y(i\PT*M ® E)—y M from the metric on T*M and E such(si,s 2 )= ^2 (si(eh,---,ei p ),s 2 (e h ,---,ei p )),Ìl

286 Weiyue DingIn order to prove Theorem we need to consider the problem of comparing theW k ' 2 norm with H k ' 2 norm of maps u £ C°°(M, N) (i.e. Sobolev section norm). Weassume that M is a closed Riemannian manifold and N is a compact Riemannianmanifold with or without boundary. It will be convenient to imbed N isometricallyinto some Euclidean space R K , and consider N as a compact submanifold of R K .Then the map u can be represented as u = (u 1 ,- • • ,u K ) with u % being globallydefinedfunctions on M. The we havek\\u\\ 2 Wk,2 = 5^ H^Hi 2 'ì=0whereWD'iiWh = 52 n DaU i L 2 ,and D denotes the covariant derivative for functions on M. The H k ' 2 norm ofu is defined similarly, only we need to replace D by V, where V is the covariantderivative for sections of the bundle u*(TN) over M ( For simplicity we also writeV« = Du). In [4] Ding and Wang obtained the following lemma.Lemma 2.2 Assume that k > ro/2. Then there exists a constantsuch that for all u £ C°°(M, N),C=C(N,k)kIIDull^-i.2 < C£||Vu||/r*-i.2, ( 2 - 3 )t=iandk||Vu|| ff *-i.2 < C52 \\Du\\\ vk^, 2 . (2.4)3. The proof of theoremIn this section we prove the local existence of smooth solutions for the initialvalue problem of the Sehrödinger flowu t = J(U)T(U), (u(;0) = Uo£C°°(M,N).{'We need to employ an approximate procedure and solve first the following perturbedproblemu t = er(u) + J(U)T(U), . >u(;0) = uo£C 00 [à(M,N),'where e > 0 is a small number.The advantage of (3.2) is that the equation with e > 0 is uniformly parabolic.Hence the initial value problem has a unique smooth solution u f _ £ C°°(M x[0,T f ],N) for some T f _ > 0. The problem is then to obtain a uniform positive

On the Sehrödinger Flows 287lower bound T of T f , and uniform bounds for various norms of u f _(t) in suitablespaces for t in the time interval [0,T). (Since we shall use L 2 estimates, the normsare W k ' 2 (M,N)— norms for all positive integer k.) Once we get these bounds it isclear that the u f _ subconverge to a smooth solution of (3.1) as e —¥ 0.Now let u = u f _ be a solution of (3.2), then it is easy to see that the energyE(u(tj) is uniformly bounded for t £ [0,T f ), i.e.E(u(tj) < E(u 0 ). (3.3)In the following we will make estimations on L 2 —norms of all covariant derivativesV k u (k = 2,3, •• •).Lemma 3.1 Let mo = [m/2] + 1, where [q] denotes the integral part of a positivenumber q, and let «o £ C°°(M,N). There exists a constant T = T(\\UO\\H'Ô+ 1 - 2 )> 0, independent of e £ [0,1], such that ifu £ C°°(M x [0,T e ]) is a solution of (3.1)with e £ (0,1] thenT f _>T(\\Vuo\\Hô.' 2 )andfor all k > mo-IIVu(t)\\ H k,2 < C(k, ||Vuo||/r*.0 t £ [0, T]Proof Fix a k > mo, and let I be any integer with 1 of length I, i.e. a = (cti, • • •, aj). Then we have for t 0 always holds, such that(b, c,d,e) = h > • • • > Je, I + 1 > 3ì > 1, ii + • • • + je = l + 3, s > 3. (3.7)

288 Weiyue DingFor the first term in the right hand side of (3.5), we may use the equation (3.2) togetVaViVt« = V a Vi(er(«) + J(U)T(U))= eVaViVfcVfc« + J(«)VaViVfcVfc« (3.8)where we have used the integrability of the complex structure J of the Kahlermanifold N. By exchanging the orders of covariant differentiation as above, we getfrom (3.5) and (3.8)VtVaVj« = eVfcVfcVaVj« + J(«)VfcVfcV a Vi« + Qwhere Q satisfies (3.6-3.7). Substituting this into (3.4) and integrating by part wethen have^l|VaV,«||l 2= / (-e|VV a Vi«| 2 -(VfcVaVi«, J(«)VfcVaVi«) + (VaVi«,Q)).J MNote that the first integrand is non-positive and the second vanishes, so we have by(3.6)IVaVé«!! 2 , < c(i,M)52 f |v' +1 u||VJ Mand consequentlyd_dtV l+1 u\\ 2 L2

On the Sehrödinger Flows 289(2) if 3i < hwhere A =A(m,l)./

290 Weiyue Dingwhich satisfies the estimates in Lemma 3.1. It follows from Proposition 2.2 that,for any k > 0 and e £ (0,1], there holdsmax \\u t \\ W k.2 {M) v 2 (M,N), we may alwaysselect a sequence of C°° maps from M into N, denoted by u»o, such thatUio —• «o in W k ' 2 ,as i —^ oo.This together with the definition of covariant differential leads to||V«jo||iî fe -i. 2 —H|Vuo||iffc-i.2, as i —¥ oo.Thus, there exists a unique, smooth solution m, defined on time interval [0,TJ, ofthe Cauchy problem (3.1) with « 0 replaced by «,o- Furthermore, it is not difficultto see from the arguments in Lemma 3.1 that if i is large enough, then there existsa uniform positive lower bound of T t , denoted by T, such that the following holdsuniformly with respect to large enough i:sup \\Vui(t)\\ H k-i,2te[o,T]< C(T, ||Vuo||ijfc-i.2).It follows from Lemma 2.2 and the last inequality thatsup ||I?«j(t)|| H /fe-i,2 2 (M,Nj)upon extracting a subsequence and re-indexing if necessary. It is easy to verify that« is a strong solution to (3.1) (see [4]).Remark For the Sehrödinger flow from an Euclidean space into a Kahlermanifold, in [4] we obtained similar local existence results.4. Some problems1. For the one-dimensional case, i.e. dim M = 1, we conjecture the Sehrödingerflows should exist globally whenever the target N is a compact Kahler manifold.This is still open, and is supported by the result with N being Hermitian locallysymmetric([ll]).

On the Sehrödinger Flows 291The result by Terng and Uhlenbeck [2] shows that for some special targets (e.g.complex Grassmannians), the Sehrödinger flows are bi-Hamiltonnian integrable systems.In their work, they assume that M = R 1 , and their result can be generalizedto compact Hermitian symmetric spaces (cf. [12]). An interesting open problem is,for these special targets, whether or not the Sehrödinger flows are bi-Hamiltonniansystems if M = S 1 .2. For higher dimensional cases, i.e. dim M > 2, we believe that the Sehrödingerflow may develop finite-time singularities. There are however no such examplesknown by now.3. All present results in the study of the Sehrödinger flows depend on theglobal estimates for the solutions. We do not know if one can find some kind oflocal estimates for the solutions. It has been well known from the research of variousgeometric flows that local estimates are important for the analysis of singularities.It is therefore desirable to develop some new methods to attack the question beforeany serious advance can be made for the study of the Sehrödinger flows.References[i[2[3;[4;[5;[6;[7;[s;[9;[10;[n[12;W. Y. Ding and Y. D. Wang, Sehrödinger flows of maps into symplectic manifolds,Science in China A, 41(7)(1998), 746^755.C. T. Terng and K. Uhlenbeck, Sehrödinger flows on Grassmannians, math.DG/9901086.N. Chang, J. Shatah and K. Uhlenbeck, Sehrödinger maps, Comm. Pure Appi.Math., 53(2000), 590^602.W. Y. Ding and Y. D. Wang, Sehrödinger flows into Kahler manifolds, Sciencein China A, 44(11)(2001), 1446^1464.L. D. Landau and E. M. Lifshitz, On the theory of the dispersion of magneticpermeability in ferromagnetic bodies, Phys. Z. Sowj. 8(1935), 153; reproducedin Collected Papers of L. D. Landau, Pergaman Press, New York, 1965, 101-114.T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations,Springer-Verlag, Berlin-Heidelberg-New York, 1982.J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math.Soc, 20 (1988), 385^524.L. Fadeev and L. A. Takhatajan, Hamiltonian Methods in the Theory of Solitons,Springer-Verlag, Berlin-Heidelberg-New York, 1987.Y. Zhou, B. Guo and S. Tan, Existence and uniqueness of smooth solution forsystem of ferromagnetic chain, Science in China A, 34(1991), 257^266.P. Sulem, C. Sulem and C. Bardos, On the continuous limit for a system ofclassical spins, Commun. Math. Phys., 107 (1986), 431-454.P. Pang, H. Wang, Y. D. Wang, Sehrödinger flow on Hermitian locally symmetricspaces, to appear in Comm. Anal. Geom. .B. Dai, Ph. D. dissertation of NUS (2002).

ICM 2002 • Vol. II • 293^302Differential Geometryvia Harmonic FunctionsP.Li-;*AbstractIn this talk, I will discuss the use of harmonic functions to study the geometryand topology of complete manifolds. In my previous joint work with Luen-fai Tarn, wediscovered that the number of infinities of a complete manifold can be estimated bythe dimension of a certain space of harmonic functions. Applying this to a completemanifold whose Ricci curvature is almost non-negative, we showed that the manifoldmust have finitely many ends. In my recent joint works with Jiaping Wang, wesuccessfully applied this general method to two other classes of complete manifolds.The first class are manifolds with the lower bound of the spectrum Ai(A-f) > 0 andwhose Ricci curvature is bounded byRie M >Ai(A-f).TO — 1The second class are stable minimal hypersurfaces in a complete manifold with nonnegativesectional curvature. In both cases we proved some splitting type theoremsand also some finiteness theorems.2000 Mathematics Subject Classification: 53C21, 58J05.Keywords and Phrases: Harmonic function, Ricci curvature, Minimal hypersurface,Parabolic manifold.1. IntroductionIn 1992, the author and Luen-fai Tam [12] discovered a general method todetermine if a complete, non-compact, Riemannian manifold have finitely manyends.An end is simply defined to be an unbounded component of the complimentof a compact set in the manifold. If the number of ends is finite, their techniquealso provides an estimate on the number of ends. In particular, they applied thismethod to prove that a certain class of manifolds must have finitely many ends.*Department of Mathematics, University of California, Irvine, CA 92697-3875, USA. E-mail:pli@math.uci.edu

294 P. LiTheorem 1 (Li-Tarn). Let M mbe a complete, non-compact, manifold withRìCM(X)> —k(r(x)),where k(r) is a continuous non-increasing function satisfyingj.ro-1 ^r~j ^r oThen there exists a constant 0 < C(m, k) < oo depending only on m and k, suchthat, M has at most C(m, k) number of ends.Since a manifold with non-negative Ricci curvature will satisfy the hypothesis,this theorem can be viewed as a perturbed version of the splitting theorem [4] ofCheeger-Gromoll. A weaker version of the above theorem for manifolds with nonnegativeRicci curvature outside a compact set was also independently proved byCai [1].In some recent work of Jiaping Wang and the author, they successfully appliedthe general theory of determining the number of ends to other situations. Thepurpose of this note is to give a quick overview of the theory and its applicationsto manifolds with positive spectrum and minimal hypersurfaces.2. General theoryThroughout this article, we will assume that (M m , ds 2 M) is an m-dimensional,complete, non-compact Riemannian manifold without boundary. In terms of localcoordinates (x\,x 2 ,... ,x m ), if the metric is given by/Vf — Qì'i ^"^ï£vX j ythen the Laplacian is defined byA- — — ( r—y/g dxi V V1 dxjwhere (g %: >) = (#y) _1 and g = det(^y). A function is said to be harmonic on M ifit satisfies the Laplace equationA/(*) = 0for all x £ M.In order to state the general theorem, it is necessary for us to define thefollowing spaces.Definition 1. Letn D (M) = {/1 A/ = 0, U/H«, < oo, /" |V/| 2 < oo}JMbe the space of bounded harmonic functions with finite Dirichlet integral defined onM.

Differential Geometry via Harmonic Functions 295Definition 2. Letn + (M) = ({f\Af = 0,f>0})be the space spanned by the set of positive harmonic functions defined on M.Definition 3. Let%'(M) = ({/ | A/ = 0, bounded from one side on each end})be the space spanned by the set of harmonic functions defined on M, which has theproperty that each one is bounded either from above or below on each end.Observe that these spaces are monotonically contained in each other, i.e.,n D (M) cn+(M) cn'(M).Yet us also recalled the following potential theoretic definition.Definition 4. An end E of M is non-parabolic if it admits a positive Green'sfunction with Neumann boundary condition on dE. Otherwise, E is said to beparabolic.It is important to note that if M has at least one non-parabolic end, then Madmits a positive Green's function. In this case, we say that M is non-parabolic.The interested reader can refer to [11] for more detail descriptions. Let us now statethe general theorem in [12].Theorem (Li-Tam). Let M be a complete, non-compact manifold without boundary.Then there exists a subspace K. C %'(M), such that, dim£ is equal to thenumber of ends of M.Moreover, if M is non-parabolic , then the subspace K. can be taken to be in% + (M). Also there exists another subspace KM C %D(M), such that, dimK-N isequal to the number of non-parabolic ends of M.At this point, it is important to point out that even though an estimate onthe dimension of the spaces %'(M), % + (M), or %D(M) will imply an estimate onthe number of ends of corresponding type, however, in general, these spaces can bebigger than K. or KM- Hence to effectively use the above theorem, one should usethe constructive argument in the proof of the theorem to give an estimate on K. andKM directly. Indeed, this was the case in the proof of Theorem 1. This is also truefor the two applications stated in the subsequence sections.3. Manifolds with positive spectrumA complete manifold (M, ds 2 M) is conformally compact if M is topologically amanifold with boundary given by DM. Moreover, there is a background metric dsôn (M, DM) such thatds%i = p dso,

296 P. Liwhere p is a defining function for DM satisfying the conditionsandp = 0 on dMdp#0on ÔM.A direct computation reveals that the sectional curvature, KM, of the completemetric ds 2 has asymptotic value given byK M ~ -\dp\ 2 ,near dM. Hence if (M,ds 2 M) is also assumed to be Einstein withRìCM = — (m - 1),thenK M (x) ~ -1,as x —ï oo.In 1999, Witten-Yau [19] proved a theorem concerning the AdS/CFT correspondence,which effectively ruled out the existence of worm holes. It is also a veryinterestingtheorem in Riemannian geometry.Theorem (Witten-Yau). Let M m be a conformally compact, Einstein manifoldof dimension at least 3. Suppose the boundary dM of M has positive Yamabeconstant, thenH m - 1 (M,Z) = 0.In particular, this implies that dM is connected and M must have only 1 end.Shortly after, Cai-Galloway [2] relaxed the assumption of Witten-Yau by assumingthe boundary dM has non-negative Yamabe constant. We would also liketo point out that by a theorem of Schoen [17], a compact manifold has non-negativeYamabe constant is equivalent to the fact that it is conformally equivalent to amanifold with non-negative scalar curvature.In his Stanford thesis, X. Wang [18] generalized the Witten-Yau, Cai-Gallowaytheorem by studying L 2 harmonic 1-forms.Theorem (Wang). Let M m be a conformally compact manifold of dimension atleast 3. Suppose the Ricci curvature of M is bounded byRìCM > — (m — 1)and the lower bound of the spectrum of the Laplacian Ai (M) has a positive lowerbound given byXi(M)> (m-2),

then eitherorDifferential Geometry via Harmonic Functions 297(1) M has no non-constant L 2 -harmonic 1-forms, i.e.,H 1 (L 2 (M)) = 0;(2) M = R x N with the warped product metricds 2 M = dt 2 + cosh" tds 2 N,where (N,ds 2 N) is a compact manifold with RìCM > —(m — 2). Moreover, Ai (M) =m — 2.To see that this is indeed a generalization of the theorems of Witten-Yau andCai-Galloway, one uses a theorem of Mazzeo [16] asserting that on a conformallycompact manifoldH 1 (L 2 (Mj)-H 1 (M,dM).By a standard exact sequence argument, the conclusion that H 1 (L 2 (Mj) = 0 impliesthat M has only 1 end. In addition to this, one also uses a theorem of Lee [10] givinga lower bound on Ai for conformally compact, Einstein manifold with non-negativeYamabe constant on dM.Theorem (Lee). Let M be a conformally compact, Einstein manifold withRìCM = — (TO — 1).Suppose that dM has non-negative Yamabe constant, thenXi(M)>^l.! — 1 ^2Sincem 4 ' > m — 2, Wang's theorem implies the theorems of Witten-Yauand Cai-Galloway. Observe that the warped product case in Wang's theorem hasnegative Yamabe constant on dM.At this point, let us also recall a theorem of Cheng [5] stating that:Theorem (Cheng). Let M be a complete manifold withRìCM > —(TO — 1),then(m - l) 2Ai(Af)

298 P. LiTheorem 2 (Li-Wang). Let M m be a complete manifold with dimension m > 3.Suppose the Ricci curvature of M is bounded byandRìCM > —(TO — 1)Ai(Af)>m-2,then either(1) M has only 1 end with infinite volume;or(2)M = R x N with the warped product metricds 2 M = dt 2 + cosh" tds 2 N,where (N,ds 2 N) is compact with RìCM > —(TO — 2). Moreover, Xi(M) = m — 2.It is worth noting that this theorem implies that when the lower bound forAi (M) of Cheng is achieved, then either(Y)M has only 1 end with infinite volume,or(2)Af = R x N is the warped product and m = 3.Also, since all the ends of a conformally compact manifold must have infinitevolume, Theorem 2 is, in fact, a generalization of the theorems of Witten-Yau, Cai-Galloway, and Wang. It is also interesting to note that without the conformallycompactness assumption, it is possible to have finite volume ends as indicated byfollowingexample.Example 1. Let M m = R x N" 1^1 with the warped product metricwhere N is a compact manifold withds M = dt" + exp(2t) ds" N ,Rie M > 0.A direct computation shows that M has Ricci curvature bounded byandRìCM > —(TO — 1)Ai(Af)>m-2.In fact, when m = 3, Ai (M) = 1. Obviously M has two ends. One end E hasinfinite volume growth withVE(T) ~ C exp((m — 1) r),

Differential Geometry via Harmonic Functions 299while the other end e has finite volume with volume decay given byVe(oo) — V e (r) ~ C exp( — (m — 1) r).We would like to point out that the pair of conditionsRìCM > — (TO — 1) 1~ > (1)Ai(Af) >ro-2 Jwis equivalent to the pair of conditions^. TO — 1 ., ,.Rtc M > »AiM)m - 2 y (2)Ai (M) >0.On the other hand, the pair of conditionsTH, ~ 1mc M >-—x^M) } (3)Ai(Af)=0are equivalent to the single assumption thatRie M > 0,because the condition Ai(M) = 0 is a consequence of the curvature assumption.Taking this point of view, Theorem 2 can be viewed as an analogue to thesplitting theorem of Cheeger-Gromoll. Similarly to the fact that Theorem 1 is aperturbed version of the Cheeger-Gromoll splitting theorem, the following theoremin [14] is a perturbed version of Theorem 2.Theorem 3 (Li-Wang). Let M m be a complete manifold with m > 3. SupposeBp(R) C M is a geodesic ball such thatRic M >-(m-l) on M\B P (R)and the lower bound of the spectrum of the Dirichlet Laplacian on M \ B P (R) isbounded byXi(M\B p (Rj) >ro-2 + efor some e > 0. Then there exists a constant 0 < C(m,R,a,v,e) < oo dependingonly on m, R, a = ìnf Bp (3R) RìCM, V = inf xeBp (2R)V x (R), and e, so that thenumber of infinite volume ends of M is at most C(m,R,a,v,e).In both Theorem 2 and Theorem 3, the authors only managed to estimate thenumber of infinite volume ends by estimating the number of non-parabolic ends. Infact, when a manifold has positive spectrum, they proved that an end must eitherbe non-parabolic with exponential volume growth, or it must be parabolic and finitevolume with exponential volume decay. Moreover, these growth and decay estimatescan be localized at each end.

300 P. LiTheorem 4 (Li-Wang). Let M be a complete, non-compact, Riemannian manifold.Suppose E is an end of M given by a unbounded component of M \ B P (R),where B P (R) is a geodesic ball of radius R centered at some fixed point p £ M.Assume that the lower bound of the spectrum Xi (E) of the Dirichlet Laplacian onE is positive. Then as r —t oo, either(1) E is non-parabolic and has volume growth given byV E (r) > dexp(2y/\ 1 (E)r)for some constant C\ > 0;or(2) E is parabolic and has finite volume with volume decay given byV(E) - V E (r) < C 2 exp(-2v%(£7)r)for some constant C2 > 0.In particular, if Ai (M) > 0, then M must have exponential volume growthgiven byV p (r) > Ci exp(2V%(M)r).Both the volume growth and the volume decay estimates are sharp. For example,the growth estimate is achieved by the hyperbolic m-space, H. Also, inExample 1 when dimension m = 3, the infinite volume end achieves the sharp volumegrowth estimate and the finite volume end achieves the sharp volume decayestimate.It is also interesting to point out that the sharp volume growth estimateis previously not known for manifolds with Ai (M) > 0.4. Minimal hypersurfacesLet us recall that the well-known Bernstein's theorem (Bernstein, Fleming,Almgren, DeGiorgi, Simons) asserts that an entire minimal graph M m c R TO+1must be linear if m < 7. Moreover, the dimension restriction is necessary as indicatedby the examples of Bombieri, DeGiorgi, and Guisti. Since minimal graphsare necessarily area minimizing and hence stable (second variation of the area functionalis non-negative), Fischer-Colbrie and Schoen [8] considered a generalizationof Bernstein's theorem in this category. They proved that a complete, oriented,immersed, stable minimal surface in a complete manifold with non-negative scalarcurvature must be conformally equivalent to either C or R x S 1 . Moreover, if theambient manifold is R 3 then the minimal surface must be planar. This special casewas independently proved by do Carmo and Peng [6].Later, Fischer-Colbrie [7] studied the structure of minimal surfaces with finiteindex. Recall that a minimal surface has finite index means that there are only afinite dimension of variations such that the second variations of the area functionalis negative. In this case, Fischer-Colbrie proved that a complete, oriented, immersed,minimal surface with finite index in a complete manifold with non-negative

Differential Geometry via Harmonic Functions 301scalar curvature must be conformally equivalent to a compact Riemann surface withfinitely many punctures. In particular, M must have finitely many ends. The specialcase when N = R 3 was also independently proved by Gulliver [9]. It is in thespirit of the number of ends that Cao, Shen and Zhu [3] found a higher dimensionalstatement for stable minimal hypersurfaces in R TO+1 .Theorem (Cao-Shen-Zhu). Let M m C R TO+1 be a complete, oriented, immersed,stable minimal hypersurface in R m+1 , then M must have only 1 end.This theorem is recently generalized to minimal hypersurfaces with finite indexby the author and Jiaping Wang [13].Theorem 5 (Li-Wang). Let M m c R TO+1 be a complete, oriented, immersed,minimal hypersurface with finite index in R m+1 , then M must have finitely manyends.In another paper [15], they also considered complete, properly immersed, stable(or with finite index) minimal hypersurfaces in a complete, non-negatively curvedmanifold.Theorem 6 (Li-Wang). Let M m c N m+1 be a complete, oriented, properly immersed,stable, minimal hypersurface. Suppose N is a complete manifold with nonnegativesectional curvature. Then either(1) M has only 1 end;or(2) M = R x S with the product metric, where S is a compact manifold withnon-negative sectional curvature. Moreover, M is totally geodesic in N.Theorem 7 (Li-Wang). Let M m c N m+1 be a complete, oriented, properly immersed,minimal hypersurface with finite index. Suppose N is a complete manifoldwith non-negative sectional curvature. Then M must have finitely many ends.It is interesting to point out that in the case when M = R x S, the manifold isparabolic. In this case, it is necessary to estimate the space K. rather than K.'. Again,the crucial point is to follow the construction of K. and obtain sufficient estimateson the functions in K. so that analytic techniques can be applied. In the case ofTheorem 5, since the ambient manifold is R TO+1 and hence the ends of M must allbe non-parabolic, it is sufficient to estimate the space K.' as stated in Theorem 2.References[1] M. Cai, Ends of Riemannian manifolds with nonnegative Ricci curvature outsidea compact set, Bull. AMS 24 (1991), 371-377.[2] M. Cai and G. J. Galloway, Boundaries of zero scalar curvature in the ADS/CFTcorrespondence, Adv. Theor. Math. Phys. 3 (1999), 1769-1783.[3] H. Cao, Y. Shen, and S. Zhu, The structure of stable minimal hypersurfacesin R"+\ Math. Res. Let. 4 (1997), 637-644.

302 P. Li[4] J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegativeRicci curvature, J. Diff. Geom. 6 (1971), 119-128.[5] S. Y. Cheng, Eigenvalue Comparison theorems and its Geometric Application,Math. Z. 143 (1975), 289-297.[6] M. do Carmo and C. K. Peng, Stable complete minimal surfaces in R 3 areplanes, Bull. AMS 1 (1979), 903-906.[7] D. Fischer-Colbrie, On complete minimal surfaces with finite Morse index inthree manifolds, Invent. Math. 82 (1985), 121-132.[8] D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimalsurfaces in 3-manifolds of non-negative scalar curvature, Comm. Pure Appi.Math. 33 (1980), 199-211.[9] R. Gulliver, Index and total curvature of complete minimal surfaces., Geometricmeasure theory and the calculus of variations (Areata, Calif., 1984),Proc. Sympos. Pure Math. 44, Amer. Math. Soc.,, Providence, RI., 1986,pp. 207-211.[10] J. Lee, The spectrum of an asymptotic hyperbolic Einstein Manifold, Comm.Anal Geom,. 3 (1995), 253-271.[11] P. Li, Curvature and function theory on Riemannian manifolds, Survey inDifferential Geometry "In Honor of Atiyah, Bott, Hirzebruch, and Singer",vol. VII, International Press, Cambridge, 2000, 71-111.[12] P. Li and L. F. Tam, Harmonic functions and the structure of complete manifolds,J. Diff. Geom. 35 (1992), 359-383.[13] P. Li and J. Wang, Minimal hypersurfaces with finite index, Math. Res. Let.9 (2002), 95-103.[14] P. Li and J. Wang, Complete manifolds with positive spectrum, J. Diff. Geom.58 (2001), 501-534.[15] P. Li and J. Wang, Stable minimal hypersurfaces in a nonnegatively curvedmanifold, Preprint.[16] R. Mazzeo, The Hodge cohomology of a conformally compact metric, J. Diff.Geom. 28 (1988), 309-339.[17] R. Schoen, Conformai deformation of a Riemannian metric to constant scalarcurvature, J. Diff. Geom. 20 (1984), 479-495.[18] X. Wang, On conformally compact Einstein manifolds, Math. Res. Let. 8(2001), 671-688.[19] E. Witten and S. T. Yau, Connectedness of the boundary in the AdS/CFTcorrespondence, Adv. Theor. Math. Phys. 3 (1999), 1635-1655.

ICM 2002 • Vol. II • 303-313Index Iteration Theory for SymplecticPaths with Applications to NonlinearHamiltonian SystemsYiming Long*AbstractIn recent years, we have established the iteration theory of the index forsymplectic matrix paths and applied it to periodic solution problems of nonlinearHamiltonian systems. This paper is a survey on these results.2000 Mathematics Subject Classification: 58E05, 70H05, 34C25.Keywords and Phrases: Iteration theory, Index, Symplectic path, Hamiltoniansystem, Periodic orbit.Since P. Rabinowitz's pioneering work [35] of 1978, variational methods havebeen widely used in the study of existence of solutions of Hamiltonian systems. Buthow to study the geometric multiplicity and stability of periodic solution orbitsobtained by variational methods has kept to be a difficulty problem. For examplelet x = x(i) be a r-periodic solution of a Hamiltonian systemx(t) = JH'(x(tj), Vt€R. (0.1)The ro-th iteration x m of x is defined by induction m — 1 times via x(t + r) = x(t)for t > 0. It runs m-times along the orbit of x. Geometrically these iterationsproduce the same solution orbit of (0.1), but they are different as critical pointsof corresponding functionals. This multiple covering phenomenon causes majordifficulties in the study.A natural way to study solution orbits found by variational methods is to studythe Morse-type index sequences of their iterations. But when one studies generalHamiltonian systems, the Morse indices of the critical points of the correspondingfunctional are always infinite. To overcome this difficulty, in their celebratedpaper [6] of 1984, C. Conley and E. Zehnder defined an index theory for any nondegeneratepaths in Sp(2n) with n > 2, i.e., the so called Conley-Zehnder index*Nankai Institute of Mathematics, Nankai University, Tianjin 300071, China.longym@nankai.edu.cnE-mail:

304 Yiming Longtheory. This index theory was further defined for non-degenerate paths in Sp(2)by E. Zehnder and the author in [33] of 1990. The index theory for degeneratelinear Hamiltonian systems was defined by C. Viterbo in [39] and the author in [20]of 1990 independently. In [25] of 1997, this index was extended to any degeneratesymplectic matrix paths.Motivated by the iteration theories for the Morse type index theories establishedby R. Bott in 1956 and by I. Ekeland in 1980s, in recent years the authorextended the index theory mentioned above, introduced an index function theory forsymplectic matrix paths, and established the iteration theory for the index theoryof symplectic paths. Applying this index iteration theory to nonlinear Hamiltoniansystems, interesting results on periodic solution problems of Hamiltonian systemsare obtained. Here a brief survey is given on these subjects. Readers are referredto the author's recent book [30] for further details.1. Index function theory for symplectic pathsAs usual we define the symplectic group by Sp(2n) = {M £ GL(R 2 ") | M T JM= J}, where J = \ J, J is the identity matrix on R", and M T denotes thetranspose of M. For OJ £ U, the unit circle in the complex plane C, we define thew-singular subset in Sp(2n) by Sp(2n)2, = {M £ Sp(2n) |a! _ "det(7(r) — OJ I) = 0}.Here for any M £ Sp(2n)°, we define the orientation of Sp(2n)2, at M by thepositive direction -f^Mexp(tJ)\t=o- Since the fundamental solution of a generallinear Hamiltonian system with continuous symmetric periodic coefficient 2n x 2nmatrix function B(t),x(t) = JB(t)x(t), Vt€R, (1.1)is a path in Sp(2n) starting from the identity, for r > 0 we define the set ofsymplectic matrix paths by V T (2n) = {7 £ C([0,r],Sp(2n)) | T(0) = I}- For anytwo path £ and n : [0,r] —¥ Sp(2n) with £(r) = n(0), as usual we define n * Ç(t)by Ç(2t) if 0 < t < T/2, and n(2t — r) if r/2 < t < r. We define a special pathC:[0,r]->Sp(2n)bydt) = diag(2 - -,..., 2 - -, (2 - -)-\..., (2 - -)" 1 ), for 0 < t < T.T T T TDefinition 1. (cf. [27]) For any r > 0, OJ £ U, and 7 £ V T (2n), we definethe uj-nullity of 7 byVu(l) = dim c ker c (7(r) -UJI). (1.2)7/7 is OJ non-degenerate, i.e., ^(7) = 0, we define the oj-index of 7 by the intersectionnumberM7) = [Sp(2 0, we let Td) be the set of all open neighborhoods0/7 in V T (2n), and defineiu(l) = sup inf{i u (ß) I ß £ U, v u (ß) = 0}. (1.4)

Index Iteration Theory for Symplectic Paths 305Then we call (4,(7), ^(7)) £ Z x {0,1,..., 2n} the index function of 7 atoj.The relation of this index (ii(^),vi(^j) with the Morse index of r-periodicsolutions of the problem (1) was proved by C. Conley, E. Zehnder, and the authorin [6], [33], and [20] (cf. Theorem 6.1.1 of [30]).2. Iteration theory of the index for symplectic pathsGiven a path 7 £ V T (2n), its iteration is defined inductively by *y(t + T) =7(1)7(7-) for t > 0, i.e.,1 m (t)=l(t^JT) 1 (Ty, JT

306 Yiming Longform matrices, which we call basic normal forms. Correspondingly by the homotopyinvariance and symplectic additivity of the index theory, the computations in (2.3)are reduced to iterations of those paths in Sp(2) or Sp(4) whose end points areone of the 10 basic normal form matrices. The study of the index for iterationsof any symplectic paths is carried out for paths in Sp(2) via the R 3 -cylindricalcoordinate representation of Sp(2), then for hyperbolic and elliptic paths in Sp(2n).This yields the precise iteration formula obtained in [29] of the index theory for anysymplectic path 7 £ V T (2n) in terms of the basic norm form decomposition of 7(7-),(z(7, l),i/( 7 ,1)), and the iteration time TO.For any M £ Sp(2n), its splitting numbers at an OJ £ U is defined in [27] byS M (OJ)= Hm t u (±v=Te) (7)-t u (7), (2.5)via any 7 £ V T (2n) satisfying 7(7-) = M. Then it is proved that the splittingnumbers of M at OJ can be characterized algebraically.Motivated by the precise iteration formulae of [29], the following second indexiteration formula of any symplectic path is established by C. Zhu and the author.Here we denote by (z( 7 ,TO), 1/(7, TO)) = (ii(7), ^1(7)).Theorem 3 (cf. [34]). For any r > 0, 7 £ V T (2n), and m £ N, there holds:*(7,m) = m(*(7,l) + S+(l)-C(M))n,2 52 EAs M (e^=Tl> )^(S M (i) + C(Mj), (2.6)9G(0,2TT)2TTwhere M = 7(r), C(M) = XÔ

Index Iteration Theory for Symplectic Paths 307Here we define e(M) to be the total multiplicity of eigenvalues of M on U and callit the elliptic height of M.A consequence of the iteration inequality (2.8) together with the necessary andsufficient conditions for any equality in (2.8) to hold for some TO yields a new proofof the following theorem of D. Dong and the author on controlling the iterationtime TO via indices:Theorem 5 (cf. [7]). For any 7 £ V T (2n) and m £ N, suppose z( 7 ,m) < n+1,i(l, 1) > n > an d v(~i, 1) > 1- Then TO = 1.Note also that the inequality (2.9) yields a way to estimate the ellipticity ofsolution orbits of Hamiltonian systems obtained by variational methods via theiriterated indices.In order to study the properties of solution orbits of the system (0.1) on agiven energy hypersurface, when the number of orbits is finite, we need to studycommonproperties of any given finite family of symplectic paths 7j- £ V Tj (2n) with1 < 3 < Q- This leads to the following common index jump theorem of C. Zhuand the author proved in [34]. For any 7 £ V T (2n), its m-th index jump Gmd) isdefined to be the open interval Gmd) = (i(l, m ) + v(^,'rn) — 1,1(7, TO + 2)).Theorem 6 (cf. [34]). Let 7j- £ V Tj (2n) with 1 < j < q satisfying*( 7i )>0, *( 7i ,l)>n, l

308 Yiming Long3.1. Prescribed minimal period solution problemIn [35] of 1978, P. Rabinowitz posed a conjecture on whether the Hamiltoniansystem possesses periodic solutions with prescribed minimal period when the Hamiltonianfunction satisfies his superquadratic conditions. This conjecture is studiedby D. Dong and the author as an application of our index iteration theory. Notethat for a non-constant r-periodic solution x of the autonomous system (0.1), thecondition on the nullity in Theorem 5 always holds. Thus Theorem 5 yields:Theorem 7 (cf. [7]). For any non-constant r-periodic solution x of (0.1),denote its minimal period by T/TO for some m £ N. Suppose i(x\[ 0 , T }, 1) < n + 1and n < i(x\[o jT / m ], 1). Then TO = 1, i.e., r is the minimal period of x.Here the first estimate on the index holds if x is obtained by minimax or minimizationmethods, and the second estimate on the index holds if the Hamiltonianfunction H is convex in a certain weak sense along the orbit of x. This result revealsthe intrinsic relationship between the minimal period of a periodic solution and itsindices, and unifies all the results on Rabinowitz's conjecture under various convexityconditions. Specially, it recovers the famous theorem of I. Ekeland and H. Hoferin 1985 (cf. [11]) who solved Rabinowitz's conjecture for convex superquadraticHamiltonian systems.3.2. Periodic points of the Poincaré map of Lagrangian systemson toriIn 1984, C. Conley stated a conjecture on whether the Poincaré map of any1-periodic time dependent Hamiltonian system defined on the standard torus T 2n =R 2 "/Z 2 " always possesses infinitely many periodic points which are produced bycontractible periodic solutions of the corresponding Hamiltonian system on T 2n .A celebrated partial answer to this conjecture was given by D. Salamon and E.Zehnder in 1992 (cf. [37]) for a large class of symplectic manifolds on which everycontractible integer periodic solution of the Hamiltonian system has at least oneFloquet multiplier not equal to 1. So far Conley conjecture is still open and seemsfar from being completely understood.In [28], we studied the Lagrangian system version of this conjecture. Considerd— Li(t,x,x)-L x (t,x,x) = 0, x£R n , (3.1)where L, x and L, x denote the gradients of L with respect to x and x respectively.The main result is the following:Theorem 8 (cf. [28]). Suppose the Lagrangian function L satisfies(LI) L(t,x,p) = \A(t)p-p+ V(t,x), where \A(t)p-p > A|p| 2 for all (t,p) £R x R" and some fixed constant X > 0.(L2) A £ C 3 (K,£ s (K n )), V £ C 3 (RxR",R), both A and V are 1-periodic inall of their variables, where £ s (R n ) denotes the set ofnxn real symmetric matrices.Then the Poincaré map \P of the system (3.1) possesses infinitely many periodicpoints on TT n produced by contractible integer periodic solutions of the system(3.1).

Index Iteration Theory for Symplectic Paths 309In the proof of Theorem 8, the above inequality (2.7) plays a crucial role. Bythis inequality, at very high iteration level, a global homological injection map canbe constructed which maps a generator of a certain non-trivial local critical groupto a nontrivial homology class [a] in a global homology group, if the number ofcontractible integer periodic solution towers of the system (3.1) is finite. But on theother hand, by a technique of V. Bangert and W. Klingenberg in [3], it is shownthat this homology class [a] must be trivial globally. This contradiction then yieldsthe conclusion of Theorem 8.3.3. Closed characteristics on convex compact hypersurfacesDenote the set of all compact strictly convex C 2 -hypersurfaces in R 2 " by%(2n). For S £ %(2n) and x £ S, let Afe(x) be the outward normal unit vector atx of S. We consider the problem of finding r > 0 and a curve x £ C 1 ([0,r],R 2 ")such thatx(t) = JNj;(x(t)), x(t)£Z, WeR, , .X(T) = x(0).['A solution (T,X) of the problem (3.1) is called a closed characteristic on S. Twoclosed characteristics (T,X) and (a,y) are geometrically distinct, if x(R) ^ J/(R).We denote by T(S) the set of all geometrically distinct closed characteristics (r, x)on S with r being the minimal period of x. Note that the problem (3.1) can bedescribed in a Hamiltonian system version and solved by variational methods. Aclosed characteristic (T,X) is non-degenerate, if 1 is a Floquet multiplier of x ofprecisely algebraic multiplicity 2, and is elliptic, if all the Floquet multipliers of xare on U. Let # A denote the total number of elements in a set A.This problem has been studied for more than 100 years since at least A. M.Liapunov in 1892. A long standing conjecture on the multiplicity of closed characteristicsis whether*JÇ£)>n, VE€ft(2n). (3.2)The first break through on this problem in the global sense was made by P. Rabinowitz[35] and A. Weinstein [40] in 1978. They proved # T(£) > 1 for allS £ %(2n). Besides many results under pinching conditions, in 1987-1988, I.Ekeland-L. Lassoued, I. Ekeland-H. Hofer, and A, Szulkin proved # T(S) > 2 forall S £ %(2n) and n > 2. In 1998, H. Hofer, K. Wysocki, and E. Zehnder provedin [14]: # T(£) = 2 or +oo for every S £ H(4). In recent years C. Liu, C. Zhu, andthe author gave the following answers to the conjecture (3.2):Theorem 9 (cf. [34]). There holds#T(E) > [|] + 1, VS G H(2n), (3.3)where [a] = maxjfc £ Z\k < a} for any a £ R. Moreover, if all the closedcharacteristics on S are non-degenerate, then # T(S) > n.Theorem 10 (cf. [19]). For any S £ %(2n), if S is symmetric with respectto the origin, i.e., x £"£ implies —a: £ S, then # T(S) > n.Very recently, Y. Dong and the author further proved the following result.

310 Yiming LongTheorem 11 (cf. [8]). Let E £ %(2n) be P-symmetric with respect to theorigin, i.e., x £ 'S implies Px £ E, where P = diag(—I n - k ,I k , —I n^k,I k ) for somefixed integer k £ [0,n — 1]. Let E(fc) = {(x,y) £ (R k ) 2 | (0,x, 0,y) £ E}. Suppose*T(S(k)) n - 2k.Proof of Theorem 11 depends on a new index iteration theory for symplecticpaths iterated by the formula 7(£ + r) = P / y(t)P , y(T) for t > 0.The second long standing conjecture on closed characteristics is whether therealways exists at least an elliptic closed characteristic on any E £ %(2n). Up to theauthor's knowledge, the existence of one elliptic closed characteristic on E e %(2n)was proved by I. Ekeland in 1990 when E is v^-pmched by two spheres, and byG.-F. Dell'Antonio, B. D'Onofrio, and I. Ekeland in 1992 when E is symmetric withrespect to the origin. Recently using an enhanced version of the iteration estimate(2.9) on the elliptic height, based on results in [29] the following result was furtherproved by C. Zhu and the author.Theorem 12 (cf. [34]). For E £ H(2n), suppose # T(E) < +oo. Then thereexists at least an elliptic closed characteristic on E. Moreover, suppose n > 2 and# T(E) < 2[n/2]. Then there exist at least two elliptic elements in T(E).The main ingredient in the proofs of Theorems 9 to 12 is our index iterationtheory mentioned above. To illustrate this method, we briefly describe below themain idea in the proof of (3.3) in Theorem 9. Because each closed characteristic onE corresponds to infinitely many critical values of the related dual action functional,our way to solve the problem is to study how the index intervals of iterated closedcharacteristics cover the set of integers 2N — 2 + n to count the number of closedcharacteristics on E. Suppose q = # J"(E) < +oo. In the proof of the multiplicityclaim(3.3) of Theorem 9, the most important ingredient is the following estimates:q > # ((2N-2 + n)nn? =1 & roj -i( 7 *,.))># ((2N-2 + n)n[2A r -Ki,2A r + K 2 ])Tl> [3] + !' ( 3 - 4 )The first inequality in (3.4) is a new version of the Liusternik-Schnirelman theoreticalargument at the iterated index level, which distinguishes solution orbitsgeometrically instead of critical points only as usual methods do. The second inequalityin (3.4) uses the common index jump Theorem 6. The last inequality in(3.4) uses the Morse theoretical approach. Roughly speaking, the common indexjump theorem picks up as many as possible points of 2N — 2 + n in the interval[2N - Ki,2N + K 2 ] C C\ q - =1 G2m,j-i(lxj), which yields a lower bound for # T(E).As usual, a hypersurface E c R 2 " is star-shaped if the tangent hyperplane atany x £'S does not intersect the origin. Closed characteristics on E can be definedby (3.1) too. In this case, the result &JCZ) > 1 was proved by P. Rabinowitz in [35]of 1978. Then multiplicity results were proved under certain pinching conditions onstar-shaped E. Recently, the following result for the free case was proved by X. Huand the author:Theorem 13 (cf. [15]). Let 'S be a star-shaped compact C 2 -hypersurface inR 2 ". Suppose all the closed characteristics on E and all of their iterates are non-

Index Iteration Theory for Symplectic Paths 311degenerate. Then # T(E) > 2. Moreover, ifn = 2 and # T(E) < +00 further holds,then there exist at least two elliptic closed characteristics on E.Here the crucial point is to prove i(x, 1) > n when (r, x) is the only geometricallydistinct closed characteristic on E. This conclusion is proved by using ourindex iteration theory and an identity of non-degenerate closed characteristics onE proved by C. Viterbo in 1989.Because of Theorem 9 and other indications, we suspect that the followingholds:{ # T(E) I E G H(2n)} = {k £ Z | [|] + 1 < k < n} U {+00}. (3.5)We also suspect that closed orbits of the Reeb field on a compact contact hypersurfacesin a symplectic manifold may have similar properties.Many other problems related to iterations of periodic solution orbits are stillopen, for example, the Seifert conjecture on the existence of at least n brake orbitsfor the given energy problem of classical Hamiltonian systems on R" (cf. [38], [1]and the references there in), and the conjecture on the existence of infinitely manygeometrically distinct closed geodesies on every compact Riemannian manifold (cf.[2] and the solution for S 2 by J. Franks and V. Bangert). We believe that our indexiteration theory for symplectic paths and the methods we developed to establishand apply it to nonlinear problems will have the potential to play more roles in thestudy on these problems and in other mathematical areas.Acknowledgements. The author sincerely thanks the 973 Program of MOST,NNSF, MCME, RFDP, PMC Key Lab of MOE of China, S. S. Chern Foundation,CEC of Tianjin, and Qiu Shi Sci. Tech. Foundation of Hong Kong for their supportsin recent years.References[1] A. Ambrosetti, V. Benci, & Y. Long, A note on the existence of multiple brakeorbits. Nonlinear Anal. TMA. 21 (1993), 643-649.[2] V. Bangert, Geodetische Linien auf Riemannschen Mannigfaltigkeiten. Jber.D. Dt. Math.-Verein. 87 (1985), 39-66.[3] V. Bangert & W. Klingenberg, Homology generated by iterated closedgeodesies. Topology. 22 (1983), 379-388.[4] R. Bott, On the iteration of closed geodesies and the Sturm intersection theory.Comm. Pure Appi. Math. 9 (1956), 171-206.[5] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems.Birkhäuser. Basel. (1993).[6] C. Conley & E. Zehnder, Morse-type index theory for flows and periodic solutionsfor Hamiltonian equations. Comm. Pure Appi. Math. 37 (1984), 207-253.[7] D. Dong & Y. Long, The iteration formula of the Maslov-type index theorywith applications to nonlinear Hamiltonian systems. Trans. Amer. Math. Soc.349 (1997), 2619-2661.[8] Y. Dong & Y. Long, Closed characteristics on partially symmetric convex hypersurfacesin R 2 ". (2002) Preprint.

312 Yiming Long[9] I. Ekeland, Une théorie de Morse pour les systèmes hamiltoniens convexes.Ann. IHP. Anal, non Linéaire. 1 (1984), 19-78.[10] I. Ekeland, Convexity Methods in Hamiltonian Mechanics. Springer. Berlin.1990.[11] I. Ekeland & H. Hofer, Periodic solutions with prescribed period for convexautonomous Hamiltonian systems. Invent. Math. 81 (1985), 155-188.[12] I. Ekeland & H. Hofer, Convex Hamiltonian energy surfaces and their closedtrajectories. Comm. Math. Phys. 113 (1987), 419-467.[13] J. Han & Y. Long, Normal forms of symplectic matrices (II). Acta Sci. Nat.Univ. Nankai. 32 (1999) 30-41.[14] H. Hofer, K. Wysocki, & E. Zehnder, The dynamics on three-dimensionalstrictly convex energy surfaces. Ann. of Math. 148 (1998) 197-289.[15] X. Hu & Y. Long, Multiplicity of closed characteristics on non-degenerate starshapedhypersurfaces in R 2 ". Sciences in China (2002) to appear.[16] C. Liu & Y. Long, An optimal increasing estimate for iterated Maslov-typeindices. Chinese Sci. Bull. 42 (1997), 2275-2277.[17] C. Liu & Y. Long, Iteration inequalities of the Maslov-type index theory withapplications. J. Diff. Equa. 165 (2000) 355-376.[18] C. Liu & Y. Long, Iterated index formulae for closed geodesies with applications.Science in China. 45 (2002) 9-28.[19] C. Liu, Y. Long, & C. Zhu, Multiplicity of closed characteristics on symmetricconvex hypersurfaces in R 2 ". Math. Ann. (to appear).[20] Y. Long, Maslov-type index, degenerate critical points, and asymptoticallylinearHamiltonian systems. Science in China (Scientia Sinica). Series A. 7(1990), 673-682. (Chinese edition), 33 (1990), 1409-1419. (English edition).[21] Y. Long, Index Theory of Hamiltonian Systems with Applications. SciencePress. Beijing. 1993. (In Chinese).[22] Y. Long, The minimal period problem for classical Hamiltonian systems witheven potentials. Ann. Inst. H. Poincaré. Anal, non linéaire. 10 (1993), 605-626.[23] Y. Long, The minimal period problem of periodic solutions for autonomoussuperquadratic second order Hamiltonian systems. J. Diff. Equa. Ill (1994),147-174.[24] Y. Long, On the minimal period for periodic solutions of nonlinear Hamiltoniansystems. Chinese Ann. of Math. 18B (1997), 481-484.[25] Y. Long, A Maslov-type index theory for symplectic paths. Top. Meth. Noni.Anal. 10 (1997), 47-78.[26] Y. Long, Hyperbolic closed characteristics on compact convex smooth hypersurfaces.J. Diff. Equa. 150 (1998), 227-249.[27] Y. Long, Bott formula of the Maslov-type index theory. Pacific J. Math. 187(1999), 113-149.[28] Y. Long, Multiple periodic points of the Poincaré map of Lagrangian systemson tori. Math. Z. 233 (2000) 443-470.[29] Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticityof closed characteristics. Advances in Math. 154 (2000), 76-131.

Index Iteration Theory for Symplectic Paths 313[30] Y. Long, Index Theory for Symplectic Paths with Applications. Progress inMath. 207, Birkhäuser. Basel. 2002.[31] Y. Long & T. An, Indexing the domains of instability for Hamiltonian systems.NoDEA. 5 (1998) 461-478.[32] Y. Long & D. Dong, Normal forms of symplectic matrices. Acta Math. Sinica.16 (2000) 237-260.[33] Y. Long & E. Zehnder, Morse theory for forced oscillations of asymptoticallylinearHamiltonian systems. In Stoc. Proc. Phys. and Geom. S. Albeverio et al.ed. World Sci. (1990) 528-563.[34] Y. Long & C. Zhu, Closed characteristics on compact convex hypersurfaces inR 2 ". Annals of Math. 155 (2002) 317-368.[35] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems. Comm. PureAppi. Math. 31 (1978) 157-184.[36] P. H. Rabinowitz, Minimax methods in critical point theory with applicationsto differential equations. CBMS Regional Conf. Ser. in Math. no.65. Amer.Math. Soc. 1986.[37] D. Salamon & E. Zehnder, Morse theory for periodic solutions of Hamiltoniansystems and the Maslov index. Comm. Pure and Appi. Math. 45. (1992). 1303-1360.[38] H. Seifert, Periodischer Bewegungen mechanischen système. Math. Z. 51(1948), 197-216.[39] C. Viterbo, A new obstruction to embedding Lagrangian tori. Invent. Math.100 (1990), 301-320.[40] A. Weinstein, Periodic orbits for convex Hamiltonian systems. Ann. of Math.108. (1978). 507-518.

ICM 2002 • Vol. II • 315-321Some Applications of Collapsingwith Bounded CurvatureAnton Petrunin*AbstractIn my talk I will discuss the following results which were obtained injoint work with Wilderich Tüschniann.1. For any given numbers m, C and D, the class of m-dimensional simplyconnected closed smooth manifolds with finite second homotopy groups whichadmit a Riemannian metric with sectional curvature \K\ < C and diameter< D contains only finitely many diffeomorphism types.2. Given any rn and any S > 0, there exists a positive constant io =io(m,,S) > 0 such that the injectivity radius of any simply connected compactm-dimensional Riemannian manifold with finite second homotopy group andRicci curvature Rie >S,K

316 Anton PetruninTheorem A. Given any ö > 0, there exists a positive constant io = io($) >0 such that the injectivity radius of any simply connected compact 3-dimensionalRiemannian manifold with Ricc> 5, K < 1, is bounded from below by io-Moreover they made a conjecture that this result should be also true for higherdimensions. Later on some new examples of manifolds with positively pinched curvaturewere found by Alloff and Wallach, Eschenburg and Bazaikin ([AW], [E], [B])which disprove this conjecture in general, but since then closely related conjecturesappeared on almost each list of open problems in Riemannian geometry. Thetheorem which we proved can be formulated as follows:Theorem B. Given any m and any ö > 0, there exists a positive constant io =io(m,ö) > 0 such that the injectivity radius of any simply connected compact m-dimensional Riemannian manifold with finite second homotopy group and Ricc> ö,K < 1, is bounded from below by io(m,ö).Theorem B generalizes the Burago-Toponogov Theorem A to arbitrary dimensionsand is also in even dimensions interesting, since there is no Synge theoremfor positive Ricci curvature. For sectional curvature pinching a similar result wasobtained independently by Fang and Rong [FR].Now I will turn to one proof of this statement which is described in the appendixof [PT] (The main part of paper contains an other proof). This proof makesuse of a generalized notion of Riemannian manifold, which was also described byGromov in the end of section 8 + of [G3], and employs a "tangential" version ofGromov-Hausdorff convergence. Here I will just give an informal analogy whichdescribes this notion. The formal aspects and all further details can be found in[PT].One may think about a manifold as a set of charts and glueing mappings. Fora Riemannian manifold, denoting the disjoint union of all charts with the pulledback metrics by (U,g), the set of all glueing maps defines an isometric pseudo-groupaction by a pseudogroup G on (U,g). Here is the definition of a pseudogroup action:Definition. A pseudogroup action (or pseudogroup of transformations) on a manifoldM is given by a set G of pairs of the form p = (D p ,p), where D p is an opensubset of M and p is a homeomorphism D p —t M, so that the following propertieshold:(1) p,q £ G implies po q = (q^1 (D p n q(D p j),po q) £ G;(2) p £ G implies p^1= (p(D p ),p^1)£ G;(3) (M, id) £ G;(4) if P is a homeomorphism from an open set D c M into M and D =(j a D a , where D a are open sets in M, then the property (D,p) £ G is equivalent to(D a ,p\i) a ) £ G for any a.We call the pseudo-group action natural if in addition the following is true:(i)' If (D,p) £ G and p can be extended as a continuous map to a boundarypoint x £ dD, then there is an element (D',p r ) £ G such that x £ D', D c D' andP'\D =P-To form a manifold this action must be in addition properly discontinuous

Some Applications of Collapsing with Bounded Curvature 317and free. If it just properly discontinuous then we obtain an orbifold. In the caseof a general (isometric!) pseudogroup action we obtain a it Riemannian megafold(cf. [PT]). The megafold which is obtained this way will be denoted by (M,g) =((U,g):G).Now we come to the main notion of this section:Definition. A sequence of Riemannian megafolds (M n ,g n ) is said to Grothendieck-Lipschitz converge (GL-converge) to a Riemannian megafold (M,g) if there arerepresentations (M n ,g n ) = ((U n ,g n ) : G n ) and (M,g) = ((U,g) : G) such that(a) The (U n ,g n ) Lipschitz converge to (U,g), and(b) For some sequence e n —¥ 0 there is a sequence of e ±l!n -bi-Lipschitz homeomorphismsh n : (U n ,g n ) —¥ (U,g) such that the pseudogroup actions on {(U n ,g n )}converge (with respect to the homeomorphisms h n ) to a pseudogroup action on{(.U,g)}.I.e., for any converging sequence of elements p nk £ G nk (U nk ,M nk ) there existsa sequence p n £ G n which converges to the same local isometry on U, and thepseudogroup of all such limits, acting on U, coincides with the pseudogroup actionG(U,M).Here are two simple examples of GL-convergence:Consider the sequence of Riemannian manifolds S^ x R, which for e —¥ 0Gromov-Hausdorff converge to R. Then this sequence converges in the GL-topologyto a Riemannian megafold M, which can described as follows: It is covered by onesingle chart U = R 2 , and the pseudogroup G simply consists of all vertical shifts ofR 2 . I.e., M is nothing but (R 2 : R) where R acts by parallel translations. (Notethat (R 2 : R) ^ R 2 /R, these megafolds even have different dimensions!)The Berger spheres, as they Gromov-Hausdorff collapse to S 2 , converge inGrothendieck-Lipschitz topology to the Riemannian megafold (S 2 x R : R). HereR acts by parallel shifts of S 2 x R.Notice that a Riemannian metric on a megafold ((U,g) : G) defines a pseudometricon the set of G orbits. In particular one has that the diameter of aRiemannian megafold is well defined. Now here is the basic result, whose proof isobvious from the definitions:Theorem C. The set of Riemannian m-manifolds (megafolds) with bounded sectionalcurvature \K\ < 1 and diameter < D is precompact (compact) in theGrothendieck-Lipschitz topology.Now let us state some natural questions which arise from this theorem:1. Which Riemannian megafolds can be approximated by manifolds withbounded curvature and diameter?Note that the infinitesimal motions of the pseudogroup G give rise to a Liealgebra of Killing fields on a megafold (U, g) from which one can recover an isometriclocal action of a connected Lie group on (U,g). Yet us call this group G 0 . ThenG 0 is obviously an invariant of the megafold, i.e., does not depend on a particularrepresentation (U : G). It follows now from [CFG] that if (M,g) is a limit of

318 Anton PetruninRiemannian manifolds with bounded curvature, then G 0 (M) must be nilpotent. Adirect construction moreover shows that this condition is also sufficent.(Note that since a pure A r -structure on a simply connected manifold is givenby a torus action, one also has the following: If a megafold can be approximatedby simply connected manifolds with bounded curvature, then G 0 (M) = R k .)2. How can one recover the Gromov-Hausdorff limit space from a Grothendieck-Lipschitz limit?Let M = ((U,g) : G) be a GL-limit of Riemannian manifolds. The GH-limitis the space of G orbits with the induced metric, in other words: The Gromov-Hausdorff limit is nothing but (U,g)/G.Riemannian megafolds are actually not that general objects as they mightseem at first sight. Indeed, given a Riemannian megafold (M,g) we can considerits orthonormal frame bundle (FM,g), equipped with the induced metric. Nowconsider some representation of it, say, (FM,g) = ((U,g) : G). Then the G pseudogroupaction is free on U, so that its closure G also acts freely. Therefore thecorresponding factor, equipped with the induced metric, is a Riemannian manifoldY = (U/G,g), and there is a Riemannian submersion (FM,g) —t (U/G,g) whosefibre is G 0 /Y 0 , where F 0 is a dense subgroup of G 0 (Roughly speaking, F 0 is generatedby the intersections of G 0 and G). If we assume that M is simply connected,then G 0 = R k and F 0 is the homotopy sequence image of n 2 (Y). In particular, thedimension of the free part of n 2 (Y) is at least k + 1.Notice that for Riemannian megafolds one can define the de Rham complexjust as well as for manifolds. (In fact I am not aware of a single notion or theoremin Riemannian geometry which does not admit a straightforward generalization toRiemannian megafolds!) From the above characterization of Riemannian megafoldsit is not hard to obtain the following:Theorem D. Let M n be a sequence of compact simply connected Riemannianm-manifolds with bounded curvatures and diameters and H% R (M n ) = 0 whichGrothendieck-Lipschitz converges to a Riemannian megafold (M,g).Then M is either a Riemannian manifold and the manifolds M n converge toM in the Lipschitz sense, or H^R(M) ^ 0.It is in particular straightforward to show that ïfRïcc(M) > 0, then H^R(M) =0. Moreover, a Grothendieck-Lipschitz limit of manifolds with uniformly boundedsectional curvatures and Ricc> ö > 0 is a Riemannian megafold with Ricc> ö > 0.Now we can prove Theorem B: Assume it is wrong. Then we can find acollapsing sequence of simply connected manifolds with finite 7i2 and positive Riccipinching,and we obtain a megafold with H^R ^ 0 as a GL-limit. Applying theBochner formula for 1-forms on this megafold, we obtain a contradiction.2. Finiteness theoremsThe following result appeared as a co-product of the theorem above, and itcame as a nice surprise. Let me first formulate this finiteness results from [PT]:

Some Applications of Collapsing with Bounded Curvature 319Theorem E (The ^-Finiteness Theorem). For given m, C and D, there isonly a finite number of diffeomorphism types of simply connected closed m-dimensionalmanifolds M with finite second homotopy groups which admit Riemannianmetrics with sectional curvature \K(M)\ < C and diameter diam(M) < D.Theorem F (A "classification" of simply connected closed manifolds). Forgiven m, C and D, there exists a finite number of closed smooth simply connectedmanifolds E t with finite second homotopy groups such that any simply connectedclosed m-dimensional manifold M admitting a Riemannian metric with sectionalcurvature \K(M)\ < C and diameter diam(M) < D is diffeomorphic to a factorspace M = Ei/T ki , where 0 < fcj = 62 (M) = dim E t — m and T ki acts freely onEi.Here is a short account of other finiteness results which only require volume,curvature, and diameter bounds: For manifolds M of a given fixed dimension m,the conditions• vol(M) > v > 0, \K(M)\ < C and diam(Af) < D imply finiteness of diffeomorphismtypes (Cheeger ([C]) 1970); this conclusion continues to hold forvol(M) >v>0,j M \R\ m / 2 < C, |Ric M | < C, diam(Af) < D (Anderson andCheeger ([AC1]) 1991);• vol(M) > v > 0, K(M) > C diam(Af ) < D imply finiteness of homeomorphismtypes (Grove-Petersen ([GP]) 1988, Grove-Petersen-Wu ([GPW]) 1990);Perelman, ([Pe]) 1992) (if in addition m > 4, these conditions imply finitenessof diffeomorphism types) and Lipschitz homeomorphism types (Perelman, unpublished);• K(M) > C and diam(Af) < D imply a uniform bound for the total Bettinumber (Gromov [Gl] 1981).The 7T2-Finiteness Theorem requires two-sided bounds on curvature, but nolower uniform volume bound. Thus, in spirit it is somewhere between Cheeger'sFiniteness and Gromov's Betti number Theorem.Each of the above results has (at least) two quite different proofs, the originalone and one which uses Alexandrov techniques. (For Gromov's Betti numbertheorem we made such a proof recently, jointly with V. Kapovich and it turned outthat one can even give an upper estimate for the total number of critical pointsof a Morse function on such a manifold, which due to the Morse inequality is astronger condition.) Let me now explain roughly this second way of proving of suchtheorems:I will take Cheeger's theorem as an example: Assume it is wrong. Thenthere is an infinite number of non-diffeomorphic manifolds with bounded curvature,diameter and a lower bound on the volume. Then due to Gromov's compactnesstheorem a subsequence of them has a limit. Then, due to the volume bound, thislimit space has the same dimension, and is in fact just little worse than Riemannian;it is a manifold with a smooth structure and curvature bounded in the sense ofAlexandrov. Then one only has to prove the stability result, i.e. one has to provethat starting from some big number all manifolds are diffeomorphic to the limit

320 Anton Petruninspace. In the case of two-sided curvature bound it is really simple, and for justlower curvature bound it is already a hard theorem of Perelman, but still it worksalong this lines.Now for both of these proofs it is very important to have a uniform lowerpositive volume bound to prevent collapsing. In fact, if one removes this boundthen it is not hard to construct infinite sequence of non-diffeomorphic manifolds.This holds for two-sided bounded as well as for lower curvature bound. And if wewould try to prove it the same way as before we would get a limit space of possiblysmallerdimension. Therefore the stability result can not hold this way.This partly explains why Theorem E looks a bit surprising, we add one topologicalcondition and get real finiteness result. The proof can go along the samelines. Take a sequence of nondiffeomorphic Riemannian manifold (M n ,g n ), by Gromov'scompactness theorem we have a limit space (for some subsequence) X. Thesequence must collapse, otherwise the same arguments as before would work. Sincethe M n are simply connected, from [CFG] we have that collapsing takes place alongsome r*-orbits of some T fc -action.Now assume for simplicity that X is a manifold and n 2 (M n ) = 0. Then allM n are diffeomorphic to T k bundles over X. Since the M n are simply connected sois X. Therefore the diffeomorphism type of M n depends only on the Euler class e nwhich in this case can be interpreted as the following mapping:0 = 7r 2 (M„) -• 7T 2 (X) -% m(T k ) -• 7Ti(M„) = 0.Therefore e n isan isomorphism between two groups and up to automorphisms ofT k all possible Euler classes e n are the same. In particular, for large n all M n arediffeomorphic.That is not quite a proof since we had made quite strong assumptions on theway. But it turns out that the general case can be ruled out using a few alreadystandardtricks from [CFG] and [GK], namely, by passing to the frame bundlesFM n and by conjugating group actions.References[AW] S. Aloff; N. R. Wallach, An infinite family of 7-manifolds admitting positivelycurved Riemannian structures, Bull. Amer. Math. Soc. 81 (1975),93-97.[B] Ya. V. Basaikin, On a certain family of closed 13-dimensional manifolds ofpositive curvature, Siberian Mathematical Journal 37:6 (1996).[BT] Y. Burago; V. A. Toponogov, On three-dimensional Riemannian spaces withcurvature bounded above, Matematicheskie Zametki 13 (1973), 881-887.[C] J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math.92 (1970), 61-74.[CFG] J. Cheeger; K. Fukaya; M. Gromov, Nilpotent structures and invariant metricson collapsed manifolds, J. A.M.S 5 (1992), 327-372.

Some Applications of Collapsing with Bounded Curvature 321[Es] J.-H. Eschenburg, New examples of manifolds with strictly positive curvature,Invent, math. 66 (1982), 469-480.[FR] F. Fang; X. Rong, Positive Pinching, volume and second Betti number,G AFA (Geometric and functional analysis 9 (1999).[Gl] M. Gromov, Curvature, diameter and Betti numbers, Comment. Math.Helv. 56 (1981), 179-195.[G2] M. Gromov, Stability and Pinching, Seminare di Geometria, Giornate diTopologia e geometria delle varietà. Università degli Studi di Bologna(1992), 55-97.[G3] M. Gromov, "Metric Structures for Riemannian and Non-Riemannianspaces", Birkhäuser, Basel, (1999).[GK] K. Grove; H. Karcher, How to conjugate C 1 -close group actions, Math. Z.132 (1973), 11-20.[GPW] K. Grove; P. Petersen; J. Wu, Controlled topology in geometry, Invent.Math. 99 (1990), 205-213; Erratum: Invent. Math. 104 (1991), 221-222.[KS1] W. Klingenberg; T. Sakai, Injectivity radius estimates for 1/4-pinched manifolds,Arch. Math. 34 (1980), 371-376.[PRT] A. Petrunin; X. Rong; W. Tuschmann, Collapsing vs. Positive Pinching,G AFA (Geometric and functional analysis) 9 (1999), 699-735.[PT] A. Petrunin and W. Tuschmann, Diffeomorphism finiteness, positive pinching,and second homotopy, GAFA (Geometric and functional analysis) 9(1999), 736-774.[PT2] A. Petrunin and W. Tuschmann, Asymptotical flatness and cone structureat infinity, Math. Ann. 321 (2001), 775-788.

ICM 2002 • Vol. II • 323-338Collapsed Riemannian Manifoldswith Bounded Sectional Curvature*Xiaochun RongtAbstractOne of the most important developments in Riemannian geometry overthe last two decades is the structure theory of Cheeger-Fukaya-Gromov, formanifolds M n of bounded sectional curvature, say |KM»| < 1, which are sufficientlycollapsed. Roughly, M n is called e-collapsed, if it appears to havedimension less than n, unless the metric is rescaled by a factor > e _1 . For example,a very thin cylinder is very collapsed (although its curvature vanishesidentically).If one fixes e and in addition, a bound, d, on the diameter, then in eachdimension, there are only finitely many manifolds, which are not e-collapsed.The basic result of collapsing theory states the existence of a constant, e(n) > 0,such that a manifold which is e-collapsed, for e < e(n), has a particular kind ofsingular fibration structure with flat (or "almost flat" ) fibers. The fibers lie inthe e-collapsed directions.The first nontrivial collapsing with bounded curvature, arose in a sequenceof metrics on the 3-sphere constructed by M. Berger. The first majorresult on the collapsed manifolds (still a corner stone of the theory) is M.Gromov's description of "almost flat manifolds" i.e. manifolds admitting a sequenceof metrics with curvature and diameter going to zero. Gromov showedthat such manifolds are infranilmanifolds.We will survey the main development of the collapsing theory and itsapplications to Riemannian geometry since 1990. The common starting pointis the existence of the above mentioned singular fibration structure. Manynew geometrical and topological constraints of collapsed metrics have beendiscovered that are accompanied with new ideas and techniques as well astools from related fields, and light has been shed on some classical problemsand conjectures, which do not, on the face of it, involve collapsing. Substantialprogress has been made on manifolds with non-positive curvature, on positivelypinched manifolds, collapsed manifolds with an a priori diameter bound, andsubclasses whose members satisfy additional topological conditions e.g. 2-connectedness.2000 Mathematics Subject Classification: 53C.*Supported partially by NSF Grant DMS 0203164 and a research found from Beijing NormalUniversity.'Rutgers University, New Brunswick, NJ 08903, USA and Beijing Normal University, Beijing100875, China. E-mail: rong@math.rutgers.edu

324 Xiaochun RongOne of the most important developments in Riemannian geometry over the lasttwo decades is the structure theory of Cheeger-Fukaya-Gromov for manifolds M nof bounded sectional curvature, say |secM»| < 1, which are sufficiently collapsed.Roughly, M n is called e-collapsed, if it appears to have dimension less than n, unlessthe metric is rescaled by a factor > e _1 .For scaling reasons, collapsing and boundedness of tend to oppose one another.Nevertheless, very collapsed manifolds with bounded curvature do in fact exist. Forexample, a very thin cylinder is very collapsed, although its curvature vanishesidentically.If one fixes e and in addition, a bound, d, on the diameter, then in eachdimension, there only finitely many manifolds, which are not e-collapsed; see [Ch].The basic result of collapsing theory states the existence of a constant e(n) > 0,such that a manifold which is e-collapsed, for e < e(n), has a particular kind ofsingular fibration structure with flat (or "almost flat") fibers. The fibers lie in thee-collapsed directions; see [CGI,2], [CFG], [Ful-3].The first nontrivial example of a collapsing sequence with bounded curvature(described in more detail below) was constructed by M. Berger in 1962; see [CFG].The first major result on the collapsed manifolds (still a cornerstone of the theory)was M. Gromov's characterization of "almost flat manifolds" i.e. manifolds admittinga sequence of metrics with curvature and diameter going to zero. Gromovshowed that such manifolds are infranil. Later in [Ru], they were shown to actuallybe nilmanifolds; compare [GMR].We will survey the development of collapsing theory and its applications toRiemannian geometry since 1990; compare [Fu4]. The common starting point forall of these is the above mentioned singular fibration structure. However, new techniqueshave been introduced and tools from related fields have been brought in. As aconsequence, light has been shed on some classical problems and conjectures whosestatements do not involve collapsing. Specifically, substantial progress has beenmade on manifolds with nonpositive curvature, on positively pinched manifolds,collapsed manifolds with an a priori diameter bound, and subclasses of manifoldswhose members satisfy additional topological conditions e.g. 2-connectedness.1. Collapsed manifolds of bounded sectionalcurvatureConvention: unless otherwise specified, "collapsing" refers to a sequence ofRiemannian manifolds with sectional curvature bounded in absolute value by oneand injectivity radii uniformly converge to zero, while "convergence" means "convergencewith respect to the Gromov-Hausdorff distance.Recall that a map from a metric space (X,dx) to a metric space (Y,dy) iscalled an e-Gromov-Hausdorff approximation, if of f(X) is e-dense in Y and if\dx(x,x') — dY(f(x),f(x'j)\ < e. The Gromov-Hausdorff distance between two(compact) metric spaces is the infimum of e as above, for all possible e-Gromov-Hausdorff approximations from X to Y and vice versa. (To be more precise, oneshould say "pseudo-distance", since isometric metric spaces have distance zero.)

Collapsed Riemannian Manifolds with Bounded Sectional Curvature 325The collection of all compact metric spaces is complete with respect to the Gromov-Hausdorff distance.a. Flat manifolds, collapsing by scaling and torus actionsFor fixed (M,g), the family, {(M, e 2 g)} converges to a point as e —¥ 0. However,if the curvature is not identically zero, it blows up. On the other hand, forany compact flat manifold, (M,g), the the manifolds, (M,e 2 g) continue to be flat.More generally, if (M,g) is a (possibly nonflat) manifold with an isometric torusr*-action for which all T^-orbits have the same dimension, then one obtains a collapsingsequence by rescaling g along the orbits i.e. by putting g f = e 2 go ® g^,where go is the restriction of g to the tangent space of a T fc -orbit and g^ is theorthogonal complement. A computation shows that g f _ has bounded sectional curvatureindependent of e. The collapse constructed by Berger in 1962 was of thistype. In his example, M 3 is the unit 3-sphere and the S 1 action is by rotation inthe fibers of the Hopf fibration S 1 —¥ S 3 —¥ S 2 . The limit of this collapse is the2-sphere with a metric of constant curvature = 4; see [Pet].More generally, a collapsing construction has been given by Cheeger-Gromovfor manifolds which admit certain mutually compatible local torus actions (possiblyby tori of different dimensions) for which all orbits have positive dimension; seethe notion of F-structure given below and (1.2.1). As above, for each individuallocal torus action, one obtains locally defined collapsing sequence. The problemis to patch together these local collapsings. If the orbits are not all of the samedimension, the patching requires a suitable scaling of the metric (by a large constant)in the transition regions between orbits of different dimensions; see [CGI]. Hence,in contrast to the Berger example, in general the diameters of such nontriviallypatched collapsings necessarily go to infinity.b. Almost flat manifolds and collapsing by inhomogeneous scalingAlthough a compact nilmanifold (based on a nonablian nilpotent Lie group)admits no flat metric, a sequence metrics on such a manifold which collapses to apoint can be constructed by a suitable inhomogeneous scaling process; see [Grl].As an example, regard a compact nilmanifold M 3 as the total space of a principlecircle bundle over a torus. A canonical metric g on M 3 splits into horizontal andvertical complements, g = gh®9h- Then g f = (egu) ® (e 2 g^) has bounded sectionalcurvature independent of e, while (M 3 ,g e ) converges to a point. The inhomogeneityof the scaling is essential in order for the curvature to remain bounded; compareTheorem 3.4.c. Positive rank F-structure and collapsed manifoldsThe notion of an F-structure may be viewed as a generalization of that ofa torus action. An F-structure J 7 on a manifold is defined by an atlas T ={(\'i,Ui,T ki )}, satisfying the following conditions:(1.1.1) {Ui} is a locally finite open cover for M.(1.1.2) 7T, : Vi —¥ Ui is a finite normal covering and V» admits an effective torusr fci -action such that it extends to a TTI(Uì) K T fci -action.

326 Xiaochun Rong(1.1.3) If Ui fl Uj ^ 0, then 7r _1 i (L/j n Uj) and 7TJ 1 (L/J n Uj) have a common finitecovering on which the lifting T fci — and T kj -actions commute.If ki = k, for all i, then T is called pure. Otherwise, T is called mixed. Thecompatibility condition, (1.1.3), implies that M decomposes into orbits, (an orbitat a point is the smallest set containing all the projections of the T ki -orbits atthe point.) The minimal dimension of all such orbits is called the rank of T. Anorbit is called regular, if it has a tubular neighborhood in which the orbits form afibration. Otherwise, it is called singular. An F-structure T is called polarized if allT ki -actions are almost free. An F-structure is called injective (resp. semi-injective)if the inclusion of any orbit to M induces an injective (resp. nontrivial) map on thefundamental groups.A Cr-structure is an injective F-structure with an atlas that satisfies two additionalproperties: i) V» = D» x T ki and T ki acts on V» by the multiplication, ii) IfUi fl Uj ^ 0, then fc» < kj or vice versa; see [Bui]. This notion arises in the contextof nonpositive curvature.A metric is called an ^-invariant (or simply invariant), if the local T^'-actionsare isomeric. For any F-structure, there exists an invariant metric.A manifold may not admit any nontrivial F-structure; compare Corollary 2.5.In fact, a simple necessary condition for a closed manifold M 2n to admit a positiverank F-structure is the vanishing of its Euler characteristic; see [CGI].A necessary and sufficient condition for the existence of a collapsing sequenceof metrics is the existence of an F-structure of positive rank; see [CGI], [CG2].Theorem 1.2 (Collapsing and F-structure of positive rank). ([CGI,2]) LetM be a manifold without boundary.(1.2.1) If M admits a positive rank (resp. polarized) F-structure, then M admits acontinuous one-parameter family of invariant metrics g f _ such that \sec 9e \ < 1 andthe injectivity radius (resp. volume) of g f _ converges uniformly to zero as e —¥ 0.(1.2.2) There exists a constant e(n) (the critical injectivity radius) such that ifM n admits a metric g with \sec g \ < 1 and the injectivity radius is less than e(n)everywhere, then M admits a positive rank F-structure almost compatible with themetric.The F-structure in (1.2.2) is actually a substructure of a so called nilpotentKilling structure on M whose orbits are infra-nilmanifolds; see [CFG] and compareto Theorem 3.5. Such an infra- nilmanifold orbit at a point contains all sufficientlycollapseddirections of the metric; the orbit of its sub F-structure, which is definedby the 'center' of the infra-nilmanifold, only contains the most collapsed directionscomparable to the injectivity radius at a point. A unsolved problem pertaining tonilpotent structures is whether a collapse as in (1.2.1) can be constructed for whichthe diameters of the nil-orbits converge uniformly to zero (as holds for F-structures).The construction of the F-structure in (1.2.2) relies only on the local geometry.Hence, (1.2.2) can be applied to a collapsed region in a complete manifoldof bounded sectional curvature. In this way, for such a manifold, one obtains athick-thin decomposition, in which the thin part carries an F-structure of positiverank; see [CFG].

Collapsed Riemannian Manifolds with Bounded Sectional Curvature 327Theorem 1.2 has been the starting point for many subsequent investigationsof collapsing in various situations. The guiding principle is that additional geometricalproperties of a collapsing should be mirrored in properties of its associatedF-structure, which in turn, puts constraints on the topology. For instance, if a collapsingsatisfies additional geometrical conditions such as: i) volume small, ii) uniformlybounded diameter, iii) nonpositive curvature, iv) positive pinched curvature,v) bounded covering geometry i.e. the injectivity radii of the Riemannian universalcovering has a uniform positive lower bound, then one may expect correspondingtopological properties of the F-structure such as: i) existence of a polarization,ii) pureness, iii) existence of a Cr-structure, iv) the existence of a circle orbit, v)injective F-structure. Results on such correspondences and their applications willoccupy the rest of this paper.d. Topological invariants associated to a volume collapseThe existence of a sufficiently (injectivity radius) collapsed metric as in (1.2.2)imposes constraints on the underlying topology. For instance, the simplicial volumeof M vanishes; see [Gr3]. As mentioned earlier, for a closed M 2n , the Eulercharacteristic of M 2n also vanishes; see [CFG].In this subsection, we focus on some topological invariants associated to certain(partially) volume collapsed metrics: the minimal volume, the L 2 -signature and thelimiting n-invariant; see below.The minimal volume, MinVol(AT), of M, is the infimum of the volumes over allcomplete metrics with |SCCM| < 1- Clearly, MinVol(M) is a topological invariant.Gromov conjectured that there exists a constant e(n) > 0 such that Min Vol (M n )

328 Xiaochun RongTheorem 1.4 (Rationality of geometric signature). ([Ro3]) If an open completemanifold, M 4 , of finite volume has bounded covering geometry outside a compactsubset, then the integral of the Hirzebruch signature form over M 4 is a rationalnumber.The main idea is to show that M 4 admits a polarized F-structure T outsidesome compact subset and an exhaustion by compact submanifolds, Mf, such thatthe restriction of T to the boundary of M is injective. The integral over M 4 is thelimit of the integrals over Mf, to which we apply the Atiyah-Patodi-Singer formulato reduce to showing the rationality of the limit of the ^-invariant terms. By makinguse of the special property of T and Theorem 1.5 below, we are able to concludethat the limit of the ^-invariant term is rational.Cheeger-Gromov showed that if a sequence of volume collapsed metrics ona closed manifold N 4 "-^1have bounded covering geometry, then the sequence ofthe associated ^-invariants converges and the limit is independent of the particularsequence of such metrics. They conjectured that the limit is rational.Theorem 1.5 (Rationality of limiting ^-invariants). ([Rol]) If a closed manifoldN 3 admits a sequence of volume collapsed metrics with bounded covering geometry,then N 3 admits an injective F-structure and the limit of the n-invariantsis rational.The idea is to show that N 3 admits an injective F-structure T. For an injectiveF-structure, the collapsing constructed in (1.2.1) has bounded covering geometryand may be used to compute the limit. Results from 3-manifold topology play arole in the proof of the existence of the injective F-structure.2. Collapsed manifolds with nonpositive sectionalcurvatureA classical result of Preismann says that for a closed manifold M n with negativesectional curvature, any abelian subgroup of the fundamental group is cyclic.By bringing in the discrete group technique, Margulis showed that if the metric isnormalized such that — 1 < SCCM» < 0, then there exists at least one point at whichthe injectivity radius is bounded below by a constant e(n) > 0.The study of the subsequent study of collapsed manifolds with — 1 < sec < 0may be viewed as an attempt to describe the special circumstances under whichthe conclusions of the Preismann and Margulis theorem can fail, if the hypothesisis weakened to nonpositive curvature; see [Bul-3], [CCR1,2], [Eb], [GW], [LY], [Sc].A collapsed metric with nonpositive curvature tends to be rigid in a precisesense; see (2.2.1) and (2.2.2). Namely, there exists a canonical Cr-structure whoseorbits are flat totally geodesic submanifolds. Of necessity, the construction of thisCr-structure is global. By contrast, the construction of less precise (but more generallyexisting) F-structure is local; see [CG2].Let M n = M n /Y, where M n denotes the universal covering space of M n withthe pull-back metric. A local splitting structure on a Riemannian manifold is a F-equivariant assignment to each point (of an open dense subset of M n ) a specified

Collapsed Riemannian Manifolds with Bounded Sectional Curvature 329neighborhood and a specified isometric splitting of this neighborhood, with a nontrivialEuclidean factor. Hence, a necessary condition for a local splitting structureis the existence of a plane of zero curvature, at every point of M n . A local splittingstructure is abelian if the projection to M n of every nontrivial Euclidean factor asabove is a closed embedded flat submanifold, and n addition, if two projected leavesintersect, then one of them is contained in the other.Theorem 2.1 (Abelian local splitting structure and Cr-structure). ([CCR1])Let M n be a closed manifold of —1 < SCCM < 0.(2.1.1) If the injectivity radius is smaller than e(n) > 0 everywhere, then M n admitsan abelian local splitting structure.(2.1.2) If M n admits an abelian local splitting structure, then it admits a compatibleCr-structure, whose orbits are the flat submanifolds (projected leaves) of the abelianlocal splitting structure. In particular, MinVol(M n ) = 0.Theorem 2.1 was conjectured by Buyalo, who proved the cases n = 3,4; see[Bul-3], [Sc].Let x £ M n . Yet Y f (x) ^ 1 denote the subgroup of F generated by those 7whose displacement function, öj(x) = d(x, , y(xj), satisfies d(x, , y(xj) < e. (In theapplication, e is small.) If all Y f (x) are abelian, then the minimal sets, {Min(F e (£))},of the Yf(x) give the desired abelian local splitting structure in (2.1.1). In general,Yf(x) is only Bieberbach. Then, a crucial ingredient in (2.1.1) is the existence ofa 'canonical' abelian subgroup of Y f _ (x) of finite index consisting of those elementswhich are stable in the sense of [BGS]. In spirit, the proof of (2.1.2) is similar to theconstruction in [CG2], but the techniques used are quite different.The following are some specific questions pertaining to abelian local splittingstructures:(2.2.1) If some metric g on M of nonpositive sectional curvature has an abelianlocal splitting structure, does every nonpositively curved metric also have such astructure?(2.2.2) If M has a Cr-structure, does every any nonpositively curved metric on Mhave a compatible local splitting structure?Note that an affirmative answer to (2.2.1) and (2.2.2) would imply a kind ofsemirigidity. It would imply that all nonpositively curved metrics on M are alikein a precise sense.Theorem 2.3 (F-structure and local splitting structure). ([CCR2]) Let X n ,M n be closed manifolds such that X n admits a nontrivial F-structure. Let f : X n —tM n have nonzero degree. Then every metric of nonpositive sectional curvature onM n has a local splitting structure.We conjecture that if an F-structure has positive rank, then the local splittingstructure is abelian. This conjecture, whose proof would provide an affirmativeanswer to (2.2.2), has been verified in dimension 3 and in some additional specialcases; see [CCR2].We conclude this section with two consequences of Theorem 2.3.

330 Xiaochun RongCorollary 2.4 (Generalized Margulis Lemma). ([CCR2]) Let M n be a closedmanifold of nonpositive sectional curvature. If the Ricci curvature is negative atsome point, then for every metric with \sec\ < 1, there is a point with injectivityradius > ö(n) > 0.Another consequence is a geometric obstruction for a nontrivial F-structure.Corollary 2.5 (Nonexistence of F-structure). ([CCR2]) If a closed manifoldM admits a metric of nonpositive sectional curvature such that the Ricci curvatureis negative at some point, then M does not admit a nontrivial F-structure.3. Collapsed manifolds with bounded sectionalcurvature and diameterIn this section, we discuss the class of collapsed manifolds of bounded sectionalcurvature whose diameters are also bounded. By the Gromov's compactness theorem,any sequence of such collapsed manifolds contains a convergent subsequence;see [GLP]. Hence, without loss of the generality, we only need to consider convergentcollapsing sequences.(3.1) Let M GH > X denote a sequence of closed manifolds converging to a compactmetric space X such that |secMf | < 1 and dim(X) < n.Main Problem 3.2. For i large, investigate relations between geometry and topologyof M and that of X. The following are some specific problems and questions.(3.2.1) Find topological obstructions for the existence of M as in (3.1).(3.2.2) To what extent is the topology of the M in (3.1) stable when i is sufficientlylarge?(3.2.3) Under what additional conditions is it true that {M} as in (3.1) containsa subsequence of constant diffeomorphism type? If all M are diffeomorphic, thento what extent do the metrics converge?Note that by the Cheeger-Gromov convergence theorem, the above problemsare well understood in the noncollapsed situation dim(X) = n.Theorem 3.3 (Convergence). ([Ch], [GLP]) Let Aff ^-y X be as in (3.1)except dim(X) = n. Then for i large, M is diffeomorphic to some fixed M nwhich is homeomorphic to X and there are diffeomorphisms, fi : M n —t M, suchthat the pulled back metrics, f*(gi), converge to a metric, g^, in the C 1 '®-topology(0

Collapsed Riemannian Manifolds with Bounded Sectional Curvature 331Theorem 3.4 (Almost flat manifolds). ([Grl]) Let Aff ^^y X be as in (3.1).If X is a point, then a finite normal covering space of M of order at most c(n) isdiffeomorphic to a nilmanifold N n /Y'i (i large), where N n is the simply connectednilpotent group.Theorem 3.4 can be promoted to a description of convergent collapsing sequence,of manifolds, M, as in (3.1). As mentioned following Theorem 1.2, anysufficiently collapsed manifold admits a nilpotent Killing structure; see [CFG]. Herea bound on diameter forces the nilpotent Killing structure to be pure.For a closed Riemannian manifold M n , its frame bundle F(M n ) admits acanonical metric determined by the Riemannian connection up to a choice of a biinvariantmetric on 0(n). A fibration, N/Y —t F(M n ) —t Y, is called 0(n)-invariantif the 0(n)-action on F(M n ) preserves both the fiber N/Y (a nilmanifold) and thestructural group. By the 0(n)-invariance, 0(n) also acts on the base space Y. Acanonical metric is invariant if its restriction on each N/Y is left-invariant. A purenilpotent Killing structure on M is an 0(n)-invariant fibration on F(M n ) for whichthe canonical metric is also invariant.Theorem 3.5 (Fibration). ([CFG]) Let Aff ^ yX be as in (3.1). Then F(M?)equipped with canonical metrics contains a convergent subsequence, F(M")GH > Y,and F(M") admits an 0(n)-invariant fibration N/Y'i —t F(M") —t Y for which thecanonical metric is ei-close in the C 1 sense to some invariant metric, where e, —¥ 0.The following properties are crucial for the study of particular instances ofcollapsing as in (3.1).Proposition 3.6. Let Aff ^^y X be as in (3.1).(3.6.1) (Regularity) ([Ro5]) For any e > 0, M admits an invariant metric gi suchthat min(secM?) — e < sec (M?,g i ) < max(secMp) + e for i large.(3.6.2) (Equivariance) ([PT], [GK]) The induced 0(n)actions on Y from the O(reactionon F(M") are C 1 -close and therefore are all 0(n)-equivariant for i large.f. Obstructions to collapsing with bounded diameterTheorem 3.7 (Polarized F-structure and vanishing minimal volume).([CR2]) Let M"GH > X be as in (3.1). Then the F-sub structure associated to thepure nilpotent Killing structure on M contains a (mixed) polarized F-structure. Inparticular, MinVol(Mf) = 0.Theorem 3.7 may be viewed as a weak version of the Gromov's gap conjecture.Note that the associated F-structure on M may not be polarized. The existenceof a polarized substructure puts constraints on the singularities of the structure.Theorem 3.8 (Absence of symplectic structure). ([FR3]) Let Mf ^ ^ X beas in (3.1). Ifm(M) is finite, then M does not support any symplectic structure.The proof of Theorem 3.8 includes a nontrivial extension of the well knownfact that any S 1 -action on a closed simply connected symplectic manifold whichpreserves the symplectic structure has a nonempty fixed point set.

332 Xiaochun RongA geometric obstruction to the existence of a collapsing sequence in (3.1) isprovided by:Theorem 3.9 (Geometric collapsing obstruction). ([Ro7]) Let Mfbe as in (3.1). Then limsup(maxM?* RìCM?) > 0.GH ) XA key ingredient in the proof is a generalization of a theorem of Bochner assertingthat a closed manifold of negative Ricci curvature admits no nontrivial invariantpure F-structure (Bochner's original theorem only guarantees the nonexistence of anontrivial isometric torus action.)Theorem 3.10 (Pure injective F-structure). ([CRI]) Let Mf ^ h X be asin (3.1). If Mf has bounded covering geometry and m(Mf) is torsion free, thenfor i large Mf admits a pure injective F-structure.g. The topological and geometric stabilityIn this subsection, we address Problems (3.2.2) and (3.2.3). Observe thatby the Gromov's Betti number estimate, [Gr2], the sequence in (3.1) contains asubsequence whose cohomology groups, fl»(M",Q), are all isomorphic. On theother hand, examples have been found showing that {H*(Mf, 2 and after passing to a subsequence, the q-th homotopygroup TTq(Mf) are all isomorphic, provided that ir q (Mf) are finitely generated (e.g.sec,Mf > 0 or iri(Mf) is finite).Note that in contrast to the Betti number bound, Theorem 3.11 does not holdif upper bound on the sectional curvature is removed; see [GZ].We now discuss sufficient topological conditions for diffeomorphism stability.Consider the sequence of fibrations, N/Y'i —t F(Mf) —t Y, associated to (3.1). Onewould like to know when all N/Y'i are diffeomorphic.Proposition 3.12. ([FR4]) Let Mf ^^y X be as in (3.1). If m(Mf) containsno free abelian group of rank two, then N/Y'i is diffeomorphic to a torus.In low dimensions, we have:Theorem 3.13 (Diffeomorphism stability—low dimensions). ([FR3], [Tu])For n < 6, let Mf ^ ^ X be as in (3.1). If in(Mf) = 1, then there is a subsequenceall whose members are diffeomorphic.Note that for n > 7, one cannot expect Theorem 3.13; see [AW]. Hence,additional restrictions are required in higher dimensions. Observe that if Mf are2-connected, then all T k —t F(Mf) —t Y are equivalent as principle T fc -bundles. Inparticular all F(Mf) are diffeomorphic.Using (3.6.2), Petrunin-Tuschmann showed that the equivalence can be chosenthat is also 0(n)-equivariant, and concluded the diffeomorphism stability for twoconnectedmanifolds; see [PT]. For the special case in which the Mf are positively

Collapsed Riemannian Manifolds with Bounded Sectional Curvature 333pinched, the same conclusion was obtained independently in [FRI] via a differentapproach.We introduce a topological condition which when Mf is simply connected,reduces to the assumption that iT2(Mf) is finite. In the nonsimply connected casehowever, there are manifolds with n 2 (M) infinite, which satisfy our condition.Let M denote the universal covering of M. For a homomorphism, p : m (M) —tÂut(Z k ), the semi-direct product, M x ni (M) %> k , is a bundle of p(7Ti(Af))-moduleswhich can be viewed as a local coefficient system over M. We denote it by Z*.Let b q (M,Zp) denote the rank of the cohomology group, H q (M,'L k p), with the localcoefficient system Z*. We refer to the integerb q (M,Z k ) = max {&,(M,Z*)}as the q-th twisted Betti number of M. Clearly, b q (M, Z fc ) is a topological invariantof M. Moreover, k • b 2 (M,Z) X be as in (3.1) with k = n — dim(X). Assume that Mf satisfies:(3.14.1) TTi(Mf) is a torsion group with torsion exponents uniformly bounded fromabove.(3.14.2) The second twisted Betti number b 2 (Mf,Z k ) = 0.Then there are diffeomorphisms, fi, from M n to (a subsequence of) {Mf}, suchthat the distance functions ofpullback metrics, f*(gi), on M n , converge to a pseudometricdoo in C°-norm. Moreover, M n admits a foliation with leaves diffeomorphicto flat manifolds (that are not necessarily compact) and a vector V tangent to a leafif and only if \\V\\ 9i —¥ 0.The proof of Theorem 3.14 is quite involved.Finally, we mention that J. Lott has systematically investigated the analyticaspects for a collapsing in (3.1); for details, see [Lol-3].4. Positively pinched manifoldsIn this section, we further investigate a subclass of the class of collapsed manifoldswith bounded diameter: collapsed manifolds with pinched positive sectionalcurvature; see [AW], [Ba], [Es], [Pü] for examples.In the spirit of Theorem 3.4, we first give the following classification result.Theorem 4.1 (Maximal collapse with pinched positive curvature). ([R08])Let Mf ^^y X be as in (3.1) such that sec M ? > ö > 0. Then dim(X) > ^"" di life 0 ""and equality implies that Mf ~ S n /Z qi (a lens space), where Mf —t Mf is acovering space of order < ^4^-.By Theorem 3.5, (3.6.1) and Proposition 3.12, the proof of Theorem 4.1 reducesto the classification of positively curved manifolds which admit invariant pure F-structures of maximal rank; see [GS].

334 Xiaochun RongTheorem 4.2 (Positive pinching and almost cyclicity of m).([Ro6]) LetMfGH ) X be as in (3.1) such that secM? > ö > 0. Then for i sufficiently large,iTi(Mf) has a cyclic subgroup whose index is less than w(n).By Theorem 3.5 and (3.6.1), the following result easily implies Theorem 4.2.Theorem 4.3 (Symmetry and almost cyclicity of m). ([Ro6]) Let M n bea closed manifold of positive sectional curvature. If M n admits an invariant pureF-structure, then m(M n ) has a cyclic subgroup whose index is less than a constantw(n).In the special case of a free isometric action, from the homotopy exact sequenceassociated to the fibration, S 1 —¥ M n —t M n /S 1 , together with the Synge theorem,one sees that m(M n ) is cyclic. The proof of the general case is by induction on nand is rather complicated.We now consider the injectivity radius estimate. Klingenberg-Sakai and Yauconjectured that the infimum of the injectivity radii of all 5-pinched metrics onM n is a positive number which depends only on Ö and the homotopy type of themanifold. By a result of Klingenberg, this conjecture is easy in even dimensions. Inodd dimensions it is open.Theorem 4.4 (Noncollapsing). ([FR4]; compare [FRI], [PT]) For n odd, letM n be a closed manifold satisfying 0 < ö < secM < 1 and \m(M n )\ < c. If~ IT—1b(M n ,Z~î~) = 0, then the injectivity radius of M n is at least e(n,ö,c) > 0.If Theorem 4.4 were false, then by Theorem 3.14 and (3.6.1) one could assumethe existence of a sequence, (M,gì)GH > X, with Ô/2 < sec Si < 1, such that thedistance functions of the metrics gi also converge. In view of the following theoremthis would lead to a contradiction.Theorem 4.5 (Gluing). ([PRT]) Let (M,g t ) ^^y X as in (1.3). If the distancefunctions of gi converge to a pseudo-metric, then lim inf (min sec gi ) < 0.Yet fi : (M, gì)GH > X denote an e, Gromov-Hausdorff approximation, wheree, —¥ 0. For an open cover {Bj} for X by small (contractible) balls, the assumptionon the distance functions implies (roughly) that the tube, Cy = ff 1 (Bj), is a subsetof M independent of i. Clearly, the universal covering Cy of Cy is noncompact.The idea is to glue together the limits of the Cy (modulo some suitable group ofisometries with respect to the pullback metrics) to form a noncompact metric spacewith curvature bounded below by liminf(minsec Si ) in the comparison sense; see[BGP], [Pe]. On the other hand, the positivity of the curvature implies that thespace so obtained would have to be compact.The above results on ^-pinched manifolds may shed a light on the topology ofpositively curved manifolds. It is tempting to make the following conjecture (whichseems very difficult).Conjecture 4.6. Let M n denote a closed manifold of positive sectional curvature.

Collapsed Riemannian Manifolds with Bounded Sectional Curvature 335(4.6.1) (Almost cyclicity) m(M n ) has a cyclic subgroup with index bounded by aconstant depending only on n.(4.6.2) (Homotopy group finiteness) For q > 2, n q (M n ) has only finitely manypossibleisomorphism classes depending only on n and q.(4.6.3) (Diffeomorphism finiteness) If ir q (M n ) = 0 (q = 1,2), then M n can haveonly finitely many possible diffeomorphism types depending only on n.Note that (4.6.1)-(4.6.3) are false for nonnegatively curved spaces. By theresults in this section, Conjecture 4.6 would follow from an affirmative answer tothe following:Problem 4.7 (Universal pinching constant). ([Be], [Ro5]) Is there a constant0 < ö(n) « 1 such that any closed n-manifold of positive sectional curvatureadmits a #(n)-pinched metric?A partial verification of (4.6.2) is obtained by [FR2].Theorem 4.8. ([FR2]) Let M n denote a closed manifold of positive sectional curvature.For q > 2, the minimal number of generators for n q (M n ) is less thanc(q,n).Previously, by Gromov the minimal number of generators of m (M n ) is boundedabove by a constant depending only on n.References[AW] S. Aloff; N. R. Wallach, An infinite family of 7-manifolds admitting positivecurved Riemannian structures, Bull. Amer. Math. Soc. 81 (1975), 93-97.[BGS] W. Ballmann; M. Gromov; Schroeder, Manifolds of nonpositive curvature,Basel:Birkhäuser, Boston, Basel, Stuttgart, (1985).[Ba] Ya. V. Bazaïkin Y, On a family of 13-dimensional closed Riemannianmanifolds of positive curvature, Sibirsk. Mat. Zh. 37 (in Russian), ii;English translation in Siberian Math. J. 6 (1996), 1068-1085.[Be] M. Berger, Riemannian geometry during the second half of the twentiethcentury, University lecture series 17 (2000).[BGP] Y. Burago; M. Gromov; Perel'man, A.D. Alexandov spaces with curvaturebounded below, Uspekhi Mat. Nauk 47:2 (1992), 3-51.[Bui] S. Buyalo, Collapsing manifolds of nonpositive curvature I, Leningrad Math.J., 5 (1990), 1135-1155.[Bu2] S. Buyalo, Collapsing manifolds of nonpositive curvature II, Leningrad[Bu3]Math. J., 6 (1990), 1371-1399.S. Buyalo, Three dimensional manifolds with Cr-structure,, Some Questionsof Geometry in the Large, A.M.S. Translations 176 (1996), 1-26.[CCR1] J. Cao; J. Cheeger; X. Rong, Splittings and Cr-structure for manifolds withnonpositive sectional curvature, Invent. Math. 144 (2001), 139-167.[CCR2] J. Cao; J. Cheeger; X. Rong, Partial rigidity of nonpositively curved manifolds(To appear).

336 Xiaochun Rong[Ch] J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math.92 (1970), 61-75.[CFG] J. Cheeger; K. Fukaya; M. Gromov, Nilpotent structures and invariant metricson collapsed manifolds, J. A.M.S 5 (1992), 327-372.[CGI] J. Cheeger; M. Gromov, Collapsing Riemannian manifolds while keepingtheir curvature bound I, J. Diff. Geom 23 (1986), 309-364.[CG2] J. Cheeger; M. Gromov, Collapsing Riemannian manifold while keepingtheir curvature bounded II, J. Differential Geom 32 (1990), 269-298.[CG3] J. Cheeger; M. Gromov, On the characteristic numbers of complete manifoldsof bounded curvature and finite volume, H. E. Rauch Mem Vol I(Chavel and Farkas, Eds) Springer, Berlin (1985), 115-154.[CG4] J. Cheeger; M. Gromov, Bounds on the von Neumann dimension of L 2 -cohomology and the Gauss-Bonnet theorem for open manifolds, J. Diff.Geom 21 (1985), 1-34.[CRI] J. Cheeger; X. Rong, Collapsed manifolds with bounded diameter and boundedcovering geometry, Geome.Funct. Anal 5 No. 2 (1995), 141-163.[CR2] J. Cheeger; X. Rong, Existence of polarized F-structure on collapsed manifoldswith bounded curvature and diameter, Geome. Funct. Anal 6, No.3(1996), 411-429.[Eb] P. Eberlein, A canonical form for compact nonpositively curved manifoldswhose fundamental groups have nontrivial center, Math. Ann. 260 (1982),23-29.[Es] J.-H Eschenburg, New examples of manifolds with strictly positive curvature,Invent. Math 66 (1982), 469-480.[FRI] F. Fang; X. Rong, Positive pinching, volume and homotopy groups, Geom.Funct. Anal 9 (1999), 641-674.[FR2] F. Fang; X. Rong, Curvature, diameter, homotopy groups and cohomologyrings, Duke Math. J. 107 No.l (2001), 135-158.[FR3] F. Fang; X. Rong, Fixed point free circle actions and finiteness theorems,Comm. Contemp. Math (2000), 75-86.[FR4] F. Fang; X. Rong, The twisted second Betti number and convergence ofcollapsing Riemannian manifolds, To appear in Invent. Math (2002).[Fui] K. Fukaya, Collapsing Riemannian manifolds to ones of lower dimension,J. Diff. Geome 25 (1987), 139-156.[Fu2] K. Fukaya, Collapsing Riemannian manifolds to ones of lower dimensionII, J. Math. Soc. Japan 41 (1989), 333-356.[Fu3] K. Fukaya, A boundary of the set of the Riemannian manifolds with boundedcurvature and diameters, J. Diff. Geome 28 (1988), 1-21.[Fu4] K. Fukaya, Hausdorff convergence of Riemannian manifolds and its applications,Recent Topics in Differential and Analytic Geometry (T. Ochiai,ed), Kinokuniya, Tokyo (1990).[GMR] P. Granaat; M. Min-Oo; E. Ruh, Local structure of Riemannian manifolds,Indiana Univ. Math. J 39 (1990), 1305-1312.[GW] D. Gromoll; J. Wolf, Some relations between the metric structure and thealgebraic structure of the fundamental group in manifolds of nonpositive

Collapsed Riemannian Manifolds with Bounded Sectional Curvature 337curvature, Bull. Am. Math. Soc, 4 (1977), 545-552.[Grl] M. Gromov, Almost flat manifolds, J. Diff. Geom 13 (1978), 231-241.[Gr2] M. Gromov, Curvature diameter and Betti numbers, Comment. Math. Helv56 (1981), 179-195.[Gr3] M. Gromov, Volume and bounded cohomology, I.H.E.S. Pul. Math. 56(1983), 213-307.[GLP] M. Gromov, J. Lafontaine; P.Pansu, Structures métriques pour les variétésriemannienes, CedicFernand Paris (1981).[GK] K. Grove; H. Karcher, How to conjugate C 1 -close actions, Math. Z 132(1973), 11-20.[GS] K. Grove, C. Searle, Positively curved manifolds with maximal symmetryrank,J. Pure Appi. Alg 91 (1994), 137-142.[GZ] K. Grove; W. Ziller, Curvature and symmetry of Milnor spheres, Ann. ofMath 152 (2000), 331-367.[LY] B. Lawson; S. T. Yau, On compact manifolds of nonpositive curvature, J.Diff. Geom., 7 (1972).[Loi] J. Lott, Collapsing and differential form Laplacian: the case of a smoothlimit space, Duke Math. J (To appear).[Lo2] J. Lott, Collapsing and differential form Laplacian: the case of a smoothlimit space, Preprint (2002).[Lo3] J. Lott, Collapsing and Dirac-type operators, Geometriae Delicata, Specialissue on partial differential equations and their applications to geometryand physics (To appear).[Pe] G. Perel'man, A.D. Alexandrov spaces with curvature bounded below II,preprint.[Pet] P. Petersen, Riemannian geometry, GTM, Springer-Verlag Berlin HeidelbergNew York 171 (1997).[PRT] A. Petrunin; X. Rong; W. Tuschmann, Collapsing vs. positive pinching,Geom. Funct. Anal 9 (1999), 699-735.[PT] A. Petrunin; W. Tuschmann, Diffeomorphism finiteness, positive pinching,and second homotopy, Geom. Funct. Anal 9 (1999).[Pü] T. Püttmann, Optimal pinching constants of odd dimensional homogeneousspaces, Invent. Math 138 (1999), 631-684.[Rol] X. Rong, The limiting eta invariant of collapsed 3-manifolds, J. Diff. Geom37 (1993), 535-568.[Ro2] X. Rong, The existence of polarized F-structures on volume collapsed 4-manifolds, Geom. Funct. Anal 3, No.5 (1993), 475-502.[Ro3] X. Rong, Rationality of geometric signatures of complete 4-manifolds, Invent.Math 120 (1995), 513-554.[Ro4] X. Rong, Bounding homotopy and homology groups by curvature and diameter,Duke. Math. J 2 (1995), 427-435.[Ro5] X. Rong, On the fundamental group of manifolds of positive sectional curvature,Ann. of Math 143 (1996), 397-411.[Ro6] X. Rong, The almost cyclicity of the fundamental groups of positively curvedmanifolds, Invent. Math 126 (1996), 47-64.

338 Xiaochun Rong[Ro7] X. Rong, A Bochner Theorem and applications, Duke Math. J 91, No.2(1998), 381-392.[Ro8] X. Rong, Collapsed manifolds with pinched positive sectional curvature, J.Diff. Geom 51 No.2 (1999), 335-358.[Ru] E. Ruh, Almost flat manifolds, J. Diff. Geome. 17 (1982), 1-14.[Sc] V. Schroeder, Rigidity of nonpositively curved graph manifolds, Math. Ann.274 (1986), 19-26.[Tu] W. Tuschmann, Geometric diffeomorphism finiteness in low dimensions andhomotopy finiteness, Math. Ann 322 (2002), 413-420.

ICM 2002 • Vol. II • 339-349Complex Hyperbolic Triangle Groups*Richard Evan Schwartz 1 'AbstractThe theory of complex hyperbolic discrete groups is still in its childhoodbut promises to grow into a rich subfield of geometry. In this paper I willdiscuss some recent progress that has been made on complex hyperbolic deformationsof the modular group and, more generally, triangle groups. Theseare some of the simplest nontrivial complex hyperbolic discrete groups. Inparticular, I will talk about my recent discovery of a closed real hyperbolic3-manifold which appears as the manifold at infinity for a complex hyperbolicdiscrete group.2000 Mathematics Subject Classification: 53.Keywords and Phrases: Complex hyperbolic space, Discrete groups, Trianglegroups, Deformations.1. IntroductionA basic problem in geometry is the deformation problem. One starts with afinitely generated group F, a Lie group G\, and a larger Lie group G2 D G\. Givena discrete embedding p 0 : Y —t G\ one asks if po fits inside a family p t : Y —t G 2 ofdiscrete embeddings. Here discrete embedding means an injective homomorphismonto a discrete set.A nice setting for the deformation problem is the case when G\ and G2 areisometry groups of rank one symmetric spaces, X\ and X 2 , and F is isomorphic to alattice in G\. If Xi = H 2 , the hyperbolic plane, and X 2 = H 3 , hyperbolic 3-space,then we are dealing with the classic and well-developed theory of quasifuchsiangroups.The (p, q,r)-reflection triangle group is possibly the simplest kind of latticein Isom(iT'). This group is generated by reflections in the sides of a geodesictriangle having angles n/p, n/q, n/r (subject to the inequality 1/p+l/q+l/r < 1.)We allow the possibility that some of the integers are infinite. For instance, the(2,3, oo)-reflection triangle group is commensurable to the classical modular group.*Supported by N.S.F. Research Grant DMS-0072706.îDepartment of Mathematics, University of Maryland, College park, MD 20742, USA. E-mail:res@math.umd.edu

340 Richard Evan SchwartzThe reflection triangle groups are rigid in Isom(iì 3 ), in the sense that any twodiscrete embeddings of the same group are conjugate. We are going to replace H 3by CH", the complex hyperbolic plane. In this case, we get nontrivial deformations.These deformations provide an attractive problem, because they furnish some of thesimplest interesting examples in the still mysterious subject of complex hyperbolicdeformations. While some progress has been made in understanding these examples,there is still a lot unknown about them.In §2 we will give a rapid introduction to complex hyperbolic geometry. In§3 we will explain how to generate some complex hyperbolic triangle groups. In §4we will survey some results about these groups and in §5 we will present a morecomplete conjectural picture. In §6 we will indicate some of the techniques we usedin proving our results.2. The complex hyperbolic planeThe book [8] is an excellent general reference for complex hyperbolic geometry.Here are some of the basics.C 2 ' 1 is a copy of the vector space C 3 equipped with the Hermitian formn(U, V) = -u 3 v 3 + Y^ ujVj- (1)Here U = (u\,u 2 ,uz) and V = (v\,v 2 ,vz). A vector V is called negative, null, orpositive depending (in the obvious way) on the sign of (V, V). We denote the set ofnegative, null, and positive vectors, by A r _, N 0 and N + respectively.C" includes in complex projective space CP" as the affine patch of vectorswith nonzero last coordinate. Let [ ] : C 2 ' 1 — {0} —ï CP 2 be the projectivizationwhose formula, expressed in the affine patch, is3=1[(vi,v 2 ,v 3 )] = (vi/v 3 ,v 2 /v 3 ). (2)The complex hyperbolic plane, CH", is the projective image of the set of negativevectors in C 2 ' 1 . That is, CH 2 = [AT_]. The ideal boundary of CH 2 is the unitsphere S 3 = [N 0 ]- If [X],[F] £ CH n the complex hyperbolic distance ß([X],[F])satisfiesQ([X],[Y}) = 2cosh- 1 y/6(X7ni 6(X,Y) = j ^ ^ y j - (3)Here X and Y are arbitrary lifts of [X] and [Y]. See [8, 77]. The distance we definedis induced by an invariant Riemannian metric of sectional curvature pinched between— 1 and —4. This Riemannian metric is the real part of a Kahler metric.SU(2,1) is the Lie group of ( , ) preserving complex linear transformations.PU(2,1) is the projectivization of SU(2,1) and acts isometrically on CH". Themap SU(2,1) —t PU(2,1) is a 3-to-l Lie group homomorphism. The group ofholomorphic isometries of CH" is exactly PU(2,1). The full group of isometries

Complex Hyperbolic Triangle Groups 341of CH" is generated by PU(2,1) and by the antiholomorphic map (01,02,23) —^(~Zi,Z 2 ,Z 3 ).An element of PU(2,1) is called elliptic if it has a fixed point in CH". Itis called hyperbolic (or loxodromic) if there is some e > 0 such that every point inCH" is moved at least e by the isometry. An element which is neither elliptic norhyperbolic is called parabolic.CH" has two different kinds of totally geodesic subspaces, real slices andcomplex slices. Every real slice is isometric to CH" n R" and every complex slice isisometric to CH 2 n C 1 . The ideal boundaries of real and complex slices are called,respectively, H-circles and C-circles. The complex slices naturally implement thePoincaré model of the hyperbolic plane and the real slices naturally model the Kleinmodel. It is a beautiful feature of the complex hyperbolic plane that it containsboth models of the hyperbolic plane.3. Reflection triangle groupsThere are two kinds of reflections in lsom(CH"). A real reflection is ananti-holomorphic isometry conjugate to the map (z,w) —¥ (z,w). The fixed pointset of a real reflection is a real slice. We shall not have much to say about theexplicit computation of real reflections, but rather will concentrate on the complexreflections.A complex reflection is a holomorphic isometry conjugate to the involution(z,w) —¥ (z,—w). The fixed point set of a complex reflection is a complex slice.There is a simple formula for the general complex reflection: Let C £ N + . Givenany U £ C 2 ' 1 define2(U,C) dIc(U) = Û + -çëjçfC. (4)le is a complex reflection.We also have the formulaU M V = («3W2 — «2^3, U1V3 — U3V1, U1V2 — u 2 vi). (5)This vector is such that (U, U S V) = (V, U S V) = 0. See [8, p. 45].Equations 4 and 5 can be used in tandem to rapidly generate triangle groupsdefined by complex reflections. One picks three vectors Vi,V2,V2 £ AT_. Next, welet Cj = Vj-i M Vj+i- Indices are taken mod 3. Finally, we let fi = Ic r Thecomplex reflection fi fixes the complex line determined by the points [Yfi-i] and[Vj + i]. This, the group (Ii,l2,h) is a complex-reflection triangle group determinedby the triangle with vertices [Vi], [V2], [V3].Here is a quick dimension count for the space of (p, q, r)-triangle groups generatedby complex reflections. We can normalize so that [Vi] = 0. The stabilizerof 0 in PU(2,1) acts transitively on the unit tangent space at 0. We cantherefore normalize so that [V2] = (s,0) where s £ (0,1). Finally, the isometries(z, w) —¥ (z, exp(i9)iv) stabilize both [Vi] and [V2]. Applying a suitable isometry wearrange that [V3] = (t+iu, v) where t,u,v £ (0,1). We cannot make any further normalizations,so the space of triangles in CH" mod isometry is 4-real dimensional.

342 Richard Evan SchwartzEach of the three angles (p, q, r) puts 1 real constraint on the triangle. For instance,the p-angle places the constraint that (Iihy is the identity. Since 4 — 3 = 1, we seeheuristically that the space of (p, q, r)-complex reflection triangle groups is 1-realdimensional.The argument we just gave can be made rigorous, and extends to the casewhen some of the integers are infinite. (In this case the corresponding vectors arenull rather than negative.) In the (oo, oo, oo)-case, the parameter is the angularinvariant arg((Vi, V 2 )(V 2 , \'z)(V3, V\)). Compare [10].This 1-dimensionality of the deformation space makes the (p, q, r)-trianglegroups an especially attractive problem to study. Indeed, there is a completelycanonicalpath of deformations. The starting point for the path of deformations isthe case when the vectors have entirely real entries. (That is, u = 0.) In this case,the three complex reflections stabilize the real slice R" n CH".4. Some resultsTo obtain a deformation of the (p, q, r)-reflection triangle group we choose aslice, either real or complex, and a triple of reflections, either real or complex, whichrestrict to the reflections in the sides of a (p, q, r)-geodesic triangle in the slice. Apriori there are 4 possibilities, given that the slice and the reflection types can beeither real or complex. These choices lead to different outcomes.If we start with complex reflections stabilizing a complex slice, the group hasorder 2, because the reflections will all stabilize the same slice.A more interesting case involving complex slices is given by:Theorem 4.1 [8] po : Y -^yIsom(CH") stabilizes a complex slice and acts on thisslice with compact quotient then any nearby representation pt also stabilizes a complexslice.Goldman's theorem applies to any co-compact lattice po(Y). In the case oftriangle groups, which are rigid in H", it says that any nearby representation isconjugate the original. In contrast:Theorem 4.2 [4, 12] There is a 1-parameter family pt.(Y(2,3,00j) of discrete faithfulrepresentations of the modular group having the property that po stabilizes a realslice and pi stabilizes a complex slice. For every parameter the generators are realreflections.Thus, in the case of non-cocompact triangle groups, two of the remaining3 cases can be connected. In their paper, Falbel and Koseleff claim that theirtechnique works for Y(p, q, 00) when max(p, q) = 4. For higher values of p and q itis not known what happens.The remaining case occurs when we start with complex reflections stabilizinga real slice. This is the case we discussed in the previous section. Henceforth werestrict our attention to this case.Goldman and Parker introduced this topic and studied the case of the idealtriangle group F(oo,oo,oo). They found that there is a 1-real parameter family of

Complex Hyperbolic Triangle Groups 343non-conjugate representations, {pt, t £ (^00,00)}. Once again p 0 stabilizes a realslice. Paraphrasing their more precise formulation:Theorem 4.3 [10] There are symmetric neighborhoods I C J of 0 such that pt isdiscrete and faithful if t £ I and not both discrete and faithful if t $ J.J consists of the parameter values t such that the element pt.(hhh) is not anelliptic element. For t $ J, this element is elliptic. If it has finite order then therepresentation is not faithful; if it has infinite order then the representation is notdiscrete. The (very slightly) smaller interval J is the interval for which their proofworks. They conjectured that pt should be discrete and faithful iff t £ J.We proved the Goldman-Parker conjecture, and sharpened it a bit.Theorem 4.4 [16] pt is discrete and faithful if and only if t £ J. Furthermore, ptis indiscrete if t $ J.The group L = p s (Y (00,00,00)), when s £ d J is especially beautiful. We callthis group the last ideal triangle group. (There are really two groups, one for eachendpoint of J, but these are conjugate.) This group seems central in the studyof complex hyperbolic deformations of the modular group. For instance, Falbeland Parker recently discovered that L arises as the endpoint of a certain family ofdeformations of the modular group, using real reflections. See [5] for details.Recall that L, like all discrete groups, has a limit set Q(L) c S 3 and a domainof discontinuity A(L) = S 3 — Q(L). The quotient A(L)/L is a 3-dimensionalorbifold, commonly called the orbifold at infinity.Theorem 4.5 [17] A(L)/Lis commensurable to the Whitehead link complement.The Whitehead link complement is a classic example of a finite volume hyperbolic3-manifold. The surprise in the above result is that a real hyperbolic3-manifold makes its appearance in the context of complex hyperbolic geometry.One might wonder about analogues of Theorem 4.4 for other triangle groups.Below we will conjecture that the space of discrete embeddings is a certain interval.In his thesis [22], Justin Wyss-Gallifent studied some special cases of this question.He made a very interesting discovery concerning the (4,4,00) triangle group:Theorem 4.6 [22] Let S be the set of parameters t for which the representationPt(F(4,4,00)) is discrete (but not necessarily injective). Then S contains isolatedpoints and, in particular, is not an interval.There seems to be an interval J of discrete embeddings and, outside of J, anextra countable sequence {tj} of parameters for which p tj is discrete but not anembedding. This sequence accumulates on the endpoints of J.Motivated by [17] I wanted to produce a discrete complex hyperbolic groupwhose orbifold at infinity was a closed hyperbolic 3-manifold. The extra representationsfound by Wyss-Gallifent seemed like a good place to start. Unfortunately,there is a cusp built into the representations of the (4,4,00) triangle groups.

344 Richard Evan SchwartzInstead, I considered the (4,4,4)-groups, and found that the extra discretedeformations exist. pt(Y(4,4,4)) seems to be discrete embedding iff all the elementsof the form pt(fiJjfiJk) are not elliptic. Here i,j, k are meant to be distinct. (Forall these parameters, the element p t (filjlk) is still a loxodromic element.) Thereis a countable collection i 5 , te,... of parameters such that pt ó (lì fi li fi) has order j.All these representations seem discrete. For ease of notation we set pj = pt ó •For j = 5,6,7,8,12 we can show by arithmetic means that pj is discrete. Therepresentation p 5 was too complicated for me to analyze and pg has a cusp. Thesimplest remaining candidate is p-j.Theorem 4.7 [18] G = p7(F(4,4,4)) is a discrete group. The orbifold at infinityA(G)/G is a closed hyperbolic 3-orbifold.In the standard terminology, A(G)/G is the orbifold obtained by labelling thebraid (AB) 15 (AB^2) 3 with a 2. Here A and B are the standard generators of the3-strand braid group.A spherical CR structure on a 3-manifold is a system of coordinate charts intoS 3 whose transition functions are restrictions of complex projective transformations.Kamishimaand Tsuboi [13] produced examples of spherical CR structures on Seifertfibered 3-manifolds, but our example in theorem 4.7 gives the only known sphericalCR structure on a closed hyperbolic 3-manifold. We think that Theorem 4.7 holdsforali j = 8,9,10....Concerning the specific topic of triangle groups generated by complex reflections,I think that not much else is known. Recently a lot of progress has beenmade in understanding triangle groups generated by real reflections. See [3] and [4].There has been a lot of other great work done recently on complex hyperbolic discretegroups, for instance [1], [2], [9], [20], [21]. Also see the references in Goldman'sbook [f8].5. A conjectural pictureWe will consider the 1-parameter family p t (p, q, r) of representations of the(p, q, r)-reflection triangle group, using complex reflections. We arrange that postabilizes a real slice. We choose our integers so that p < q < r. We let I p , I q , J rbe the generators of the reflection triangle group. The notation is such that I p isthe reflection in the side of the triangle opposite p, etc. DefineWA = Ipfilqfi] Wß = Iplqlr- (6)Conjecture 5.1 The set oft for which pt.(p,q,r) is a discrete embedding is theclosed interval consisting of the parameters t for which neither Pì(WA) nor pt(Wß)is elliptic.We call the interval of Conjecture 5.1 the critical interval.We say that the triple (p, q, r) has type A if the endpoints of the critical intervalcorrespond to the representations when WA is a parabolic element. In other words,WA becomes elliptic before WB- We say otherwise that (p,q,r) has type B.

Complex Hyperbolic Triangle Groups 345Conjecture 5.2 The triple (p,q,r) has type A if p < 10 and type B if p> 13.The situation is rather complicated when p £ {10,11,12,13}. Our Java applet[19] lets the user probe these cases by hand, though the roundoff error makes a fewcases ambiguous. The extra deformation, which was the subject of Theorem 4.7,seems part of a more general pattern.Conjecture 5.3 // (p,q,r) has type A then there is a countable collection of parametersti,t2,h... for which ptj(p,q,r) is infinite and discrete but not injective. If(p, q, r) has type B then all infinite discrete representations pt (p, q, r) are embeddingsand covered by Conjecture 5.1.The proviso about the infinite image arises because there always exists an extremelydegeneraterepresentation of Y(p, q, r) onto Z/2. The generators are all mapped tothe same complex reflection.In summary, there seems to be a critical interval J, such the representationsPt (p, q, r) are discrete embeddings iff t £ I. Depending on the endpoints of J, thereare either no additional discrete representations, or a countable collection of extradiscrete representations.It is interesting to see what happens as t moves to the boundary of J fromwithin J. We observed a certain kind of monotonicity to the way the representationvaries. Let F be the abstract (p,q,r) triangle group. For any word W £ Y, letWt = Pt(W). We will concentrate on the case when W is an infinite word. Fort £ I, the element Wt is (conjecturally) either a parabolic or loxodromic. Let X(Wt)be the translation length of Wt-Conjecture 5.4 As t increases monotonically from 0 to dì, the quantity X(Wt)decreases monotonically for all infinite words W.Conjecture 5.4 is closely related to some conjectures of Hanna Sandler [15]about the behavior of the trace function in the ideal triangle case. I think thatthere is some fascinating algebra hiding behind the triangle groupsîn the form ofthe behavior of the trace function—but so far it is unreachable.6. Some techniques of proofIf G Clsom(X), one can try to show that G is discrete by constructing afundamental domain for G. One looks for a set F C I such that the orbit G(F)tiles X. This means that the translates of F only intersect F in its boundary.The Poincaré theorem [B, §9.6] gives a general method for establishing the tilingproperty of F based on how certain elements of G act on dF.When X = H n , one typically builds fundamental domains out of polyhedrabounded by totally geodesic codimension-1 faces. When X = CH n , the situationis complicated by the absence of totally geodesic codimension-1 subspaces. Themost natural replacement is the bisector. A bisector is the set of points in CH nequidistant between two given points. Mostow [14] used bisectors in his analysis

346 Richard Evan Schwartzof some exceptional non-arithmetic lattices in Isom(CiT'), and Goldman studiedthem extensively in [8]. (See Goldman's book for additional references on paperswhich use bisectors to construct fundamental domains.)My point of view is that there does not seem to be a "best" kind surface touse in constructing fundamental domains in complex hyperbolic space. Rather, Ithink that one should be ready to fabricate new kinds of surfaces to fit the problemat hand. It seems that computer experimentation often reveals a good choice ofsurface to use. In what follows I will give a quick tour of constructive techniques.Consider first the deformations Gt = Pt(oo, 00,00) of the ideal triangle group,introduced in [10]. According to [16] these groups are discrete for t £ [0,T]. Herer is the critical parameter where the product of the generators is parabolic. Itis convenient to introduce the Clifford torus. Thinking of CH" as the open unitball in C", the Clifford torus is the subset T = {\z\ = \w\} C S 3 . AmazinglyT has 3 foliations by C-circles: The horizontal foliation consists of C-circles ofthe form {(z,w)\z = zo}- The vertical foliation consists of C-circles of the form{(z,w)\w = wo}- The diagonal foliation consists of C-circles having the form{(z,w)\ z = Àow}.Recall that Gt is generated by 3 complex reflections. Each of these reflectionsfixes a complex slice and hence the bounding C-circle. One can normalize so thatthe three fixed C-circles lie on the Clifford torus, one in each of the foliations.Passing to an index 2 subgroup, we can consider a group generated by 4 complexreflections: Two of these reflections, H\ and H 2 , fix horizontal C-circles hi and h 2and the other two, V\ and V 2 , fix vertical C-circles v\ and 1)2.The ideal boundary of a bisector is called a spinal sphere. This is an embedded2-sphere which is foliated by C-circles (and also by i2-circles.) We can find aconfiguration of 4-spinal spheres S(l,v), S(2,v), S(l,h) and S(2,h). Here S(j,v)contains Vj as part of its foliation and S(j,h) contains hj as part of its foliation.The map Hj stabilizes S(j, h) and interchanges the two components of S 3 — S(j, h).Analogous statements apply to the Vs.The two spheres S(h,j) are contained in the closure of one component ofS 3 —T and the two spheres S(v,j) are contained in the closure of the other. Whenthe parameter t is close to 0 these spinal spheres are all disjoint from each other,excepting tangencies, and form a kind of necklace of spheres. Given the way theelements Hj and Vj act on our necklace of spheres, we see that we are dealing withthe usual picture associated to a Schottky group. In this case the discreteness of thegroup is obvious.As the parameter increases, the two spinal spheres S(v, 1) and S(v,2) collide.Likewise, S(h, 1) and S(h, 2) collide. Unfortunately, the collision parameter occursbefore the critical parameter. For parameters larger than this collision parameter,we throw out the spinal spheres and look at the action of G on the Clifford torusitself. (This is not the point of view taken in [10] but it is equivalent to what theydid.)Let H be the subgroup generated by the reflections Hi and H 2 • One finds thatthe orbit H(T) consists of translates of T which are disjoint from each other exceptfor forced tangencies. Even though H is an infinite group, most of the elements in

Complex Hyperbolic Triangle Groups 347H move T well off itself, and one only needs to take care in checking a short finitelist of words in H. Once we know how H acts on T we invoke a variant of theping-pong lemma to get the discreteness.At some new collision parameter, the translates of the Clifford torus collidewith each other. Again, the collision parameter occurs before the critical parameter.This is where the work in [16] comes in. I define a new kind of surface called a hybridcone. A hybrid cone is a certain surface foliated by arcs of H-circles. These arcsmake the pattern of a fan: Each arc has one endpoint on the arc of a C-circleand the other endpoint at a single point common to all the arcs. I cut out twotriangular patches on the Clifford torus and replace each patch by a union of threehybrid cones. Each triangular patch is bounded by three arcs of C-circles; so thatthe hybrid cones are formed by connecting these exposed arcs to auxilliary pointsusing arcs of H-circles. In short, I put some dents into the Clifford torus to make itfit better with its ff-translates, and the I apply the ping-pong lemma to the dentedtorus.I also use hybrid cones in [17], to construct a natural fundamental domainin the domain of discontinuity A(L) for the last ideal triangle group L. In thiscase, the surfaces fit together to make three topological spheres, each tangent tothe other two along arcs of H-circles. The existence of this fundamental domainlets me compute explicitly that A(L)/L is commensurable to the Whitehead linkcomplement.Falbel and Zocca [6] introduce related surfaces called C-spheres, which arefoliated by C-circles. These surfaces seem especially well adapted to groups generatedby real reflections. See [3] and [4]. Indeed, Falbel and Parker construct adifferent fundamental domain for L using C-spheres. See [5].To prove Theorem 4.7 in [18] I introduce another method of constructingfundamental domains. My proof revolves around the construction of a simplicialcomplex Z C C 2 ' 1 . The vertices of Z are canonical lifts to C 2 ' 1 of fixed points ofcertain elements of the group G = p7(F(4,4,4)). The tetrahedra of Z are Euclideanconvex hulls of various 4-element subsets of the vertices. Comprised of infinitelymanytetrahedra, Z is invariant under the element hfih- Modulo this element Zhas only finitely many tetrahedra.Recall that [ ] is the projectivization map. Let [Z 0 ] = [Z] n S 3 . I deduce thetopology of the orbifold at infinity by studying the topology of [Z 0 ] • To show thatmy analysis of the topology at infinity is correct, I show that one component F ofCH" — [Z] has the tiling property: The G-orbit of F tiles CH". Now, Z is anessentially combinatorial object, and it not too hard to analyze the combinatoricsand topology of Z in the abstract. The hard part is showing that the map Z —¥ [Z]is an embedding. Assuming the embedding, the combinatorics and topology of Zare reproduced faithfully in [Z], and I invoke a variant of the Poincaré theorem.After making some easy estimates, my main task boils down to showing thatthe projectivization map [ ] is injective on all pairs of tetrahedra within a large butfinite portion of Z. Roughly, I need to check about 1.3 million tetrahedra. Thesheer number of checks forces us to bring in the computer. I develop a technique forproving, with rigorous machine-aided computation, that [ ] is injective on a given

348 Richard Evan Schwartzpair of tetrahedra.A novel feature of my work is the use of computer experimentation andcomputer-aided proofs. This feature is also a drawback, because it only allowsfor the analysis of examples one at a time. To make this analysis automatic Iwould like to see a kind of marriage of complex hyperbolic geometry and computation.On the other hand, I would greatly prefer to see some theoretical advances indiscreteness-proving which would eliminate the computer entirely.References[i[2[3;[4;[s;[6;[9[io;[n[12:[is;[w;[15'[16[17;[18D. Allcock, J. Carlson, D. Toledo, The Moduli Space of Cubic Threefolds, J.Alg. Geom. (to appear).P. Deligne and G.D. Mostow, Commensurabilities among Lattices in PU(l,n),Annals of Mathematics Studies 132, Princeton University Press (1993).E. Falbel and P.-V. Koseleff, Flexibility of the Ideal Triangle Group in ComplexHyperbolic Geometry, Topology 39(6) (2000), 1209^1223.E. Falbel and P.-V. Koseleff, A Circle of Modular Groups, preprint 2001.E. Falbel and J. Parker, The Moduli Space of the Modular Group in ComplexHyperbolic Geometry, Math. Research Letters (to appear).E. Falbel and V. ZOCCA, A Poincaré's Fundamental Polyhedron Theorem forComplex Hyperbolic Manifolds, J. reine angew Math. 516 (1999), 133^158.W. Goldman Representations of fundamental groups of surfaces, in "Geometryand Topology, Proceedings, University of Maryland 1983^1984", J. Alexanderand J. Harer (eds.), Lecture Notes in Math. Vol. 1167 (1985), 95^117.W. Goldman, Complex Hyperbolic Geometry, Oxford Mathematical Monographs,Oxford University Press, (1999).W. Goldman, M. Kapovich and B. Leeb, Complex Hyperbolic Surfaces HomotopyEquivalent to a Riemann surface, Communications in Analysis andGeometry 9 (2001), 61^95.W. Goldman and J. Parker, Complex Hyperbolic Ideal Triangle Groups, J. reineagnew Math. 425 (1992), 71^86.W. Goldman and J. Millson, Local Rigidity of Discrete Groups Acting on ComplexHyperbolic Space, Inventiones Mathematicae 88 (1987), 495^520.N. Gusevskii and J.R. Parker, Complex Hyperbolic Representations of SurfaceGroups and Toledo's Invariant, preprint (2001).Y. Kamishima and T. Tsuboi, CR Structures on Seifert Manifolds, Invent.Math. 104 (1991) 149^163.G.D. Mostow, On a Remarkable Class of Polyhedra in Complex HyperbolicSpace, Pac. Journal of Math 86 (1980) 171^276.H. Sandler, Trace Equivalence in SU(2,Y), Geo Dedicata 69 (1998) 317^327.R. E. Schwartz, Ideal Triangle Groups, Dented Tori, and Numerical Analysis,Annals of Math 153 (2001).R.E. Schwartz, Degenerating the Complex Hyperbolic Ideal Triangle Groups,Acta Mathematica 186 (2001).R. E. Schwartz, Real Hyperbolic on the Outside, Complex Hyperbolic on the

Complex Hyperbolic Triangle Groups 349Inside, Invent. Math (to apear).[19] R. E. Schwartz, Applet 29 (2001) http://www.math.umd.edu/~res.[20] Y. Shalom, Rigidity, Unitary Representations of Semisimple Groups, and FundamentalGroups of Manifolds with Rank One Transformation Group, Annalsof Math 152 (2000) 113-182.[21] D. Toledo, Representations of Surface Groups on Complex Hyperbolic Space,Journal of Differential Geometry 29 (1989) 125^133.[22] J. Wyss-Gallifent, Discreteness and Indiscreteness Results for Complex HyperbolicTriangle Groups, Ph.D. Thesis, University of Maryland (2000).

ICM 2002 • Vol. II • 351^360Fukaya Categories and DeformationsPaul Seidel*AbstractIt is widely believed that the right "cycles" for symplectic geometry areLagrangian submanifolds of symplectic manifolds (see for instance Weinstein's1981 survey). This can be given several different meanings, depending onthe kind of symplectic geometry one is interested in. In one direction, thedevelopment of Floer cohomology for Lagrangian submanifolds, culminatingin recent work of Fukaya, Oh, Ohta and Ono, has led to the definition of a"Fukaya category" associated to a symplectic manifold. I want to look at therelation between the Fukaya category of an affine variety M C C^ and thatof its projective closure M C CP N . This can be set up as a "deformationproblem" in the abstract algebraic sense.2000 Mathematics Subject Classification: 57R17, 57R56, 18E30.Soon after their first appearance [7], Fukaya categories were brought to theattention of a wider audience through the homological mirror conjecture [14]. Sincethen Fukaya and his collaborators have undertaken the vast project of laying downthe foundations, and as a result a fully general definition is available [9, 6]. The taskthat symplectic geometers are now facing is to make these categories into an effectivetool, which in particular means developing more ways of doing computations in andwith them.For concreteness, the discussion here is limited to projective varieties which areCalabi-Yau (most of it could be carried out in much greater generality, in particularthe integrability assumption on the complex structure plays no real role). Thefirst step will be to remove a hyperplane section from the variety. This makesthe symplectic form exact, which simplifies the pseudo-holomorphic map theoryconsiderably. Moreover, as far as Fukaya categories are concerned, the affine piececan be considered as a first approximation to the projective variety. This is a fairlyobviousidea, even though its proper formulation requires some algebraic formalismof deformation theory. A basic question is the finite-dimensionality of the relevant* Centre de Mathématiques, Ecole Polytechnique, F-91128 Palaiseau, France. E-mail:seidel@math.polytechnique.fr

352 Paul Seideldeformation spaces. As Conjecture 4 shows, we hope for a favourable answer inmany cases. It remains to be seen whether this is really a viable strategy forunderstanding Fukaya categories in interesting examples.Lack of space and ignorance keeps us from trying to survey related developments,but we want to give at least a few indications. The idea of working relativeto a divisor is very common in symplectic geometry; some papers whose viewpointis close to ours are [12, 16, 3, 17]. There is also at least one entirely different approachto Fukaya categories, using Lagrangian fibrations and Morse theory [8, 15, 4].Finally, the example of the two-torus has been studied extensively [18].1. Symplectic cohomologyWe will mostly work in the following setup:Assumption 1. X is a smooth complex projective variety with trivial canonicalbundle, and D a smooth hyperplane section in it. We take a suitable small openneighbourhood U D D, and consider its complement M = X\U. Both X and M areequipped with the restriction of the Fubini-Study Kahler form. Then M is a compactexact symplectic manifold with contact type boundary, satisfying ci(M) = 0.Consider a holomorphic map u : S —t X, where S is a closed Riemann surface.The symplectic area of u is equal (up to a constant) to its intersection number withD. When counting such maps in the sense of Gromov-Witten theory, it is convenientto arrange them in a power series in one variable t, where the t k term encodesthe information from curves having intersection number k with D. The t° termcorresponds to constant maps, hence is sensitive only to the classical topology of X.Thus, for instance, the small quantum cohomology ring QH*(X) is a deformationof the ordinary cohomology H*(X).As we've seen, there are only constant holomorphic maps from closed Riemannsurfaces to M = X \ D. But one can get a nontrivial theory by using puncturedsurfaces, and deforming the holomorphic map equation near the punctures throughan inhomogeneous term, which brings the Reeb dynamics on dM into play. Thiscan be done more generally for any exact symplectic manifold with contact typeboundary, and it leads to the symplectic cohomology SH*(M) of Cieliebak-Floer-Hofer [2] and Viterbo [26, 27]. Informally one can think of SH*(M) as the Floercohomology HF*(M \ dM, H) for a Hamiltonian function H on the interior whosegradient points outwards near the boundary, and becomes infinite as we approachthe boundary. For technical reasons, in the actual definition one takes the directlimit over a class of functions with slower growth (to clarify the conventions: ourSH k (M) is dual to the FH 2n^k(M) in [26]). The algebraic structure of symplecticcohomology is different from the familiar case of closed M, where one has large quantumcohomology and the WDVV equation. Operations SH*(M)® P -t SH*(M)® q ,for p > 0 and q > 0, come from families of Riemann surfaces with p + q punctures,together with a choice of local coordinate around each puncture. The Riemannsurfaces may degenerate to stable singular ones, but only if no component of the

Fukaya Categories and Deformations 353normalization contains some of the first p and none of the last q punctures. Thismeans that if we take only genus zero and q = 1 then no degenerations at all areallowed, and the resulting structure is that of a Batalin-Vilkovisky (BV) algebra[10]. For instance, let M = D(T*L) be a unit cotangent bundle of an oriented closedmanifold L. Viterbo [27] computed that SH*(M) = ff„_»(AL) is the homology ofthe free loop space, and a reasonable conjecture says that the BV structure agreeswith that of Chas-Sullivan [1].Returning to the specific situation of Assumption 1, and supposing that U hasbeen chosen in such a way that the Reeb flow on dM becomes periodic, one can usea Bott-Morse argument [19] to get a spectral sequence which converges to SH*(M).The starting term isEPi = i H9 ( M ) P=°> (1)1\Hi+ 3 P(dM)p 2 (andappealing to hard Lefschetz, which will be the only time that we use any algebraicgeometry) one hasdimSH 2 (M) of exact Lagrangian submanifold in it. Such submanifolds L havethe property that there are no non-constant holomorphic maps u : (£, 9S) —t (M, L)for a compact Riemann surface S, hence a theory of "Gromov-Witten invariantswith Lagrangian boundary conditions" would be trivial in this case. To get somethinginteresting, one removes some boundary points from S, thus dividing theboundary into several components, and assigns different L to them. The part ofthis theory where S is a disk gives rise to the Fukaya A^-category J(M).The basic algebraic notion is as follows. An A^-category A (over some field,let's say Q) consists of a set of objects Ob A, and for any two objects a gradedQ-vector space of morphisms hom A (Xo,Xi), together with composition operationsp A : hom A (X 0 ,Xi) —y hom A (X 0 , Xi)[l],p 2 A : hom A (Xi,X 2 ) ® hom A (X 0 ,Xi) —y hom A (Xo,X 2 ),p? A : hom A (X 2 ,X 3 ) hom A (Xi,X 2 ) ® hom A (X 0 ,Xi) ->—>hom A (X 0 ,X 3 )[-l], ....These must satisfy a sequence of quadratic "associativity" equations, which ensurethat p} A is a differential, p A a morphism of chain complexes, and so on. Note that by

354 Paul Seidelforgetting all the p A with d > 3 and passing to p3r cori0mo l°gy i n degree zero, oneobtains an ordinary Q-linear category, the induced cohomological category H°(A)- actually, in complete generality H°(A) may not have identity morphisms, but wewill always assume that this is the case (one says that A is cohomologically unital).In our application, objects of A = y(M) are closed exact Lagrangian submanifoldsL c M \ dM, with a bit of additional topological structure, namely agrading [14, 22] and a Spin structure [9]. If L 0 is transverse to L\, the space ofmorphisms hom A (Lo,Li) = CF(L Q ,Li) is generated by their intersection points,graded by Maslov index. The composition p A counts "pseudo-holomorphic (d+ 1)-gons", which are holomorphic maps from the disk minus d + 1 boundary points toM. The sides of the "polygons" lie on Lagrangian submanifolds, and the corners arespecified intersection points; see Figure 1. There are some technical issues having todo with transversality, which can be solved by a small inhomogeneous perturbationof the holomorphic map equation. This works for all exact symplectic manifoldswith contact type boundary, satisfying ci = 0, and is quite an easy construction bytoday'sstandards, since the exactness condition removes the most serious problems(bubbling, obstructions).Figure 1:It is worth while emphasizing that, unlike the case of Gromov-Witten invariants,each one of the coefficients which make up p A depends on the choiceof perturbation. Only by looking at all of them together does one get an objectwhich is invariant up to a suitable notion of quasi-isomorphism. To get somethingwhich is well-defined in a strict sense, one can descend to the cohomological categoryiJ°(3 r (Af )) (which was considered by Donaldson before Fukaya's work) whosemorphisms are the Floer cohomology groups, with composition given by the "pairof-pants"product; but that is rather a waste of information.At this point, we must admit that there is essentially no chance of computingJ(M) explicitly. The reason is that we know too little about exact Lagrangiansubmanifolds; indeed, this field contains some of the hardest open questions insymplectic geometry. One way out of this difficulty, proposed by Kontsevich [14],is to make the category more accessible by enlarging it, adding new objects in aformal process, which resembles the introduction of chain complexes over an additive

Fukaya Categories and Deformations 355category. This can be done for any A^-category A, and the outcome is called theAoo-category of twisted complexes, Tw(A). It contains the original Aoo-categoryas a full subcategory, but this subcategory is not singled out intrinsically, andvery different A can have the same Tw(A). The cohomological category D b (A) =H°(Tw(Aj), usually called the derived category of A, is triangulated (passage tocohomology is less damaging at this point, since the triangulated structure allowsone to recover many of the higher order products on Tw(A) as Massey products).For our purpose it is convenient to make another enlargement, which is Karoubi oridempotent completion, and leads to a bigger A^-category Tw 7! (A) D Tw(A) andtriangulated category D n (A) = H°(Tw n (Aj). The main property of D n (A) is thatfor any object X and idempotent endomorphism n : X —t X, n 2 = n, there is adirect splitting X = im(n) ® ker(n). The details, which are not difficult, will beexplained elsewhere.3. Picard-Lefschetz theoryWe will now restrict the class of symplectic manifolds even further:Assumption 2. In the situation of Assumption 1, suppose that X is itself a hyperplanesection in a smooth projective variety Y, with Xy — Gy(—X). Moreover,X = X 0 should be part of a Lefschetz pencil of such sections {X z }, whose base locusis D = X 0 fl XQO .This gives a natural source of Lagrangian spheres in M, namely the vanishingcycles of the Lefschetz pencil. Recall that to any Lagrangian sphere S one canassociate a Dehn twist, or Picard-Lefschetz monodromy map, which is a symplecticautomorphism rg. The symplectic geometry of these maps is quite rich, and containsinformation which is not visible on the topological level [20, 21, 22]. The action ofTs on the Fukaya category is encoded in an exact triangle in Tw(3 r (Af )), of the form^rs(L) (3)HF*(S,L)

356 Paul SeidelTheorem 3. Si,...,S m are split-generators for D 7 '('J(Mj). This means that anyobject ofTw 7 '('J(Mj) can be obtained from them, up to quasi-isomorphism, by repeatedlyforming mapping cones and idempotent splittings.4. Hochschild cohomologyThe Hochschild cohomology HH*(A,A) of an A^-category A can be definedby generalizing the Hochschild complex for algebras in a straightforward way, ormore elegantly using the A^-category fun(A,A) of functors and natural transformations,as endomorphisms of the identity functor. A well-known rather impreciseprinciple says that "Hochschild cohomology is an invariant of the derived category".In a rigorous formulation which is suitable for our purpose,HH*(A,A) h. HH*(Tw n (A),Tw n (Aj). (4)This is unproved at the moment, because Tw 7! (A) itself has not been consideredin the literature before, but it seems highly plausible (a closely related result hasbeen proved in [13]). Hochschild cohomology is important for us because of itsrole in deformation theory, see the next section; but we want to discuss its possiblegeometric meaning first.Let M be as in Assumption 1 (one could more generally take any exact symplecticmanifold with contact type boundary and vanishing ci). Then there is a natural"open-closed string map" from the symplectic cohomology to the Hochschildcohomology of the Fukaya category:SH*(M) —• HH*($(M),$(Mj). (5)This is defined in terms of Riemann surfaces obtained from the disk by removing oneinterior point and an arbitrary number of boundary points. Near the interior point,one deforms the holomorphic map equation in the same way as in the definitionof SH*(M), using a large Hamiltonian function; otherwise, one uses boundaryconditionsas for J(M). Figure 2 shows what the solutions look like.HH* (A, A) for any A carries the structure of a Gerstenhaber algebra, and onecan verify that (5) is a morphism of such algebras. Actually, since SH*(M) is aBV algebra, one expects the same of HH*('J(M),'J(Mj). This should follow fromthe fact that 7(M) is a cyclic A^-category in some appropriate weak sense, butthe story has not yet been fully worked out (two relevant papers for the algebraicside are [25] and [24]).Conjecture 4. If M is as in Assumption 2, (5) is an isomorphism.Assumption 2 appears here mainly for the sake of caution. There are a numberof cases which fall outside it, and to which one would want to extend the conjecture,but it is not clear where to draw the line. Certainly, without some restriction onthe geometry of M, there can be no connection between the Reeb flow on dM andLagrangian submanifolds?

Fukaya Categories and Deformations 357Lagrangian submanifoldsPeriodic orbit of theHamiltonianFigure 2:5. Deformations of categoriesThe following general definition, due to Kontsevich, satisfies the need for adeformation theory of categories which should be applicable to a wide range of situations:for instance, a deformation of a complex manifold should induce a deformationof the associated differential graded category of complexes of holomorphicvector bundles. By thinking about this example, one quickly realizes that sucha notion of deformation must include a change in the set of objects itself. TheAQO-formalism, slightly extended in an entirely natural way, fits that requirementperfectly. The relevance to symplectic topology is less immediately obvious, but itplays a central role in Fukaya, Oh, Ohta and Ono's work on "obstructions" in Floercohomology [9] (a good expository account from their point of view is [5]).For concreteness we consider only A^-deformations with one formal parameter,that is to say over Q[[t]]. Such a deformation £ is given by a set Ob £ of objects,and for any two objects a space homcfiXo, Xi) of morphisms which is a free gradedQ[[r]]-module, together with composition operations as before but now including a0-ary one: this consists of a so-called "obstruction cocycle"p\ £ hom\(X,X) (6)for every object X, and it must be of order t (no constant term). There is a sequenceof associativity equations, extending those of an A^-category by terms involvingp\. Clearly, if one sets t = 0 (by tensoring with Q over Q[[r]]), p% vanishes and theoutcome is an ordinary A^-category over Q. This is called the special fibre anddenoted by £ sp . One says that £ is a deformation of £ sp .A slightly more involved construction associates to £ two other A^-categories,the global section category £. g i and the generic fibre £. gen , which are defined overQ[[t]] and over the Laurent series ring Q[t _1 ][[t]], respectively. One first enlarges £to a bigger A^-deformation £ c by coupling the existing objects with formal connections(the terminology comes from the application to complexes of vector bundles).Objects of £ c are pairs (X,a) consisting of X £ Ob £ and an a e hom\(X, X)

358 Paul Seidelwhich must be of order t. The morphism spaces remain the same as in £, butall the composition maps are deformed by infinitely many contributions from theconnection. For instance,A*e„ = Me + ß\( a ) + ß\( a , a ) + • • • £ hom\ ((X,a), (X,aj) = hom\(X,X). (7)Egi C £ c is the full A^-subcategory of objects for which (7) is zero; and £. gen isobtained from this by inverting t. The transition from £ sp to £ s j and £. gen affectsthe set of objects in the following way: if for some X one cannot find an a suchthat (7) vanishes, then the object is "obstructed" and does not survive into £ s j; ifon the other hand there are many different a, a single X can give rise to a wholefamily of objects of £ s j. Finally, two objects of £. gen can be isomorphic even thoughthe underlying objects of £ sp aren't; this happens when the isomorphism involvesnegative powers of t.The classification of A^-deformations of an A^-category A is governed by itsHochschild cohomology, or rather by the dg Lie algebra underlying HH* +1 (A, A), inthe sense of general deformation theory [11]. We cannot summarize that theory here,but as a simple example, suppose that HH 2 (A,A) = Q. Then a nontrivial .4 œ -deformation of A, if it exists, is unique up to equivalence and change of parametert >-¥ f(t) (to be accurate, f(t) may contain roots of t, so the statement holds overQ[[t, fil 2 , fil 3 ,...]]). The intuitive picture is that the "versai deformation space"has dimension < 1, so that any two non-constant arcs in it must agree up toreparametrization.In the situation of Assumption 1, the embedding of our exact symplectic manifoldM into X should give rise to an A^-deformation J (M c X). We say "should"because the details, which in general require the techniques of [9], have not been carriedout yet. Roughly speaking one takes the same objects as in J(M) and the samemorphism spaces, tensored with Q[[t]], but now one allows "holomorphic polygons"which map to X, hence may intersect the divisor D. The numbers of such polygonsintersecting D with multiplicity k will form the t k term of the composition mapsin J(M c X). Because there can be holomorphic discs bounding our Lagrangiansubmanifolds in X, nontrivial obstruction cocycles (6) may appear.The intended role of J(M c X) is to interpolate between 'J(M), which wehave been mostly discussing up to now, and the Fukaya category ^(X) of the closedsymplectic manifold X as defined in [9, 6]. The t° coefficients count polygons whichare disjoint from D, and these will automatically lie in M, so that•J(M c X) sp~ -J(M).The relation between the generic fibre and ^(X) is less straightforward. First of all,J(M c X) gen will be an A^-category over Q[£ -1 ,£]], whereas 7(X) is defined overthe Novikov ring A t . Intuitively, one can think of this difference as the consequenceof a singular deformation of the symplectic form. Namely, if one takes a sequence ofsymplectic forms (all in the same cohomology class) converging towards the current[D], the symplectic areas of holomorphic discs u would tend to the intersectionnumber u • D. A more serious issue is that J(M c X) gen is clearly smaller than

Fukaya Categories and Deformations 3597(X), because it contains only Lagrangian submanifolds which lie in M. However,that difference may disappear if one passes to derived categories:Conjecture 5. In the situation of Assumption 2, there is a canonical equivalenceof triangulated categoriesD*(Ï(M C X) gen ® Q[t -i] [[t] ] A t ) - D*(f(X)).In comparison with the previous conjecture, Assumption 2 is far more importanthere. The idea is that there should be an analogue of Theorem 3 for D*(9r(X)),saying that this category is split-generated by vanishing cycles, hence by objectswhich are also present in J(M c X).To pull together the various speculations, suppose that Y = CP" +1 for somen > 3; X c Y is a hypersurface of degree n + 2; and D c X is the intersectionof two such hypersurfaces. Then D 7 '('J(Mj) is split-generated by finitelymanyobjects, hence Tw 7 '('J(Mj) is at least in principle accessible to computation.Conjecture 4 together with (2), (4) tells us that HH 2 (7(M),'3 : (Mj) =ëHH 2 (Tw 7 '('J(Mj),Tw 7 '('J(Mjj) is at most one-dimensional, so an A,»-deformationof Tw 7 '('J(Mj) is unique up to a change of the parameter t. From this deformation,Conjecture 5 would enable one to find D*(9r(X)), again with the indeterminacy inthe parameter (fixing this is somewhat like computing the mirror map).Acknowledgements. Obviously, the ideas outlined here owe greatly to Fukaya andKontsevich. The author is equally indebted to Auroux, Donaldson, Getzler, Joyce,Khovanov, Smith, and Thomas (an incomplete list), all of whom have influenced histhinking considerably. The preparation of this talk at the Institute for AdvancedStudy was supported by NSF grant DMS-9729992.References[1] M. Chas and D. Sullivan, String topology, Preprint math.GT/9911159.[2] K. Cieliebak, A. Floer, and H. Hofer, Symplectic homology II: a general construction,Math. Z. 218 (1995), 103^122.[3] Ya. Eliashberg, A. Gi ventai, and H. Hofer, Introduction to symplectic fieldtheory, Geom. Funct. Anal. Special Volume, Part II (2000), 560^673.[4] K. Fukaya, Asymptotic analysis, multivalued Morse theory, and mirror symmetry,Preprint 2002.[5] , Deformation theory, homological algebra, and mirror symmetry,Preprint, December 2001.[6] , Floer homology and mirror symmetry II, Preprint 2001.[7] , Morse homotopy, .4 œ -categories, and Floer homologies, Proceedings ofGARC workshop on Geometry and Topology (H. J. Kim, ed.), Seoul NationalUniversity, 1993.[8] K. Fukaya and Y.-G. Oh, Zero-loop open strings in the cotangent bundle andMorse homotopy, Asian J. Math. 1 (1998), 96^180.[9] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono, Lagrangian intersection Floertheory - anomaly and obstruction, Preprint, 2000.

360 Paul Seidel[10] E. Getzler, Batalin-Vilkovisky algebras and 2d Topological Field Theories, Commun.Math. Phys 159 (1994), 265^285.[11] W. Goldman and J. Millson, The deformation theory of the fundamental groupof compact Kahler manifolds, IHES Pubi. Math. 67, 43^96.[12] E.-N. Ionel and T. Parker, Gromov-Witten invariants of symplectic sums,Math. Res. Lett. 5 (1998), 563^576.[13] B. Keller, Invariance and localization for cyclic homology of DG algebras, J.Pure Appi. Alg. 123 (1998), 223^273.[14] M. Kontsevich, Homological algebra of mirror symmetry, Proceedings of theInternational Congress of Mathematicians (Zürich, 1994), Birkhäuser, 1995,120^139.[15] M. Kontsevich and Y. Soibelman, Homological mirror symmetry and torusfibrations, Symplectic geometry and mirror symmetry, World Scientific, 2001,203^263.[16] A.-M. Li and Y. Ruan, Symplectic surgery and Gromov-Witten invariants ofCalabi-Yau 3-folds, Invent. Math. 145 (2001), 151-218.[17] P. Ozsvath and Z. Szabo, Holomorphic disks and topological invariants forrational homology three-spheres, Preprint math.SG/0101206.[18] A. Polishchuk and E. Zaslow, Categorical mirror symmetry: the elliptic curve,Adv. Theor. Math. Phys. 2 (1998), 443-170.[19] M. Pozniak, Floer homology, Novikov rings and clean intersections, NorthernCalifornia Symplectic Geometry Seminar, Amer. Math. Soc, 1999, 119^181.[20] P. Seidel, Floer homology and the symplectic isotopy problem, Ph.D. thesis,Oxford University, 1997.[21] , Lagrangian two-spheres can be symplectically knotted, J. DifferentialGeom. 52 (1999), 145-171.[22] , Graded Lagrangian submanifolds, Bull. Soc. Math. France 128 (2000),103^146.[23] , A long exact sequence for symplectic Floer cohomology, Preprintmath.SG/0105186.[24] D. Tamarkin and B. Tsygan, Noneommutative differential calculus, homotopyBV algebras and formality conjectures, Preprint math.KT/0002116.[25] T. Tradier, Infinity-inner-products on A-infinity-algebras, Preprint math.-AT/0108027.[26] C. Viterbo, Functors and computations in Floer homology with applications,Part I, Geom. Funct. Anal. 9 (1999), 985^1033.[27] , Functors and computations in Floer homology with applications, PartII, Preprint 1996.

ICM 2002 • Vol. II • 361^369Heat Kernels and the Index Theorems onEven and Odd Dimensional Manifolds*Weiping ZhangAbstractIn this talk, we review the heat kernel approach to the Atiyah-Singer indextheorem for Dirac operators on closed manifolds, as well as the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. We alsodiscuss the odd dimensional counterparts of the above results. In particular,we describe a joint result with Xianzhe Dai on an index theorem for Toeplitzoperators on odd dimensional manifolds with boundary.2000 Mathematics Subject Classification: 58G.Keywords and Phrases: Index theorems, heat kernels, eta-invariants, Toeplitzoperators.1. IntroductionAs is well-known, the index theorem proved by Atiyah and Singer [ASI] in1963, which expresses the analytically defined index of elliptic differential operatorsthrough purely topological terms, has had a wide range of implications in mathematicsas well as in mathematical physics. Moreover, there have been up to nowmany different proofs of this celebrated result.The existing proofs of the Atiyah-Singer index theorem can roughly be dividedinto three categories:(i) The cobordism proof: this is the proof originally given in [ASI]. It usesthe cobordism theory developed by Thom and modifies Hirzebruch's proof of hisSignature theorem as well as his Riemann-Roch theorem;(ii) The if-theoretic proof: this is the proof given by Atiyah and Singer in[AS2]. It modifies Grothendieck's proof of the Hirzebruch-Riemann-Roch theoremand relies on the topological if-theory developed by Atiyah and Hirzebruch. TheBott periodicity theorem plays an important role in this proof;*Partially supported by the MOEC and the 973 Project.ÎNankai Institute of Mathematics, Nankai University, Tianjin 300071, China. E-mail:weiping@nankai.edu.cn

362 Weiping Zhang(iii) The heat kernel proof: this proof originates from a simple and beautifulformula due to Mckean and Singer [MS], and has closer relations with differentialgeometry as well as mathematical physics. It also lead directly to the importantAtiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary.In this article, we will survey some of the developments concerning the heatkernel proofs of various index theorems, including a recent result with Dai [DZ2]on an index theorem for Toeplitz operators on odd dimensional manifolds withboundary.2. Heat kernels and the index theorems on evendimensional manifoldsWe start with a smooth closed oriented 2n-dimensional manifold M and twosmooth complex vector bundles E, F over M, on which there is an elliptic differentialoperator between the spaces of smooth sections, D + : Y(E) —t Y(F).If we equip TM with a Riemannian metric and E, F with Hermitian metricsrepectively, then Y(E) and Y(F) will carry canonically induced inner products.Let £>_ : Y(F) —t Y(E) be the formal adjoint of D + with respect to theseinner products. Then the index of D + is given byindD+ = dim (kerD + ) — dim (kerD_). (2-1)It is a topological invariant not depending on the metrics on TM, E and F.The famous Mckean-Singer formula [MS] says that ind D + can also be computedby using the heat operators associated to the Laplacians £>_£>+ and £>+£>_.That is, for any t > 0, one hasind D + = Tr [exp (-tD_D+)] - Tr [exp (-tD+D_)]. (2.2)By introducing the Z 2 -graded vector bundle E ® F and setting D = ( ® D -we can rewrite the difference of the two traces in the right hand side of (2.2) as asingle "supertrace" as follows,ind D + = Tr s [exp (^tD 2 )] , for any t > 0. (2.2)'Let Pt(x,y) be the smooth kernel of exp(—tD 2 ) with respect to the volumeform on M. For any / £ Y(E ® F), one hasexp (^tD 2 ) f(x) = [ Pt(x,ij)f(ij)dy. (2.3)JMIn particular,Tr s [exp (^tD 2 )] = / Tr 8 [P t (x,x)]dx. (2.4)JM

Heat Kernels and the Index Theorems • • • 363Now, for simplicity, we assume that the elliptic operator D is of order one.Then by a standard result, which goes back to Minakshisundaram and Pleijiel [MP],one has that when t > 0 tends to 0,Pt(x,x) = (a- n + a- n+ it +••• + a Q t n + o x (t n j), (2.5)(4nt) nwhere a* G End ((E © F) x ), i = -n,..., 0.By (2.2)', (2.4) and (2.5), and by taking t > 0 small enough, one deduces that/ Tr 8 [ai]dx = 0, -n

364 Weiping Zhang3. The index theorem for Dirac operators on evendimensional manifolds with boundaryLet M be a smooth compact oriented even dimensional spin manifold with(nonempty) smooth boundary dM. Then dM is again oriented and spin.Let g be a metric on TM. Yet g T9M be its restriction on TdM. Weassume for simplicity that g is of product structure near the boundary dM.Yet S (TX) = S + (TX) ® S-(TX) be the Z 2 -graded Hermitian vector bundle of(TX, g TX )-spinors.Since now M has a nonempty boundary dM, the associated Dirac operatorD + : Y(S + (TMj) —t T(S-(TM)) is not elliptic. To get an elliptic problem, oneneeds to introduce an elliptic boundary condition for D + , and this was achieved byAtiyah, Patodi and Singer in [APS]. It is remarkable that this boundary condition,to be described right now, is global in nature.First of all, the Dirac operator D + induces canonically a formally self-adjointfirst order elliptic differential operatorDQM '• Y (S^(TM)\QM) —ï Y (S^(TM)\QM) ,which is called the induced Dirac operator on the boundary dM.Clearly, the L 2 -completion of S+(TM)\QM admits an orthogonal decompositionL 2 (S+(TM)\ dx )= 0 E x , (3.1)AeSpec(-DeM)where E\ is the eigenspace of A.Let L 2 >0(S + (TM)\QM) denote the direct sum of the eigenspaces E\ associatedto the eigenvalues A > 0. Let P>o denote the orthogonal projection fromL 2 (S+(TM)\QM) to L 2 >0(S + (TM)\QM)- We call P>o the Atiyah-Patodi-Singer projectionassociated to DQM, to emphasize its role in [APS].Then by [APS], the boundary problem(D + ,P>o) :{u:u£ Y(S+(TMj), P> 0 (U\ 9 M) = 0} -> Y(S-(TMj), (3.2)is Fredholm. We call this elliptic boundary problem the Atiyah-Patodi-Singerboundary problem associated to D + . We denote by ind(D+,P>o) the index ofthe Fredholm operator (3.2).The Atiyah-Patodi-Singer index theorem The following identity holds,ind (D+,P>o)= / Äl -^— J -rj(D dM ). (3.3)The boundary correction term TJ(DQM) appearing in the right hand side of(3.3) is a spectral invariant associated to the induced Dirac operator DQM on dM.It is defined as follows: for any complex number s £ C with Re(s) > dim M, definen(D dM ,s)= 2^ —nrji—• ( 3 - 4 )AeSpec(-DeM)

Heat Kernels and the Index Theorems • • • 365By using the heat kernel method, one can show easily that ï](DQM,S) can be extendedto a meromorphic function on C, which is holomorphic at s = 0. Following[APS], we then define_ dim (ker £> eM ) + »} (-DOM, 0)n (D dM ) = 2 ( 3 - 5 )and call it the (reduced) eta invariant of DQM-The eta invariants of Dirac operators have played important roles in manyaspectsof topology, geometry and mathematical physics.In the next sections, we will discuss the role of eta invariants in the heat kernelapproaches to the index theorems on odd dimensional manifolds.4. Heat kernels and the index theorem on odd dimensionalmanifoldsLet M be now an odd dimensional smooth closed oriented spin manifold. Letg be a Riemannian metric on TM and S(TM) the associated Hermitian vectorbundle of (TAf,^)-spinors. 1 In this case, the associated Dirac operator D :Y(TM) —t Y(TM) is (formally) s elf-adjoint. 2 Thus, one can proceed as in Section3 to construct the Atiyah-Patodi-Singer projectionP>o • L 2 (S(TMj) -+ Ll 0 (S(TMj).Now consider the trivial vector bundle C* over M. We equip C* with thecanonical trivial metric and connection. Then F> 0 extends naturally to an orthogonalprojection from L 2 (S(TM) C^) to L% 0 (S(TM) C^) by acting as identityon C^. We still denote this extension by P>o-On the other hand, letg:M -• U(N)be a smooth map from M to the unitary group U(N). Then g can be interpretedas automorphism of the trivial complex vector bundle C N . Moreover g extendsnaturally to an action on L 2 (S(TM) ® C*) by acting as identity on L 2 (S(TMj).We still denote this extended action by g.With the above data given, one can define a Toeplitz operator T g as follows,T g = P>ogP>o • Lio (S(TM) C") —• L\ 0 (S(TM) ® C") . (4.1)The first important fact is that T g is a Fredholm operator. Moreover, it isequivalent to an elliptic pseudodifferential operator of order zero. Thus one cancompute its index by using the Atiyah-Singer index theorem [AS2], as was indicatedin the paper of Baum and Douglas [BD], and the result isindT, = - (A(TAf)ch(

366 Weiping Zhangwhere ch(g) is the odd Chern character associated to g.There is also an analytic proof of (4.2) by using heat kernels. For this onefirst applies a result of Booss and Wojciechowski (cf. [BW]) to show that thecomputation of ind T g is equivalent to the computation of the spectral flow of thelinear family of self-adjoint elliptic operators, acting of Y(S(TM) ® C N ), whichconnects D and gDg^1.The resulting spectral flow can then be computed byvariationsof ^-invariants, where the heat kernels are naturally involved.The above ideas have been extended in [DZ1] to give a heat kernel proof of afamily extension of (4.2).5. An index theorem for Toeplitz operators on odddimensional manifolds with boundaryIn this section, we describe an extension of (4.2) to the case of manifolds withboundary, which was proved recently in my paper with Xianzhe Dai [DZ2]. Thisresult can be thought of as an odd dimensional analogue of the Atiyah-Patodi-Singerindex theorem described in Section 3.This section is divided into three subsections. In Subsection 4.1, we extend thedefinition of Toeplitz operators to the case of manifolds with boundary. In Subsection4.2, we define an ^-invariant for cylinders which will appear in the statementof the main result to be described in Subsection 4.3.5.1. Toeplitz operators on manifolds with boundaryLet M be an odd dimensional oriented spin manifold with (nonempty) boundarydM. Then dM is also oriented and spin. Let g be a Riemannian metric onTM such that it is of product structure near the boundary dM. Yet S(TM) be theHermitian bundle of spinors associated to (M,g). Since dM ^ 0, the Dirac operatorD : Y(S(TMj) —t Y(S(TMj) is no longer elliptic. To get an elliptic operator,one needs to impose suitable boundary conditions, and it turns out that again wewill adopt the boundary conditions introduced by Atiyah, Patodi and Singer [APS].Let DQM '• Y(S(TM)\QM) —t Y(S(TM)\QM) be the canonically induced Diracoperator on the boundary dM. Then DQM is elliptic and (formally) self-adjoint.For simplicity, we assume here that DQM is invertible, that is, ker DOM = 0.Let PdM,>o denote the Atiyah-Patodi-Singer projection from L 2 (S(TM)\QM)to L 2 >Q(S(TM)\QM). Then (D, PQM;>Q) forms a self-adjoint elliptic boundary problem.We will also denote the corresponding elliptic self-adjoint operator by Dp 8M>0 .Yet L 2 P >Q(S(TMj) be the space of the direct sum of eigenspaces of nonnegativeeigenvalues of Dp aM >0 . Yet Pp BM >0 >o denote the orthogonal projectionfrom L 2 (S(TM)) to L 2 PgM^; 0 (S(TM)).Now let C^ be the trivial complex vector bundle over M of rank N, whichcarries the trivial Hermitian metric and the trivial Hermitian connection. We extendPp BMi>0 ,>o to act as identity on C*.Let g : M —¥ U(N) be a smooth unitary automorphism of C N . Then g extendsto an action on S(TM) ® C^ by acting as identity on S(TM).

Heat Kernels and the Index Theorems • • • 367Since g is unitary, one verifies easily that the operator gP$M,>o9^1 is an orthogonalprojection on L 2 ((S(TM)(E)C N )\QM), and that gPdM,>o9^1 ^ PdM,>o is apseudodifferentialoperator of order less than zero. Moreover, the pair (D,gP$M,>o9^1)forms a self-adjoint elliptic boundary problem. We denote its associated ellipticself-adjoint operator by D gPgM >og -i-Yet L 2 p ! >0(S(TM )®C JV ) be the space of the direct sum of eigenspacesof nonnegative eigenvalues of D gPgM >og -i • Yet P g p BM > oS -\>o denote the orthogonalprojection from L 2 (S(TM) ® C k ) to L 2 gPgM >og_^>Q (S(TM) ® C*).Clearly, if « e L 2 (S(TM)® C*) verifies -POM,>O(«|OM) = 0, then gs verifiesgPdM,>og^ ((gs)\ 9 M) = 0.Definition 5.1 The Toeplitz operator T g is defined byT 9 = PgPeM.ôg- 1 ,>o9 P PeM.>o,>0 '•L h M ,>_ 0 ,>o (S(TM) ® C») -+ L 2 gPgM>_ og^^0 (S(TM) ® C») -One verifies that T g is a Fredholm operator. The main result of this sectionevaluates the index of T g by more geometric quantities.5.2. An ^-invariant associated to gWe consider the cylinder [0,1] x dM. Clearly, the restriction of g on dMextends canonically to this cylinder.Let -D|[o,i]xOM be the restriction of D on [0,1] x dM. We equip the boundaryconditionPdM,>o at {0} x dM and the boundary condition Id — gPdM,>o9^1 at{1} x dM. Then (-D|[O,I]XOMJ PBM;>Q, Id — gP$M,>o9^1)forms a self-adjoint ellipticboundary problem. We denote the corresponding elliptic self-adjoint operator by-PeM,>o,S-PeM,>oS _1 'Let n(D PgM >0}9 p BM >og - 1 is ) be the ^-function of s £ C which, when Re(s) >>0, is defined by(n \ - Y^ sS n ( A )f l \ u PdM.xsigPaM.xsg- 1 1 s ) — / „ i x Is 'A#0 ' 'where A runs through the nonzero eigenvalues of D PgM >0 , g p BM >og - 1 -It is proved in [DZ2] that under our situation, n(D PgM >0}9 p BM >og -i,s) can beextended to a meromorphic function on C which is holomorphic at s = 0.Yet rj(Dp BM >Q}9 p BM >oS - 1 ) be the reduced ^-invariant defined by^ \ D PBM.yo^gPBM.yog- 1 )_ dfmker Î>p eM ,> 0 , s p eM ,> oS -iJ + n (-Dp eM ,> 0 , s p eM ,> oS -i)

- T p ( P d M , > 0 , 9 P d M , > 0 9 ~1; P M ) •368 Weiping Zhang5.3. An index theorem for T gYet V be the Levi-Civita connection associated to the Riemannian metricg. Yet R = (V) 2 be the curvature of V. Also, we use d to denotethe trivial connection on the trivial vector bundle C* over M. Then g^1dgis aF(End(C JV )) valued 1-form over M.Yet ch(g,d) denote the odd Chern character form (cf. [Z]) of (g,d) defined by(dimM-l)/2 , . n+in=0Yet VM denote the Calderón projection associated to D on M (cf. [BW]). ThenVM is an orthogonal projection on L 2 ((S(TM) ® C N )\$M), and that VM — PdM,>ois a pseudodifferential operator of order less than zero.Let T ß (PßM,>o,9PdM,>o9^1 ,PM) £ Z be the Maslov triple index in the senseof Kirk and Lesch [KL, Definition 6.8].We can now state the main result of [DZ2], which generalizes an old result ofDouglas and Wojciechowski [DoW], as follows.Theorem 5.2 The following identity holds,i- ^ / R\ / \indT s = - J^ A [-1^] ch(9,d) + rj (l>p eM .> 0 , fl p eM .> ofl -i)The following immediate consequence is of independent interests.Corollary 5.3 The numberis an integer.IMjTM— / R\ / \•4 {-^jch^' d ) -îW.äo.fliW.äofl- 1 )The strategy of the proof of Theorem 5.2 given in [DZ2] is the same as thatof the heat kernel proof of (4.2). However, due to the appearance of the boundarydM, one encounters new difficulties. To overcome these difficulties, one makes useof the recent result on the splittings of n invariants (cf. [KL]) as well as some ideasinvolved in the Connes-Moscovici local index theorem in noneommutative geometry[CM] (see also [CH]). Moreover, the local index calculations appearing near dM ishighly nontrivial. We refer to [DZ2] for more details.References[APS] M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry andRiemannian geometry I. Proc. Cambridge Philos. Soc. 77 (1975), 43^69.

Heat Kernels and the Index Theorems • • • 369[ASI] M. F. Atiyah and I. M. Singer, The index of elliptic operators on compactmanifolds. Bull. Amer. Math. Soc. 69 (1963), 422-133.[AS2] M. F. Atiyah and I. M. Singer, The index of elliptic operators I. Ann. ofMath. 87 (1968), 484-530.[BD]P. Baum and R. G. Douglas, if-homology and index theory, in Proc. Sympos.Pure and Appi. Math., Vol. 38, 117-173, Amer. Math. Soc. Providence,1982.[BGV] N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dirac operators.Grundlagen der Math. Wissenschften Vol. 298. Springer-Verlag, 1991.[BW] B. Booss and K. Wojciechowski, Elliptic Boundary Problems for Dirac Operators,Birkhäuser, 1993.[CH][CM]S. Chern and X. Hu, Equivariant Chern character for the invariant Diracoperator. Michigan Math. J. 44 (1997), 451—173.A. Connes and H. Moscovici, The local index formula in noneommutativegeometry. Geom. Funct. Anal. 5 (1995), 174-243.[DZ1] X. Dai and W. Zhang, Higher spectral flow. J. Funct. Anal. 157 (1998),432-469.[DZ2] X. Dai and W. Zhang, An index theorem for Toeplitz operators on odddimensional manifolds with boundary. Preprint, math.DG/0103230.[DoW] R. G. Douglas and K. P. Wojciechowski, Adiabatic limits of the r\ invariants:odd dimensional Atiyah-Patodi-Singer problem. Commun. Math. Phys. 142(1991), 139-168.[KL] P. Kirk and M. Lesch, The ^-invariant, Maslov index, and spectralflow for Dirac type operators on manifolds with boundary. Preprint,math.DG/0012123.[MP] S. Minakshisundaram and A. Pleijel, Some properties of the eigenfunctionsof the Laplace operator on Riemannian manifolds. Canada J. Math. 1 (1949),242-256.[MS] H. Mckean and I. M. Singer, Curvature and eigenvalues of the Laplacian. J.[P][Yu][Z]Diff. Geom. 1 (1967), 43-69.V. K. Patodi, Curvature and eigenforms of the Laplace operator. J. Diff.Geom. 5 (1971), 251-283.Y. Yu, The Index Theorem and the Heat Equation Method. Nankai Tracksin Mathematics Vol. 2. World Scientific, Singapore, 2001.W. Zhang Lectures on Chern- Weil Theory and Witten Deformations. NankaiTracks in Mathematics Vol. 4. World Scientific, Singapore, 2001.

Section 5. TopologyMladen Bestvina: The Topology of Out(F n ) 373Yu. V. Chekanov: Invariants of Legendrian Knots 385M. Furuta: Finite Dimensional Approximations in Geometry 395Emmanuel Giroux: Géométrie de Contact: de la Dimension Trois vers lesDimensions Supérieures 405Lars Hesselholt: Algebraic K-theory and Trace Invariants 415Eleny-Nicoleta Ionel: Symplectic Sums and Gromov- Witten Invariants 427Peter Teichner: Knots, von Neumann Signatures, and Grope Cobordism 437Ulrike Tillmann: Strings and the Stable Cohomology of Mapping ClassGroups 447Shicheng Wang: Non-zero Degree Maps between 3-Manifolds 457

ICM 2002 • Vol. II • 373-384The Topology of Out(F n )Mladen Bestvina*AbstractWe will survey the work on the topology of Out(F n ) in the last 20 yearsor so. Much of the development is driven by the tantalizing analogy withmapping class groups. Unfortunately, Out(F n ) is more complicated and lesswell-behaved.Culler and Vogtmann constructed Outer Space X n , the analog of Teichmüllerspace, a contractible complex on which Out(F n ) acts with finitestabilizers. Paths in X n can be generated using "foldings" of graphs, an operationintroduced by Stallings to give alternative solutions for many algorithmicquestions about free groups. The most conceptual proof of the contractibilityof X n involves folding.There is a normal form of an automorphism, analogous to Thurston's normalform for surface homeomorphisms. This normal form, called a "(relative)train track map", consists of a cellular map on a graph and has good propertieswith respect to iteration. One may think of building an automorphismin stages, adding to the previous stages a building block that either growsexponentially or polynomially. A complicating feature is that these blocks arenot "disjoint" as in Thurston's theory, but interact as upper stages can mapover the lower stages.Applications include the study of growth rates (a surprising feature of freegroup automorphisms is that the growth rate of / is generally different fromthe growth rate of / _1 ), of the fixed subgroup of a given automorphism, andthe proof of the Tits alternative for Out(F n ). For the latter, in addition totrain track methods, one needs to consider an appropriate version of "attractinglaminations" to understand the dynamics of exponentially growingautomorphisms and run the "ping-pong" argument. The Tits alternative isthus reduced to groups consisting of polynomially growing automorphisms,and this is handled by the analog of Kolchin's theorem (this is one instancewhere Out(F n ) resembles GL„(Z) more than a mapping class group).Morse theory has made its appearance in the subject in several guises.The original proof of the contractibility of X n used a kind of "combinatorial"Morse function (adding contractible subcomplexes one at a time and studyingthe intersections). Hatcher-Vogtmann developed a "Cerf theory" for graphs.This is a parametrized version of Morse theory and it allows them to provehomological stability results. One can "bordify" Outer Space (by analogywith the Borei-Serre construction for arithmetic groups) to make the action'Department of Mathematics, University of Utah, USA. E-mail: bestvina@math.utah.edu

374 Mladen Bestvinaof Out(F n ) cocompact and then use Morse theory (with values in a certainordered set) to study the connectivity at infinity of this new space. The resultis that Out(F n ) is a virtual duality group.Culler-Morgan have compactified Outer Space, in analogy with Thurston'scompactification of Teichmüller space. Ideal points are represented by actionsof F n on R-trees. The work of Rips on group actions on R-trees can be usedto analyze individual points and the dynamics of the action of Out(F n ) on theboundary. The topological dimension of the compactified Outer Space and ofthe boundary have been computed. The orbits in the boundary are not dense;however, there is a unique minimal closed invariant set. Automorphisms withirreducible powers act on compactified Outer Space with the standard NorthPole - South Pole dynamics. By first finding fixed points in the boundary ofOuter Space, one constructs a "hierarchical decomposition" of the underlyingfree group, analogous to the Thurston decomposition of a surface homeomorphism.The geometry of Outer Space is not well understood. The most promisingmetric is not even symmetric, but this seems to be forced by the nature ofOut(F n ). Understanding the geometry would most likely allow one to proverigidity results for Oiit(F„).2000 Mathematics Subject Classification: 57M07, 20F65, 20E08.Keywords and Phrases: Free group, Train tracks, Outer space.1. IntroductionThe aim of this note is to survey some of the topological methods developed inthe last 20 years to study the group Out(F n ) of outer automorphisms of a free groupF n of rank n. For an excellent and more detailed survey see also [69]. Stallings'paper [64] marks the turning point and for the earlier history of the subject thereader is referred to [55]. Out(F n ) is defined as the quotient of the group Aut(F n )of all automorphisms of F n by the subgroup of inner automorphisms. On one hand,abelianizing F n produces an epimorphism Out(F n ) —t Out(Z n ) = GL n (Z), and onthe other hand Out(F n ) contains as a subgroup the mapping class group of anycompactsurface with fundamental group F n . A leitmotiv in the subject, promotedby Karen Vogtmann, is that Out(F n ) satisfies a mix of properties, some inheritedfrom mapping class groups, and others from arithmetic groups. The table belowsummarizes the parallels between topological objects associated with these groups.Outer space is not a manifold and only a polyhedron, imposing a combinatorialcharacter on Out(F n ).2. Stallings' FoldsA graph is a 1-dimensional cell complex. A map / : G —¥ G' between graphs issimplicial if it maps vertices to vertices and open 1-cells homeomorphically to open

The Topology of Out(F n 375MappingOut(F n)GL„(Z)algebraicclass groups(arithmetic groups)propertiesTeichmüllerCuller-Vogtmann'sGL n(R)/O nfiniteness propertiesspaceOuter space(symmetric spaces)cohomological dimensionThurstontrain trackJordangrowth ratesnormal formrepresentativenormal formfixed points (subgroups)Harer'sbordification ofBorei-SerreBieri-EckmannbordificationOuter spacebordificationdualitymeasuredK-treesflag manifoldKolchin theoremlaminations(Furstenberg boundary)Tits alternativeHarvey's?Titsrigiditycurve complexbuilding1-cells. The simplicial map / is a fold if it is surjective and identifies two edges thatshare at least one vertex. A fold is a homotopy equivalence unless the two edgesshare both pairs of endpoints and in that case the induced homomorphism in mcorresponds to killing a basis element.Theorem 1 (Stallings [63]). A simplicial map f : Gnected graphs can be factored as the compositionG' between finite con-G — (jr0 —y (jrx Gk G'where each Gì —¥ Gj+i is a fold and Gk —¥ G' is locally injective (an immersion).Moreover, such a factorization can be found by a (fast) algorithm.In the absence of valence 1 vertices the last map Gk —¥ G' can be thoughtof as the core of the covering space of G' corresponding to the image in m of /.The following problems can be solved algorithmically using Theorem 1 (these wereknown earlier, but Theorem 1 provides a simple unified argument). Let F be a freegroup with a fixed finite basis.• Find a basis of the subgroup H generated by a given finite collection hi,--- ,huof elements of F.• Given w £ F, decide if w £< hi, • • • ,hu >•• Given w £ F, decide if w is conjugate into < hi,- • • ,hu >•• Given a homomorphism : F —t F' between two free groups of finite rank,decide if is injective, surjective.• Given finitely generated H < F decide if it has finite index.• Given two f.g. subgroups Hi,H 2 < F compute HiriH 2 and also the collectionof subgroups Hi fl flf where g £ F. In particular, is Hi malnormal?• Represent a given automorphism of F as the composition of generators ofAut(F) of the following form:Signed permutations: each a, maps to a, or to a,Change of maximal tree: cti H> cti, H> a±i1 a, or a, H> 0,% (i > 1).Todd-Coxeter process [65].

376 Mladen Bestvina3. Culler-Vogtmann's Outer spaceFix the wedge of n circles R n and a natural identification 7ri(i? n ) — F n inwhich oriented edges correspond to the basis elements. Thus any £ Out(F n ) canbe thought of as a homotopy equivalence R n —¥ R n . A marked metric graph is apair (G,g) where• G is a finite graph without vertices of valence 1 or 2.• g : R n —¥ G is a homotopy equivalence (the marking).• G is equipped with a path metric so that the sum of the lengths of all edgesis 1.Outer space X n is the set of equivalence classes of marked metric graphs underthe equivalence relation (G, g) ~ (G',g r ) if there is an isometry h : G —¥ G' suchthat gh and g' are homotopic [28].If a is a loop in R n we have the length function l a : X n —t R where l a (G,g) isthe length of the immersed loop homotopic to g(ct). The collection {l a } as a rangesover all immersed loops in R n defines an injection X n —t R°° and the topologyon X n is defined so that this injection is an embedding. X n naturally decomposesinto open simplices obtained by varying edge-lengths on a fixed marked graph. Thegroup Out(F n ) acts on X n on the right via(G,g)4>=(G,g4>).Theorem 2 (Culler-Vogtmann [28]). X n is contractible and the action ofOut(F nis properly discontinuous (with finite point stabilizers). X n equivariantly deformationretracts to a (2n — 3) -dimensional complex (n > 1).If (G,g) and (G',g r ) represent two points of X n , there is a "difference ofmarkings" map h : G —¥ G' such that hg and g' are homotopic. Representing h as acomposition of folds (appropriately interpreted) leads to a path in X n from (G,g)to (G',g r ). Arranging that these paths vary continuously with endpoints leads to aproof of contractibility of X n [66],[60],[71].Corollary 3. The virtual cohomological dimension vcd(Out(F n j) = 2n — 3 (n > 1).Theorem 4 (Culler [26]). Every finite subgroup of Out(F n ) fixes a point of X n .Outer space can be equivariantly compactified [27]. Points at infinity arerepresented by actions of F n on R-trees.4. Train tracksAny

The Topology of Out(F n ) 377Otherwise, we say that is irreducible.A cellular map / : G —¥ G is a train track map if for every k > 0 the map/* : G —¥ G is locally injective on every open 1-cell. For example, homeomorphismsare train track maps and Culler's theorem guarantees that every £ Out(F n ) offinite order has a representative / : G —¥ G which is a homeomorphism. Moregenerally, we haveTheorem 5 (Bestvina-Handel [12]). Every irreducible outer automorphism can be represented as a train track map f : G —¥ G.Any vertex v £ G has a cone neighborhood, and the frontier points can bethought of as "germs of directions" at v. A train track map (or any cellular mapthat does not collapse edges) / induces the "derivative" map Df on these germs(on possibly different vertices). We declare two germs at the same vertex to beequivalent (and the corresponding "turn" illegal) if they get identified by somepower of Df (and otherwise the turn is legal). An immersed loop in G is legal ifevery turn determined by entering and then exiting a vertex is legal. It followsthat / sends legal loops to legal loops. This gives a method for computing thegrowth rate of , as follows. The transition matrix (ay) of / (or more generally of acellular map G —¥ G that is locally injective on edges) has ay equal to the number oftimes that the /-image of j t h edge crosses i th edge. Applying the Perron-Frobeniustheorem to the transition matrix, one can find a unique metric structure on G suchthat / expands lengths of edges (and also legal loops) by a factor A > 1. For aconjugacy class 7 in F n the growth rate is defined asGR(4>,'--/) = limsuplog(||ç!i* ! (7)||)/fcwhere ||7|| is the word length of the cyclically reduced word representing 7. Growthrates can be computed using lengths of loops in G rather than in R n .Corollary 6. If is irreducible as above, then either 7 is a -periodic conjugacyclass, or GR(

378 Mladen BestvinaDefinition 8. A cellular map / : G —¥ G on a finite graph with no vertices ofvalence 1 that does not collapse any edges is a relative train track map if there is afiltration0 = G 0 C • • • C G m = Ginto f-invariant subgraphs with the following properties. Denote by H r the closureof G r \ G r -i, and by M r the part of the transition matrix corresponding to H r .Then M r is the zero matrix or an irreducible matrix. If M r is irreducible and thePerron-Frobenius eigenvalue A r > 1 then:• the derivative Df maps the germs in H r to germs in H r ,• if a is a nontrivial path in G r _i with endpoints in G r _i C\H r then f(ct), afterpulling tight, is also a nontrivial path with endpoints in G r _i n H r , and• every legal path in H r is mapped to a path that does not cross illegal turns inH r .As an example, consider the automorphism a H> a,b >-¥ ab,c >-¥ caba^1b^1d,d H> dbcd represented on the rose R±. The strata are 0 C Gi = {a} C {a, 6} C G.Hi and H 2 have A = 1 while H 3 has A3 > 1. The following is an analog of Thurston'snormal form for surface homeomorphisms.Theorem 9. [12] Every automorphism of F n admits a relative train track representative.Consequently, automorphisms of F n can be thought of as being built frombuilding blocks (exponential and non-exponential kinds) but the later stages areallowed to map over the previous stages. This makes the study of automorphismsof F n more difficult (and interesting) than the study of surface homeomorphisms.Other non-surface phenomena (present in linear groups) are:• stacking up non-exponential strata produces (nonlinear) polynomial growth,• the growth rate of an automorphism is generally different from the growthrate of its inverse.5. Related spaces and structuresUnfortunately, relative train track representatives are far from unique. As areplacement, one looks for canonical objects associated to automorphisms that canbe computed using relative train tracks. There are 3 kinds of such objects, all stemmingfrom the surface theory: laminations, R-trees, and hierarchical decompositionsofF n [59].Laminations. Laminations were used in the proof of the Tits alternative forOut(F n ). To each automorphism one associates finitely many attracting laminations.Each consists of a collection of "leaves", i.e. biinfinite paths in the graph G,or alternatively, of an F„-orbit of pairs of distinct points in the Cantor set of endsof F n . A leaf £ can be computed by iterating an edge in an exponentially growingstratum H r . The other leaves are biinfinite paths whose finite subpaths appear assubpaths of I. Some of the attracting laminations may be sublaminations of other

The Topology of Out(F n ) 379attracting laminations, and one focuses on the maximal (or topmost) laminations.It is possible to identify the basin of attraction for each such lamination. Let %be any subgroup of Out(F n ). Some of the time it is possible to find to elementsf,g £ % that attract each other's laminations and then the standard ping-pongargument shows that < f,g >= F 2 . Otherwise, there is a finite set of attractinglaminations permuted by %, a finite index subgroup WoCH that fixes each of theselaminations and a homomorphism ( "stretch factor" ) H 0 —* A to a finitely generatedabelian group A whose kernel consists entirely of polynomially growing automorphisms.There is an analog of Kolchin's theorem that says that finitely generatedgroups of polynomially growing automorphisms can simultaneously be realized asrelative train track maps on the same graph (the classical Kolchin theorem saysthat a group of unipotent matrices can be conjugated to be upper triangular, orequivalently that it fixes a point in the flag manifold). The main step in the proofof the analog of Kolchin's theorem is to find an appropriate fixed R-tree in theboundary of Outer space. This leads to the Tits alternative for Out(F n ):Theorem 10 (Bestvina-Feighn-Handel [9],[10],[7]). Any subgroup H ofOut(F n ]either contains F 2 or is virtually solvable.A companion theorem [8] (for a simpler proof see [1]) is that solvable subgroupsof Out(F n ) are virtually abelian.R-trees. Points in the compactified Outer space are represented as Fractionson R-trees. It is then not surprising that the Rips machine [5], which is used tounderstand individual actions, provides a new tool to be deployed to study Out(F n ).Gaboriau, Levitt, and Lustig [37] and Sela [59] find another proof of Theorem 7.Gaboriau and Levitt compute the topological dimension of the boundary of OuterSpace [36]. Levitt and Lustig show [51] that automorphisms with irreducible powershave the standard north-south dynamics on the compactified Outer space. Guirardel[43] shows that the action of Out(F n ) on the boundary does not have dense orbits;however, there is a unique minimal closed invariant set. For other applications ofR-trees in geometric group theory, the reader is referred to the survey [2].Cerf theory. An advantage of Aut(F n ) over Out(F n ) is that there is a naturalinclusion Aut(F n ) —t Aut(F n+ i). One can define Aider Space AX n similarly toOuter space, except that all graphs are equipped with a base vertex, which is allowedto have valence 2. The degree of the base vertex v is 2n — valence(w). Denote by D kthe subcomplex of AX n consisting of graphs of degree < k. Hatcher-Vogtmann [47]develop a version of Cerf theory and show that D k is (k — l)-connected. Since thequotient D k /Aut(F n ) stabilizes when n is large, one sees that (rational) homologyHi(Aut(F n j) also stabilizes when n is large (n > 3i/2). Hatcher-Vogtmann showthat the same is true for integral homology and in the range n > 2i + 3. They alsomake explicit computations in low dimensions [49] and all stable rational homologygroups Hi vanish for i < 7.Bordification. The action of Out(F n ) on Outer space X n is not cocompact. Byanalogywith Borei-Serre bordification of symmetric spaces [14] and Harer's bordificationof Teichmüller space [44], Bestvina and Feighn [6] bordify X n , i.e. equivariantlyadd ideal points so that the action on the new space BX n is cocompact.This is done by separately compactifying every simplex with missing faces in X n

380 Mladen Bestvinaand then gluing these together. To see the idea, consider the case of the thetagraphin rank 2. Varying metrics yields a 2-simplex a without the vertices. As asequence of metrics approaches a missing vertex, the lengths of two edges convergeto 0. Restricting a metric to these two edges and normalizing so that the totallength is 1 gives a point in [0,1] (the length of one of the edges), and a way tocompactify a by adding an interval for each missing vertex. The compactified ais a hexagon. This procedure equips the limiting theta graph with a metric thatmay vanish on two edges, in which case a "secondary metric" is defined on theirunion. In general, a graph representing a point in the bordification is equipped witha sequence of metrics, each defined on the core of the subgraph where the previousmetric vanishes.Lengths of curves (at various scales) provide a "Morse function" on BX nwith values in a product of [0,oo)'s with the target lexicographically ordered. Thesublevel and superlevel sets intersect each cell in a semi-algebraic set and it ispossible to study how the homotopy types change as the level changes. A distinctadvantage of BX n over the spine of X n (an equivariant deformation retract) is thatthe change in homotopy type of superlevel sets as the level decreases is very simple- via attaching of cells of a fixed dimension.Theorem 11 (Bestvina-Feighn [6]). BX n and Out(F n ) are (2n — 5)-connectedat infinity, and Out(F n ) is a virtual duality group of dimension 2n — 3.Mapping tori. If ) satisfies quadratic isoperimetric inequality for all .Geometry. Perhaps the biggest challenge in the field is to find a good geometrythat goes with Out(F n ). The payoff would most likely include rigidity theoremsfor Out(F n ). Both mapping class groups and arithmetic groups act isometricallyon spaces of nonpositive curvature. Unfortunately, the results to date for Out(F n )are negative. Bridson [15] showed that Outer space does not admit an equivariantpiecewise Euclidean CAT(0) metric. Out(F n ) (n > 2) is far from being CAT(0)[17],[40].An example of a likely rigidity theorem is that higher rank lattices in simpleLie groups do not embed into Out(F n ). A possible strategy is to follow the proofin [11] of the analogous fact for mapping class groups. The major missing piece ofthe puzzle is the replacement for Harvey's curve complex; a possible candidate isdescribed in [48].

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ICM 2002 • Vol. II • 385-394Invariants of Legendrian KnotsYu. V. Chekanov*AbstractWe present two different constructions of invariants for Legendrian knots inthe standard contact space R 3 . These invariants are defined conibinatorially,in terms of certain planar projections, and are useful in distinguishing Legendrianknots that have the same classical invariants but are not Legendrianisotopie.2000 Mathematics Subject Classification: 57R17.Keywords and Phrases: Legendrian submanifold, Legendrian knot.1. Introduction1.1. Legendrian knotsA smooth knot L in the standard contact space (R 3 ,a) = ({(q,p,«)}, du—pdq)is called Legendrian if it is everywhere tangent to the 2-plane distribution ker(a)(or, in other words, if the restriction of a to L vanishes). Two Legendrian knotsare Legendrian isotopie if the they can be connected by a smooth path in thespace of Legendrian knots (or, equivalently, if one can be sent to another by adiffeomorphism g of R 3 such that g*a = (pa, where (p > 0). In order to visualizea knot in R 3 , it is convenient to project it to a plane. In the Legendrian case, thecharacter of the resulting picture will depend on the choice of the projection. Theuseful two are: the Lagrangian projection n: R 3 —¥ R 2 , (q,p,u) >-¥ (q,p), and thefront projection a: R 3 —¥ R 2 , (p,q,u) >-¥ (q,u). In Figure 1, two projections of thesimplest Legendrian knot (unknot) are shown.We say that a Legendrian knot L C R 3 is 7r-generic if all self-intersectionsof the immersed curve n(L) are transverse double points. We can represent a n-generic Legendrian knot L by its (Lagrangian) diagram: the curve n(L) C R 2 ,at every crossing of which the overpassing branch (the one with the greater valueof u) is marked. Of course, not every abstract knot diagram in R 2 is a diagram of* Moscow Center for Continuous Mathematical Education, B. Vlasievsky per. 11, Moscow119002, Russia. E-mail: chekanov@mccme.ru

386 Yu. V. Chekanovplu I,Figure 1: Lagrangian projection and front projectiona Legendrian knot, or is oriented diffeomorphic to such (it requires a bit of extrawork to check whether a given diagram corresponds to a Legendrian knot, cf. [1]).Given a Legendrian knot Lcl 3 , its a-projection, or front, a(L) C R 2 is asingular curve with nowhere vertical tangent vectors. Its singularities, generically,are semi-cubic cusps and transverse double points. We say that L is a-generic if,moreover, all self-intersections of a(L) have different ^-coordinates. Every closedplanar curve with these types of singularities and nowhere vertical tangent vectorsis a front of a Legendrian knot. Note that there is no need to explicitly indicatethe type of a crossing of a front: the overpassing branch (the one with the greatervalue of p) is always the one with the greater slope.1.2. Classical invariantsThe so-called classical invariants of an oriented Legendrian knot L are definedas follows. The first of them is, formally, just the smooth isotopy type of L. TheThurston-Bennequin number ß(L) of L is the linking number (with respect to theorientation defined by a Ada) between L and s(L), where « is a small shift along theu direction. The Maslov number m(L) (which actually is an invariant of Legendrianimmersion) is twice the rotation number of the projection of L to the (q,p) plane(or, equivalently, the value of the Maslov 1-cohomology class on the fundamentalclass of L). The change of orientation on L changes the sign of m(L) and preservesß(L). The Thurston-Bennequin number of a 7r-generic Legendrian knot L can becomputed by counting the crossings of its Lagrangian diagram n(L) with signs:0(L) = #(X)-#(X)(where the q axis is horizontal and the p axis is vertical). In terms of the frontprojection, the classical invariants can be computed as follows. The Maslov numberof a CT-generic oriented Legendrian knot L is the number of the right cusps of thefront a(L), counted with signs depending on the orientations:m(L) = # ( » - # ( » .The Thurston-Bennequin number of L is the number of crossings of a(L) countedwith signs minus half the total number of cusps (= the number of right cusps):w = #(X) + #(X)-#(X)-#(X)-#(HFor the Legendrian knot shown in Figure 1, we have m = 0, ß = — 1.

Invariants of Legendrian Knots 3871.3. New invariants and classification resultsIt is easy to show that every smooth knot admits a Legendrian realization. Anatural question to ask is whether there exists a pair of Legendrian knots whichhave the same classical invariants but are not Legendrian isotopie The answer ispositive:Theorem 1.1. [1, 2] There exist Legendrian knots L,L' (see Figure 2 on p. 390,Figure 6 on p. 393) that have the same classical invariants (smooth knot type ï>2,m = 0, ß = 1) but are not Legendrian isotopie.In the next two sections, we present two combinatorial constructions of Legendrianknot invariants. The first one associates to the Lagrangian projection ofa Legendrian knot a differential graded algebra (DGA). The second constructiondeals with decompositions of the front projection into closed curves. Each of the twoprovides a proof of Theorem 1.1. The invariants do not change when the orientationof the knot reverses, so essentially they are invariants of non-oriented Legendrianknots. It should be mentioned that these constructions also produce, with minormodifications, invariants of Legendrian links.The number of Legendrian knots with given classical invariants is known to befinite [3]. Eliashberg and Fraser gave a classification of Legendrian realization forsmooth unknots [5, 6]. It turned out that smooth unknots are Legendrian simplein the sense that the Legendrian isotopy types of their Legendrian realizations aredetermined by the classical invariants. Etnyre and Honda obtained a classificationof Legendrian realization for torus knots and the figure eight knot [8]. Again, thesesmooth knot types proved to be Legendrian simple. The 52 is the simplest knottype for which the classification is not known. By Theorem 1.1, the type 52 is notLegendrian simple. Conjecturally, two Legendrian knots of smooth type 52 withthe same classical invariants are Legendrian isotopie unless they form the pair L, L'from Theorem 1.1. Several interesting examples of knots with coinciding classicalinvariants but not Legendrian isotopie were constructed by Ng [14, 15].2. DGA of a Legendrian knot2.1. DefinitionsIn this section, we associate with every 7r-generic Legendrian knot L a DGA(A,d) over Z/2Z ([1]; a similar construction was also given by Eliashberg). ThisDGA is related to the symplectic field theory introduced by Eliashberg, Givental,and Hofer in [7] (see [10]).Let {ai ,•••, a n } be the set of crossings of Y = n(L). Define A to be the tensoralgebra (free associative unital algebra) T(ai,..., a n ) with generators cti,..., a n .The grading on A takes values in the group Z/m(L)Z and is defined as follows.Given a crossing a,j, consider the points z + ,z_ £ L such that ir(z + ) = ir(z-) = UJand the «-coordinate of z + is greater than the «-coordinate of z— These pointsdivide L into two pieces, 71 and 72, which we orient from z + to z_. We can assume,

388 Yu. V. Chekanovwithout loss of generality, that the intersecting branches are orthogonal at a. Then,for e £ {1,2}, the rotation number of the curve 7r(7 e ) has the form N e /2 + 1/4,where N e £ Z. Clearly, Ni —N 2 is equal to ±m(L). Hence Ni and N 2 represent thesame element of the group F = Z/m(L)Z, which we define to be the degree of a,j.We are going to define the differential d. For every natural k, fix a (curved)convex fc-gon 11^ c R 2 whose vertices #§, • • •, x t-iare numbered counter-clockwise.The form dq A dp defines an orientation on R 2 . Denote by Wk(Y) the collection ofsmooth orientation-preserving immersions /: II & —t R 2 such that f(dllk) C Y. Notethat / £ Wk(Y) implies f(x k ) £ {cti,..., a n }. Consider the set of nonparametrizedimmersions Wk(Y), which is the quotient of Wk(Y) by the action of the group{g £ DiS + (Ilk)\g(x k ) = x k }. The diagram Y divides a neighbourhood of each ofits crossings into four sectors. We call positive two of them which are swept outby the underpassing curve rotating counter-clockwise, and negative the other two(the sectors are marked in Figure 1). For each vertex x k of the polygon II*, asmooth immersion / £ Wk(Y) maps its neighbourhood in 11^ to either a positiveor a negative sector; we shall say that x k is, respectively, a positive or a negativevertex for /. Define the set W£(Y) to consist of immersions / £ Wk(Y) such thatthe vertex XQ is positive for /, and all other vertices are negative. Let W£(Y,aj) ={/ G W+(Y) | f(x k ) = afi}. Denote A x = {en,... ,a n } 1/21 c A, A k = (Ai)® k .Then A = (Bfl 0 Ai. Let d = ^k>o®k, where dk(A t ) £ A i+k -i- Definedk(aj) = Yl f( x i)---f( x k)feW+ +1 (Y, aj )(for k = 0, we have do(a,j) = #(1A' 1 + (F, UJ)), and extend d to A by linearity and theLeibniz rule. The following theorem says that (A, d) is indeed a DGA:Theorem 2.1. The differential d is well defined. We have deg(9) = —1 and d 2 = 0.Define the (l-th, where I £ Y) stabilization of a DGA (T(ai,... ,a n ),d) tobe the DGA (T(ai,... ,a n ,a n +i,a n + 2 ),d), where deg(a„ + i) = I, deg(a„ + 2) =/ —1, d(a n+ i) = a n+ 2, and d acts on ai,...,a n as before. An automorphism ofT(ai,..., a n ) is called elementary if it sends a, to a, + v, where v does not involvea,, and fixes a,j for j ^ i. Two DGAs (T(ai,... ,a n ),d), (T(ai,... ,a n ),d') arecalled tame isomorphic if one can be obtained from another by a composition ofelementary automorphisms; they are called stable tame isomorphic if they becometame isomorphic after (iterated) stabilizations.Theorem 2.2. Let (A,d), (A',d r ) be the DGAs of (n-generic) Legendrian knotsL,L'. If L and L' are Legendrian isotopie then (A,d) and (A',d r ) are stabletame isomorphic. In particular, the homology rings H(A,d) = ker (9)/im (9) andH(A',d') = ker (9)/im (9) are isomorphic as graded rings.The hard part in the proof of Theorem 2.1 is to show that 9 2 = 0. The proof ofthis fact mimics, in a combinatorial way, the classical gluing-compactness argumentof the Floer theory (cf. [11]). The proof of Theorem 2.2 involves a careful study of

Invariants of Legendrian Knots 389the behaviour of the DGA associated with a Legendrian knot when its Lagrangiandiagram goes through elementary bifurcations (Legendrian Reidemeister moves).It turns out that one cannot replace the coefficient ring Z/2Z by Z: in somesense, our homology theory is not oriented. However, the construction describedabove can be modified to associate with a Legendrian knot L a DGA graded by Z andhaving Z[s, s^1](where deg(s) = m(Lj) as a coefficient ring [10]. After reducing thegrading to Z/ro(L)Z, and applying the homomorphism Z[s, s^1]—¥ Z/2Z sendingboth s and 1 G Z to 1 G Z/2Z, this Z^s^-DGA becomes the Z/2Z-DGA of theknot L.2.2. Poincaré polynomialsHomology rings of DGAs can be hard to work with. We are going to define aneasily computable invariant I, which is a finite subset of the group monoid No[F],where No = {0,1,...}, F = Z/ro(L)Z. Assume that do = 0. Then 9 2 = 0. Since9(Ai) c Ai, we can consider the homology H(Ai,di) = ker(9i| J 4 1 )/im(9i| J 4 1 ),which is a vector space graded by the cyclic group F. Define the Poincaré polynomialP {A , d) £No[Y]byP (A,d)(t) = J2dim(H X (Ai,di))t X ,Aerwhere H\(Ai,di) is the degree À homogeneous component of H(Ai,di). Definethe group Auto (A) to consist of graded automorphisms of A such that for each i £{1,... ,n} we have #(a,) = a, + c», where c, G A 0 = Z/2Z. (of course, c, = 0 whendeg(ctj) 7^ 0). Consider the set Uo(A,d) consisting of automorphisms g £ Auto(A)such that (9 s )o = 0 (where d 9 = g^1o 9 o g). DefineI(A,d) = {P {A9g) \g£Uo(A,d)}.Since Auto (A) has at most 2" elements, this invariant is not hard to compute. Wecan associate with every (7r-generic) Legendrian knot L the set I(L) = I(AL,8L)-Note that P( — l) = ß(L) for P £ I(L). One can show that J is an invariant ofstable tame DGA isomorphism. Hence Theorem 2.2 implies the followingCorollary 2.3. If L is Legendrian isotopie to L' then I(L) =I(L').The set I(L) can be empty (cf. Section 4) but no examples are known whereI(L) contains more than one element. Also, for all known examples of pairs L, L' ofLegendrian knots with coinciding classical invariants we have F(l) = P'(l), whereP £ 1(E), P' £ I(L'). Other, more complicated invariants of stable tame isomorfismwere developed and applied to distinguishing Legendrian knots in [15].2.3. Examplesla. Let (A,d) = (T(ai,... ,a®),d) be the DGA of the Legendrian knot L givenin Figure 2. We have m(L) = 0, ß(L) = 1, deg(a,) = 1 for i < 4, deg(as) = 2,deg(ag) = —2, deg(a,) = 0 for i > 7, d(ai) = 1 + a-j + a-jaâ^, d(a 2 ) = 1 + ag +a 5 a 6 a 9 , 8(0,3) = 1 + a s a 7 , d(afi) = 1 + a s a 9 , d(a t ) = 0 for i > 5.

390 Yu. V. ChekanovFigure 2: Lagrangian projections of two Legendrian 52 knotslb. Let (A',d) = (T(ai,... ,a®),d) be the DGA of the Legendrian knot L' givenin Figure 2. We have m(L') = 0, ß(L') = 1, deg(a,) = 1 for i < 4, deg(a,) = 0for i > 5, 9(ai) = 1 + a 7 + a 5 + a 7 a 6 a 5 + a 9 a s a 5 , d(a 2 ) = 1 + a 8 + a 5 a 6 a 9 ,9(03) = 1 + a%a,T, 9(04) = 1 + a%ag, 9(a») = 0 for i > 5.An explicit computation shows that I(L) = {t^ + ^ + t 2 }, I(L') = fôfl + t 1 }and hence Theorem 1.1 follows from Corollary 2.3.Figure 3: Lagrangian projection of a Legendrian 62 knot2. [14] Let (A, 9) = (T(ai,..., an), 9) be the DGA of the Legendrian knot K givenin Figure 3. We have m(K) = 0, ß(K) = -7, deg(aj) = 1 for i £ {1,2,7,9,10},deg(a,) = 0 for i £ {3,4}, deg(a,) = —1 for i £ {5,6,8,11}; 9(ai) = 1 + 0100503,d(a 2 ) = 1 + a 3 + a 3 a 6 aio + a 3 ana 7 , d(a i ) = a 5 + an + ana 7 a 5 , d(a 6 ) = ana 8 ,d(a 7 ) = a$aio, d(ag) = 1 + aioan, d(a,i) = 0 for i £ {3,5,8,10,11}. Denoteby K the 'Legendrian mirror' of if — the image of K under the map (q,p,u) >-¥(—q,p, —u). The Legendrian knots K,K have the same classical invariants. However,they are not Legendrian isotopie, and it is possible to distinguish them bymeansof their DGAs. There exist homology classes £+,£- in the graded homologyring H(A,d) such that deg(£ + ) = 1, deg(£_) = —1, and £+£- = 1 (choose£ + = [aio], C- = [ a n])- It follows from the definitions that the DGA for K is obtainedfrom (A, 9) by applying the anti-automorphism reversing the order of gener-

Invariants of Legendrian Knots 391ators in all monomials. Thus, if K and K are Legendrian isotopie then the gradedhomology ring H (A, 9) is anti-isomorphic to itself, and there exist £+,£!_ G H (A, 9)such that deg(£|_) = 1, deg(£L) = —1, £-£+ = 1. But one can check that suchclasses do not exist (see [14, 15] for details) and hence K and K are not Legendrianisotopie. Note that 'first order invariants' such as Poincaré polynomials are uselessin distinguishing Legendrian mirror knots.3. Admissible decompositions of fronts3.1. DefinitionsIn this section, we present the invariants of Legendrian knots constructed in [2].These invariants are defined in terms of the front projection.Given a a-generic oriented Legendrian knot L, denote by C(L) the set ofits points corresponding to cusps of a(L). The Maslov index p:L\ C(L) —¥ Y =Z/ro(L)Z is a locally constant function, uniquely defined up to an additive constantby the following rule: the value of p jumps at points of C(L) by ±1 as shown inFigure 4. We call a crossing of S = a(L) Maslov if p takes the same value on bothits branches.ß = i+l~^^ / \ ^~-li = i+l[1 = i-^ \ / ^-// = iFigure 4: Jumps of the Maslov index near cuspsAssume that S = a(L) is a union of closed curves Xi,..., X n that have finitelymanyself-intersections and meet each other at finitely many points. Then we callthe unordered collection {Xi,... ,X n } a decomposition of S. A decomposition{Xi,...,X n } is called admissible if it satisfies certain conditions, which we aregoing to define. The first two are as follows:(1) Each curve Xi bounds a topologically embedded disk: X t = dB t .(2) For each i £ {l,...,n}, q £ R, the set Bi(q) = {« G R | (q,u) £ Bi} is eithera segment, or consists of a single point « such that (q,u) is a cusp of S, or isempty.Conditions (1) and (2) imply that each curve X t has exactly two cusps (and hencethe number of curves is half the number of cusps). Each X t is divided by cuspsinto two pieces, on which the coordinate q is a monotone function. Near a crossingx £ XiCi Xj, the decomposition of S may look in one of the three ways representedin Figure 5. Conditions (1) and (2), in particular, rule out the decomposition shownin Figure 5a. We call the crossing point x switching if X t and Xj are not smoothnear x (Figure 5b), and non-switching otherwise (Figure 5c).(3) If (qo, u) £ Xi fl Xj (i ^ j) is switching then for each q ^ q 0 sufficiently closeto qo the set Bi(q) fl Bj(q) either coincides with Bi(q) or Bj(q), or is empty.

392 Yu. V. ChekanovFigure 5: Local decompositions(4) Every switching crossing is Maslov.We call a decomposition admissible if it satisfies Conditions (l)-(3), and gradedadmissible if it also satisfies Condition (4). Denote by Adm(S) (resp. Adm + (S))the set of admissible (resp. graded admissible) decompositions of S. Given D £Adm(S), denote by Sw(£>) the set of its switching points. Define 0(D) = #(-D)) —#(Sw(D)).Theorem 3.1. If a-generic Legendrian knots L,L' c R 3 are Legendrian isotopiethen there exists a one-to-one mapping g: Adm(

Invariants of Legendrian Knots 393Figure 6: Fronts of two Legendrian 52 knotsgoing to show that #(Adm + (S)) = 1, #(Adm + (S')) = 2, and hence Theorem 1.1is a consequence of Theorem 3.1. Assume that D £ Adm(S). Consider the curveXi £ D containing the piece of S indicated by the lower arrow. Being applied toXi, Conditions (1) and (2) imply that C2,C3 G Sw(D). Similarly, looking at thecurve X 2 £ D containing the piece of S indicated by the upper arrow, we concludethat C4,C5 G Sw(D). If one of the crossings ci,c$ is switching, so is the other. Theneither Sw(D) = {02,03,04,05} or Sw(D) = {ci,c 2 ,C3,Ci,C5,ce}. It is not hard tocheck that both decompositions are admissible but only the first one is graded. Thus#(Adm + (S)) = 1. Arguing similarly, one can find that #(Adm(S')) = 2, wherethe admissible decompositions Di,D 2 are defined by Sw(Di) = {02,03,04,05},Sw(£>2) = {ci,c 2 ,03,04,05,cfi}, and are both graded.4. Instability of invariantsThere are two stabilizing operations, S_ and S + , on Legendrian isotopy classesof oriented Legendrian knots, defined as follows. Given an oriented Legendrian knotL, we perform one of the operations shown in Figure 7 in a small neighbourhood of apoint on L. One can check that, up to Legendrian isotopy, the resulting Legendrianknot S±(L) does not depend on the choices involved, and the operations S-,S+commute. An important observation is that two Legendrian knots L, L' have thesame classical invariants if and only if they are stable Legendrian isotopie in thesense that there exist n_,n + G No such that S"~(S" + (L)) is Legendrian isotopietoS"-(S+ + (L')) [13].p.1 °-u. -/^ q5 4FA«AS-PnUkÖFigure 7: StabilizationsThus the invariants constructed in Sections 2 and 3 cannot be stable. In fact,they fail already after the first stabilization. The homology ring H of the DGAcorresponding to S±(L) vanish, and the set I(S±(Lj) is empty. This can be easily

394 Yu. V. Chekanovderived from the fact that the DGA of S±(L) can be obtained from the DGAof L by adding a new generator a such that d(a) = 1. The front of S±(L) has noadmissible decompositions because Conditions (1) and (2) cannot hold for the curveXi containing the newly created cusps.Studying Legendrian realizations of non-prime knots, Etnyre and Honda constructed,for each m, examples of Legendrian knots that have the same classicalinvariants but are not Legendrian isotopie even after m stabilizations [9]. Theirproof uses the classification of Legendrian torus knots given in [8]. It is an openproblem to find invariants distinguishing those knots, or any pair of stabilized knotswith the same classical invariants.References[1] Yu. V. Chekanov, Differential algebra of Legendrian links, to appear in InventionesMathematicae.[2] Yu. V. Chekanov & P. E. Pushkar, in preparation.[3] V. Colin, E. Giroux & K. Honda, On the coarse classification of tight contactstructures, Preprint, 2002.[4] Ya. Eliashberg, A theorem on the structure of wave fronts and its applicationin symplectic topology, Funct. Anal. Appi, 21 (1987), 227-232.[5] Ya. Eliashberg, Legendrian and transversal knots in tight contact 3-manifolds,In: Topological methods in modern mathematics (Stony Brook, NY, 1991),Publish or Perish, 1993, 171-193.[6] Ya. Eliashberg & M. Fraser, Classification of topologically trivial Legendrianknots, In: Geometry, topology, and dynamics (Montreal, PQ, 1995), CRMProc. Lecture Notes, 15, AMS, Providence, 1998, 17-51.[7] Ya. Eliashberg, A. Givental, & H. Hofer, An introduction to symplectic fieldtheory, Geom. Funct. Anal. (2000), Special Volume, Part II, 560-673.[8] J. Etnyre & K. Honda, Knots and contact geometry I: torus knots and thefigure eight knot, Preprint, 2000, math.GT/0006112.[9] J. Etnyre & K. Honda, Knots and Contact Geometry II: Connected Sums,Preprint, 2002, math.GT/0205310.[10] J. Etnyre, L. Ng, & J Sabloff, Invariants of Legendrian knots and coherentorientations, Preprint, 2001, math.GT/0101145.[11] A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math.Phys., 120, 1989, 575-611.[12] D. Fuchs, Private communication, 2001.[13] D. Fuchs & S. Tabachnikov, Invariants of Legendrian and transverse knots inthe standard contact space, Topology, 36 (1997), 1025-1053.[14] L. Ng, Legendrian mirrors and Legendrian isotopy, Preprint, 2000,math.GT/0008210.[15] L. Ng, Computable Legendrian invariants, Preprint, 2001, math.GT/0011265.

ICM 2002 • Vol. II • 395-403Finite Dimensional Approximationsin GeometryM. Furata*AbstractIn low dimensional topology, we have some invariants defined by using solutionsof some nonlinear elliptic operators. The invariants could be understoodas Euler class or degree in the ordinary cohomology, in infinite dimensionalsetting. Instead of looking at the solutions, if we can regard some kind ofhomotopy class of the operator itself as an invariant, then the refined versionof the invariant is understood as Euler class or degree in cohomotopy theory.This idea can be carried out for the Seiberg-Witten equation on 4-dimensionalmanifolds and we have some applications to 4-dimensional topology.2000 Mathematics Subject Classification: 57R57.Keywords and Phrases: Seiberg-Witten, 4-manifold, Finite dimensionalapproximation.1. IntroductionThe purpose of this paper is to review the recent developments in a formalframework to extract topological information from nonlinear elliptic operators.We also explain some applications of the idea to 4-dimensional topology byusingthe Seiberg-Witten theory.A prototype is the notion of index for linear elliptic operators. In this introductionwe explain this linear case. Later we mainly explain the Seiberg-Wittencase.Let D : Y(E°) —t r(£' 1 ) be an linear elliptic operator on a close manifold X.The index mdD is defined to bemdD= dimKerD — dimCofcerD.We can extend this definition as follows. Take any decomposition D = L ® L' :V° ® W° —¥ V 1 ® W 1 . such that L : V° —¥ V 1 is a linear map between two finite*Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1, Komaba, Meguro-ku,Tokyo 153-8914, Japan. E-mail: furuta@ms.u-tokyo.ac.jp

396 M. Furatadimensional vector spaces and that L' : W° —¥ W 1 is an isomorphism betweeninfinite dimensional vector spaces. Then we haveindD = dim V° — dim V 1 .It is easy to check that the right-hand-side is independent of the choice of thedecomposition. For example we have decomposition satisfying V° = KerD, V 1 =CokerD, L = 0, which gives the former definition of the index.An important property of ind D is its invariance under continuous variation ofD. This property is closely related to the above well-definedness.Another way to understand this property is to consider the whole space ofFredholm maps. Then the given map D sits in the space and the indD is nothingbut the label of the connected component containing D.In other words, there are presumably three possible attitudes:1. The essential data is "supported" on KerD and CokerD.2. It is convenient to look at "some" finite dimensional approximation. L : V° —¥V 1 .3. The essential data is the whole map D : Y(E°) —t F (I? 1 ).When one considers a family of elliptic operators and tries to define the indexof the family, it is not enough to look at their kernels and cokernels.It is tempting to regard the finite dimensional approximation as a topologicalversion of the notion of "low energy effective theory" in physics. In this story, thewhole map D would be regarded as a given original theory.In this paper we explain a nonlinear version of the notion of index which isformulated by using finite dimensional approximations.2. Non-linear casesWhile every elliptic operator on a closed manifold has its index as topologicalinvariant, it is quite rare that a nonlinear elliptic operator gives some topologicalinvariant.We have three examples of this type of invariants: the Donaldson invariant,the Gromov-Ruan-Witten invariant and the Seiberg-Witten invariant. Moreover,the Casson invariant is regarded a variant of the Donaldson invariant. Some otherfinite type invariants for 3-manifolds are also supposed to be related to these kindsof invariant [37].Even for these cases, however, it is not obvious how to proceed to obtainnonlinear version of index in full generality.Let us first give several examples of finite dimensional approximations.1. C. Conley and E. Zehnder solved the Arnold conjecture for torus by reducinga certain variational problem to a finite dimensional Morse theory [10].2. Casson's definition of the Casson invariant. Taubes gave an interpretation ofthe Casson invariant via gauge theory [41]. In other words, Casson's constructiongave a finite dimensional approximation of the gauge theoretical setting

Finite Dimensional Approximations in Geometry 397by Taubes. (The statement of the Atiyah-Floer conjecture could be regardedas a partial finite dimensional approximation along fibers.)3. Seiberg-Witten equation. The moduli space of Seiberg-Witten equation isknown to be compact for closed 4-manifolds. This enables us to globalize theKuranishi construction to obtain finite dimensional approximations [18], [3].4. Seiberg-Witten-Floer theory C. Manolescu and P.B. Kronheimer defined Floerhomotopy type for Seiberg-Witten theory, which is formulated as spectrum[32] [27].5. Kontsevish explained an idea to define invariants of 3-manifolds by usingconfiguration spaces. This idea was realized by Fukaya [13], Bott-Cattaneo [5][6], and Kuperberg-Thurston [28]. Formally the configuration spaces appearas finite approximations of certain path spaces.3. Kuranishi constructionWhile the index is regarded as the infinitesimal information of a nonlinearelliptic operator, its local information is given by the Kuranishi map, which hasbeen used to describe local structure in various moduli problems [29].A few years ago the Arnold conjecture was solved in a fairly general setting andthe Gromov-Ruan-Witten invariant was defined for general symplectic manifolds.These works were done by several groups independently [14], [31], [34], [38]. A keyof their arguments was to construct virtual moduli cycle over Q.In their case, the point is to glue local structure to obtain some global datato define invariants. Since their invariants are defined by evaluating cohomologyclasses, it was enough to have the virtual moduli cycle.4. Global approximationThe notion of Fukaya-Ono's Kuranishi structure or Ruan's virtual neighborhoodis defined as equivalence class of collections of maps, which define the modulispace. The collection of maps is necessary because the moduli space as topologicalspace is not enough to recover the nature of the singularity on it.The data depend on the choice of various choice of auxiliary data. Whenwe change the data, the change of the moduli space is supposed to be given by acobordism, even with the extra structure we have to look at.Suppose we would like to regard this structure itself as an invariant. Thenwe have to identify the place where the invariant lives. Since cobordism classesare identified by Pontrjagin-Thom construction, what we need would be a certainstable version of Pontrjagin-Thom construction.In the case of symplectic geometry or Donaldson's theory, this constructionhas not been done. A main problem seems to describe a finite dimensional approximationof the ambient space where the compactification of the moduli spacelies. (The same problem occurs for Kotschick-Morgan conjecture.) Since the compactificationis fairly complicated, it is not straightforward to identify the finitedimensional approximation.

398 M. FurataHowever in the Seiberg-Witten case, the moduli spaces are known to be compactfor closed 4-manifolds and it is not necessary to take any further compactifications.Let us briefly recall the Seiberg-Witten equation for a closed Spin 0 manifoldX. For simplicity we assume bfiX) = 0. Let W = W° ® W 1 be the spinor bundleand A be the space of connections on det W° = det W 1 . Then the Seiberg-Wittenequation is given by a mapY(W°) xA^Y(W V ) x F(A+),where F(A + ) is the self-dual 2-forms for a fixed Riemannian metric. This is an[/(l)-equivariant map. The inverse image of 0 divided by S 1 is the moduli space,which is known to be compact.A finite approximation of the above map is defined by global version of theKuranishi construction. The approximation is a proper [/(l)-equivariant mapCao 0 Edo _^ Cai & Rdifor some natural numbers co,ci,do and di. The differences Co — ci and do — didepends only on the topology of X and its spin c -structure.The invariant we have is the stable homotopy class of the above [/(l)-equivariantproper map, or equivalently, the [/(l)-equivariant map from the sphere S(C a ° ®R rf ° )to the sphere S(C ai ® R rfl ).S. Bauer and the author pointed out that the invariant constructed above is arefinement of the usual Seiberg-Witten invariant [3].5. 4-dimensional topology and Seiberg-WittentheoryWe explain some applications of the finite dimensional approximation to 4-dimensional topology.(1) Bauer's connected sum formula [2]Suppose X is the connected sum of X 0 and Xi. If the neck of the connectedsum is long enough, it is known that the moduli space of the solution of the Seiberg-Witten equation (or anti-self-dual equation) for X is identified with the product ofthe moduli spaces for X 0 and Xi. When Xi = CP 2 , then this gives the blowing-upformula. When ò + (X 0 ),ò _ (Xi) > 1, this gives vanishing of the Seiberg-Witten(or the Donaldson) invariant of X = X 0 #Xi. Bauer essentially showed that theproduct formula holds true for the virtual neighborhood of the moduli spaces, if weuse Ruan's terminology. In the language of stable maps between spheres, "product"becomes "join". In particular Bauer's formula gives the blowing-up formula for therefined invariant. When 6 + (X 0 ),6 _ (Xi) > 1, the join is torsion. It is, however, notnecessary zero. In this way Bauer gave many new examples of 4-manifolds whichare homeomorphic but not diffeomorphic to each other.Ishida-Lebrun [24] [25] obtained some applications of the connected sum formulato Riemannian geometry.

Finite Dimensional Approximations in Geometry 399(2) Intersection form of spin 4-manifoldsWhen 4-manifold is spin, we have certain extra symmetry, and the place wherethe invariant lives is a set of Fm(2)-equivariant stable maps [18].When X is a closed spin 4-manifold with 6i(X) = 0, the Seiberg-Witten mapfor the spin structure is a Fm(2)-equivariant map formally given byH°° ® R°° -• M 00 ® R°°,where R is the non-trivial 1-dimensional real representation space of Pin(2). andH is the 4-dimentional real irreducible representation space of Pin(2). Yet Z/4 bethe subgroup of Pin(2) generated by an element in Pin(2) \ U(Y). The differencesof the power oo's are given by the index of some elliptic operators.A finite dimensional approximation is given by a Fm(2)-equivariant propermapfor some co,ci,do,disatisfyingC0_ Cl = _?î^), do-d 1 = b + (X).This existence implies some inequality between the signature and the second Bettinumber.To obtain the inequality explicitly we can use the following results.Theorem Suppose k > 0 and k = a mod 4 for a = 0,1,2, or 3. Thenthere does not exist a G-equivariant continuous map from S(M k+x ® IF) to S(W ®R 2fc+a ^1+y ). for the following G and a'.1. (B. Schmidt [39] see also [40] [11] [33]) G = Z/4 and a' = a for a = 1,2,3.2. (F - Y.Kametani [20]) G = Pin(2) and a' = 3 for a = 0.From the above non-existence results, we have the following inequality, whichis a partial result towards the 11/8-conjecture b + > 3\sign (X)/16|.Theorem Let X be a closed spin 4-manifold with sign(X) = —16k < 0. Ifk = a mod 4 for a = 0,1,2 or 3, then we have b + > 2k + b, where a' = a if a = 1,2,3and a' = 3 if a = 0.Equivariant version and V-manifold version can be formulated similarly [7],[12], [16], [1]. There are some applications of these extended versions:1. C. Bohr [4] and R. Lee - T.-J. Li [30] investigated the intersection forms ofclosed even 4-manifolds which are not spin.2. Y. Fukumoto, M. Ue and the author [16] [15] [17] [42], and N. Saveliev [36]investigated homology cobordims groups of homology 3-spheres.When 6i > 0, we can construct another closed spin 4-manifold with 6i = 0 withoutchanging the intersection form. It implies that we can assume 6i = 0 to obtainrestriction on the intersection form. However when the intersection form on H 1 (X)is non-trivial, we may have a stronger restriction. Y. Kametani, H. Matsue, N. Minamiand the author found that such a phenomenon actually occurs if there areai,«2,«3,«4 £ -ff 1 (X,Z) such that (n a *>[^]) is odd [22].

400 M. Furata6. Seiberg-Witten-Floer homotopy typeRecently C. Manolescu and P. B. Kronheimer extends the above formulationfor closed 4-manifolds to the relative version [32], [27]. Let us explain their theorybriefly.We mentioned that Conley-Zehnder used a finite dimensional approximation ofa Morse function on an infinite dimensional space to approach the Arnold conjecturefor torus. Following this line, Conley exteded the notion of Morse index and definedthe Conley index for compact isolated set [9]. The Conley index is not a number,but a homotopy type of spaces. Floer extracted some information from the Conleyindex just by looking at some finite dimensional skeleton of the Conley index undersome assumption. Floor's formulation has the advantage that the Floer homologyis defined even when the Conley index is not rigorously defined.On the other hand R. L. Cohen, J. D. S. Jones and G. B. Segal tried to definecertain stable homotopy type directly which should be an extended version of theConley index [8]. They called it the Floer homotopy type. At that time the Floerhomology was defined only for the Donaldson theory and the Gromov-Ruan-Wittentheory. In these theories the moduli spaces are non-compact in general. This causea serious difficulty to carry out their program.In the Seiberg-Witten theory, we have a strong compactness for the modulispaces. Manolescu and Kronheimer succeeded to construct the Floer homotopytype as spectra for the Seiberg-Witten theory by using this compactness.They also defined relative invariant for 4-manifolds with boundary is also definedand it extends the invariant in [3].7. Concluding remarksThe idea of finite dimensional approximation is closely related to the notionof "low energy effective theory" in physics. Actually the approximation should beregarded just as a part of the vast notion which we can deal with rigorously ormathematically.Since Witten's realization of Donaldson theory as a TQFT, the formal relationbetween mathematically regorous definition of invariants and their formal pathintegral expressions has suggested many things. For instance, the well-definednessof the Donaldson invariant is based on the fact that the formal dimension of themoduli space increases when the instanton number goes up. This fact seems equivalentto the other fact that the pure Yang-Mills theory is asymptitotically free (forN=2 SUSY theory) and its renormalized theory does exists.In the case of the finite dimensional approximations of Seiberg-Witten theory,the suspension maps give relations between many choices of approximations. If weuse some generalized cohomology theories to detect our invariants, the suspensionmaps induces the Thom isomorphisms, or integrations along fibers. If we comparethis setting with physics, the family of integrations look quite similar to the renormalizationgroup. It seems the Thom classes which play the role of vacua. In thissense, one could say that the family of finite approximations is a topological version

Finite Dimensional Approximations in Geometry 401of the renormlization group. This topological setting is very limitted. It, however,has one advantage: Usually the path integral expression is supposed to take valuesin real or complex numbers. On the other hand our invariants could take values intorsions.Let us conclude this survey by giving three open problems.1. What is the correct formulation of the geography of spin 4-manifolds with61 > 0? (If the intersection on H 1 is complicated enough, then sign (X)would have stronger restriction.)2. When an oriented closed 3-manifold is a link of isolated algebraic singularpoint, construct a canonical Galois group action on some completion of theFloer homotopy type of Kronheimer-Manolescu. (This problem was suggestedby a hand-written manuscript by D. Johnson in which Casson-type invariantswere defined.)3. The Seiberg-Witten map is quadratic. Extract non-topological informationfrom this structure. (Is it possible to approach the 11/8-conjecture from thispoint of view?)References[i[2[3;[4;[5;[6;[7;t«:[9[10:[n[12:[13;D.J. Acosta, A Furuta-like inequality for spin orbifolds and the minimal genusproblem, Topology Appi. 114 (2001) 9H06.S. Bauer, A stable cohomotopy refinement of Seiberg-Witten invariants: II,math.DG/0204267S. Bauer and M. Furuta, A stable cohomotopy refinement of Seiberg-Witteninvariants: I, math.DG/0204340C. Bohr, On the signatures of even 4-manifolds, Math. Proc. Cambridge Philos.Soc. 132 (2002), 453-169.R. Bott & A. S. Cattaneo, Integral invariants of 3-manifolds. J. DifferentialGeom. 48 (1998), no. 1, 91-133.R. Bott & A. S. Cattaneo, Integral invariants of 3-manifolds. II, J. DifferentialGeom. 53 (1999), no. 1, 1-13.J. Bryan, Seiberg-Witten theory and Z/2 P actions on spin 4-manifolds, Math.Res. Lett. 5 (1998) 165^183.R. L. Cohen, J. D. S. Jones & G. B. Segal, Floor's infinite dimensional Morsetheory and homotopy theory, The Floer memorial Volume, Birkhäuser (1995).C. Conley Isolated invariant sets and the Morse index, Amer. Math. Soc,Provindence, 1978.C. Conley & E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjectureof V. I. Arnold. Invent. Math. 73 (1983), 33-19.M. C. Crabb, Periodicity in Z/4-equivariant stable homotopy theory, Cont.Math. 96, (1989) 109^124.F. Fang, Smooth group actions on 4-manifolds and Seiberg-Witten invariants,Internat. J. Math. 9 (1998), 957^973.K. Fukaya, Morse homotopy and Chern-Simons perturbation theory, Comm.Math. Phys. 181 (1996), no. 1, 37^90.

402 M. Furata[14] K. Fukaya & K. Ono, Arnold conjecture and Gromov-Witten invariant. Topology38 (1999), no. 5, 933^1048.[15] Y. Fukumoto, On an invariant of plumbed homology 3-spheres,J. Math. Kyoto.Univ. 40, No. 2, (2000) 379^389.[16] Y. Fukumoto & M. Furuta, Homology 3-spheres bounding acyclic 4-manifolds,Math. Res. Lett. 7, (2000), 757^766.[17] Y. Fukumoto, M. Furuta & M. Ue, W invariants and the Neumann-Siebenmanninvariants for Seifert homology 3-spheres, Topology Appi. 116 (2001), no. 3,333^369. .[18] M. Furuta, Monopole equation and the 11/8-conjecture, Math. Res. Lett. 8(2001), no. 3, 279-291.[19] M. Furuta & Y. Kametani, The Seiberg-Witten equations and equivariant e-invariants, preprint (UTMS 2001-10, University of Tokyo).[20] M. Furuta & Y. Kametani, Equivariant maps and ifO*-degree, preprint.[21] M. Furuta, Y. Kametani & H. Matsue, Spin 4-manifolds with signature=-32,Math. Res. Letters 8, (2001) 293-301.[22] M. Furuta, Y. Kametani, H. Matsue & N. Minami, Stable-homotopy Seiberg-Witten invariants and Pin bordisms, preprint (UTMS 2000-46, University ofTokyo).[23] M. Furuta, Y. Kametani & N. Minami, Stable-homotopy Seiberg-Witten invariantsfor rational cohomology K3#K3's, J. Math. Sci. Univ. Tokyo. 8 (2001)157^176.[24] M. Ishida & C. LeBrun, Spin Manifolds, Einstein Metrics, and DifferentialTopology, math.DG/0107111[25] M. Ishida & C. LeBrun, Curvature, Connected Sums, and Seiberg-Witten Theory,math.DG/0111228[26] M. Kontsevish, Feynman diagrams and low-dimensional topology, First EuropeanCongress of Mathematics, Vol. II (Paris, 1992), 97^121, Progr. Math.,120, Birkhäuser, Basel, 1994.[27] P. B. Kronheimer & C. Manolescu, Floer pro-spectra from the Seiberg-Wittenequations, math.GT/0203243[28] G. Kuperberg & D. Thurston, Perturbativi 3-manifold invariants by cut-andpastetopology, math.GT/9912167[29] M. Kuranishi, New proof for the existence of local free complete families ofcomplex structures, Conference on Complex Analysis. Springer, Minneapolis,1964.[30] R. Lee & T.-J. Li, Intersection forms of non-spin four manifolds, Math. Ann.319 (2001), no. 2, 311-318.[31] G. Liu & G. Tian, Floer homology and Arnold conjecture, J. Differential Geom.49 (1998), no. 1, 1-74.[32] C. Manolescu, Seiberg-Witten-Floer stable homotopy type of three-manifoldswith 6i = 0, math.DG/0104024[33] N. Minami, The G-join theorem - an unbased G-Freudenthal theorem, preprint.[34] Y. Ruan, Virtual neighborhoods and pseudo-holomorphic curves, Proceedingsof 6th Gokova Geometry-Topology Conference. Turkish J. Math. 23 (1999),

Finite Dimensional Approximations in Geometry 403no. 1, 161-231.[35] Y. Ruan, Virtual neighborhoods and the monopole equations, Topics in symplectic4-manifolds (Irvine, CA, 1996), 101-116, First Int. Press Lect. Ser., I,Internat. Press, Cambridge, MA, 1998.[36] N. Saveliev, Fukumoto-Furuta invariants of plumbed homology 3-spheres,preprint, (2000).[37] N. Seiberg & E. Witten, Gauge Dynamics And Compactification To ThreeDimensions, hep-th/9607163[38] B. Siebert, Gromov-Witten invariants of general symplectic manifolds, dgga/9608005[39] B. Schmidt, Ein Kriterium für die Esistenz äquivarianter Abbildungen zwischenreellen Darstellungssphären der Gruppe Pin(2), Diplomarbeit UniverstätBielefeld, 1997.[40] S. Stolz, The level of real projective spaces, Comment. Math. Helvetici 64(1989), 661^674.[41] C. H. Taubes, Casson's invariant and gauge theory, J. Differential Geom. 31(1990), no. 2, 547^599.[42] M. Ue, On the intersection forms of spin 4-manifolds bounded by spherical3-manifolds, Algebraic and Geometric Topology 1 (2001), 549^578.

ICM 2002 • Vol. II • 405-114Géométrie de Contact:de la Dimension Troisvers les Dimensions SupérieuresEmmanuel Giroux*RésuméOn décrit ici des relations entre la géométrie globale des variétés de contactcloses et celle de certaines variétés symplectiques, à savoir les variétés de Steincompactes. L'origine de ces relations est l'existence de livres ouverts adaptésaux structures de contact.2000 Mathematics Subject Classification : 57M50, 53D35.Mots clés : Structures de contact, Livres ouverts.La géométrie de contact en dimension trois a connu un essor important durantla dernière décennie grâce au développement de méthodes topologiques adéquates.Dans le prolongement des travaux de D. Bennequin [Be] et de Y. Eliashberg [EU],la théorie des « surfaces convexes » [Gii] et l'étude des rocades (bypasses) [Ho] ontmené à une classification complète des structures de contact sur quelques variétéssimples et, plus récemment, à une classification grossière sur toutes les variétéscloses [Co, HKM, CGH]. En fait, comme on essaiera de le montrer plus loin, lesstructures de contact en dimension trois sont des objets purement topologiques,un peu comme les structures symplectiques en dimension deux. En termes précis,sur toute variété close V de dimension trois, les classes d'isotopie des structuresde contact se trouvent en correspondance bijective avec les classes d'isotopie et destabilisation des livres ouverts dans V, l'opération élémentaire de stabilisation étantun plombage positif [Gi2].En dimension supérieure, des méthodes radicalement différentes permettentde mettre en évidence une correspondance similaire [GM] et, au-delà, de faire apparaîtredes liens étroits entre la géométrie globale des variétés de contact closes etcelle de certaines variétés symplectiques compactes. Les livres ouverts qu'on associeà une structure de contact sont en effet particuliers : leurs pages sont des variétés deStein compactes, leur monodromie est un difféomorphisme symplectique à supportdans l'intérieur et l'opération élémentaire de stabilisation qui les unifie est un plombagelagrangien positif. En outre, l'outil essentiel pour les construire est la théorie* Unité de Mathématiques Pures et Appliquées, École Normale Supérieure de Lyon, 46 alléed'Italie, 69364 Lyon cedex 07, France. Mél: giroux@umpa.ens-lyon.fr

406 E. Girouxdes fibres positifs que S. Donaldson a introduite et développée en géométrie symplectiquedans [Dol, Do2] et qui a été adaptée en géométrie de contact dans [IMP].A. Structures de contact et livres ouvertsDans ce texte, V désigne toujours une variété close et orientée. Les champsd'hyperplans tangents qu'on considère sur V sont coorientés, donc aussi orientéspuisque V l'est. Un tel champ £ est le noyau d'une forme a, appelée équation de £,unique à multiplication près par une fonction positive. On dit que £ est une structurede contact si da induit sur £ en tout point une forme symplectique directe, i.e. siV est de dimension impaire 2n + 1 et si a A (da) n est en tout point un élément devolume direct pour l'orientation de V.D'autre part, un livre ouvert dans V est un couple (K,9) formé des objetssuivants :- une sous-variété close K c V de codimension deux à fibre normal trivial ;- une fibration 9: V\K —¥ S 1 qui, dans un voisinage KxD 2 de K = K x {0},coïncide avec la coordonnée angulaire normale.On peut aussi voir les livres ouverts autrement. Soit : F->Fun difféomorphismed'une variété compacte égal à l'identité près du bord K = dF. Sa suspension, àsavoir la variété compacte£(F,^)=(Fx[0,l])/~,où (p,l)~Mp),0),est bordée par K x S 1- car | K = id - et la variété closeë(F,^) = S(F,^)U 0 (A:XD 2 ),possède un livre ouvert évident. En outre, tout livre ouvert (K, 9) dans V identifie Và Yi(F,4>), où F est une fibre de 9 (un peu rétrécie) et l'application de premierretour sur F d'un flot transversal aux fibres de 9 et constitué, près de K, de rotationsautour de K. Le difféomorphisme , défini seulement à conjugaison et isotopie près,est la monodromie de (K,9).Toute la discussion à venir tourne autour de la définition suivante :Définition 1 [Gi2, GM]. Une structure de contact £ sur V est dite portée par unlivre ouvert (K, 9) si elle admet une équation a ayant les propriétés suivantes :- a induit sur K une forme de contact ;- da induit sur chaque fibre F de 9 une forme symplectique ;- l'orientation de K définie par la forme de contacta coïncide avec son orientationcomme bord de la variété symplectique (F,da).Une telle forme a sera dite adaptée à (K,9).Exemple [GM]. Soit/: (C",0) —¥ (C, 0) une fonction holomorphe ayant à l'origineun point critique isolé et soit H l'hypersurface (singulière) / _1 (0). Il existe uneboule fermée lisse B autour de l'origine dans C" et un feuilletage de B \ {0} pardes sphères strictement pseudoconvexes S r , où r £ ]0,1] et Si = dB, tels que, pourr assez petit, les propriétés suivantes soient satisfaites :

Géométrie de Contact 407- la sphère S r est transversale à H, de sorte que K = H n S r est une sousvariétéclose de S r de codimension deux et à fibre normal trivial ;- l'application 9 = arg/: S r \ K —t S 1 est une fibration qui fait de (K,9) unlivre ouvert ;- le livre ouvert (K, 9) porte la structure de contact sur S r définie par le champdes tangentes complexes.Autrement dit, chaque livre ouvert donné dans la sphère par le théorème de fibrationde J. Milnor porte, à isotopie près, la structure de contact standard.B. Structures de contact et livres ouverts en dimensiontroisEn dimension trois, divers travaux ont depuis longtemps fait apparaître desconnivences entre les structures de contact et les livres ouverts sans toutefois établiraucun lien formel. Dans [TW], W. Thurston et H. Winkelnkemper construisent desformes de contact sur toute variété close V a partir d'un livre ouvert dans V.Avec les termes de la définition 1, ils démontrent en fait que tout livre ouvertdans V porte une structure de contact. Dans [Be] d'autre part, pour transformer enthéorème de géométrie de contact son résultat sur les tresses fermées, D. Bennequinmet en évidence la propriété suivante : toute courbe transversale à la structure decontact standard £o dans R 3 - structure d'équation dz + r 2 d9 = 0 - est isotope,parmi les courbes transversales, à une tresse fermée c'est-à-dire une courbe transversaleau livre ouvert formé par l'axe des z et la coordonnée angulaire 9. Or cettepropriété vient de ce que ce livre ouvert porte &>• Enfin, dans [To], I. Torisu a clairementdégagé les relations entre les livres ouverts et les configurations de théoriede Morse considérées dans [Gil] pour étudier les structures de contact convexes ausens de [EG].La première observation qui montre l'étroitesse des liens imposés par la définition1 et découle de la stabilité des structures de contact est la suivante :Proposition 2 [Gi2]. Sur une variété close de dimension trois, toutes les structuresde contact portées par un même livre ouvert sont isotopes.Quant à la question de savoir quelles structures de contact possèdent un livreouvert porteur, la réponse est simple :Théorème 3 [Gi2]. Sur une variété close de dimension trois, toute structure decontact est portée par un livre ouvert.Cependant, comme l'illustre l'exemple des fibrations de Milnor, le livre ouvertqui porte une structure de contact donnée est loin d'être unique - même à isotopieprès. Pour appréhender ce phénomène, quelques définitions sont utiles.Soit F c V une surface compacte à bord et C C F un arc simple et propre.On dit qu'une surface compacte F' c V s'obtient à partir de F par le plombagepositif (resp. négatif) d'un anneau le long de C si F' = F U A où A C V est unanneau ayant les propriétés suivantes :- A n F est un voisinage régulier de C dans F ;

408 E. Giroux- A est inclus dans une boule fermée B dont l'intersection avec F est réduite àAn F et l'enlacement des deux composantes de dA dans B vaut 1 (resp. —1).Un résultat de J. Stallings affirme que, si (K,9) est un livre ouvert dans V et siF est l'adhérence d'une fibre de 9, alors, pour toute surface F' obtenue à partirde F par le plombage d'un anneau, il existe un livre ouvert (K',9 r ) tel que K' soitle bord de F' et que F' soit l'adhérence d'une fibre de 9'. Dans la suite, on diraque le livre (K',9 r ) et l'entrelacs K' sont eux-mêmes obtenus par plombage à partirrespectivement de (K,9) et de K. En outre, on dira qu'un livre ouvert (K',9 r ) estune stabilisation d'un autre (K, 9) s'il s'obtient à partir de (K, 9) par une suite finiede plombages positifs.Théorème 4 [Gi2]. Dans une variété close de dimension trois, deux livres ouvertsquelconques qui portent une même structure de contact ont des stabilisationsisotopes.Les théorèmes 3 et 4 permettent de traduire nombre de questions sur lesstructures de contact en questions sur les livres ouverts, autrement dit sur lesdifféomorphismes des surfaces compactes à bord. En ce sens, ce sont les analoguesdes théorèmes de S. Donaldson [Do2] sur les pinceaux de Lefschetz dans les variétéssymplectiques de dimension quatre. Ils admettent cependant, à la différence deceux-ci, des démonstrations purement topologiques dont on décrit brièvement lesidées ci-dessous, après avoir introduit l'outil essentiel. On supposera le lecteur familieravec certaines notions de géométrie de contact en dimension trois (structuresde contact vrillées/tendues, invariant de Thurston-Bennequin des courbes legendriennes,surfaces ^-convexes).On appelle cellule polyédrale dans V l'image d'un polyèdre convexe compacteuclidien par un plongement topologique. Une telle cellule possède une structureaffine induite par son paramétrage et son intérieur est, par définition, l'image del'intérieur « intrinsèque » du polyèdre, c'est-à-dire de son intérieur topologique dansson enveloppe affine. Une cellulation polyédrale de V désigne ici un recouvrementfini de V par des cellules polyédrales ayant les propriétés suivantes :- les intérieurs des cellules forment une partition de V ;- le bord de chaque cellule D est une union de cellules Dj et les inclusionsDj —t D sont affines ;- les cellules de dimension deux (et moins) sont lisses, i.e. sont les images deplongements lisses.Les cellulations polyédrales ont cet avantage sur les triangulations d'être très facilesà subdiviser : toute subdivision d'un sous-complexe se prolonge trivialement. Enoutre, elles jouent un rôle clé dans la démonstration du théorème de Reidemeister-Singer donnée dans [Si], démonstration qui sert de guide pour établir le théorème 4.Esquisse de la démonstration du théorème 3. Soit £ une structure de contactsur V. On construit d'abord dans (V, £) une cellulation de contact, c'est-à-dire unecellulation polyédrale A ayant les propriétés suivantes :1) chaque cellule de dimension 1 est un arc legendrien ;2) chaque cellule de dimension 2 est ^-convexe et l'invariant de Thurston-Bennequin de son bord vaut —1 ;

Géométrie de Contact 4093) chaque cellule de dimension 3 est contenue dans le domaine d'une carte deDarboux.On épaissit ensuite le 1-squelette L de A en une surface compacte F (presque)tangente à £ le long de L et on choisit un voisinage régulier W de L assez petit pourque F = F n W soit une surface proprement plongée dans W. Quitte à prendre Wplus petit, £ admet une équation a vérifiant les conditions suivantes :- da induit sur F une forme d'aire ;- a est non singulière sur K = dF et oriente K comme le bord de (F,da).D'autre part, pour toute cellule D de dimension 2, la propriété 2) dit que le bordde D n (V \ Int W) intersecte K en deux points (à isotopie près). Il en résulte qu'ilexiste une fibration 9 : V \ K —t S 1 ayant Int F pour fibre. Quitte à rogner W, onpeut supposer que W est une union de fibres de 9 sur lesquelles da induit une formed'aire. Il reste à voir que £ est isotope, relativement à W, à une structure de contactportée par (K, 9). Le point clé est que £ est tendue sur W* = V \ Int W et que dW*est une surface ^-convexe dont le découpage est fourni par K.DEtapes de la démonstration du théorème 4. Soit A une cellulation de contactde (V, £). On dira ici qu'un livre ouvert porteur (K, 9) est associé à A si, comme dansla démonstration du théorème 3, l'une des fibres de 9 contient le 1-squelette de A etse rétracte dessus par une isotopie de contact. En imitant [Si], on montre d'abordque tout livre ouvert porteur admet une stabilisation associée à une cellulation decontact. On se ramène ainsi à considérer le cas de deux livres ouverts porteursassociés à des cellulations de contact A 0 et Ai en position générale. D'après [Si],A 0 et Ai possèdent une subdivision commune A 2 qui s'obtient, à partir de A 0comme de Ai, par des bissections. On déforme alors A 2 , relativement à l'union des1-squelettes de A 0 et Ai, en une cellulation vérifiant les propriétés 1) et 3) des cellulationsde contact et ayant des 2-cellules ^-convexes. Il suffit ensuite de subdiviserle 2-squelette de A 2 pour obtenir une cellulation de contact A et on montre pourfinir que le livre ouvert associé à A est une stabilisation de ceux associés à A 0 età Ai.DOn discute maintenant quelques corollaires des théorèmes 3 et 4.On rappelle d'abord qu'un théorème de M. Hilden et J. Montesinos affirmeque toute variété close V de dimension trois est un revêtement à trois feuillets dela sphère S 3 simplement ramifié au-dessus d'un entrelacs (simplement signifie quele degré local aux points de ramification dans V vaut deux). On obtient le mêmerésultat pour les variétés de contact closes :Corollaire 5 [Gi2]. Toute variété de contact close de dimension trois est un revêtementà trois feuillets de la sphère de contact standard (S 3 ,£o) simplement ramifiéau-dessus d'un entrelacs transversal à &>•Un autre corollaire concerne la dynamique des flots de Reeb. Un flot de Reebsur une variété de contact est un flot qui préserve la structure de contact tout enlui étant transversal et en pointant du côté positif. Un exemple typique est le flotgéodésique sur le fibre cotangent unitaire d'une variété riemannienne. Les flots deReeb d'une structure de contact donnée £ sont en bijection avec les équations de £ :à toute forme a correspond l'unique champ de vecteurs V Q qui engendre le noyau

Géométrie de Contact 411entre les invariants de Hopf correspondants.DC. Structures de contact et livres ouverts en dimensionsupérieureEn dimension supérieure à trois, les livres ouverts porteurs de structures decontact ne sont pas quelconques : leurs fibres ont une structure symplectique invariantepar la monodromie. Pour préciser ce point, quelques définitions sont utiles.Soit F une variété compacte, à bord K = dF. Une forme symplectique exacte usur Int F est convexe à l'infini s'il existe sur Int F un champ de Liouville (champ devecteurs w-dual d'une primitive de u) qui est transversal à toutes les hypersurfacesK x {t}, t £ ]0,1], où if x [0,1] est un voisinage collier de K = K x {0}. On diten outre que (Int F, u) est une variété de Weinstein [EG] s'il existe un tel champde Liouville qui, de plus, est le (pseudo) gradient d'une fonction de Morse F —t Rconstante et sans points critiques sur K. L'exemple typique de variété de Weinsteinest l'intérieur d'une variété de Stein compacte. On nomme ainsi toute variétécomplexe compacte F qui admet une fonction strictement pluri-sous-harmonique/: F —t R constante et sans points critiques sur le bord. La 2-forme iddf définitalors une structure symplectique. Il ressort en fait du travail de Y. Eliashberg [E12]que toute variété de Weinstein est symplectiquement difféomorphe à l'intérieur d'unetelle variété de Stein compacte.Si maintenant a est une forme de contact adaptée à un livre ouvert (K,9),sa différentielle da induit sur chaque fibre de 9 une structure symplectique exacteconvexe à l'infini. Celle-ci dépend du choix de a mais sa completion [EG] est biendéfinie à isotopie près. Le théorème de W. Thurston et H. Winkelnkemper et laproposition 2 s'étendent alors ainsi en grande dimension :Proposition 9 [GM]. Soit F une variété compacte avec, sur Int F, une formesymplectique exacte convexe à l'infini et soit 0 et des fonctionsSk : V —¥ C, k > 1, vérifiant les conditions suivantes :- en tout point de V,\ s k(p)\ < C) \d-Sk — ikskO,] < Ck 1 ' 2 et \d(_Sk\ < C ;

412 E. Giroux- en tout point p où \sk(p)\ < n,\d(,Sk(p)\ > rjk 1/2 .(Ici, d^Sk et d^Sk sont les parties respectivement J-linéaire et J-antilinéaire dedsu | (_•) En termes plus parlants, les fonctions su sont des sections approximativementholomorphes et équitransversales du fibre L® k —t V, où L est le fibre hermitientrivial FxC-* V muni de la connexion unitaire définie par la forme —ia.Les estimations ci-dessus entraînent d'abord que, pour \w\ < n, l'ensembleK w = sZ 1 (w) est une sous-variété et que la forme a w induite par a sur K w est uneforme de contact (voir [IMP]). En effet, a w est non singulière pour k assez grandpuisque son noyau est égal au noyau de dsk |ç et que \dtSk\ > nk 1 / 2 tandis que\®Z s k\ < C- Mieux, ces inégalités montrent que, pour k grand,le noyau de a w estproche d'un sous-espace J-complexe de £ si bien que da w y est non dégénérée.L'observation suivante est que l'application arg«/.: V \ K —t S 1 est une fibrationdont les fibres sont transversales au champ de Reeb V Q en tout point oùI s *;I > f l- Pour le voir, on note que l'estimation sur dsu — iksua implique que\d-Sk(â) — iksk\ < Ck 1 ' 2 .Ainsi, en un point p où \sk(p)\ > n et pour k assez grand, dsu(p) (V a ) est proche deiksk(p), i.e. est non nul et presque orthogonal à Sk(p)- Par suite, les sous-variétéss^1(Re ), où Re = {re %e , r > n},sont transversales au champ de Reeb V Q .Ces arguments montrent que le livre ouvert (K = K 0 , 9 = argsk), pour kassez grand, porte la structure de contact £ = ker a. Il reste à vérifier que les fibresde 9 sont des variétés de Weinstein. Pour simplifier, on prouve ci-dessous l'assertionanalogue en géométrie symplectique.DProposition 11. Soit W une variété close, u une forme symplectique entière sur Wet Hk une sous-variété symplectique de W en dualité de Poincaré avec koj et obtenuepar la construction de Donaldson [Dol], à partir d'un fibre hermitien en droites Lmuni d'une connexion unitaire de courbure —ioj. Pour k assez grand, (W \ Hk,oj)est une variété de Weinstein.Démonstration. En reprenant les arguments de [Do2], on peut supposer que Hkest le lieu d'annulation d'une section Sk '• V —¥ L® k qui vérifie, en tout point de W,\dk$k\ < c\dkSk\ avec c < —j=.v2Dans la trivialisation de L® k donnée au-dessus de W \ Hk par la section unitaire« = Sk/\$k\, la connexion est définie par une 1-forme —iX où dX = koj. Si on poseSk = pu, l'inégalité ci-dessus donne\ckp/

Géométrie de Contact 413Comme en dimension trois, le livre ouvert porteur d'une structure de contactdonnée n'est pas unique. On décrit dans [GM] une opération de plombage le longd'un disque lagrangien - dans laquelle les twists de Dehn-Seidel viennent remplacerles twists de Dehn - qui permet d'établir des analogues du théorème 4 et ducorollaire 7. Ces résultats ramènent l'étude des structures de contact à celles desdifféomorphismes symplectiques des variétés de Stein compactes qui sont l'identitéprès du bord. Ils permettent peut-être ainsi de rapprocher les travaux de Y. Eliashberg,H. Hofer et A. Givental sur la théorie symplectique des champs de ceux de, parexemple, de P. Seidel sur l'nomologie de Floer et les groupes de difféomorphismessymplectiques. On peut aussi se demander si le théorème 10 cache des obstructionsà l'existence d'une structure de contact sur les variétés closes. D'après [Qu], toutevariété close V de dimension 2n + 1 possède un livre ouvert dont chaque fibre a letype d'homotopie d'un complexe cellulaire de dimension n. Il est probable que, siV admet un champ d'hyperplans tangents muni d'une structure presque complexe,il existe un tel livre ouvert pour lequel chaque fibre est une variété presque complexeet est donc, d'après [E12], l'intérieur d'une variété de Stein compacte. Toutela difficulté serait donc vraiment de réaliser la monodromie par un difféomorphismesymplectique... Dans cet ordre d'idée, voici un corollaire concret du théorème 10obtenu par F. Bourgeois et qui montre, en réponse à une vieille question, que touttore de dimension impaire possède une structure de contact :Corollaire 12 [Bo]. Si une variété close V admet une structure de contact, V x T 2en admet une aussi.Démonstration. Soit £ une structure de contact sur V, soit a une équation de£ adaptée à un livre ouvert porteur (K,9) et soit N = K x D 2 un voisinage deK = K x {0} dans lequel 9 est la coordonnée angulaire normale. On note r lacoordonnée radiale normale dans N et on poseâ = a + f(r)(cos9dxi — sin9dx 2 ), (xi,x 2 ) £ T 2 = R 2 /Z 2 ,où la fonction/(r) vaut r pour r 2r 0 et vérifie f'(r) > 0. Un calculmontre que, si on choisit r 0 assez petit, â est une forme de contact sur V x T 2 . DRéférences[Be] D. BENNEQUIN, Entrelacements et équations de Pfaff. Astérisque 107—108(1983), 87-161.[Bo] F. BOURGEOIS, Odd-dimensionai tori are contact manifolds. Int. Math.Res. Notices (à paraître).[Co]V. COLIN, Une infinité de structures de contact tendues sur les variétéstoroïdales. Comment. Math. Helv. 76 (2001), 353^372.[CGH] V. COLIN, E. GIROUX et K. HONDA, Finitude homotopique et isotopiquedes structures de contact tendues. En préparation.[Dol] S. DONALDSON, Symplectic submanifolds and almost-complex geometry. J.Diff. Geom. 44 (1996), 666^705.[Do2]S. DONALDSON, Lefschetz pencils on symplectic manifolds. J. Diff. Geom.53 (1999), 205^236.

414 E. Giroux[EU] Y. ELIASHBERG, Classification of over-twisted contact structures on 3-manifolds. Invent. Math. 98 (1989), 623^637.[E12] Y. ELIASHBERG, Topological characterization of Stein manifolds of dimension> 2. Int. J. Math. 1 (1990), 29-16.[EG] Y. ELIASHBERG et M. GROMOV, Convex symplectic manifolds. SeveralComplex Variables and Complex Geometry (part 2), Proc. Sympos. PureMath. 52, Amer. Math. Soc. 1991, 135^162.[Gil] E. GIROUX, Convexité en topologie de contact. Comment. Math. Helv. 66(1991), 637^677.[Gi2] E. GIROUX, Structures de contact, livres ouverts et tresses fermées. Enpréparation.[GM] E. GIROUX et J.-P. MOHSEN, Structures de contact et fibrations symplectiquesau-dessus du cercle. En préparation.[Ha] J. HARER, HOW to construct all fibered knots and links. Topology 21 (1982),263^280.[Ho] K. HONDA, On the classification of tight contact structures I. Geom. Topol.4 (2000), 309^368.[HKM] K. HONDA, W. KAZEZ et G. MATIC, Convex decomposition theory. Int.Math. Res. Notices 2002, 55^88.[HWZ] H. HOFER, K. WYSOCKI et E. ZEHNDER, The dynamics on three-dimensionalstrictly convex energy surfaces. Ann. of Math. 148 (1998), 197-289.[IMP] A. IBORT, D. MARTINEZ et F. PRESAS, On the construction of contactsubmanifolds with prescribed topology. J. Diff. Geom. 56 (2000), 235^283.[LP] A. Loi et R. PIERGALLINI, Compact Stein surfaces with boundary as branchedcovers of B A . Invent. Math. 143 (2001), 325^348.[NR] W. NEUMANN et L. RUDOLPH, Unfoldings in knot theory. Math. Ann. 278(1987), 409-139 - Corrigendum : Math. Ann. 282 (1988), 349-351.[Qu] F. QuiNN, Open book decompositions and the bordism of automorphisms.Topology 18 (1979), 55^73.[Si] L. SIEBENMANN, Les bissections expliquent le théorème de Reidemeister-Singer. Prépublication 1979 (Orsay).[To] I. ToRisu, Convex contact structures and fibered links in 3-manifolds. Int.Math. Res. Notices 2000, 441-154.[TW] W. THURSTON et H. WINKELNKEMPER, On the existence of contact forms.Proc. Amer. Math. Soc. 52 (1975), 345^347.

ICM 2002 • Vol. II • 415-125Algebraic Jf-theory and Trace InvariantsLars Hesselholt*(Dedicated to lb Madsen on his sixtieth birthday)AbstractThe cyclotomic trace of Bökstedt-Hsiang-Madsen, the subject ofBökstedt's lecture at the congress in Kyoto, is a map of pro-abelian groupsK-fiA) ^.TR;(A;p)from Quillen's algebraic A"-theory to a topological refinement of Connes' cyclichomology. Over the last decade, our understanding of the target and itsrelation to A"-theory has been significantly advanced. This and possible futuredevelopment is the topic of my lecture.The cyclotomic trace takes values in the subset fixed by an operator Fcalled the Frobenius. It is known that the induced mapK*(A,Z/p v ) -^ TR;(A;P,Z/P V ) F=1is an isomorphism, for instance, if A is a regular local F p -algebra, or if Ais a henselian discrete valuation ring of mixed characteristic (0,p) with aseparably closed residue field. It is possible to evaluate A"-theory by meansof the cyclotomic trace for a wider class of rings, but the precise connectionbecomes slightly more complicated to spell out.The pro-abelian groups TR*(A;p) are typically very large. But they comeequipped with a number of operators, and the combined algebraic structure isquite rigid. There is a universal example of this structure — the de Rham-Wittcomplex — which was first considered by Bloch-Deligne-Illusie in connectionwith Grothendieck's crystalline cohomology. In general, the canonical mapW.Q q A^TR- q (A-p)is an isomorphism, if q < 1, and the higher groups, too, can often be expressedin terms of the de Rham-Witt groups. This is true, for example, if A is aregular F p -algebra, or if A is a smooth algebra over the ring of integers in alocal number field. The calculation in the latter case verifies the Lichtenbaum-Quillen conjecture for focal number fields, or more generally, for henseliandiscrete valuation fields of geometric type.2000 Mathematics Subject Classification: 19D45, 19D50, 19D55, 11S70,14F30, 55P91.* Massachusetts Institute of Technology, Cambridge, MA 02139, USA. E-mail:larsh@math.mit.edu

416 L. Hesselholt1. Algebraic üC-theoryThe algebraic if-theory of Quillen [30], inherently, is a multiplicative theory.Trace invariants allow the study of this theory by embedding it in an additivetheory. It is possible, by this approach, to evaluate the if-theory (with coefficients)of henselian discrete valuation fields of mixed characteristic. We first recall theexpected value of the if-groups of a field k.The groups K*(k) form a connected anti-commutative graded ring, there is acanonical isomorphism I: k* ^y Kfik), and £(x) • 1(1 — x) = 0. One defines theMilnor if-groups K^(k) to be the universal example of this algebraic structure [29].The canonical map K^(k) —¥ K q (k) is an isomorphism, if q < 2. Yet us now fix theattention on the if-groups with finite coefficients. (The rational if-groups, while ofgreat interest, are of a rather different nature [11, 12].) The groups if» (k,Z/m) forman anti-commutative graded Z/m-algebra, at least if v 2 (m) ^ 1,2 and vz(m) ^ 1.And if ß m C k, there is a canonical liftingyb /ßm-—-—>Ki(k),K 2 (k,Z/m)which to a primitive roth root of unity ( associates the Bott element 6ç. Hence, inthis case, there is an additional map of graded rings Sz/ m (p m ) —* K*(k,Z/m). TheBeilinson-Licthenbaum conjectures predict that the combined mapiff (k) 8zS z/ro (fa) -+K.(k,Z/m)be an isomorphism of graded rings [1, 26]. The case TO = 2 V follows from thecelebrated proof of the Milnor conjecture by Voevodsky [34]. We here considerthe case of a henselian discrete valuation field of mixed characteristic (0,p) withp odd and m = p v [20, 14]. The groups K^(k)/m typically are non-zero in onlyfinitely many degrees. Hence, above this range, the groups K*(k,Z/m) are twoperiodic.All rings (resp. graded rings, resp. monoids) considered in this paperare assumed commutative (resp. anti-commutative, resp. commutative) and unitalwithout further notice.2. The de Rham-Witt complexLet V be a henselian discrete valuation ring with quotient field K of characteristiczero and residue field k of odd characteristic p. (At this writing, we furtherrequire that V be of geometric type, i.e. that V be the henselian local ring atthe generic point of the special fiber of a smooth scheme over a henselian discretevaluation ring V 0 C V with perfect residue field.) A first example of a trace map isprovided by the logarithmic derivative^MKf(K)^œ {VM)

Algebraic if-theory and Trace Invariants 417which to the symbol {cti,. • • ,a q } associates the form dlogai... dloga g . The righthand side is the de Rham complex with log poles in the sense of Kato [25] : A logring (A,M) is a ring A and a map of monoids a: M —t (A, •); a log differentialgraded ring (E*,M) is a differential graded ring E* together with maps of monoidsa: M —t (E 0 ,-) and dlog: M —t (E 1 ,-^) such that do dlog = 0 and such thatda(a) = a(a)dlogafor all a £ M; the de Rham complex Q? A M) is the universal logdifferential graded ring with underlying log ring (A,M). We will always considerthe ring V with the canonical log structurea: M = VnK*

418 L. Hesselholt(i) a pro-log differential graded ring (ET, M E) and a map of pro-log rings(ii) a map of pro-log graded ringssuch that ÀF = FA and such thatA: (W.(A),M)->(E?,M E );F: ET -> ET_i,Fdlog„ a = dlog„_i a, for all a £ M,FdA[a]„ = A[a]^Z 1 dA[a]„_i, for all a £ A;(iii) a map of pro-graded modules over the pro-graded ring ET,V: F*ET_i -+ET,such that XV = VX, FV = p and FdV = d.A map of Witt complexes over (A, M) is a map of pro-log differential gradedrings which commutes with the maps A, F and V. Standard category theory showsthat there exists a universal Witt complex over (A,M). This, by defintion, is thede Rham-Witt complex W. W A M) . (The canonical maps W.(A) —t W. 0,9 A M) and01 A M) —t Wi 01 i M) are isomorphisms, so the construction really does combinedifferential forms and Witt vectors.) We lift the logarithmic derivative to a mapiff(if)Î^Q» l/jM)which to the symbol {ai,...,a q } associates dlog n cti...dlog n a q . This trace mapbetter captures the Milnor A'-groups. Indeed, the following result was obtained incollaboration with Thomas Geisser [14]:Theorem 2.1 Suppose that p p » C K and thatk is separably closed. Then the tracemap induces an isomorphism of pro-abelian groupsKM(K)/p«^(WMl ViM) /p«) F=1 .To prove this, we first show that W n ii q , v M) /p has a (non-canonical) k-vectorspace structure and find an explicit basis. The dimension isdim fc {W n ü\ ViM) lp) =n-le Y,Pwhere \k : k p \ = p r and e the ramification index of Ä". It is not difficult to see thatthis is an upper bound for the dimension. The proof that it is also a lower bound ismore involved and uses a formula for the de Rham-Witt complex of a polynomialextension by Madsen and the author [19]. We then evaluate the kernel of 1 — Fand compare with the calculation of Ki M (K)/p by Kato [24, 4]. The assumption

Algebraic ÜT-theory and Trace Invariants 419that the residue field k be separably closed is not essential. In the general case, oneinstead has a short-exact sequence0 -+ (W. 0^M) ® Mp.) F=1 -+ K(K)/p* -+ (W. 0^M) /p") F=1 -+ 0,where the superscript (resp. subscript) "F = 1" indicates Frobenius invariants(resp. coinvariants).We discuss a global version of theorem 2.1. Let V u be a henselian discretevaluation ring with quotient field K 0 of characteristic zero and perfect residue fieldfco of odd characteristic p. Let X be a smooth Vp-scheme, and let i (resp. j) denotethe inclusion of the special (resp. generic) fiber as in the cartesian diagramX" >X< -Spec K 0c/> Spec V u

420 L. HesselholtThe canonical map from the de Rham complexnlv,M)Ê q (V\K)is compatible with the trace maps and is an isomorphism, if q < 2. The topologicalDennis trace, again, is far from injective. This can be rectified by a constructionwhich, in retrospect, can be seen as incorporating Witt vectors. The result is thecyclotomic trace of Bökstedt-Hsiang-Madsen [6] which we now recall. The readeris referred to [20, 15, 10] for details.The topological Dennis trace, we recall, is defined as the map of homotopygroups induced from a continuous map of spacesK(C) -+ THH(C).As a consequence of Connes' theory of cyclic sets, the right hand space is equippedwith a continuous action by the circle group T. Moreover, the image of the tracemap is point-wise fixed by the T-action. LetTR"(C;_p)=THH(

Algebraic ÜT-theory and Trace Invariants 421We briefly outline the steps in the proof: We proved in [15] that the sequence0 -• if g (fc,Z/p») -• TR g (fc;p,Z/p») i-4 TR g (fc;p,Z/p») -• 0is exact. This uses [4, 16, 18]. Given this, the theorem by McCarthy [28] thatfor nilpotent extensions, relative ÜT-theory and relative topological cyclic homologyagree, and the continuity results of Suslin [32] for ÜT-theory and Madsen and theauthor [21] for TR show that also the sequence0 -+ K q (V,Z/p v ) -+ TR'(F;p,Z/p») i=4 TR'(F;p,Z/p») -+ 0is exact. Theorem 3.1 follows by comparing the localization sequence of Quillen [30]• • • -+ K q (k,Z/p») A K q (V,Z/p») ^ K q (K,Z/p») -+ • • •to the corresponding sequence by Madsen and the author [20]• • • -+ TR^(fc;p,Z/p») 4 TR n q(V ;p,1/p v ) ^ TR g (F|if;p,Z/p») -+••..Again, the assumption in the statement of theorem 3.1 that the residue field k beseparably closed is not essential. The general statement will be given below. It isalso not necessary for theorem 3.1 to assume that V be of geometric type.4. The Tate spectral sequenceIf G is a finite group and X a G-space, it is usually not possible to evaluatethe groups 7r»(X G ) from knowledge of the G-modules 7r»(X). At first glance, thisis the problem that one faces in evaluating the groupsTR n q(C ] p) =Tx q (TEE(C) c^).However, the mapping fiber of the structure map TR"(C;p) —^ TR" _1 (C;p), it turnsout, is given by the Borei construction ML (C p —i, THH(C)) whose homotopy groupsare the abutment of a (first quadrant) spectral sequenceElt = H s (C p „-i,TEE t (Cj)=> 7r J+t H.(C p »-i,THH(C)).This suggests that the groups TR q (C;p) can be evaluated inductively starting fromthe case n = 1. However, it is generally difficult to carry out the induction step.In addition, the absence of a multiplicative structure makes the spectral sequenceabove difficult to solve. The main vehicle to overcome these problems, first employedby Bökstedt-Madsen in [7], is the following diagram of fiber sequencesH.(C p »-i,THH(C)) >TR n (C;p) »TR^^p)rfH. (C p »-i, THH(C)) • B (C p »-i, THH(C)) • É(C p »-i, THH(C))

422 L. Hesselholttogether with a multiplicative (upper half-plane) spectral sequenceE 2 }t = H- s (C pn -i,TEE t (Cj)^ 7r,+tlî(C p ,.-i,THH(C))starting from the Tate cohomology of the (trivial) C p »-i-module TEE t .(C). Thelower fiber sequence is the Tate sequence; see Greenlees and May [17] or [20]. Infavorable cases, the maps F and F induce isomorphisms of homotopy groups innon-negative degrees. Indeed, this is true in the case at hand (if k is perfect). Thedifferential structure of the spectral sequenceElt = H- s (C pn^,TEE t (y\K,1/pj)^ 7r s+t (H(G pn - 1 ,THH(F|if)),Z/p)was determined in collaboration with lb Madsen [20] in the case where the residuefield fcis perfect. This is the main calculational result of the work reported here. Thefollowing result, for perfect fc, is a rather immediate consequence. The extension tonon-perfect fc is given in [19].Theorem 4.1 Suppose that ß p v c K. Then the canonical map is an isomorphismof pro-abelian groupsW. 0* (VM) ® z S Z/P .(AV) ^ TR;(F|if;p,Z/p»).We can now state the general version of theorem 3.1 which does not requirethat the residue field fc be separably closed. The second tensor factor on the lefthand side in the statement of theorem 4.1 is the symmetric algebra on the Z/p»-module ß p v, which is free of rank one. Spelling out the statement for the group indegree q, we get an isomorphism of pro-abelian groups0 W. 0\- 2^ ® ßp ^ TR'(F|if ;p, Z/p»).In the case of a separably closed residue field, theorem 3.1 idenfies the Frobeniusfixed set of the common pro-abelian group with K q (K,Z/p v ). In the general case,one has instead a short-exact sequence0 _• 0 (W. 0^M2s® np) F=1 -+ K q (K, Z/p») -+ 0 (W. 0\- 2^ ® ßp) F -U 0valid for all integers q. (There is a similar sequence for the topological cyclic homologygroup TCg(V\K;p,Z/p v ) [20] which includes the summand "s = 0" on theleft.) Comparing with the general version of theorem 2.1, we obtain the followingresult promised earlier [20, 14].Theorem 4.2 Suppose that ß p v c K. Then the canonical mapis an isomorphism.if» M (if) ®z S Z/P .(AV) ^ K*(K,Z/p v )

5. Galois descentAlgebraic ÜT-theory and Trace Invariants 423We now assume that the residue field fc be perfect. In homotopy theoreticterms, theorem 4.1 states that the pro-spectrum TR'(V|Ä";p) is equivalent to the( — l)-connected cover of its localization with respect to complex periodic ÜT-theory,see [8]. This suggests the possibility of completely understanding the homotopytype of this pro-spectrum. We expect that this, in turn, is closely related to thefollowing question. Let K be an algebraic closure of K with Galois group GK, andlet V be the integral closure of V in K. (The ring V is a valuation ring with valuegroup the additive group of rational numbers.)Conjecture 5.1 If k is perfect then for all q > 0, the canonical mapT'R-(V\K;p,Q P /Z p ) -+ TR q (V\K;p,Q P /% P f Kbe an isomorphism of pro-abelian groups and that the higher continuous cohomologygroups Hônt (GK,TR^(V\K;p,Qp/Zp)) vanish.It follows from Tate [33] that the groups ü* ont (Gif,TR l g (I / |A > ;p,Q p )) vanishfor i > 0 and q > 0. One may hope that these methods will help shed some lighton the structure of the groups fl'* ont (G r ü-,TR(V|Ä';p,Q p /Z p )). We now describethe structure of these G/f-modules; proofs will appear elsewhere.The group TR q (V\K;p,Qp/Z p ) is divisible, if q > 0, and uniquely divisible,if q > 0 and even. The Tate module T P TR(V|Ä";p) is a free module of rank oneover TR%(V\K;p,Zp), and the canonical map an isomorphism:STR 0 »(v|if ;P ,z p )(4TR?(V1if;p)) ^ TK(V\K;p,Z p )(note that TR^(V\K;p,Q p /Z p ) ^ TR n q(V\R;p,Z p )®%IZ p ). We note the formalanalogy with the results on K*(K) by Suslin [31, 32].The structure of the ring TR^(V|Ä";p,Z p ) = W n (V) is well-understood (unlikethat of W n (Vj): Following Fontaine [13], we let R v be the inverse limit of thediagram V/p 4- V/p 4- • • • with the Frobenius as structure map. This is a perfectF p -algebra and an integrally closed domain whose quotient field is algebraicallyclosed.There is a surjective ring homomorphism 9 n : W(R V ) -» W n (V) whosekernel is a principal ideal. If e = {e(")}„>i is a compatible sequence of primitivep» _1 st roots of unity considered as an element of Ry, and if e n is the unique p"throot of e, then ([e] — l)/([e»] — 1) is a generator. Moreover, as n varies, the maps9 n constitute a map of pro-rings compatible with the Frobenius maps.The Bott element 6 €j „ £ T P TR(V|Ä";p) determined by the sequence e is nota generator (so the statement of theorem 4.1 is not valid for K). Instead there is agenerator a f _ : „ such that 6 €j „ = ([e„] — l)-a ( , n . The structure map of the pro-abeliangroup T p TRi(V|Ä";p) (resp. the Frobenius) takes a f _ : „ to ([e n _i] — l)/([e»] — 1) •a f _ : n-i (resp. to a ei „_i), and the action of the Galois group is given bycr _ / x [e»] - !a t,n ~ X\°~) r cr] _ i 'a t,n,

424 L. Hesselholtwhere \'- GK —^ Aut(^p°°) = Z* is the cyclotomic character.Acknowledgments The research reported here was supported in part by grantsfrom the National Science Foundation and by an Alfred P. Sloan Fellowship.References[i[2:[3;[4;[5;[6;[7;t«:[9[10:[n[12:[13;[14[15[16[ir[18[19A. A. Beilinson, Height pairing between algebraic cycles, FJ-theory, arithmeticand geometry (Moscow, 1984-1986), Lecture Notes in Math., vol. 1289,Springer-Verlag, 1987, 1-25.P. Berthelot, Cohomologie cristalline des schémas de caractéristique p > 0,Lecture Notes in Math., vol. 407, Springer-Verlag, 1974.S. Bloch, Algebraic K-theory and crystalline cohomology, Pubi. Math. I.H.E.S.47 (1977), 187-268.S. Bloch and K. Kato, p-adic etale cohomology, Pubi. Math. IHES 63 (1986),107-152.M. Bökstedt, Topological Hochschild homology, Preprint 1985, UniversitätBielefeld.M. Bökstedt, W.-C. Hsiang, and I. Madsen, The cyclotomic trace and algebraicK-theory of spaces, Invent. Math. Ill (1993), 465-540.M. Bökstedt and I. Madsen, Topological cyclic homology of the integers, K-theory (Strasbourg, 1992), Astérisque, vol. 226, 1994, 57-143.A. K. Bousfield, The localization of spectra with respect to homology, Topology18 (1979), 257-281.K. Dennis, Algebraic K-theory and Hochschild homology, Algebraic FJ-theory,Evanston, IL., 1976 (unpublished lecture).B. I. Dundas and R. McCarthy, Topological Hochschild homology of ring functorsand exact categories, J. Pure Appi. Alg. 109 (1996), 231-294.J. L. Dupont, Algebra of polytopes and homology of flag complexes, Osaka J.Math. 19 (1982), 599-641.J. L. Dupont and C.-H. Sah, Scissors congruences, II, J. Pure Appi. Alg. 25(1982), 159-195.J.-M. Fontaine, Le corps des périodes p-adiques, Périodes p-adiques (Séminairede Bures, 1988), Astérisque, vol. 223, 1994, 59-111.T. Geisser and L. Hesselholt, On the K-theory of a henselian discrete valuationfield with non-perfect residue field, Preprint 2002., Topological cyclic homology of schemes, FJ-theory (Seattle, 1997),Proc. Symp. Pure Math., vol. 67, 1999, 41-87.T. Geisser and M. Levine, The K-theory of fields in characteristic p, Invent.Math. 139 (2000), 459-493.J. P. C. Greenlees and J. P. May, Generalized Tate cohomology, vol. 113, Mem.Amer. Math. Soc, no. 543, 1995.L. Hesselholt, On the p-typical curves in Quillen's K-theory, Acta Math. 177(1997), 1-53.L. Hesselholt and I. Madsen, On the de Rham-Witt complex in mixed characteristic,Preprint 2002.

Algebraic F'-theory and Trace Invariants 425[20] , On the K-theory of local fields, Ann. Math, (to appear).[21] , On the K-theory of finite algebras over Witt vectors of perfect fields,Topology 36 (1997), 29-102.[22] O. Hyodo and K. Kato, Semi-stable reduction and crystalline cohomology withlogarithmic poles, Périodes p-adiques, Astérisque, vol. 223, 1994, 221-268.[23] L. Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Ann. Scient.Éc. Norm. Sup. (4) 12 (1979), 501-661.[24] K. Kato, Galois cohomology of complete discrete valuation fields, Algebraic F'-theory, Part II (Oberwolfach, 1980), Lecture Notes in Math., vol. 967, Springer,Berlin-New York, 1982, 215-238.[25] , Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry,and number theory, Proceedings of the JAMI Inaugural Conference (Baltimore,1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, 191-224.[26] S. Licthenbaum, Values of zeta-functions at non-negative integers, Numbertheory, Lecture Notes in Math., vol. 1068, Springer-Verlag, 1983, 127-138.[27] R. McCarthy, The cyclic homology of an exact category, J. Pure Appi. Alg. 93(1994), 251-296.[28] , Relative algebraic K-theory and topological cyclic homology, ActaMath. 179 (1997), 197-222.[29] J. Milnor, Algebraic K-theory and quadratic forms, Invent. Math. 9 (1970),318-344.[30] D. Quillen, Higher algebraic K-theory I, Algebraic F'-theory I: Higher K-theories (Battello Memorial Inst., Seattle, Washington, 1972), Lecture Notesin Math., vol. 341, Springer-Verlag, 1973.[31] A. A. Suslin, On the K-theory of algebraically closed fields, Invent. Math. 73(1983), 241-245.[32] , On the K-theory of local fields, J. Pure Appi. Alg. 34 (1984), 304-318.[33] J. Tate, p-divisible groups, Proc. conf. local fields (Driebergen, 1966), Springer-Verlag, 1967, 158-183.[34] V. Voevodsky, The Milnor conjecture, Preprint 1996, Max Planck Institut,Bonn.[35] F. Waldhausen, Algebraic K-theory of topological spaces. II, Algebraic topology(Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer-Verlag, 1979, 356-394.[36] , Algebraic K-theory of spaces, Algebraic and geometric topology (NewBrunswick, N.J., 1983), Lecture Notes in Math., vol. 1126, Springer-Verlag,1985, 318-419.

ICM 2002 • Vol. II • 427-436Symplectic Sums andGromov-Witten InvariantsEleny-Nicoleta IoneFAbstractGromov-Witten invariants of a symplectic manifold are a count of holomorphiccurves. We describe a formula expressing the GW invariants of asymplectic sum X#Y in terms of the relative GW invariants of X and V.This formula has several applications to enumerative geometry. As one application,we obtain new relations in the cohomology ring of the moduli space ofcomplex structures on a genus g Riemann surface with n marked points.2000 Mathematics Subject Classification: 57R17, 53D45, 14N35.1. Gromov-Witten invariantsA symplectic structure on a closed smooth manifold X 2N consists of a closed,non-degenerate 2-form OJ. Gromov's idea [8] was that one could obtain informationabout the symplectic structure on X by studying holomorphic curves. For thatone needs to introduce an almost complex structure, which is an endomorphismJ £ End(TX) with J 2 = —Id. Such a J is compatible with u if the bilinearform g(v,w) = OJ(V,JW) defines a Riemannian metric on TX. For a fixed symplecticstructure, the space of compatible almost complex structures is a nonempty,contractible space.One then considers the moduli space of J-holomorphic maps from Riemannsurfaces into X. Constraints are imposed on the maps, requiring the domain tohave a certain form and the image to pass through geometric representatives offixed homology classes in X. When the right number of constraints are chosenthere will be finitely many maps satisfying those constraints; the (oriented) count ofthese maps will give the corresponding Gromov-Witten invariant. In general, thereare several technical difficulties one must overcome to get a well-defined Gromov-Witten invariant. The foundations of this theory began with [8], [24], [25] and havebeen developed since then by the efforts of a large group of mathematicians (see,* Department of Mathematics, University of Wisconsin-Madison, Madison, Wl 53706, USA.E-mail: ionel@math.wisc.edu

428 Eleny-Nicoleta Ionelfor example, the references in [15] and [22]). Here we present a brief overview ofthe technical setup.Consider (X, UJ) a symplectic manifold. For each compatible almost complexstructure J and perturbation v one considers maps / : C —¥ X from a genus gRiemann surface C with n marked points which satisfy the pseudo-holomorphicmap equation df = v and represent a fixed homology class A = [/] £ H 2 (X). Theset of such maps (modulo reparametrizations), together with their limits, forms thecompact space of stable maps M g , n (X,A). For each stable map f : C —¥ X, thedomain determines a point in the Deligne-Mumford moduli space M g , n of genusg Riemann surfaces with n marked points (see also §3). The evaluation at eachmarked point determines a point in X. All together, this gives a natural mapM g , n (X,A)^M g , n xX n .For generic (J,v) the image of this map carries a fundamental homology class[GIA'x,/i, s ,»] which is defined to be the Gromov-Witten invariant of (X,UJ). Thedimension of this homology class, given by an index computation, isdimÄ4 r Sj„(X, A) = 2ci(TX)A + (dimX - 6)(1 - g) + 2n.A cobordism argument shows that the homology class [GWx,/t, s ,n] is independentof generic (J, v) and moreover depends only on the isotopy class of the symplecticform UJ. Frequently, the Gromov-Witten invariant is thought of as a collection ofnumbers obtained by evaluating the homology class [GWx,/t, s ,n] on a basis of thedual cohomology group. For complex algebraic manifolds these symplectic invariantscan also be defined by algebraic geometry, and in important cases the invariantsare the same as the counts of curves that are the subject of classical enumerativealgebraic geometry.The next important question is to find effective ways of computing the GWinvariants. One useful technique is the method of 'splitting the domain'. Anytimewe have a relation in the cohomology of M g , n it pulls back to a relation (sometimestrivial) between the GW invariants of a symplectic manifold X. As an example,suppose that the constraints imposed on the domain of the holomorphic curves areboundary classes in H*(M g: „) (as defined in section 3 below). One then obtainsrecursive relations which relate such GW invariant to invariants of lower degree orgenus. This method was first used by Kontsevich and Ruan-Tian [25] to determinerecursively the genus 0 invariants of the projective spaces P". These recursiverelations follow from the observation that in the Deligne-Mumford space Mo,4 — P 1each boundary class corresponds to a point, and are thus all homologous to eachother.In joint work with Thomas H. Parker, the author established a general formuladescribing the behavior of GW invariants under the operation of 'splitting the target'([14], [15], [16]). Because we work in the context of symplectic manifolds the naturalsplitting of the target is the one associated with the symplectic cut operation andits inverse, the symplectic sum. The next section describes the symplectic sumoperation and the main ingredients entering the sum formula for GW invariants.

Symplectic Sums and Gromov-Witten Invariants 4292. Symplectic sumsThe operation of symplectic sum is defined by gluing along codimension twosubmanifolds (see [7], [21]). Specifically, let X be a symplectic manifold with acodimension two symplectic submanifold V. Given a similar pair (Y, V) with asymplectic identification between the two copies of V and a complex anti-linearisomorphism between the normal bundles NxV and NyV of V in X and in Y wecan form the symplectic sum X#yY.Perhaps it is in more natural to describe the symplectic sum not as a singlemanifold but as a family Z —¥ D over the disk depending on a parameter X £ D.For À 7^ 0 the fibers Z\ are smooth and symplectically isotopie to X#yY while thecentral fiber Z 0 is the singular manifold X Uy Y. In a neighborhood of V the totalspace Z is NxV ® NyV and the fiber Z\ is defined by the equation xy = X wherex and y are coordinates in the normal bundles NxV and NyV — (NxV)*. Thefibration Z —¥ D extends away from V as the disjoint union of X x D and Y x D.Our overall strategy for proving the symplectic sum formula for GW invariants[16] is to relate the pseudo-holomorphic maps into Z\ for À small to pseudoholomorphicmaps into Z 0 - One expects the stable maps into the sum to be pairsof stable maps into the two sides which match in the middle. A sum formula thusrequires a count of stable maps in X that keeps track of how the curves intersectV.So the first step is to construct Gromov-Witten invariants for a symplecticmanifold (X,UJ) relative to a codimension two symplectic submanifold V. Theseinvariants were introduced in a separate paper with Thomas H. Parker [15] andwere designed for use in symplectic sum formulas. Of course, before speaking ofstable maps one must extend the almost complex structure J and the perturbationv to the symplectic sum. To ensure that there is such an extension we require thatthe pair (J, v) be V-compatible. The precise definition is given in section §6 of [15],but in particular for such pairs V is a J-holomorphic submanifold — somethingwhich is not true for generic J. The relative invariant gives counts of stable mapsfor these special V-compatible pairs. Such counts are in general different fromthose associated with the absolute GW invariants described in the first section ofthis note.Restricting to V-compatible pairs has repercussions. Any pseudo-holomorphicmap f : C —¥ V into V then automatically satisfies the pseudo-holomorphic mapequation into X. So for V-compatible (J,v), stable maps may have domain componentswhose image lies entirely in V, so they are far from being transverse to V.Worse, the moduli spaces of such maps can have dimension larger than the dimensionof M g:n (X, A). We circumvent these difficulties by restricting attention to thestable maps which have no components mapped entirely into V. Such 'V-regular'maps intersect V in a finite set of points with multiplicity. After numbering thesepoints, the space of V-regular maps separates into components labeled by vectorss = (si,..., Si), where £ is the number of intersection points and Sk is the multiplicityof the k th intersection point. Each (irreducible) component M.^ (X, A) ofV-regular stable maps is an orbifold; its dimension depends of g, n, A and on thevector of multiplicities s.

430 Eleny-Nicoleta IonelNext key step is to show that the space of V-regular maps carries a fundamentalhomology class. For this we construct an orbifold compactification A4 (X,A),the space of V-stable maps. The relative invariants are then defined in exactly thesame way as the GW invariants. We consider the natural mapMln,,(X, A) -+ M g , n+i x X n x V e . (2.1)The new feature is the last factor (the evaluation at the £ points of contact withV) which allows us to constrain how the images of the maps intersect V. Thus therelative invariants give counts of V-stable maps with constraints on the complexstructure of the domain, the images of the marked points, and the geometry of theintersection with V. There is one more complication: to be useful for a symplecticsum formula, the relative invariant should record the homology class of the curve inX \ V rather than in X. This requires keeping track of some additional homologydata which is intertwined with the intersection data, as explained in [15].We now return to the discussion of the symplectic sum formula. As previouslymentioned, the overall strategy is to relate the pseudo-holomorphic maps intoZ 0 , which are simply maps into X and Y which match along V, with pseudoholomorphicmaps into Z\ for À close to zero. For that we consider sequences ofstable maps into the family Z\ of symplectic sums as the 'neck size' À —¥ 0. Theselimit to maps into the singular manifold Z 0 = X Uy Y. A more careful look revealsseveral features of the limit maps.First of all, if the limit map fi : Co —t Z 0 has no components in V then f 0has matching intersection with V on X and Y side. For such a limit map fi allits intersection points with V are nodes of the domain Co- Ordering this nodeswe obtain a sequence of multiplicities s = (si,..., S() along V. But it turns outthat the squeezing process is not injective in general. For a fixed A^ 0 there are\s\ = si •... • s.( many stable maps into Z\ close to fi-Second, connected curves in Z\ can limit to curves whose restrictions to Xand Y are not connected. For that reason the GW invariant, which counts stablecurves from a connected domain, is not the appropriate invariant for expressing asum formula. Instead one should work with the 'Gromov-Taubes' invariant GT,which counts stable maps from domains that need not be connected. Thus we seeka formula of the general formGTx# v y = GT X * GT Y (2.2)where * is the operation that adds up the ways curves on the X and Y sides matchand are identified with curves in Z\. That necessarily involves keeping track ofthe multiplicities s and the homology classes. It also involves accounting for thelimit maps which have components in V; such maps are not counted by the relativeinvariant and hence do not contribute to the left side of (2.2).Finally, we need to consider limit maps which have components mapped entirelyin V. We deal with that possibility by squeezing the neck not in one region,but several regions. As a result, the formula (2.2) in general has an extra term

Symplectic Sums and Gromov-Witten Invariants 431called the S-matrix which keeps track of how the genus, homology class, and intersectionpoints with V change as the images of stable maps pass through the neckregion. One sees these quantities changing abruptly as the map passes through theneck — the maps are "scattered" by the neck. The scattering occurs when someof the stable maps contributing to the GT invariant of Z\ have components thatlie entirely in V in the limit as À —¥ 0. Those maps are not V-regular, so are notcounted in the relative invariants of X or Y. But this complication can be analyzedand related to the relative invariants of the ruled manifold P(A r x V ® C).Putting all these ingredients together, we can at last state the main result of[16].Theorem 2.1 Let Z be the symplectic sum of (X, V) and (Y, V) and fix a decompositionof the constraints a into ax on the X side and ay on the Y side. Thenthe GT invariant of Z is given in terms of the relative invariants of (X, V) and(Y,V) byGT z (a) = GTx'(ax) * S v * GT%(a Y ) (2.3)where * is the convolution operation and Sv is the S-matrix defined in [16].Several applications of this formula are described in the next two sections (seealso [16] for more applications). But the full strength of the symplectic sum theoremhas not yet been used.A.-M. Li and Y. Ruan also have a sum formula [18]. Eliashberg, Givental, andHofer are developing a general theory for invariants of symplectic manifolds gluedalong contact boundaries [3]. Jun Li has recently adapted our proof to the algebraiccase [19].3. Relations in H*(Aig, n )A smooth genus g curve with n marked points is stable if 2g — 2 + n > 0.The set of such curves, modulo diffeomorphisms, forms the moduli space A4 g , n .The stability condition assures that the group of diffeomorphisms acts with finitestabilizers, and so A4 g , n has a natural orbifold structure. Its Deligne-Mumfordcompactification A4 g , n is a projective variety. Elements of A4 g , n are called stablecurves; these are connected unions of smooth stable components G, joined at ddouble points with a total of n marked points and Euler characteristic \ = 2 — 2g+d.The compactification A4 g , n is also an orbifold, and in fact Looijenga proved thatit has a finite degree cover which is a smooth manifold. In any event, the rationalcohomology of A4 g , n satisfies Poincaré duality. Throughout this section we workonly with rational coefficients.There are several maps between moduli spaces of stable curves. First, there isa projection 7r, : M g: n+i —¥ A4 g , n that forgets the marked point xi (and collapsesthe components that become unstable). Second, we can consider the attaching mapsthat build a boundary stratum in A4 g , n . For each topological type of a stable curvewith d nodes, with components G, of genus gi and n, marked points the attachingmap £ at the d nodes takes UiA4 g^ni onto a boundary stratum of A4 g , n .

432 Eleny-Nicoleta IonelWe focus next on three kinds of natural classes in H*(A4 g , n ) (or the Chowring). For each i between 1 and n let L t —¥ A4 g , n denote the relative cotangentbundle to the stable curve at the marked point xi. The fiber of F, over a point G =(S, xi,..., x n ) £ Mg : „, is the cotangent space to S at x», and its first Chern classipi is called a descendant class. So there are n descendant classes ifi,..., ip n , one foreach marked point. Next, there are tautological (or Mumford-Morita-Miller) classesKo,Ki,... obtained from powers of descendants by the formula K a = (7r n +i)»(^+ì)for each a > 0 (where 7r» denotes the push forward map in cohomology definedusing the Poincaré duality). Finally, the Poincaré dual of a boundary stratum iscalled a boundary class. These three kinds of natural classes are all algebraic andeven dimensional; we define their degree to be their complex dimension.One natural — and difficult — problem is to describe the structure of thecohomology rings of A4 g , n and A4 g , n - This arises from a different perspective as wellsince H*(Mg : „) is also the cohomology of the mapping class group (for more details,see Tillman's I.CM. talk). In genus zero Keel [17] determined the cohomology ringof Mo,n in terms of generators (which are boundary classes) and relations. Forhigher genus far less is known about the cohomology ring.In this section we will instead focus on finding relations in the cohomology ring.For example, in genus 0 all relations come from the "4-point relation", essentiallythat in the cohomology of Mo,4 — P 1 the four ip t classes as well as the threeboundary classes are all cohomologous (all being Poincaré dual to a point). Ingenus 1 it is also known that ifi is equal to 1/12 of the boundary class in A4i,i-One might wonder whether in higher genus all the ip classes come from the boundary.That turns out not to be true in genus g > 2, but in genus 2 Mumford [23] found arelation in A4 2 ,i expressing ip 2 as a combination of boundary classes. Several yearsago, Getzler [6] found a similar relation for ipiip2 in A4 2 , 2 and he conjectured thatthis pattern would continue in higher genus. In fact,Theorem 3.1 When g > 1, any product of descendant or tautological classesof degree at least g (or at least g — 1 when n = 0) vanishes when restricted toH*(M g , n ,Q).This result was proved by the author in [11]. It extends an earlier result ofLooijenga [20], who proved that a product of descendant classes of degree at leastg + n — 1 vanishes in the Chow ring A* (C) of the moduli space C of smooth genusg curves with n not necessarily distinct points.The idea of proof of Theorem 3.1 is simple. We start with the moduli spaceyd,g,n of degree d holomorphic maps from smooth genus g curves with n markedpoints to S 2 which have a fixed ramification pattern over r marked points in thetarget. We then consider its relative stable map compactification yd,g,n (closelyrelatedto the space of admissible covers [9]). The space yd,g,n has an orbispacestructure and it comes with two natural maps st and q that record respectively thedomain and the target of the cover.yd,g,n___V V_ (3.1)Mg,nA4o,r

Symplectic Sums and Gromov-Witten Invariants 433A simple way to get relations in the cohomology of A4 g , n is to pull back by q knownrelations in the cohomology of Mo,r, and then push them forward by st.To begin with, note that the diagram above provides several other naturalclasses in A4 g , n : for each choice of ramification pattern, st*y,i, g ,n defines a cyclein A4g,n- The most useful ones turn out to be the "2-point ramification cycles",for which all but at most two of the branch points are simple. Pushing forwardsuch cycles by the attaching map of a boundary stratum gives a generalized 2-pointcycle.To prove Theorem 3.1, we choose a degree d of the cover and a 2-point ramificationcycle yd,g,n in such a way that the stabilization map st : yd,g,n —* ^g,nhas finite, nonzero degree. The key step is the following proposition.Proposition 3.2 The Poincaré dual of any degree m product of descendant andtautological classes can be written as a linear combination of generalized 2-pointramification cycles of codimension m.But the codimension of a 2-point ramification cycle is at most g. A simpledegeneration argument proves that the cycles of codimension exactly g vanish on•Mg,n, thus implying Theorem 3.1.There are three main ingredients in the proof of Proposition 3.2. First, therelative cotangent bundle to the domain is related to the pullback of the relativecotangent bundle to the target, so we can express the descendant classes in thedomain via descendant classes in the target. Second, the target has genus zero and(nontrivial) products of descendants in A4o, r are Poincaré dual to boundary cyclesD. This means that we can relate a product of descendants on the domain to cyclesof type st*q*D. Finally, a degeneration formula, which is essentially a consequenceof the symplectic sum Theorem 2.1, expresses cycles of type st*q*D in terms of2-point ramification cycles.The degree g in Theorem 3.1 is the lowest degree in which some monomialin descendants would vanish on A4 g , n (see the discussion in [10]). However, thereare lower degree polynomial relations in descendent and tautological classes. Forexample, if we restrict our attention to the moduli space A4 g of smooth genus gcurves then the subring generated by the tautological classes is called the tautologicalring R*. Looijenga's result [20] implies that R* = 0 for * > g — 1 and Faber [4]made the followingConjecture 3.3 The classes Ki,...,K[ s /3] generate the tautological ring R*.We refer the reader to [4] for the full conjecture.It turns out that techniques similar to those of Theorem 3.1 produce severalother sets of relations between tautological classes. One such set of relations impliesthat, for each a > [g/3], the class K a can be written as polynomial in lower degreetautological classes, as required by Faber's conjecture. A detailed proof will appearin [11].4. Further applications

434 Eleny-Nicoleta IonelThere are other applications of the sum formula (2.3). One such applicationconsidered in [16] begins with the following simple observation. Given any symplecticmanifold X with a codimension 2 symplectic submanifold V, we can write X asa (trivial) symplectic sum X#yFy where Py is the ruled manifold W(NxV ® C)and V is identified with its infinity section. We can then obtain recursive formulasfor the GW invariants of X by moving constraints from one side to the other andapplying the symplectic sum formula.In [15] we used this method to obtain both (a) the Caporaso-Harris formula forthe number of nodal curves in P 2 [2], and (b) the "quasimodular form" expression forthe rational enumerative invariants of the rational elliptic surface [1]. In hindsight,our proof of (a) is essentially the same as that in [2]; using the symplectic sumformula makes the proof considerably shorter and more transparent, but the keyideasare the same. Our proof of (b), however, is completely different from that ofBryan and Leung in [1].We end with another interesting application of the Symplectic Sum Theorem2.1. For each symplectomorphism / of a symplectic manifold X, one can form thesymplectic mapping cylinderX f = X x R x S x /Z (4.1)where the Z action is generated by (x, s, 9) H> (f(x), s + 1,9). In a joint paper [13]with T. H. Parker we regarded Xf as a symplectic sum and computed the Gromovinvariants of the manifolds Xf and of fiber sums of the Xf with other symplecticmanifolds. The result is a large set of interesting non-Kähler symplectic manifoldswith computational ways of distinguishing them. In dimension four this gives asymplectic construction of the 'exotic' elliptic surfaces of Fintushel and Stern [5].In higher dimensions it gives many examples of manifolds which are diffeomorphicbut not 'equivalent' as symplectic manifolds.More precisely, fix a symplectomorphism / of a closed symplectic manifold X,and let /»* denote the induced map on flfe(X;Q). Note that Xf fibers over thetorus T 2 with fiber X. If det (i — /*i) = ±1 then there is a well-defined sectionclass T. Our main result of [13] computes the genus one Gromov invariants of themultiples of this section class. These are the particular GW invariants that, indimension four, CH. Taubes related to the Seiberg-Witten invariants (see [27] and[12]).Theorem 4.1 If det (I—f*i) = ±1, the partial Gromov series of Xf for the sectionclass T is given by the Lefschetz zeta function of f in the variable t = tr-'Gr T (X f ) = Cf(t) IL odd det ( j - tf*k)life even det ( J ~ */**) 'When Xf is a four-manifold, a wealth of examples arise from knots. Associatedto each fibered knot K in S 3 is a Riemann surface S and a monodromydiffeomorphism /#- of S. Taking f = f K gives symplectic 4-manifolds XK of thehomology type of S 2 x T 2 withGr(X K )-ÄK{tT)(i^t T y

Symplectic Sums and Gromov-Witten Invariants 435where AK(t) = det(i — i/*i) is the Alexander polynomial of K and T is the sectionclass.We can elaborate on this construction by fiber summing Xf with other 4-manifolds. For example, let E(n) be the simply-connected minimal elliptic surfacewith fiber F and holomorphic Euler characteristic n. Then E(l) is the rationalelliptic surface and K3 = E(2). Forming the fiber sum of XK with E(n) along thetori T = F, we obtain a symplectic manifoldE(n,K) = E(n)# F=T X K .homeomorphic to E(n). In fact, for fibered knots K, K' of the same genus there isa homeomorphism between E(n, K) and E(n, K') preserving the periods of UJ andthe canonical class K. For n > 1 we can compute the full (not just partial) Gromovseries.Proposition 4.2 For n > 2, the Gromov and Seiberg-Witten series of E(n,K)areGr(E(n,Kj) = SW(E(n,Kj) = A K (t F ) (1 - t F ) n - 2 . (4.2)Thus fibered knots with distinct Alexander polynomials give rise to symplecticmanifolds E(n, K) which are homeomorphic but not diffeomorphic. In particular,there are infinitely many distinct symplectic 4-manifolds homeomorphic to E(n).Fintushel and Stern [5] have independently shown how (4.2) follows from knot theoryand results in Seiberg-Witten theory.References[1] J. Bryan and N.-C Leung, The enumerative geometry of K3 surfaces andmodular forms, J. Amer. Math. Soc. 13 (2000), 371-410.[2] L. Caporaso and J. Harris, Counting plane curves in any genus, Invent. Math.131 (1998), 345^392.[3] Y. Eliashberg, A. Givental and H. Hofer, Introduction to Symplectic Field Theory,G AFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, Special Volume,Part II, 560^673.[4] C. Faber, A conjectural description of the tautological ring of the moduli spaceof curves, Moduli of curves and abelian varieties, 109^129, Aspects Math., E33,Vieweg, Braunschweig, 1999.[5] R. Fintushel and R. Stern, Knots, Links and fiManifolds, Invent. Math. 134(1998), 363-100.[6] E. Getzler, Topological recursion relations in genus 2, Integrable systems andalgebraic geometry (Kobe/Kyoto, 1997), 73^106, World Sci. Publishing, RiverEdge, NJ, 1998.[7] R. Gompf, A new construction of symplectic manifolds, Annals of Math., 142(1995), 527^595.[8] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math.82 (1985), 307^347.

436 Eleny-Nicoleta Ionel[9] J. Harris, I. Morrison, Moduli of curves, Graduate Texts in Math, vol 187,Springer-Verlag, 1998.[10] E. Ionel, Topological recursive relations in H 29 (A4 g , n ), to appear in Invent.Math.[11] E. Ionel, On relations in the tautological ring of A4 g , in preparation.[12] E. Ionel and T. H. Parker, The Gromov invariants of Ruan-Tian and Taubes,Math. Res. Lett. 4 (1997), 521^532.[13] E. Ionel and T. H. Parker, Gromov Invariants and Symplectic Maps, Math.Annalen, 314, 127^158 (1999).[14] E. Ionel and T. H. Parker, Gromov-Witten Invariants of Symplectic Sums,announcement, Math. Res. Lett., 5(1998), 563^576.[15] E. Ionel and T. H. Parker, Relative Gromov-Witten Invariants, to appear inAnnals of Math.[16] E. Ionel and T. H. Parker, The Symplectic Sum Formula for Gromov-WittenInvariants, preprint, math.SG/0010217.[17] S. Keel, Intersection theory of moduli space of stable n-pointed curves of genuszero, Trans. Amer. Math. Soc. 330(1992), 545^574.[18] A.-M. Li, Y. Ruan, Symplectic surgery and Gromov-Witten invariants ofCalabi-Yau 3-folds, Invent. Math. 145 (2001), 151-218.[19] Jun Li, A Degeneration formula of GW-invariants, preprint,math.AG/0110113.[20] E. Looijenga, On the tautological ring of A4 g , Invent. Math. 121(1995), 411-419.[21] J. McCarthy and J.Wolfson, Symplectic Normal Connect Sum, Topology, 33(1994) 729^764.[22] D. McDuff and D. Salamon, J-holomorphic curves and quantum cohomology,A.M.S., Providence, R.I., 1994.[23] D. Mumford, Towards an enumerative geometry of the moduli space of curvesin Arithmetic and geometry II (éd. M. Artin and J. Tate), Progress in Math,vol 36, Birkhäuser, Basel, 1983.[24] T. H. Parker and J. Wolfson, Pseudo-holomorphic maps and bubble trees, Jour.Geometric Analysis, 3 (1993) 63^98.[25] Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. DifferentialGeom. 42 (1995), 259^367.[26] Y. Ruan and G. Tian, Higher genus symplectic invariants and sigma modelscoupled with gravity, Invent. Math. 130 (1997), 455^516.[27] C. H. Taubes, Counting pseudo-holomorphic curves in dimension four, J. Diff.Geom. 44 (1996), 818^893.

ICM 2002 • Vol. II • 437-146Knots, von Neumann Signatures,and Grope Cobordism*Peter TeiehnerÂbstractWe explain new developments in classical knot theory in 3 and 4 dimensions,i.e. we study knots in 3-space, up to isotopy as well as up to concordance.In dimension 3 we give a geometric interpretation of the Kontsevich integral(joint with Jim Conant), and in dimension 4 we introduce new concordanceinvariants using von Neumann signatures (joint with Tim Cochran and KentOrr). The common geometric feature of our results is the notion of a gropecobordism.2000 Mathematics Subject Classification: 57M25, 57N70, 46L89.Keywords and Phrases: Knot, Signature, von Neumann algebra, Concordance,Kontsevich integral, Grope.1. IntroductionA lot of fascinating mathematics has been created when successful tools aretransferred from one research area to another. We shall describe two instancesof such transfers, both into knot theory. The first transfer realizes commutatorcalculus of group theory by embedded versions in 3- and 4-space, and producesmany interesting geometric equivalence relations on knots, called grope cobordismin 3-space and grope concordance in 4-space. It turns out that in 3-space thesenew equivalence relations give a geometric interpretation (Theorem 2) of Vassiliev'sfinite type invariants [21] and that the Kontsevich integral [17] calculates the newtheory over Q (Theorem 3).In 4-space the new equivalence relations factor naturally through knot concordance,and in fact they organize all known concordance invariants in a wonderfulmanner (Theorem 5). They also point the way to new concordance invariants (Theorem6) and these are constructed using a second transfer, from the spectral theoryof self-adjoint operators and von Neumann's continuous dimension [20].* Partially supported by an NSF-grant and the Max-Planck Gesellschaft.tUniversity of California in San Diego, 9500 Oilman Drive, La Jolla, CA 92093-0112, USA.E-mail: teichner@math.ucsd.edu

438 P. Teichner1.1. A geometric interpretation of group commutatorsTo explain the first transfer into knot theory, recall that every knot bounds aSeifert surface (embedded in 3-space), but only the trivial knot bounds an embeddeddisk. Thus all of knot theory is created by the difference between a surface and adisk. The new idea is to filter this difference by introducing a concept into knottheory which is the analogue of iterated commutators in group theory. Commutatorsarise because a continuous map : S 1 —¥ X extends to a map of a surface if and onlyif 4> represents a commutator in the fundamental group mX. Iterated commutatorscan similarly be expressed by gluing together several surfaces. Namely, there arecertain finite 2-complexes (built out of iterated surface stages) called gropes by-Cannon [1], with the following defining property: : S 1 —¥ X represents an elementin the fc-th term of the lower central series of 7TiX if and only if it extends to acontinuous map of a grope of class k. Similarly, there are symmetric gropes whichgeometrically implement the derived series of niX, see Figures 2. and 3.Gropes, therefore, are not quite manifolds but the singularities that arise are ofa very simple type, so that these 2-complexes are in some sense the next easiest thingafter surfaces. Two sentences on the history of the use of gropes in mathematics arein place, compare [11, Sec.2.11]. Their inventor Stan'ko worked in high-dimensionaltopology, and so did Edwards and Cannon who developed gropes further. BobEdwards suggested their relevance for topological 4-manifolds, where they wereused extensively, see [11] or [12]. It is this application that seems to have createda certain " Angst " of studying gropes, so we should point out that the only reallydifficultpart in that application is the use of infinite constructions, i.e. when theclass of the grope goes to infinity.One purpose of this note is to convince the reader that (finite) gropes are avery simple, and extremely powerful tool in low-dimensional topology. The pointis that once one can describe iterated commutators in mX by maps of gropes, onemight as well study embedded gropes in 3-space (respectively 4-space) in order toorganize knots up to isotopy (respectively up to concordance). In Section 2. we shallexplain joint work with Jim Conant on how gropes embedded in 3-space lead to ageometric interpretation of Vassiliev's knot invariants [21] and of the Kontsevichintegral [17].1.2. von Neumann signatures and knot concordanceIn Section 3. we study symmetric gropes embedded in 4-space, and explain howthey lead to a geometric framework for all known knot concordance invariants andbeyond. More precisely, we explain our joint work with Tim Cochran and KentOrr [6], where we define new concordance invariants by inductively constructingrepresentations into certain solvable groups G, and associating a hermitian formover the group ring ZG to the knot K, which is derived from the intersection formof a certain 4-manifold with fundamental group G and whose boundary is obtainedby 0-surgery on K. This intersection form represents an element in the Cappell-Shaneson F-group [2] of ZG and we detect it via the second transfer from a differentarea of mathematics: The standard way to detect elements in Witt groups like the

Knots, von Neumann Signatures, and Grope Cobordism 439F-group above is to construct unitary representations of G, and then consider thecorresponding (twisted) signature of the resulting hermitian form over C. It turnsout that the solvable groups G we construct do not have any interesting finitedimensional representations, basically because they are "too big" (e.g. not finitelygenerated),a property that is intrinsic to the groups G in question because theyare "universal solvable" in the sense that many 4-manifold groups (with the rightboundary) must map to G, extending the given map of the knot group.However, every group G has a fundamental unitary representation given by£ 2 G, the Hilbert space of square summable sequences of group elements with complexcoefficients. The resulting (weak) completion of CG is the group von Neumannalgebra AfG. It is of type Ifi because the map Y^ a g g >-¥ cti extends from CG togive a finite faithful trace on AfG.The punchline is that hermitian forms over the completion AfG are mucheasier to understand than over CG because they are diagonalizable (by functionalcalculus of self-adjoint operators). Here one really uses the von Neumann algebra,rather than the G*-algebra completion of CG because the functional calculus mustbe applied to the characteristic functions of the positive (respectively negative) realnumbers, which are bounded but not continuous.The subspace on which the hermitian form is positive (respectively negative)definite has a continuous dimension, which is the positive real number given by thetrace of the projection onto that subspace. As a consequence, one can associate toevery hermitian form over AfG a real valued invariant, the von Neumann signature.In [6] we use this invariant to construct our new knot concordance invariants, anda survey of this work can be found in Section 3. It is not only related to embeddedgropes in 4-space but also to the existence of towers of Whitney disks in 4-space.Unfortunately, we won't be able to explain this aspect of the theory, but see [6,Thm.8.12].1.3. Noneommutative Alexander modulesIn Section 3. we shall hint at how the interesting representations to our solvablegroups are obtained. But it is well worth pointing out that the methods developedfor studying knot concordance have much simpler counterparts in 3-space, i.e. if oneis only interested in isotopy invariants.A typical list of knot invariants that might find its way into a text bookor survey talk on classical knot theory, would contain the Alexander polynomial,(twisted) signatures, (twisted) Arf invariants, and maybe knot determinants. Itturns out that all of these invariants can be computed from the homology of theinfinite cyclic covering of the knot complement, and are in this sense "commutative"invariants.Instead of the maximal abelian quotient one can use other solvable quotientgroups of the knot group to obtain "noneommutative" knot invariants. The canonicalcandidates are the quotient groups G n of the derived series G^ of the knotgroup (compare Section 3. for the definition). One can thus define the higher order

440 P. TeichnerAlexander modules of a knot K to be the ZG„ + i-modulesA n (K) := Hi(S 3 \ K;ZG n+ i).The indexing is chosen so that Ao is the classical Alexander module. For n > 1these modules are best studied by introducing further algebraic tools as follows. Bya result of Strebel the groups G n are torsionfree. Therefore, the group ring ZG nsatisfies the Ore condition and has a well defined (skew) quotient field. This field isin fact the quotient field of a (skew) polynomial ring K^f^1],with K„ the quotientfield of ZfGW/G*"}] and Gi = (t) = Z. Thus one is exactly in the context of [6,Sec.2] and one can define explicit noneommutative isotopy invariants of knots. Forexample, let d n (K) be the dimension (over the field K„+i ) of the rational AlexandermoduleA(F')£gi Z G„ +1 K„ + i[t ±1 ].It is shown in [6, Prop.2.11] that these dimensions are finite with the degree of theusual Alexander polynomial being do(K). Moreover, Cochran [5] has proven thefollowing non-triviality result for these dimensions.TheoremIf K is a nontrivial knot then for n > 1 one hasdo(K)

Knots, von Neumann Signatures, and Grope Cobordism 441group Z. Moreover, this signature is the integral, over the circle, of all (Levine-Tristram) twisted signatures of the knot [7, Prop.5.1] (and is thus a concordanceinvariant). For n > 1 there is in general no such 4-manifold available and thecorresponding ^-invariants are not concordance invariants.2. Grope cobordism in 3-spaceWe first give a more precise treatment of the first transfer from group theoryto knot theory hinted at in the introduction. Recall that the fundamental groupconsists of continuous maps of the circle S 1 into some target space X, modulohomotopy (i.e. 1-parameter families of continuous maps). Quite analogously, classicalknot theory studies smooth embeddings of a circle into S 3 , modulo isotopy(i.e. 1-parameter families of embeddings). To explain the transfer, we recall thata continuous map : S 1 —¥ X represents the trivial element in the fundamentalgroup 7TiX if and only if it extends to a map of the disk, represents a commutator in mX if and only if it extends to a map of a surface(i.e. of a compact oriented 2-manifold with boundary S 1 ). The first statement hasa straightforward analogy in knot theory: K : S 1 ^y S 3 is trivial if and only if it extendsto an embedding of the disk into S 3 . However, every knot "is a commutator"in the sense that it bounds a Seifert surface, i.e. an embedded surface in S 3 .Figure 1: Gropes of class 3, with one respectively two boundary circlesRecall from the introduction that gropes are finite 2-complexes defined by thefollowing property: : S 1 —¥ X represents an element in the fc-th term niXk of thelower central series of 7TiX if and only if it extends to a continuous map of a grope ofclass k. Here Gk is defined inductively for a group G by the iterated commutatorsG 2 := [G,G] and G k := [G,G k -i] for k > 2.Accordingly, a grope of class 2 is just a surface, and one can obtain a grope of class kby attaching gropes of class (k — 1) to g disjointly embedded curves in the bottomsurface. Here g is the genus of the bottom surface and the curves are assumed tospan one half of its homology. This gives gropes of class k with one boundary circleas on the left of Figure 2. It's not the most general way to get gropes because of

442 P. Teichnerre-bracketing issues, and we refer to [9, Sec.2.1] for details. The boundary of a gropeis by definition just the boundary of the bottom surface, compare Figure 2.Definition 1 Two (smooth oriented) knots in S 3 are grope cobordant of class k,if there is an embedded grope of class k in S 3 (the grope cobordism J such that itsboundary consists exactly of the given knots.An embedding of a grope is best defined via the obvious 3-dimensional localmodel. Since every grope has a 1-dimensional spine, embedded gropes can then beisotoped into the neighborhood of a 1-complex. As a consequence, embedded gropesabound in 3-space! It is important to point out that if two knots Ki cobounds agrope then Ki and FJ 2 might very well be linked in a nontrivial way. Thus it is amuch stronger condition on K to assume that it is the boundary of an embeddedgrope than to say that it cobounds a grope with the unknot. For example, if Kbounds an embedded grope of class 3 in S 3 then the Alexander polynomial vanishes.Together with Stavros Garoufalidis, we recently showed [13] that the 2-loop termof the Kontsevich integral detects many counterexamples to the converse of thisstatement.In joint work with Jim Conant [9], we show that grope cobordism definesequivalence relations on knots, one for every class k £ N. Moreover, Theorem 2below implies that the resulting quotients are in fact finitely generated abeliangroups (under the connected sum operation). For the smallest values k = 2,3,4and 5, these groups are isomorphic to{0},Z/2,Zand Z x Z/2and they are detected by the first two Vassiliev invariants [10, Thm.4.2].The following theorem is formulated in terms of clasper surgery which wasintroduced independently by Habiro [15] and Goussarov [14], as a geometric answerto finite type invariants a la Vassiliev [21]. We cannot explain the definitions herebut see [9, Thm.l and 3]. We should say that the notion of a capped grope is wellknown in 4 dimensions, see [11, Sec.2]. In our context, it means that all circles atthe "tips" of the grope bound disjointly embedded disks in 3-space which are onlyallowedto intersect the bottom surface of the grope.Theorem 2 Two knots K 0 and Ki are grope cobordant of class k if and only if Kican be obtained from K 0 by a finite sequence of clasper surgeries of grope degree k(as defined below).Moreover, two knots are capped grope cobordant of class k if and only if theyhave the same finite type invariants of Vassiliev degree < k.As a consequence of this result, the invariants associated to grope cobordismare highly nontrivial as well as manageable. For example, we prove the followingresult in [10, Thm. 1.1]:Theorem 3 The (logarithm of the) Kontsevich integral (with values in B B

Knots, von Neumann Signatures, and Grope Cobordism 443Here B a

444 P. TeichnerIn the following definition we attempt to distinguish the terms "cobordant"and "concordant" in the sense that the latter refers to 4 dimensions, whereas theformer was used in dimension 3, see Definition 1. Historically, these terms were usedinterchangeably, but we hope not to create any confusion with our new distinction.Definition 4 Two oriented knots in S 3 are grope concordant of height h £ |N, ifthere is an embedded symmetric grope of height h in S 3 x [0,1] such that its boundaryconsists exactly of the given knots K t : S 1 ^y S 3 x {i}.Observe that since an annulus is a symmetric grope of arbitrary height weindeed get a filtration of the knot concordance group. This group is defined byidentifyingtwo knots which cobound an embedded annulus in S 3 x [0,1], where thereare two theories depending on whether the embedding is smooth or just topological(and locally flat). For grope concordance the smaller topological knot concordancegroup is the more natural setting: a locally flat topological embedding of a grope(defined by the obvious local model at the singular points) can be perturbed tobecome smooth. This perturbation might introduce many self-intersection pointsin the surface stages of the grope. However, these new singularities are arbitrarilysmalland thus they can be removed at the expense of increasing the genus of thesurface stage in question but without changing the height of the grope.In joint work with Tim Cochran and Kent Orr [6], we showed that all knownknot concordance invariants fit beautifully into the scheme of grope concordance!In particular, all known invariants turned out to already be invariants of gropeconcordance of height 3.5:Theorem 5 Consider two knots K t in S 3 .Then1. Ki have the same Arf invariant if and only if they are grope concordant ofheight 1.5 (or class 3).2. Ki are algebraically concordant in the sense of Levine [19] (i.e. all twistedsignatures and twisted Arf invariants agree) if and only if they are grope concordantof height 2.5.3. If Ki are grope concordant of height 3.5 then they have the same Casson-Gordon invariants [3].The third statement includes the generalizations of Casson-Gordon invariantsby Gilmer, Kirk-Livingston, and Letsche. In [6] we prove an even stronger versionof the third part of Theorem 5. Namely, we give a weaker condition for a knotK to have vanishing Casson-Gordon invariants: it suffices that K is (1.5) -solvable.All the obstruction theory in [6] is based on the definition of (h)-solvable knots,h £ |N, which we shall not give here. Suffice it to say that this definition is closerto the algebraic topology of 4-manifolds than grope concordance. In [6, Thm.8.11]we show that a knot which bounds an embedded grope of height (h + 2) in D 4 is(ft)-solvable.It should come as no surprise that the invariants which detect grope concordancehave to do with solvable quotients of the knot group. In fact, the aboveinvariants are all obtained by studying the Witt class of the intersection form of a

Knots, von Neumann Signatures, and Grope Cobordism 445certain 4-manifold M 4 whose boundary is obtained by 0-surgery on the knot. Thedifferent cases are distinguished by the fundamental group m AI, namely1. m M is trivial for the Arf invariant,2. mM is infinite cyclic for algebraic concordance, and3. m AI is a dihedral group for Casson-Gordon invariants.So the previously known concordance invariants stopped at solvable groups whichare extensions of abelian by abelian groups. To proceed further in the understandingof grope (and knot) concordance, one must be able to handle more complicatedsolvable groups. A program for that purpose was developed in [6] by giving an elaborateboot strap argument to construct inductively representations of knot groupsinto certain universal solvable groups. On the way, we introduced Blanchfield dualitypairings in solvable covers of the knot complement by using noneommutativelocalizations of the group rings in question.The main idea of the boot strap is that a particular choice of "vanishing" of theprevious invariant defines the map into the next solvable group (and hence the nextinvariant). In terms of gropes this can be expressed quite nicely as follows: picka grope concordance of height h £ N and use it to construct a certain 4-manifoldwhose intersection form gives an obstruction to being able to extend that grope toheight h.5. There is an obvious technical problem in such an approach, alreadypresentin [3]: to show that there is no grope concordance of height h.5, one needsto prove non-triviality of the obstruction for all possible gropes of height h. Oneway around this problem is to construct examples where the grope concordances ofsmall height are in some sense unique. This was done successfully in [6] for the levelabove Casson-Gordon invariants, and in [7] we even obtain the following infinitegeneration result. Let Qu be the graded quotient groups of knots, grope concordantof height h to the unknot, modulo grope concordance of height h.5. Then the resultsof Levine and Casson-Gordon show that Q 2 and Qz are not finitely generated.Theorem 6 Q4 is not finitely generated.The easiest example of a non-slice knot with vanishing Casson-Gordon invariantsis given in [6, Fig.6.5]. As explained in the introduction, the last step inthe proof of Theorem 6 is to show that the intersection form of the 4-manifold inquestion is nontrivial in a certain Witt group. Our new tool is the von Neumannsignature which has the additional bonus that it takes values in R, which is notfinitely generated as an abelian group. This fact makes the above result tractable.We cannot review any aspect of the von Neumann signature here, but see [6, Sec.5].Last but not least, it should be mentioned that we now know that for everyh £ N the groups Qu are nontrivial. This work in progress [8] uses as the mainadditional input the Cheeger-Gromov estimate for von Neumann ^-invariants [4] inorder to get around the technical problem mentioned above. It is very likely thatnon of the groups Qu, h £ ~H,h > 2, are finitely generated.

446 P. TeichnerReferences[i[2[3;[4;[5p;[rt«:[9[io;[n[12:[is;[w;[15[16[ir[18[19[2o;[21J. W. Cannon, The recognition problem: what is a topological manifold? Bull.AMS 84 (1978) 832^866.S. Cappell and J. L. Shaneson, The codimension two placement problem andhomology equivalent manifolds, Annals of Math. 99 (1974) 227^348.A. J. Casson and C. McA. Gordon, Cobordism of classical knots, Orsay notes1975, published in A la recherche de la topologie perdue, ed. Guilllou and Marin,Progress in Math. 1986.J. Cheeger and M. Gromov, Bounds on the von Neumann dimension of L 2 -cohomology and the Gauss-Bonnet Theorem for open manifolds, J. Diff. Geometry21 (1985) 1-34.T. Cochran, Noneommutative Knot theory, math.GT/02.T. Cochran, K. Orr and P. Teichner, Knot concordance, Whitney Towers, andL 2 -signatures, math.GT/9908117, to appear in the Annals of Math.T. Cochran, K. Orr and P. Teichner, Structure in the classical knot concordancegroup, math.GT/0206059.T. Cochran and P. Teichner, Von Neumann n-invariants and grope concordance,in preparation.J. Conant and P. Teichner, Grope cobordism of classical knots,math.GT/0012118.J. Conant and P. Teichner, Grope cobordism and Feynman diagrams,math.GT/02.M. Freedman and F. Quinn, The topology of fimanifolds, Princeton Math.Series 39, Princeton, NJ, 1990.M. Freedman and P. Teichner, fiManifold topology I+II, Inventiones Math. 122(1995) 509^529 and 531^557.S. Garoufalidis and P. Teichner, On Knots with trivial Alexander polynomial,math.GT/0206023.M. Goussarov, Knotted graphs and a geometrical technique of n-equivalence,St. Petersburg Math. J. 12-1 (2001).K. Habiro, Claspers and finite type invariants of links, Geometry and Topology,vol 4 (2000) 1-83.S. Harvey, Higher Order Polynomial Invariants of 3-manifolds giving lowerbound for the Thurston Norm, preprint, Rice University 2001.M. Kontsevich, Vassiliev's knot invariants, Adv. in Sov. Math. 16(2) (1993)137^150.P. Kronheimer, Minimal genus in S 1 x AT 3 , Inventiones Math. 134 (1998) 363^400.J. P. Levine, Knot cobordism groups in codimension two, Comm. Math. Helv.44 (1969) 229^244.F.J. Murray and J. von Neumann, On Rings of Operators, Annals of Math. 37(1936) 116^229.V. A. Vassiliev, Cohomology of Knot Spaces, Theory of Singularities and itsApplications, ed. V. I. Arnold, AMS, Providence 1990.

ICM 2002 • Vol. II • 447-156Strings and the Stable Cohomologyof Mapping Class GroupsUlrike Tillmann*2000 Mathematics Subject Classification: 57R20, 55P47, 32G15, 81T40.Keywords and Phrases: Elliptic cohomology, Cohomology of moduli spaces, Infiniteloop spaces, Cobordism theory.1. IntroductionTwenty years ago, Mumford initiated the systematic study of the cohomologyring of moduli spaces of Riemann surfaces. Around the same time, Harer provedthat the homology of the mapping class groups of oriented surfaces is independent ofthe genus in low degrees, increasing with the genus. The (co)homology of mappingclass groups thus stabelizes. At least rationally, the mapping class groups have thesame (co)homology as the corresponding moduli spaces. This prompted Mumfordto conjecture that the stable rational cohomology of moduli spaces is generatedby certain tautological classes that he defines. Much of the recent interest in thissubject is motivated by mathematical physics and, in particular, by string theory.The study of the category of strings led to the discovery of an infinite loop space,the cohomology of which is the stable cohomology of the mapping class groups. Weexplain here a homotopy theoretic approach to Mumford's conjecture based on thisfact. As byproducts infinite families of torsion classes in the stable cohomology aredetected, and the divisibility of the tautological classes is determined. An analysisof the category of strings in a background space leads to the formulization of aparametrized version of Mumford's conjecture.The paper is chiefly a summary of the author's work and her collaboration withlb Madsen. Earlier this year Madsen and Weiss announced a solution of Mumford'sconjecture. We touch on some of the consequences and the ideas behind this mostexciting new developement.*Math. Inst., 24-29 St. Giles, Oxford OX1 3LB, UK. E-mail: tillmann@maths.ox.ac.uk

448 U. Tillmann2. Mumford's conjectureLet Fg be an oriented, connected surface of genus g with s marked pointsand n boundary components. Let Diïï(Fg ) be its group of orientaton preservingdiffeomorphisms that fix the n boundary components pointwise and permute the smarked points. By [2], for genus at least 2, Diïï(Fg ) is homotopic to its groupof components, the mapping class group r* . Furthermore, if the surface hasboundary, r* acts freely on Teichmüller space. Hence,B mS(Fl n ) ~ BT' gin ~ M' Sin for n > l,g > 2,where A4 s gn denotes the moduli space of Riemann surfaces appropriately marked.When n = 0, the action of the mapping class group on Teichmüller space has finitestabilizer groups and the latter is only a rational equivalence.We recall Harer's homology stability theorem [4] which plays an importantrole through out the paper.Harer Stability Theorem 2.1. H*BY s g^n is independent of g and n in degrees3* < g — 1-Ivanov [5], [6] improved the stability range to 2* < g — 1 and proved a versionwith twisted coefficients. Glueing a torus with two boundary components to asurface F Sj i induces a homomorphism F Sj i —t Y g+ i : i. Let F œ := limôo F Sj i bethe stable mapping class group.Mumford [12] introduced certain tautological classes in the cohomology of themoduli spaces A4 g . Topological analogues were studied by Miller [10] and Morita[11]: Let E be the universal F-bundle over B Diff(F), and let T V E be its verticaltangent bundle with Euler class e £ H 2 E. Definef e i+1 £ H 2i B Y)ïS(F).JFHere J F denotes "integration over the fiber" - the Gysin map. Miller and Moritashowed independently that the rational cohomology of the stable mapping classgroup contains the polynomial ring on the K,.Mumford Conjecture 2.2. H^BY^; Q) ~ Q[KI ,K 2 ,...].2.1. Remark.The stable cohomology of the decorated mapping class groups is known moduloH*BY 00 because of decoupling [1]. For example, let F^ := limY s gl. The followingis a consequence of Theorem 2.1.Proposition 2.3. (BF^)+ ~ OT+ x B(E S I S v )+.Here Y + denotes Quillen's plus-construction on Y with respect to the maximalperfect subgroup of the fundamental group. It is important to note that the plusconstruction does not change the (co)homology. In particular,ff*Br œ = H*BY+ .

Strings and the Stable Cohomology of Mapping Class Groups 4493. String categoryThe category underlying the quantum mechanics of a state space X is thepath category VX. Its objects are particles represented by points in X. As timeevolves a particle sweeps out a path. Thus a morphism between particles a and 6is a continuous path in X starting at a and ending at 6. Concatenation of pathsdefines the composition in the category.{objects : a, b, • • • £ X,morphisms :TT map([0, t],X).In string theory, the point objects are replaced by closed loops in X. As timeevolves these strings sweep out a surface. Thus the space of morphisms from onestring to another is now described by a continuous map from an oriented surface Fto X. The parametrization of the path should be immaterial. To reflect this, takehomotopy orbits under the action of Diff(F). 1 Composition is given by concatenationof paths, i.e. by glueing of surfaces along outgoing and incoming boundarycomponents.To be more precise, let LX = map(S' 1 ,X) denote the free loop space on X.A cobordism F is a finite uniont>o^gi,m+mi U • • • U r gkt n k -^mkwhere n = £,«,. boundary components are considered incoming and m = Ejro,outgoing. For technical reasons we will assume m, > 0. The category of strings inX is thenSX=

450 U. Tillmann3.2. Conformai field theory.The category S := S(*) is studied in conformai field theory [15]. Its objectsare the natural numbers and its morphims are Riemann surfaces. A conformai fieldtheory (CFT) is a linear space H which is an algebra over S. Thus each element inMg : n+m, defines a linear map from H®" to H® TO . The physical states of a topologicalconformai field theory (TCFT) form a graded vector space A». Each element of thehomology H*A4 g , n +m defines a linear map from Af n to Af m .3.3. Gromov-Witten theory.Let X be a symplectic manifold. A model for the homotopy orbit spaces in thedefinition of SX is the fiber bundle map(F Sj „,X) —t M g: „(X) —t A4 g , n over theRiemann moduli space. In each fiber, F comes equipped with a complex structureand we may replace the continuous maps by the space of pseudo-holomorphic mapshol(F^,X) yielding a category S hol X. This is the category relevant to Gromov-Witten theory. Note, for X a complex Grassmannian, a generalized flag manifold, ora loop group, the degree d-component of hol(F s ,X) approximates the componentsof map(F s ,X) in homology. The categories SX and S hol X are therefore closelyrelated.4. From categories to infinite loop spacesThere is a functorial way to associate to a category C a topological space \C\, therealization of its nerve. It takes equivalences of categories to homotopy equivalences.It is a generalization of the classifying space construction of a group: \G\ = BGwhere G is identified with the category of a single object and endomorphism setG. The path-category VX of a connected space X is a many object group up tohomotopy. The underlying "group" is the space OX = map»(S' 1 ,X) of based loopsin X. Again one has \VX\ ~ B(iiX) ~ X. A functor from VX to n-dimensionalvector spaces and their isomorphisms thus defines a mapX —• BGL n C,and hence an element in the FJ-theory of X. Motivated by this, we would liketo understand the classifying space of the string category SX and its relation toelliptic cohomology.Definition 4.1.St(X) := Ü\SX\.Theorem 4.2. St is a homotopy functor from the category of topological spaces tothe category of infinite loop spaces.We recall that Z is an infinite loop space if it is homotopic to some Z 0 such thatsuccessive based spaces Z t can be found with homeomorphisms 7, : Z t ~ ÛZ i+ i.Any infinite loop space Z gives rise to a generalized homology theory ft» whichevaluated on a space Y ish*Y := 7T» lim tt l (Zi A Y).

Strings and the Stable Cohomology of Mapping Class Groups 451Infinite loop spaces are abelian groups up to homotopy in the strongest sense.The proof of Theorem 4.2 can be sketched as follows, compare [16]. SX is asymmetric monoidal category under disjoint union. Infinite loop space machinery(see for example [13]) implies that its classifying space \SX\ is a homotopy abelianmonoid in the strongest sense. But 7To|o with A4(n) = J_L> 0 BY'g : „ + i be the operad contained in the CFTcategory S. A space X is an algebra over A4 if there are compatible maps A4 (n) xX" —t X. In particular X has a monoid structure. Let QX be its group completion.QX is homotopic to X if and only if TTOX is a group.Theorem 5.1. ([18]) If X is an algebra over A4 then its group completion QX isan infinite loop space.CFT is therefore closely linked to the theory of infinite loop spaces. Thecrucial point of the proof is a decoupling result similar to Proposition 2.3. Thecorresponding statement for TCFT's implies that Getzler's Batalin-Vilkovisky algebrastructure on the physical states A» is stably trivial, see [17]. The followingexamples illustrate the strength of Theorem 5.1.Example 5.1.Let Xi be the disjoint union IL>o BTg,i- ^ nas a P r °duct induced by glueingF g: i and Fu : i to the "legs" of a pairs of pants surface F 0j 3. Miller observes in [10] thatthis induces a double loop space structure on the group completion QXi ~Zx FF+.This product extends to an Af-algebra structure. Hence Miller's double loop spacestructure extends to an infinite loop space structure. Wahl [19] proved that it isequivalent to the infinite loop spaces structure implied by Corollary 4.3.Example 5.2.Let X 2 be the disjoint union of the Borei constructionsE g := E Diff(F Sjl ) x Differ) map(F Sjl ,d;X,*).As the functions to X map the boundary to a point, they can be extended fromF g: i to F g+ i : i via the constant map. X 2 thus becomes an Af-algebra and QX 2 =Z x (linij-s-oo Eg) + is an infinite loop space. QX 2 is homotopic to St(X) when Xis simply connected.

452 U. TillmannExample 5.3.Let C(Fg : i;X) denote the space of unordered configurations in the interior ofF g: i with labels in a connected space X. Let C g be its Borei construction. Theirdisjoint union defines an Af-algebra X 3 . The following decoupling result determinesits group completion.Proposition 5.2. QX 3 ~Zx (lim^^ C g )+ ~Zx BY+, x Q(BS 1 A X+).Q = limQ 00^00 is the free infinite loop space functor and X + denotes X witha disjoint basepoint. Note the close relation with the above example. By work ofMcDuff and Bödigheimer, there is a homotopy equivalenceC(F 9il ;X)~ mar>(F gA ,d;S 2 AX).Note though that the induced Diff(F Sj i)-action from the left on the right is nontrivialon the sphere in the target space.6. Refinement of Mumford's conjectureInfinite loop spaces are relatively rare and the question arises whether Z x FF+can be understood in terms of well-known infinite loop spaces. This question wasaddressed in joint work with Madsen.Let P' be the Grassmannian of oriented 2-planes in rf+ 2 and let —Li be thecomplement of the canonical 2-plane bundle L over P'. The one-point compactification,the Thom space Th(—Lj), restricts on the subspace P' _1 to the suspensionof Th(—L/_i). Taking adjoints yields maps Th(—L/_i) —t QTh(—Lj), and we maydefineMore generally, for any space X, defineQ°°Th(^F) := lim Q'TIì(^FJ)-n°°(Th(-L)AX + ) := lim ü l (Th(-L t )/—»ooA X+).Conjecture 6.1. There is a homotopy equivalence of infinite loop spacea : St(X) ^Q°°(Tft(^F) A X+).Remarks 6.2.For X = *, Conjecture 6.1 postulates a homotopy equivalence a : Z x FF+ —tQ°°Th(^F). A proof of this has been announced by Madsen and Weiss, see Section 8.The Mumford conjecture follows from this as we will explain presently. Conjecture6.1 claims in addition that St(_) is a homology functor, i.e. 7r» St(_) is the homologytheory associated to the infinite loop space Q°°Th(^F).

Strings and the Stable Cohomology of Mapping Class Groups 453The infinite loop space Q°°Th(^F) is well-studied, more recently because ofits relation to Waldhausen F'-theory. The inclusion of —Li into the trivial bundleF © (-Li) ~ P' x rf+ 2 induces a mapw : 0°°Th(^F) —• Q(P~).UJ has homotopy fibre ii 2 Q(S°). As the stable homotopy groups of the sphereare torsion in positive dimensions, w is a rational equivalence. Let P°° —¥ BUbe the map that classifies F. By Bott periodicity, the map can be extended tothe free infinite loop space F : Qo(V°°) —¥ BU. The subscript here indicates the0-component. F has a splitting and is well known to be a rational equivalence:F*(ng°Th(-L);Q!)'"*-'* Q[ci,C2,...].The Z/p-homology of Q°°Th(^F) has recently been determined by Galatius [3].Figure 1: Surface h(F) C rf +2 with tubular neighborhood U.To define a map a comes down to defining maps from the morphism spacesof SX. (See also Example 5.2.) a is the homotopy theoretic interpretation of theformula defining K, where the wrong way map J F is replaced by the (pre)transfermap of Becker and Gottlieb. We give now an explicid description of this map.For simplicity, let F be a closed surface. Consider the space of smooth embeddingsEmb(F, rf+ 2 ). By Whitney's embedding theorem, as I —¥ oo, it may serveas a model for F Diff(F). Let(ft,/) £ Emb(F,R'+ 2 )x Difr(F) map(F,X).Choose a tubular neighborhood U of h(F) such that every t £ U can uniquely bewritten as x + v with x £ h(F) and v normal to the tangent plane T x h(F). a sends(ft, /) to the continuous function a(h, f) : S l+2 —¥ Th(^Fj) A X + defined byt H>oo if t $ U,((T x h(F),v),f(h- 1 (xj) if t = x + v£ U.In [8], a is shown to be a 3-connected map of infinite loop spaces and thetautological classes are identified. Let il(chi) £ H 2t BU denote the z-th integralChern character class. ThenKi = a* o UJ* o L*(i\chi).

454 U. Tillmann7. Splittings and (co) homological resultsThe main result of [8] is a partial splitting of the compositionUJ o a : Z x FF+ —• Q(P°?) ~ Q(S°)x Q(P°°).This is achieved by constructing a map ß from P + to Z x FF+ and then extendingit to the free infinite loop space Q(W°^) utilizing the infinite loop space structureon Z x FF+. In order to construct ß, approximate P°° ~ BS 1 by the classifyingspaces of cyclic groups Cp» for n —¥ oo, one prime p at a time, as the cyclic groupscan be mapped into suitable mapping class groups. However, this means that wehave to work with p-completions.Let Yp A denote the p-completion of Y and g £ Z p be a topological generatorof the p-adic units (g = 3 if p = 2). Denote by fi k : P°° —¥ (V°°) p the map thatrepresents k times the first Chern class in H 2 (W°°,Z P ).Theorem 7.1. [8]. There exists a map ß : (Q(S°) x Q(P°°))£ -• (Z x FF+ )£such thatÜ O Ö °^("o 2 l-gi/j*The map 1 — gfi 9 induces multiplication by 1 — g n+1 on H 2n (^°°;Z^) whichis a p-adic unit precisely if n ^ —1( mod p — 1). The following applications ofTheorem 7.1 are also found in [8]. There is a splitting Q(W°^) p ~ F 0 x • • • x F p _ 2corresponding to the idempotent decomposition of Z[Z/p x ] c Z P [Z*].Corollary 7.2. For some W p , there is a splittingof infinite loop spaces(Z x FF+ )£ ~ F 0 x • • • x F p _ 3 x W p .The Z/p-homology of Q(P+) is well-understood in terms of Dyer-Lashof operation.These are homology operations for infinite loop spaces that are formallysimilarto the Steenrod operations. For each generator a, £ H 2 i^°° = Z there isan infinite family of Z/p-homology classes freely generated by the Dyer-Lashof operations.The product F 0 x • • • x F p _ 3 contains precisely those families for whichi ^ ^l(mod p — 1), giving a huge collection of new p-torsion in H^BY^.For odd primes p, Madsen and Schlichtkrull [MS] found split surjective mapslo and l_i of infinite loop spaces such that the following diagram is commutativeQ°°Th(^F)£ — ^Q(P^)£19V , in, ,, nrnA(Z x BU)$ — ^ (Z x BU)'

Strings and the Stable Cohomology of Mapping Class Groups 455Corollary 7.3. For odd primes p and some space V p , there is a splitting of spaces(FF+ )£ ~ BU£ x V p .This gives a Z p -integral version of Miller and Morita's theorem: the polynomialalgebra Z p [ci,c 2 ,...] is a split summand of H*(BY 00 ; Z p ). The divisibility of thetautological classes K, at odd primes p can also be deduced from the above diagram.Corollary 7.4. If i = —l(modp— 1), then K, is divisible by p 1 +"p(*+ 1 ) where v p isthe p-adic valuation. Otherwise, p does not divide K,.In the light of [9] this result is sharp.8. Geometric interpretationa : FF+ —t Qg°Th(^F) is a homotopy equivalence if and only if it inducesan isomorphism in oriented cobordism theory Qf°. An element in Q|°(FF+) =0|°(FF 0O ) is a cobordism class of oriented surface bundles F —t E n+2 ^y M n . Anelement in Q|°(Qg°Th(^F)) is a cobordism class of pairs [n : E n+2 —t M n ,n] ofsmooth maps n and stable bundle surjections from FF to TT*TM. (Upto cobordismone can assume that n is a vector bundle surjection.) a maps a bundle [F —tE ^y M] to the pair [n : E —t M, Dn] where Dn denotes the differential of n.Hence, a is a homotopy equivalence if and only if each cobordism class of pairs[n : E n+2 —t M n ,n] contains a "unique" representative with n a submersion.It is this geometric formulation that underpins the solution to the Mumfordconjecture by Madsen and Weiss. A key ingredient of the proof is the Phillips-Gromov ft-principle of submersion theory: A pair (g : X —t M, g : TX —t g*TM)can be deformed to a submersion - provided X is open. F above, however, is closed.The approach taken in [9] is to replace n : E —t M by g = nopn : X = E x R —t M.Now the submersion ft-principle applies and g can be replaced by a submersion /.The proof then consists of a careful analysis of the singularities of the projectionpi'i : X —t R on the fibers of /. At a critical point it uses Harer's Stability Theorem2.1.is a homoMadsen-Weiss Theorem 8.1. The map a : Z x FF+ -¥ ii°°Th(-L)topy equivalence.References[1] C.-F. Bödigheimer & U. Tillmann, Stripping and splitting decorated mappingclass groups, Birkhäuser, Progress in Math. 196 (2001), 47^ 57.[2] C.J. Earle & J. Eells, A fibre bundle description of Teichmüller theory, J. Diff.Geom. 3 (1969), 19-13.[3] S. Galatius, Homology of Q°°SCF~ and Q°°CF~ , preprint 2002.[4] J.L. Harer, Stability of the homology of the mapping class groups of orientablesurfaces, Annals Math. 121 (1985), 215^249.

456 U. Tillmann[5] N.V. Ivanov, Stabilization of the homology of Teichmüller modular groups,Leningrad Math. J. 1 (1990), 675-691.[6] N.V. Ivanov, On the homology stability for Teichmüller mudular groups: closedsurfaces and twisted coefficients, Mapping Class Groups and Moduli Spaces ofRiemann Surfaces, Contemp. Math. 150 (1993), 149^194.[7] I. Madsen & C. Schlichtkrull, The circle transfer and K-theory, AMS ContemporaryMath. 258 (2000), 307^328.[8] I. Madsen & U. Tillmann, The stable mapping class group and Q(CP°°), Invent.Math. 145 (2001), 509^544.[9] I. Madsen & M. Weiss, Cohomology of the stable mapping class group, in preparation.[10] E.Y. Miller, The homology of the mapping class group, J. Diff. Geom. 24(1986), 1-14.[11] S. Morita, Characteristic classes of surface bundles, Invent. Math. 90 (1987),551-577.[12] D. Mumford, Towards an enumerative geometry of the moduli space of curves,Arithmetic and Geometry, M. Artin and J. Tate, editors, Progr. Math.,Birkhäuser 36 (1983), 271^328.[13] G. Segal, Categories and cohomology theories, Topology 13 (1974), 293^312.[14] G. Segal, Elliptic cohomology (after Landweber-Stong, Ochanine, Witten, andothers), Seminar Bourbaki, Astérisque 161-162 (1989), 187-201.[15] G. Segal, The definition of conformai field theory, manuscript.[16] U. Tillmann, On the homotopy of the stable mapping class group, Invent. Math.130 (1997), 257^275.[17] U. Tillmann, Vanishing of the Batalin-Vilkovisky algebra structure for TCFTs,Commun. Math. Phys 205 (1999), 283^286.[18] U. Tillmann, Higher genus surface operad detects infinite loop spaces, Math.Ann. 317 (2000), 613^628.[19] N. Wahl, Infinite loop space structure(s) on the stable mapping class group,Oxford Thesis 2001.

ICM 2002 • Vol. II • 457-168Non-zero Degree Mapsbetween 3-Manifolds*Shicheng WangfAbstractFirst the title could be also understood as "3-manifolds related by nonzerodegree maps" or "Degrees of maps between 3-manifolds" for some aspectsin this survey talk.The topology of surfaces was completely understood at the end of 19-thcentury, but maps between surfaces kept to be an active topic in the 20-thcentury and many important results just appeared in the last 25 years. Thetopology of 3-manifolds was well-understood only in the later 20-th century,and the topic of non-zero degree maps between 3-manifolds becomes activeonly rather recently.We will survey questions and results in the topic indicated by the title,present its relations to 3-manifold topology and its applications to problemsin geometry group theory, fixed point theory and dynamics.There are four aspects addressed: (1) Results concerning the existenceand finiteness about the maps of non-zero degree (in particular of degree one)between 3-manifolds and their suitable correspondence about epimorphismson knot groups and 3-manifold groups. (2) A measurement of the topologicalcomplexity on 3-manifolds and knots given by "degree one map partial order",and the interactions between the studies of non-zero degree map among 3-manifolds and of topology of 3-manifolds. (3) The standard forms of non-zerodegree maps and automorphisms on 3-manifolds and applications to minimizingthe fixed points in the isotopy class. (4) The uniqueness of the coveringdegrees between 3-manifolds and the uniqueness embedding indices (in particularthe co-Hopfian property) between Kleinian groups .The methods used are varied, and we try to describe them briefly.2000 Mathematics Subject Classification: 57M, 55C, 37E, 30F40, 20E26.0. IntroductionThe topology of surfaces was completely understood by the end of 19th century,but maps between surfaces keep to be an active topic in the 20th century, and*Supported by grants of MSTC and NSFC.fDepartment of Mathematics, Peking University, Beijing 100871, China. E-mail:swang@sxxO.math.pku.edu.cn

458 S. C. Wangsome basic results just appeared in the last 25 years, among which are the Nielsen-Thurston classification of surface automorphisms [Th3], and Edmonds' standardform for surface maps [El]. Then fine results followed, say the realization of Nielsennumber in the isotopy class of surface automorphisms by Jiang [J], and the simpleloop theorem for surface maps by Gabai [Ga2].The topology of 3-manifolds was well-understood only in the later 20th centurydueto many people's deep results, in particular Thurston's great contribution tothe geometrization of 3-manifolds, and the topic of non-zero degree maps between3-manifolds becomes active only rather recently.We will survey the results and questions in the topic indicated by the title,present its relations to 3-manifold topology and its applications to problems ingeometry group theory, fixed point theory and dynamics. The methods used arevaried, and we try to describe them briefly.For standard terminologies of 3-manifolds and knots, see the famous booksof J. Hempel, W. Jaco and D. Rolfsen. For a proper map / : M —¥ N betweenoriented compact 3-manifolds, deg(/), the degree of /, is defined in most booksof algebraic topology. A closed orientable 3-manifold is said to be geometric ifit admits one of the following geometries: H 3 (hyperbolic), PSL 2 R, H 2 x E 1 ,Sol, Nil, E 3 (Euclidean), S 2 x E 1 , and S 3 (spherical). A compact orientable 3-manifold M admits a geometric decomposition if each prime factor of M is eithergeometric or Haken. Thurston's geometrization conjecture asserts that any closedorientable 3-manifold admits a geometric decomposition. Each Haken manifoldM with dM a (possibly empty) union of tori has a Jaco-Shalen-Johannson (JSJ)torus decomposition, that is, it contains a minimal set of tori, unique up to isotopy,cutting M into pieces such that each piece is either a Seifert manifold or a simplemanifold, which admits a complete hyperbolic structure with finite volume [Th2].In the remainder of the paper, all manifolds are assumed to be compactand orientable, all automorphisms are orientation preserving, all knots are inS 3 , all Kleinian groups are classical, and all maps are proper, unless otherwisespecified.Let M and N be 3-manifolds and d > 0 an integer. We say that M d-dominates(or simply dominates) N if there is a map / : M —¥ N of degree ±d. Denote byD(M, N) the set of all possible degrees of maps from M to N. A 3-manifold M issmall if each closed incompressible surface in M is boundary parallel.Due to space limitation, quoted literature are only partly listed in the references;while the others are briefly indicated in the context.1. Existence and finitenessA fundamental question in this area (and in 3-manifold theory) is the following.Question 1.1. Given a pair of closed 3-manifolds M and N, can one decide if Md-dominates N? In particular, can one decide if M 1-dominates N?

Non-zero Degree Maps between 3-Manifolds 459The following two natural problems concerning finiteness can be considered astesting cases of Question 1.1.Question 1.2 [Ki, Problem 3.100 (Y. Rong)]. Let M be a closed 3-manifold. DoesM 1-dominate at most finitely many closed 3-manifolds?Question 1.3. Let N be a closed 3-manifold. When is \D(M,N)\ finite for anyclosed3-manifold M ?An important progress towards the solution of Question 1.2 is the following:Theorem 1.1 ([So2], [WZh2], [HWZ3]). Any closed 3-manifold 1-dominates atmost finitely many geometric 3-manifolds.Theorem 1.1 was proved by Soma when the target manifolds N admit hyperbolicgeometry [So2]. The proof is based on the argument of Thurston's originalapproach to the deformation of acylindrical manifolds. Porti and Reznikov had aquick proof of Soma's result, based on the volume of representations [Re2]. HoweverSoma's approach deserves attention as it proves that the topological types ofall hyperbolic pieces in closed Haken manifolds 1-dominated by M are finite [So3].Theorem 1.1 was proved in [WZh2] when the target manifolds admit geometriesof H 2 x E 1 , PSL 2 (R), Sol or Nil. The proof for the case of H 2 x E 1 geometryinvokesGabai's result that embedded Thurston Norm and singular Thurston Normare equal [Gal], and the proof for case of PSL 2 (R) geometry uses Brooks and Goldman'swork on Seifert Volume [BG]. Theorem 1.1 was proved in [HWZ3] when thetarget manifolds admit S 3 geometry, using the linking pairing of 3-manifolds. Notethat only finitely many 3-manifolds admit the remaining two geometries.For maps between 3-manifolds which are not necessarily orientable, there is anotion of geometric degree (See D. Epstein, Proc. London Math. Soc. 1969). It isworth mentioning that if d-dominating maps are defined in terms of geometric degree,then Rong constructs a non-orientable 3-manifold which 1-dominates infinitelymanylens spaces [Ro3]. Actually there is a non-orientable hyperbolic 3-manifoldwhich 1-dominates infinitely many hyperbolic 3-manifolds [BW1]. Such examplesdo not exist in dimension n > 3 due to Gromov's work on simplicial volume andH.C. Wang's theorem that, for any V > 0, there are at most finitely many closedhyperbolic n-manifolds of volume < V.The answer to Question 1.2 is still unknown for closed irreducible 3-manifoldsadmitting geometric decomposition. The following result is related.Theorem 1.2 ([Rol], [So3]). For any 3-manifold M there exists an integer NM,such that if M = Afi —t Mi —¥ .... —¥ Afi is a sequence of degree one maps withk > NM, and each Mi admits a geometric decomposition, then the sequence containsa homotopy equivalence.The situation for Question 1.3 can be summarized in the following theorem.

460 S. C. WangTheorem 1.3 ([Grl], [BG], [W2]). Suppose N is a closed 3-manifold admittinggeometric decomposition. Then(1) |F(Af, N)\ is finite if either a prime factor of N contains a hyperbolic piecein its JSJ decomposition, or N itself admits the geometry of PSL 2 (R).(2) \D(N, N)\ is infinite if and only if either (i) N is covered by a torus bundleover the circle or a surfacexS 1 , or (ii) each prime factor of N has a cyclic or finitefundamental group.Part (1) of Theorem 1.3 follows from the work of Gromov [Grl] and Brooks-Goldman [BW]. Part (2) can be found in [W2]. Note that if \D(N,N)\ is infiniteand D(M,N) contains non-zero integers, then \D(M,N)\ is also infinite. I suspectthat Theorem 1.3 (2) indicates a general solution to Question 1.3.There are many partial results for Question 1.1: When both AI and N areSeifert manifolds with infinite fundamental groups Rong has an algorithm to determineif AI 1-dominates N [Ro3]. When N is the Poincaré homology sphere and aHeegaard diagram of AI is given, Hayat-Legrand, Matveev and Zieschang have analgorithm to decide if AI d-dominates N [HMZ]. There are simple answers to Question1.1 in the following cases: (1) AI and N are prism spaces and d = 1 [HWZ2];(2) At = N admit geometry of S 3 and /» an automorphism on m [HKWZ]; (3) Nis a lens space. I will state (3) as a theorem, since both its statement and proofare short, and since it has rich connections with previous results and with differenttopics.Theorem 1.4 ([HWZ1], [HWZ3]). A closed 3-manifold AI d-dominates the lensspace L(p,q) if and only if there is an element a in the torsion part of Hi(M,Z)such that a® a = -^ in Q/Z, where a® a is the self-linking number of a.A direct consequence of Theorem 1.4 is the known fact that L(p, q) 1-dominatesL(m,n) if and only if p = km and n = kqc 2 mod m. This fact has at least fourdifferent proofs: using equivariant maps between spheres by de Rham (J. Math.1931) and by Olum (Ann. of Math. 1953), using Whitehead torsion by Cohen(GTM 10, 1972), using pinch in [RoW] and using linking pair in [HWZ1].Degree one maps from general 3-manifolds to some lens spaces, in particularthe RP 3 , have been studied by Bredon-Wood (Invent. Math. 1969) and by-Rubinstein (Pacific J. Math. 1976) to find one-sided incompressible surfaces, byLuft-Sjerve (Topo. Appi. 1990) to study cyclic group actions on 3-manifolds, byShastri-Williams-Zvengrowski [SWZ] in theoretical physics, by Taylor (Topo. Appi.1984) to define normal bordism classes of degree one maps, and by Kirby-Melvin(Invent. Math. 1991) to connect with new 3-manifold invariants.Degree one maps induce epimorphisms on m. There are easy examples indicatethat Question 1.2 does not have direct correspondence in the level of 3-manifoldgroups [BW1], [RWZli]. However the following related question was raised in 1970's.Question 1.4 [Ki, Problem 1.12 (J. Simon)]. Conjectures:

Non-zero Degree Maps between 3-Manifolds 461(1) Given a knot group G, there is a number NQ such that any sequence ofepimorphisms of knot groups G —¥ Gi —¥ .... —¥ G n with n > NQ contains anisomorphism.(2) Given a knot group G, there are only finitely many knot groups H forwhich there is an epimorphism G —¥ H.According to a conversation with Gonzalez-Acuna, who discussed Question 1.4with Simon before it was posed, the epimorphisms in Question 1.4 are peripheralpreserving in their minds.Theorem 1.5 ([So5], [RW]). The conjecture in Question 1.4 (1) holds if the knotcomplements involved are small. The conjecture in Question 1.4 (2) holds if theknot complements are small and the epimorphisms are peripheral preserving.The first claim is due to Soma [So5] and the second claim is in [RW]. Both ofthem invoke Culler-Shalen's work on the representation varieties of knot groups. Itis also proved that any infinite sequence of epimorphisms among 3-manifold groupscontains an isomorphism if all manifolds are either hyperbolic [So5] or Seifert fibered[RWZli]. In [RWZh], the proof uses the fact that epimorphisms between asphericalSeifert manifolds with the same m rank are realized by maps of non-zero degree.Both this fact and Question 1.4 (1) are variations of the Hopfian property.We end this section by mention that there are results about D(M, N) in [DW]for (n — l)-connected 2n-manifolds, n > 1, which are quite explicit and of interestfrom both topological and number-theoretic point of view.2. UniquenessThe following question is raised in 1970's.Question 2.1 [Ki, Problem 3.16 (W. Thurston)]. Suppose a 3-manifold AI is notcovered by (surface) x S 1 or a torus bundle over S 1 . Yet f,g : AI —t N be twocoverings, must deg(f) = deg(g)?It is known [WWu2] that Question 2.1 has positive answer if AI admits geometricdecomposition and is not a graph manifold (AI is a graph manifold if eachpiece of its JSJ decomposition is Seifert fibered.) For graph manifolds there are fourdifferent covering invariants introduced in middle 1990's by [WWu2], Luecke andWu [LWu], Neumann [N] and Reznikov [Rei] . Unfortunately all those four coveringinvariants are either vanishing or not well-defined for some non-trivial graphmanifolds. It is also known that covering degree is uniquely determined if the graphmanifold in the target is either a knot complement [LWu] or its corresponding graphis simple [WWu2, N]. The positive answer to Question 2.1 for graph manifolds wasfinally obtained in [YW], using the matrix invariant defined in [WWu2] and anelegant application of matrix theory due to Yu.

462 S. C. WangTheorem 2.1 ([WWu2], [YW]). For 3-manifolds admitting geometric decompositionand not covered by either (surface)xS 1 or a torus bundle over S 1 , coveringdegrees are uniquely determined by the manifolds involved.It is worth mentioning an interesting fact that any knot complement is nontriviallycovered by at most two knot complements and any knot complement nontrivially covers at most one knot complement. The first claim follows from thecyclic surgery theorem of Culler-Gordon-Luecke-Shalen and the positive answer tothe Smith Conjecture. The second claim is in [WWul].Question 2.1 is equivalent to asking the uniqueness of indices of finite indexembeddings between 3-manifold groups. Recently there are also some discussionson the uniqueness of indices of self-embeddings of groups. A group G is said to beco-Hopf if each self-monomorphism of G is an isomorphism.Question 2.2. Let G be either a 3-manifold group, or a Kleinian group, or a wordhyperbolic group. When is G co-Hopf?The cohopficity of groups were first considered by Baer (Bull. AMS 1944).For word hyperbolic groups it was first considered by Gromov in 1987 [Gr2, p.157],and subsequently by Rips-Sela (GAFA, 1994), Sela [Se], and Kapovich-Wise (IsrealJ. Math. 2001). Cohopficity of 3-manifold groups was first studied in 1989 by-Gonzalez-Acuna and Whitten [GWh], and then in [WWu2] and [PW]. The answerfor 3-manifolds admitting geometric decomposition with boundary either empty setor a union of tori is known [GWh], [WWu2], and partial results for 3-manifolds withboundary of high genus surfaces are in [PW]. Cohopficity of Kleinian groups wasfirst considered in 1992 in an early version of [PW], then in 1994 in an early versionof [WZhl], also by Ohshika-Potyagailo (Ann. Sci. Ecole Norm. Sup. 1998) andDelzant-Potyagailo (MPI Preprint, 2000) for high dimensional Kleinian groups.Theorem 2.2. Suppose K is a non-elementary, freely indecomposable, geometricallyfinite Kleinian group and K contains no Z ® Z subgroup. Then(l)[Se], [PW], [WZhl] K is co-Hopf if K is a group of one end.(2)[\NZhl] If the singular locus of the hyperbolic 3-orbifold H 3 /K is a 1-manifold, then K is co-Hopf if and only if no circle component of singular locusmeets a minimal splitting system of hyperbolic cone planes.The proof of Theorem 2.2 (1) in [WZhl], influenced by that of torsion freecase in [PW], use a generalization of Thurston-Gromov's finiteness theorem on theconjugacy classes of group embeddings (Delzant, Comm. Math. Helv. 1995) anda proper conjugation theorem of Kleinian groups (Wang-Zhou, Geometriae Dedicata,1995). Theorem 2.2 (2) is proved by using 3-dimensional hyperbolic orbifoldstructures and orbifold maps, which turn out to be useful geometric tools.Note that groups in Theorem 2.2 are word hyperbolic groups. Accordingto Sela ([Se] and his MSRI preprints in 1994), people once expected that a nonelementaryword hyperbolic group is co-Hopf if and only if it has one end. Selaproved this expectation for the torsion free case [Se]. Theorem 2.2 (2) and examples

Non-zero Degree Maps between 3-Manifolds 463in [WZhl] show that cohopficity phenomenon is very complicated in the torsion case.In particular there are co-Hopf word hyperbolic groups which have infinitely manyends.Inspired by Questions 2.1 and 2.2 it is natural to askQuestion 2.3. Are the indices (including the infinity) of embeddings H —t Gbetween co-Hopf groups unique?3. Interactions with 3-manifold topologyDegree one maps define a partial order on Haken manifolds and hyperbolic3-manifolds. By Gordon-Luecke's theorem knots are determined by their complements[GL]. We say that a knot K 1-dominates a knot K' if the complement of K1-dominates the complement of K'. 1-domination among knots also gives a partialorder on knots. This partial order seems to provide a good measurement of complexityof 3-manifolds and knots. The reactions of non-zero degree maps between3-manifolds and 3-manifold topology are reflected in the following very flexibleQuestion 3.1. Suppose AI and N are 3-manifolds (knots) and AI 1-dominates(d-dominates) N.(1) Is a(AI) not "smaller" than a(N) for a topological invariant a(N) when a is either the rank of m, or Gromov'ssimplicial volume, or Haken number (of incompressible surfaces), or genus of knots;a(N) is a direct summand of a(M) when a is the homology group, and a(N) is afactor of a(M) if a is the Alexander polynomial of knots. The answer to Question3.1 (1) is still unknown for many invariants of knots and 3-manifolds, for examplecrossing number, unknotting number, Jones polynomial, knot energy, and tunnelnumber, etc. Li and Rubinstein are specially interested in Question 3.1 (1) forCasson invariant in order to prove it is a homotopy invariant [LRu].There are both positive and negative answers to Question 3.1 (2), depending onthe interpretation of the problem. On the negative side, Kawauchi has constructed,using the imitation method invented by himself, degree one maps between nonhomeomorphic3-manifolds AI and N with many topological invariants identical,see his survey paper [Ka]. On the positive side, there are many results. An easyoneis that if AI d-dominates N and both AI and N are aspherical Seifert manifolds,then the Euler number of AI is zero if and only if that of N is zero [Wl]. A deeperresult is Gromov-Thurston's Rigidity theorem, which says that a degree one mapbetween hyperbolic 3-manifolds of the same volume is homotopic to an isometry[Th2]. The following are some recent results in this direction.Theorem 3.1 ([So4], [Sol]). (1) For any V > 0, suppose f : AI —t N is a degreeone map between closed hyperbolic 3-manifolds with Vol(M) < V. Then there is a

464 S. C. Wangconstant c = c(V) such that (1 — c)Vol(AI) < Vol(N) implies that f is homotopicto an isometry.(2) If AI —t N is a map of degree d between Haken manifolds such that \\M\\ =d\\N\\, then f can be homotoped to send H(M) to H(N) by a covering, where \\*\\is the Gromov norm and H(*) is the hyperbolic part under the JSJ decomposition.Theorem 3.2 ([BW1], [BW2]). (1) Let AI and N be two closed irreducible 3-manifolds with the same first Betti number and suppose AI is a surface bundle. Iff : AI —t N is a map of degree d, then N is also a surface bundle.(2) Let At and N be two closed, small hyperbolic 3-manifolds. If there is adegree one map f : AI —t N which is a homeomorphism outside a submanifoldH c N of genus smaller than that of N, then AI and N are homeomorphic.(1) and (2) of Theorem 3.1 provide a stronger version and a generalization ofGromov-Thurston's Rigidity theorem, respectively. In respect of Theorems 3.2, thefollowing examples should be mentioned: There are degree one maps between twonon-homeomorphic hyperbolic surface bundles with the same first Betti numberand between two non-homeomorphic small hyperbolic 3-manifolds [BW2]. Theconstructions of those maps are quite non-trivial. There are many applicationsof Theorem 3.2. We list two of them which are applications of Theorem 3.2 toThurston's surface bundle conjecture and to Dehn surgery respectively, where degreeone maps constructed by surgery on null-homotopic knots are involved.Theorem 3.3 ([BW1], [BW2]). (1) There are closed hyperbolic 3-manifolds AIsuch that any tower of abelian covering of AI contains no surface bundle.(2) Suppose AI is a small hyperbolic 3-manifold and that k C AI is a nullhomotopicknot, which is not in a 3-ball. If the unknotting number of k is smallerthan the Heegaard genus of M, then every closed 3-manifold obtained by a nontrivialDehn surgery on k contains an incompressible surface.4. Standard formsQuestion 4.1. What are standard forms of non-zero degree maps and of automorphismsof 3-manifolds?Sample answers to analogs of Question 4.1 in dimension 2 are that each mapof non-zero degree between closed surfaces is homotopic to a pinch followed by abranched covering [El], and each automorphism on surfaces can be isotoped to amap which is either pseudo Anosov (Anosov), or periodic, or reducible [Th3].Theorem 4.1 (Haken, Waldhausen, [E2], [Ro2]). (1) A degree one map betweenclosed 3-manifolds is homotopic to a pinch.(2) A map of degree at least three between closed 3-manifolds is homotopic toa branched covering.(3) A non-zero degree map between Seifert manifolds with infinite m is homotopicto a fiber preserving pinch followed by a fiber preserving branched covering.

Non-zero Degree Maps between 3-Manifolds 465(1) is proved by Haken (Illinois J. Math. 1966), also by Waldhausen, and aquick proof using differential topology is in [RoW]. (2) is proved by Edmonds [E2]quickly after Hilden-Montesinos's result that each 3-manifold is a 3-fold branchedcovering of 3-sphere. (3) is due to Rong [Ro2], which invokes [El]. According toconversations with D. Gabai and with M. Freedman, people are still wondering ifeach map of degree 2 between closed 3-manifolds is homotopic to a pinch followedby a double branched covering.For non-prime 3-manifolds, Cesar de Sa and Rourke claim that every automorphismis a composition of those preserving and permuting prime factors (Bull.AMS, 1979), and those so-called sliding maps. A proof is given by Hendricks andLaudenbach [HL], and by McCullough [Mc].Standard forms of automorphisms on prime 3-manifolds admitting geometricdecomposition have been studied in [JWW]. The orbifold version of Nielsen-Thurston's classification of surface automorphisms is established, i.e., each orbifoldautomorphism is orbifold-isotopic to a map which is either (pseudo) Anosov, orperiodic, or reducible. We then have the following theorem.Theorem 4.2 ([JWW]). Let AI be a closed prime 3-manifold admitting geometricdecomposition. Let f : AI —t AI be an automorphism. Let fi be the productneighborhood of the JSJ tori. Then(1) f is isotopie to an affine map if AI is a 3-torus.(2) f is isotopie to an isometry if AI is the Euclidean manifold having a Seifertfibration over RP 2 with two singular points of index 2.(3) f is isotopie to a map which preserves the torus bundle structure over1-orbifold if AI admits the geometry of Sol.(4) for all the remaining cases, f can be isotoped so that fi is invariant underf, and for each f-orbit O of the components in {fi, AI — fi}, f\0 is an isometry ifOis hyperbolic, f\0 is affine if O belongs to fi, otherwise there is a Seifert fibrationon O so that f is fiber preserving and the induced map on the orbifold is eitherperiodic, or (pseudo) Anosov, or reducible.As in dimension 2, standard forms in Theorems 4.2 are useful in the study offixed point theory and dynamics of 3-manifold automorphisms. The following is aresult in this direction.Theorem 4.3 ([JWW]). Suppose AI is a closed prime 3-manifold admitting geometricdecomposition and f : AI —t AI is an automorphism. Then(1) the Nielsen number N(f) is realized in the isotopy class of f.(2) f is isotopie to a fixed point free automorphism unless some componentof the JSJ decomposition of AI is a Seifert manifold whose orbifold is neither a2-sphere with a total of at most three holes or cone points nor a projective planewith a total of at most two holes or cone points.References[BWl] M. Boileau and S.C. Wang, Non-Zero degree maps and surface bundles over

466 S. C. WangS 1 , J. Diff. Geom. 43 (1996), 789^908.[BW2] M. Boileau and S.C.Wang, Degree one maps, incompressible surfaces andHeegaard genus, Preprint (2002).[BG] R. Brooks and W. Goldman, Volume in Seifert space, Duke Math. J. 51(1984), 529^545.[CS] M. Culler and P. Shalen, Varieties of group representations and splittingsof 3-manifolds, Ann. of Math. 117 (1983), 109^146.[DW] H. Duan and S.C.Wang, The degree of maps between manifolds, Math. Zeit.(to appear).[El] A. Edmonds, Deformation of maps to branched covering in dimension 2,Ann. of Math. 110 (1979), 113-125.[E2], Deformation of maps to branched covering in dimension 3, Math.Ann. 245 (1979), 273^279.[Gal] D. Gabai, Foliations and the topology of 3-manifolds, J. Diff. Geom. 18(1983), 479^536.[Ga2] , Simple loop theorem, J. Diff. Geom. 21 (1985), 143^149.[GL] C. Gordon and J. Luecke, Knots are determined by their complements,JAMS 2 (1989), 371-115.[GWh] F. Gonzalez-Acuna and W. Whitten, Embeddings of 3-manifold groups,Mem. AMS 474.[Grl] M. Gromov, Volume and bounded cohomology, Pubi. Math. IHES 56(1983), 5^99.[Gr2] , Hyperbolic groups, Essays in Group Theory, edited by S.M. Gersten,MSRI Pub., vol. 8, Springer-Verlag, 75^263.[HMZ] C. Hayat-Legrand, S. Matveev and H. Zieschang, Computer calculation ofthe degree of maps into the Poincaré homology sphere, Experiment. Math.10 (2001), 497^508.[HKWZ]C. Hayat-Legrand, E. Kudryavtseva, S.C.Wang and H. Zieschang, Degreesof self-mappings of Seifert manifolds with finite ni, Rend. Istit. Mat. Univ.Trieste 32 (2001), 131-147.[HWZ1]C. Hayat-Legrand, S.C.Wang and H. Zieschang, Degree one map onto lensspaces, Pacific J. Math. 176 (1996), 19^32.[HWZ2] , Minimal Seifert manifolds, Math. Ann. 308 (1997), 673^700.[HWZ3] , Any 3-manifold 1-dominates only finitely many 3-manifolds supportingS 3 geometry, Proc. AMS 130 (2002), 3117^3123.[HL] H. Hendriks and F. Laudenbach, Difféomorphismes des sommes connexesen dimension trois., Topology 23 (1984), 423—143.[J] B. Jiang, Fixed points of surface homeomorphisms, Bull. AMS 5 (1981),176^178.[JWW] B. Jiang, S.C.Wang, and Y-Q.Wu, Homeomorphisms of 3-manifolds andthe realization of Nielsen number, Comm. Anal. Geom. 9 (2001), 825^877.[Ka] A. Kawauchi, Topological imitations, Lectures at Knots 96 (ed. Shin'ichiSuzuki), World Sci. Pubi. Co., 1997, 19^37.

Non-zero Degree Maps between 3-Manifolds 467[Ki] R. Kirby, Problems in low-dimensional topology,, Geometric topology, (ed.H.Kazez), AMS/, International Press, 1997, 35—173.[LRu] W. Li and J.H. Rubinstein, Casson invariant is a homotopy invariant,Preprint (2002).[LWu] J. Luecke and Y-Q. Wu, Relative Euler number and finite covers of graphmanifolds, Geometric topology, (ed. H.Kazez), AMS/, vol. 1, InternationalPress, 1997, 80^103.[Me] D. McCullough, Mapping of reducible 3-manifolds, Proc. of the Semester inGeometric and Algebraic Topology, Warsaw, Banach Center, Pubi., 1986,61^76.[N] W.D. Neumann, Commensurability and virtual fibration for graph manifolds,Topology 36 (1997), 355^378.[PW] L. Potyagailo, and S.C. Wang, 3-manifolds with co-Hopf fundamental groups,St. Petersburg Math. J. (English Translation) 11 (2000), 861-881.[Rei] A. Reznikov, Volume of discrete groups and topological complexity of homologyspheres, Math. Ann. 306 (1996), 547^554.[Re2] , Analytic Topology, Progressin Math. Vol.201, 519^532. Birkhäuser2002.[RW] A. Reid and S.C.Wang, Non-Haken 3-manifolds are not large with respectto mappings of non-zero degree, Comm. Anal. Geom. 7 (1999), 105^132.[RWZh] A.Reid, S.C.Wang, Q.Zhou, Generalized Hopfian property, a minimal Hakenmanifold, and epimorphisms between 3-manifold groups, Acta Math. Sinica18 (2002), 157^172.[Roi] Y. Rong, Degree one maps between geometric 3-manifolds, Trans. AMS(1992).[Ro2] , Maps between Seifert fibered spaces of infinite m, Pacific J. Math.160 (1993), 143^154.[Ro3] , Degree one maps of Seifert manifolds and a note on Seifert volume,Topology Appi. 64 (1995), 191^200.[RoW] Y. Rong and S.C.Wang, The preimage of submanifolds, Math. Proc. Camb.Phil. Soc. 112 (1992), 271^279.[Sel] Z. Sela, Structure and rigidity in (Gromov) hyperbolic groups, GAFA 7(1997), 561^593.[SWZ] A. Shastri, J.G.Williams and P.Zvengrowski, Kinks in general relativity,Internat. J. Theor. Phys. 19 (1980), 1-23.[Sol] T. Soma, A rigidity theorem for Haken manifolds, Math. Proc. Camb.Phil. Soc. 118 (1995), 141-160.[So2] , Non-zero degree maps onto hyperbolic 3-manifolds, J. Diff. Geom.49 (1998), 517^546.[So3] , Sequence of degree-one maps between geometric 3-manifolds., Math.Ann. 49 (2000), 733^742.[So4] , Degree one maps between hyperbolic 3-manifolds with the samelimit, Trans. AMS 353 (2001), 2753.

468 S. C. Wang[So5] , Epimorphisms sequences between hyperbolic 3-manifold groups, Proc.AMS 130 (2002), 1221-1223.[Thl] W. Thurston, The Geometry and Topology of Three-Manifolds, PrincetonLecture Notes.[Th2] , Three dimensional manifolds, Kleinian groups and hyperbolic geometry,Bull. AMS 6 (1982), 357^388.[Th3] , On the geometry and dynamics of diffeomorphisms of surfaces,Bull. AMS 19 (1988), 417-131.[Wl] S.C.Wang, The existence of non-zero degree maps between aspherical 3-manifolds, Math. Zeit. 208 (1991), 147^160.[W2] , The ni-injectivity of self-maps of non-zero degree on 3-manifolds,Math. Ann. 297 (1993), 171-189.[WWul]S.C.Wang and Y.Q.Wu, Any knot complement covers at most one knotcomplement, Pacific J. Math. 158 (1993), 387^395.[WWu2] , Covering invariant of graph manifolds and cohopficity of 3-manifoldgroups, Proc. London Math. Soc. 68 (1994), 221^242.[WZhl] S.C.Wang and Q.Zhou, Embeddings of Kleinian groups with torsion, ActaMath. Sinica 17 (2001), 21^34.[WZh2] , Any 3-manifold 1-dominates at most finitely many geometric 3-manifolds, Math. Ann. 332 (2002), 525^535.[YW] F.Yu and S.C.Wang, Covering degrees are uniquely determined by graphmanifolds involved, Comm. Math. Helv. 74 (1999), 238^247.

Section 6. Algebraic and Complex GeometryHélène Esnault: Characteristic Classes of Flat Bundles and Determinantof the Gauss-Manin Connection 471L. Göttsche: Hilbert Schemes of Points on Surfaces 483Shigeru Mukai: Vector Bundles on a K3 Surface 495R. Pandharipande: Three Questions in Gromov-Witten Theory 503Miles Reid: Update on 3-folds 513Vadim Schechtman: Sur les Algèbres Vertex Attachées aux VariétésAlgébriques 525B. Totaro: Topology of Singular Algebraic Varieties 533

ICM 2002 • Vol. II • 471-181Characteristic Classes of Flat Bundlesand Determinant of theGauss-Manin ConnectionHélène Esnault*2000 Mathematics Subject Classification: 14C22, 14C25, 14C40,14C35, 14C99.1. IntroductionThe purpose of this note is to give a survey on recent progress on characteristicclasses of flat bundles, and how they behave in a family.2. Characteristic classesLet X be a smooth algebraic variety over a field k. In [13] and [15], we definedthe ringAD(X)= ® n AD n (X)= ® n W(X,K^d^ü n x/k A.-.^Q^1)(2.1)of algebraic differential characters. Here the Zariski sheaf K.ff is the kernel of theresidue map from Milnor F'-theory at the generic point of X to Milnor F'-theory atcodimension 1 points. More precisely, K.^ satisfies a Gersten type resolution (see[16] and [18])/Cf 4(* fc( x),.^(feW)ê ie x(i)* I ,.î(«(x))-^• • • ® œG xw iz,*K%_ a (K(x)) -•...-•® xexW i x ,,K^(K(x))).Here X^ means the free group on points in codimension a, while i x : x —ï X is theembedding. The map dlog({ai,... ,a n }) = dlogai A • • • Adloga„ from Kff(k(X))* Mathematik, Universität Essen, FB6, Mathematik, 45117 Essen, Germany. E-mail:esnault@uni-essen.de

472 Hélène Esnaultto ^klX)/k carr i es K-n toü x/k = Ker(ü- {x)^® x exîi^jnThis defines the map dlog : K.^ —¥ 0X/k-By the Gersten resolution, H n (X,K.ff) = CH n (X), the Chow group of codimensionn points. Thus one has a forgetful mapAD n (X) f^f CH n (X). (2.2)The restriction map to the generic point Spec(fc(X)) fulfillsADHX) 4 F°(X,Q x/fc /dlogO x )cAF 1 (fc(X))AD n (X) -+ H a {X,^/d^^)c^/k /dYÏ%^/k , for n> 2 (2.3)(see [2]). It is no longer injective for n > 2.The Kahler differential d : ^2xfi}—^ ^x/k cisd defines.4D»(I)4ff°(I,fì^cM ). (2.4)The ring AD(X) = ® n AD n (X)contains the subringAF(X)eisd(X) = ®„AF« lsd (X) e„ H" (X, /Cf ^1 Q x/k -+ ... -+ iifj^x) )= Ker(AF(X)4®„F°(X,Q^clsd )). (2.5)We call them the closed characters. The restriction map to Spec(fc(X)) fulfillsAF^XOcisd 4F°(X,Q x/fcjClsd /dlogO x )cAF 1 (fc(X))eisdi2n—l\ r- 0-2»—AF"(X)eisd -+ ^ft^fl^ff^W^lA). l DR ) u n DRforn>2 (2.6)(see [2]). Here H P DR is the Zariski sheaf of de Rham cohomology.If fcis the field of complex numbers C, one can change from the Zariski topologyto the analytic one. This yields a mapAD(X) = ®„AF"(X)4D(X) = ® n D n (X) = ® n if" n (X &n ,Z(n)Â x 4...Â^- 1 ). (2.7)Here D(X) is the ring of differential characters defined by Cheeger-Simons ([10]).One hasi(AF(X) c isd) C enff^-^Xan.C/Zfn)) C D(X). (2.8)It is classical that AD 1 (X) is the group of isomorphism classes of lines bundlesF with connection V. If £y £ Y(Uij,O x ) is a cocycle of F in a local frame e, of

Classes and Gauss-Manin Bundle 473F on Ui, and V(e») = a, £ Y(Ui,ii x , k ) is the local form of the connection, thendlogÇij = ctj —ai = ö(a)ij defines the Cech cocycle of ci((L, Vj). In [12], [13], [15],we generalize this class.Theorem 2.1 ([12], [13], [15]). Associated to an algebraic bundle E with connectionV (resp. with integrable connection), one has characteristic classes c n ((E,\7j) £AD n (X) (resp. £ AD n (X) c i s^). These classes satisfy the following properties:(1) The classes c»((F, V)) £ AD*(X) are functorial and additive.(2) ci((E,\7j) is the isomorphism class of (det(E),det(Vj).(3) forget (c n ((E,\7jj) = c n (E) £ CH n (X) is the Chern class of the underlyingalgebraic bundle E in the Chow group.(4) d(c n ((E,\7))) = c„(V 2 ) G F°(X,Q2» fcjClsd ) is the Chern-Weil form which isthe evaluation of the invariant polynomial c n on the curvature V 2 .(5) The restriction to the generic point c n ((E,\7)\k(x)) is the algebraic Chern-Simons invariant CS n ((E,\7j) defined in [4]. It has values in H°(X,n x/k /dlogO x ) (resp.H°(X,n x/kcisd /dlogO x )) for n = 1, and in H°(X,^xTkl^xfk) (resp.H^X^fi 1 )) for n>2.(6) //fc C C, then i(C n ((E,Vjj) £ D n (X)(resp. £ Ff^-^X^C/Z^))) is thedifferential character defined by Chern-Cheeger-Simons, denoted by c n (Ej n ) £F 2 "- 1 (X an ,C/Z(n)) i/V is integrable.If fc = C, V is flat and the underlying monodromy is finite, then the existenceof c n ((E, V))immediately implies that the Chern-Simons classes in F 2 " _1 (X an ,Q(n)/Z(nj) are in the smallest possible level of the coniveau filtration ([14]).If X is complex projective smooth, V is integrable and n > 2, we relateCS n ((E, V)) for n > 2 to the (generalized) Griffiths' group Griff"(X). It consists ofcycles which are homologous to 0 modulo those which are homologous to 0 on somedivisor([4], definition 5.1.1). For n = 2, [2] implies that GriS"(X) is the classicalGriffiths' group. For n > 2, Reznikov's theorem ([19]) (answering positively Bloch'sconjecture [3]), together with the existence of the lifting c n ((E, V)), imply that theclasses CS n ((E, V)) lie in the image Im of the global cohomology H 2n^1(X,Q(nj)in H°(X,H 2n (Q(nj). This subgroup Im maps to Griff"(X). One hasTheorem 2.2 ([4], Theorem 5.6.2).image of CS n ((E,\7j) £ GrifP(X) ®Qis the Chern class c ( /j_" s (E) ® Q of the underlying algebraic bundle E. Moreover,CS n ((E, V)) =0 if and only if c« riff (F) ® Q = 0.A relative version AD(X/S) of AD(X) is defined in [6]. We give here anexample of application.Theorem 2.3([6], Corollary 3.15). Let f : X —t S be a smooth projective familyof curves over a field fc. Let (E,\7x/s) be a bundle with a relative connection.Then there are classes c 2 ((F,V)) G AD 2 (X/S) := ïïf"(X,K. 2 ^1 Û^®^/s ) liftingthe classes c 2 (E) £ CH 2 (X). There is a trace map /» : AD 2 (X/S) -• AD 1 (S)compatible with the trace map on Chow groups /» : CH 2 (X) —t CF 1 (S). Thus

474 Hélène Esnaultf*c 2 (E, V) is a connection on the line bundle /»02(F), which depends functoriallyonon the choice of Vx/s E.We now study the behavior at 00 of CS n ((E, Vj) for n > 2 in characteristic0. Let j : X —t X be a smooth compactification of X. Recall that a de Rham classG Hp R (k(X)/k) at the generic point is called unramified if it lies in H°(X,H g DR cH« DR (k(X)/k)([2]).Theorem 2.4. We assume k to be of characteristic 0, and V to be integrable. ThenCS n ((E, Vj) is unramified for n>2.Proof. If (F, V) is regular singular, this is shown in [4], theorem 6.1.1. In general,one may argue as follows. One hasH^X^R 1 ) = Ker(F°(X,^fl 1 ) *4 ® xeX , Xl H 2 D n R 2 (x/kj).Thus it is enough to show that the residue map at each generic point at 00 dies.At a smooth point of a divisor D at 00, the residue depends only on the formalcompletion of X along D. So we may assume that V is a connection on Ox =k(D)[[x]], integrable relative to fc. By a variant (see [1], proposition 5.10.) ofLevelt's theorem ([17]) for absolute flat connections, there are a finite extensionK D k(D), and a ramified extension n : K[[x]] C FJffy]], y N = x for some N £ N\{0}such that n*(V) = ®(F ® U). Here F is integrable of rank 1 and U is integrablewith logarithmic poles along y=0. Since Res y= on*(a) = A r 7r*(Res œ= o(o;)), andHp R (k(D)/k) C H p DR(K/k), functoriality and additivity of the classes imply thatwe may assume V = F ® U on K(fixj) with K = k(D), In a local frame we havedxthe equations U = Y h S where F G GL(r,K[[x]]),E G M(r,K[[x]]) 0^, anddxL = d(f) + A h ß where / G K((x)),X £ k,ß £ SlY- with dß = 0. The explicitformula Res Tr(A(d(A))"- 1 ) G H^fi^K) of CS n ((E,Vj) £ HîK^x))) and[4], prop. 5.10 in the logarithmic case, imply thatCS n ((E,Yj) = Tr(d(f) + ^f+ ß)(d(T) + dS)"- 1 .This is the sum of 2 terms with rational coefficients, Res Tr(d(/) + /3)(d(S)" -2dxdxd(Y) — ) and Res Tr(A—(d(S)" -1 )). Both terms are obviously exact.Discussion 2.5. We assume here fc = C, V is integrable and n > 2. We consider theimage c„(Fj n | c(x) ) G H°(X,H 2n -HC/Z(n))) of CS n ((E,V)) £ H^X,^^1).When X is not compact, there is Deligne's unique algebraic (F, V) with regular singularitiesat 00 with the given underlying local system F^([ll]), but there are manyirregularconnections (F, V). The topological class C n (E^n) £ F 2 " _1 (X an ,C/Z(n))is not, a priori, extendable to X, but we have seen that its restriction to Spec(C(X))is unramified.There is on X a fundamental system of Artin neighborhoods U which aregeometrically successive fiberings in affine curves. Topologically they are Kmand their fundamental group is a successive extension of free groups in finitelymanyletters. On such an open U, the class c n (Ej n ) lies in H 2n^1(U ain ,C/Z(nj) =ff^Mtfan.t^C/Zfn)).ffff(IT

Classes and Gauss-Manin Bundle 475If U is such that m(Ua, n ,u) is isomorphic as an abstract group to 7Ti(V an ,w),where V is an Artin neighborhood on a rational variety, then Ej n \u becomes arepresentation of 7Ti(V an ,w), and since then H° (V ,H?p R ~ 1 ) = 0 for n > 2 and V C Va good compactification, one obtains c n (Ej n \c( X )) = 0,n > 2 in this case. Such anexample is provided by a product of smooth affine curves of any genus. It has thefundamental group of a product of P 1 minus finitely many points.Question 2.6. In view of the previous discussion, we may ask what complexsmooth varieties X are dominated by h :Y —¥ X proper, with Y smooth, such thatY has an Artin neighborhood, the fundamental group of which is the fundamentalgroup of an Artin neighborhood on a rational variety, or more generally of a varietyfor which H° (smooth compactification, HQH) = 0 for n > 1. We have seen thatthis would imply vanishing modulo torsion of c n (Ej n \c( X )), n > 2, or equivalentlyCS n ((E,V)) £ HOiX^-HQln))).On the other hand, if X is projective smooth, Reznikov's theorem ([19])shows vanishing modulo torsion of the Chern-Cheeger-Simons classes in Jf 2 "^1 (X an ,C/Q(n)). It is a consequence of Simpson's nonabelian Hodge theory on smooth projectivevarieties. Our classes CS n ((E, Vj) live at the generic point of X. We don'thave a nonabelian mixed Hodge theory at disposal. Yet one may ask whether it isalways true that CS n ((E, Vj) £ H°'(X,H 2n - 1 (Q(n))) for n > 2, even if many Xdon't have the topological property explained above.3. The behavior of the algebraic Chern-Simonsclasses in families in the regular singular caseThe algebraic Chern-Simons invariants CS n ((E,Vj) have been studied in afamily in [5]. Given / : X —t S a proper smooth family, and (F, V) a flat connectionon X, the Gauß-Manin bundlesF7*(Ûx/ S ®F,V x/s )carry the Gauß-Manin connection GM l (V). We give a formula for the invariantsCS„((GM*(V)-rank (V) • GAF(d))) on S, as a function of CS n ((E,Vj) and ofcharacteristic classes of /. Here (0,d) is the trivial connection.More generally, we may assume that / is smooth away from a normal crossingsdivisor T c S such that Y = / _1 (F) c X is a normal crossings divisor with theproperty that 0^., s (logF) is locally free. Then (E,V) has logarithmic poles alongY U Z where Y U Z C X is a normal crossings divisor, still with the property that0^. , s (log(F + Zj) is locally free. That is Z is the horizontal divisor of singularitiesof V. The formula involves the top Chern class Cd(ii x , s (log(Y + Zj) £ M d (X, K,d —^®iXz Ld ), rigidified by the residue maps ii x , s (log(Y + Zj) —t Oz t , as defined by T.Saito in [20]. One of its main features is that CS n ((GM*(V) — rank (V) • GM % (d)))vanishes if CS n ((E,Vj) vanishes. It is

476 Hélène EsnaultTheorem 3.1 ([5], Theorem 0.1).CSnî^yiGM^V)- rank(V) • GMfidjj)= hl) dim{X/S) f*c dim{x/S) (n x/s (log(Y + Z)),res) • CS n ((E,V)).Here • is the cup product of the algebraic Chern-Simons invariants with this rigidifiedclass, which is well defined, as well as the trace /» to S.Discussion 3.2. One weak point of the method used in [5] is that it does notallow to understand a formula for the whole invariants c n ((E,Vj), but only forCS n (E, V). Indeed, we use the explicit formula studied in [4] to compute it, whichcan't exist for the whole class in AD(X), as it in particular involves the Chernclasses of the underlying algebraic bundle F in the Chow group.4. The determinant of the Gauss-Manin connection:the irregular rank 1 caseNow we no longer assume that (F, V) is regular singular at oo. In the nexttwo sections, we reduce ourselves to the case where / : X —t S is a family of curves,and we consider only the determinant of the Gauß-Manin connection. That is weconsiderdet (GM) := ^(-l) i ci(GM iG H 1^,©! ^1 Q^) c nl (S) /diog(k(S)r.Since the determinant is recognized at the generic point of S, we replace S by itsfunction field K := k(S) in the next two sections. In other words, X/K is an affinecurve. Let X/K be its smooth compactification.In this section, we assume that the integrable connection (F, V) we start withon X has rank 1. Following Deligne's idea (see his 1974 letter to Serre publishedas an appendix to [7], we first reduce the problem of computing the determinantof the cohomology of V on the curve to the one of computing the determinant ofcohomology of an integrable invariant connection, still denoted by V, on a generalizedJacobian. More precisely, V has a divisor (with multiplicities) of irregularityî'miPi, where m, — l=irregularity of V in p t £ X \ X. On the JacobianG = Pic(X,^! rriiPi) of line bundles trivialized at the order m, at the points pi,there is an invariant connection which pulls back to V via the cycle map. On thetorsor p~ 1 (uJxŒî m iPi)) under the affine group p^1(Ox),where p : G —¥ Pic(X),one considers the hypersurface S : ^. res Pi = 0. We show that the relative invariantconnection V/K on G, while restricted to S e P~ 1 (^xŒîm ìPìj), acquires exactlyonezero which is a FJ-rational point K of G. Restricting V to this special pointyields a connection V| K on K. The formula then says that the determinant of theGauß-Manin connection is the sum of this connection V| K and of a 2-torsion term,which we describe now. In a given frame of F at a singularity pi, the local equation

Classes and Gauss-Manin Bundle 477of the connection is a, = a,-^y+lower order terms, where ti is a local parameterand a, G fc x . Then the 2-torsion connection ^dloga, G iì 1 K, k /dlogK x does notdepend on the choices and is well defined. The 2-torsion term is the sum over theirregular points of these 2-torsion connections. Summarizing, one hasTheorem 4.1 ([7], Theorem 1.1).detf^(-l) i F i (X,(Q^/if ®F,V x/ K),GAF(V)J= (-1)V| K + Yli,ro;>2T Y 1, it is

478 Hélène Esnaultno longer the case that j*E necessarily contains a rank r bundle such that in a localdti ßiformal frame of this bundle, the local equation has the shape Ai = a,—- H ^-j-,ti t iwith a, G GL(r,K[[ti]]),ßi G ifi K ® M(r,K[[ti]]). We call an integrable connection(F, V) with this existence property an admissible connection.Even if (E,V) is admissible, its determinant connection det(F, V) might havemuch lower order poles (for example trivial). This indicates that one can not extenddirectly in this form the formula 4.1. However, assuming (E,V) to be admissibleand choosing some v £ OJ X /K ® ^(X) which generates i^C^,i m iPi)a t Pi as f° rformula 4.3, the right hand side of 4.3 makes sense, if one replaces dloga, byd log det (a»). Using global methods inspired by the Higgs correspondence betweenHiggs fields and connections on complex smooth projective varieties ([21]), one isable to prove the "same" formula as 4.3 in the higher rank case on P 1 .Theorem 5.1 ([8], Theorem 1.3). If (E,V) is admissible and has at least oneirregular point, and if v £ OJ X /K ® ^(X) generates uCJ^îPi) at the points pi,thendetlj2(^y(H i (X,(n x/K ®L,Vx/K),GM i (Vj))= (-1)V| K („) + ^2 ( SU P C 1 » -y)dlogdet(ai(pij) + Tr Res^dt^1 A AA .ìThe connection Res Tr^dg,^1 A A t £ Q 1 K/dlogK x is well defined, as well asTH, 'the 2-torsion connection sup (l, —-)dlogdet(a,(p,)).However, one needs a different method in order to understand the contributionof singularities in which (F, V) is not admissible.We describe now the origin of the method contained [1]. It is based on theidea that Tate's method ([22]) applies for connections.Locally formally over the Laurent series field K((tj), E becomes a r-dimensionalvector space over K((tj). The relative connection VK((T))/K '• E —¥ i^K((t))/K ® Eis a Fredholm operator. This means that FF(VX/K),ì = 0,1 are finite dimensionalFJ-vector spaces, and that VX/K carries compact lattices to compact lattices.Let F = ®\K((ij) be the choice of a local frame. A compact lattice is aFJ-subspace of F which is commensurable to ®iFJ[[r]]. Given 0 fi^ v ^G ^K((t))lK,one composes VK((t))/K,v '•= l/^1 ° ^K((t))/K '• E —t E to obtain a Fredholmendomorphism. To a Fredholm endomorphism A : E —t E, one associates a1-dimensional FJ-vector space X(A) = det(F°(A)) ® det(F 1 (A)) _1 together withthe degree x(A) = dimF°(A) — dimF 1 (A). We call this a super-line. It doesnot refer to the topology defined by compact lattices. Then one measures how Amoves a compact lattice F c E. First for 2 lattices F and F', one takes a smallercompact lattice N c F n F' and defines det(F : F') := det(L/N) • det(F'/A r ) _1 ,where • is the tensor product of super-lines and det(L/N) has degree dim(L/N).This does not depend on the choice of N. Then one defines asymptotic superlines.The compact one is X C (A) = det(A(F) : F) -det(FnKer(A)) and the discrete one isXd(A) = det(F : A^1^)) • det(V/(L + A(\'j). They do not depend on the choice of

Classes and Gauss-Manin Bundle 479F. Taking Ofi^ v £ UJ X I K ®K(X) a rational differential form, and F m ; n the minimaloneextension of F, X the complement of the singularities of v and VX/K, h asV\-\det H*(X / K,E min ) = ® X€XXX Xd(V K(m/K:l ,)-det(E v )as a product of discrete lines.On the other hand, one has the relation X(A) = Xd(A) • A C (A) _1 . One easilycomputes that x(^K((t))/K,v) = 0,X(VK((t))/K,i>) = 1- Setting£-x(VK((t))/K,v) = X C (V K ((t)/K,v) • det(F ) _ ,this implies immediately the product formula.Theorem 5.2 ([1], (1.3.1)).det H* (X/K, F min ) = ® xe x\x e x(VK((t))/K,v)-It remains to endow the local e lines with a connection, compatible with theGauß-Manin connection on the left. One chooses a section of the vector fieldsT K /k C Tx/k an d a relative differential form v which is annihilated by this section.One applies Grothendieck's definition of a connection. The Gauß-Manin connectionis given by the infinitesimal automorphismp\detH*(X/K, F m ; n ) —t pi det H*(X/K,F m ; n ) on K ®k K, induced by r : p\E m \ n —t p 2 E m i n ,r £ T K / k , which by thechoice commutes to VX/K• By functoriality of the e lines, this defines a K ®k Khomomorphism p\e —t p\e. This is the e-connection.Theorem 5.3 ([1], Theorem 5.6). For an admissible connection of local equationA dt ß . , „ , , , . .A = a 1 -, with m> 2, the local e-connectton is—jm £ — 1dtme(Vif(( t ))/if, —) = TrRes t=0 daa _1 A + — dlogdet(a(t = 0))G Ü^JdlogK-.The restriction on the choice of v given by the commutativity constraint withsome lifting of vector fields of K is not necessary. The construction is more general.Given a relative connection V K ((t))/K, the e-lines for 0 ^ v £ uL»,, K build asuper-line bundle on the ind-scheme ^Kn t \\i K - The line bundle obeys a connectionrelative to K on ^Kn t \\i K - Formula 5.2 identifies line bundles with connectionsrelative to K, where the left hand side carries the constant connection. The choiceof an integrable lifting V of VX/K yields a lifting of the relative connection on the eline to an integrable connection relative to fc. Formula 5.2 identifies line bundles withintegrable connections where the left hand side carries the Gauß-Manin connection([1], 1.3).The e lines and connections are additive in exact sequences and compatiblewith push-downs. By a variant of Levelt's theorem for integrable formal connections,this allows to show that all connections are induced from admissible ones, for whichwe have the formula 5.3.

480 Hélène EsnaultQuestion 5.4. We don't know how to precisely relate the algebraic group viewpointdeveloped to treat the rank 1 case, and the special rational point found there, withthe polarized Fredholm line method which works in general.Acknowledgements. I thank the mathematicians I have worked with on the materialexposed in those notes. A large part of it has been jointly developed withSpencer Bloch. It is a pleasure to acknowledge the impact of his ideas on a programmeI had started earlier and we continued together. I thank Alexander Beilinson.An unpublished manuscript of his and David Kazhdan allowed me to understandcompletely one of the two constructions explained in [15]. His deep viewpointreflected in [1] changed the understanding of the formula we had as explained in[8]. I thank Pierre Deligne, whose ideas on epsilon factors have shaped much of mythinking. His letter to Serre on the rank 1 case is published as an appendix to [7],but the content of his seminar at the IHES in 1984 has not been available to me.I thank Takeshi Saito for his willingness to explain different aspects of the £ -adictheory.References[i[2:[3;[4;[*.[6;[9[io;[n[12:[is;Beilinson A., Bloch S., Esnault H., e-factors for Gauß-Manin Determinants,preprint 2001, 62 pages.Bloch S., Ogus A., Gersten's conjecture and the homology of schemes, Ann.Se. Éc.Normale Sup. IV, sér. 7 (1974), 181-201.Bloch S., Applications of the dilogarithm function in algebraic F'-theory andalgebraic geometry, Proc. int. Symp. on Alg. Geom., Tokyo 1977, 103-114(1977).Bloch S., Esnault H., Algebraic Chern-Simons theory. Am. J. of Mathematics119 (1997), 903-952.Bloch S.,Esnault H., A Riemann-Roch theorem for flat bundles, with values inthe algebraic Chern-Simons theory, Annals of Mathematics 151 (2000), 1-46.Bloch S., Esnault H., Relative Algebraic Characters, preprint 1999, 25, appearsin the Irvine Lecture Notes.Bloch S., Esnault H., Gauß-Manin determinants of rank 1 irregular connectionon curves, Math.Ann. 321 (2001), 15-87, with an addendum: the letter of P.Deligne to J.-P.Serre (Feb.74) on e-factors, 65-87.Bloch S., Esnault H., A formula for Gauß-Manin determinants, preprint 2000,37.Chern S., Simons J., Characteristic forms and geometric invariants, Ann. ofMaths User 99 (1974), 48-69.Cheeger, J., Simons J., Differential characters and geometric invariants, Geometryand Topology, Proc. Special Year College Park/Md. 1983/1984, Lect.Notes MatlL 1167 (1985), 50-80.Deligne P., Equations Différentielles à Points Singuliers Réguliers, Lect. Notesin Mathematics, 163 (1970), Springer-Verlag.Esnault H., Characteristic classes of flat bundles. Topology 27 (1988), 323-352.Esnault H., Characteristic classes of flat bundles, II. F'-theory, 6 (1992), 45-56.

Classes and Gauss-Manin Bundle 481[14] Esnault H., Coniveau of Classes of Flat Bundles Trivialized on a Finite SmoothCovering of a Complex Manifold, FJ-Theory, 8 (1994), 483-497.[15] Esnault H., Algebraic differential characters, College Park/Md. 1983/1984,Lect. Notes Math. 1167 (1985), 50-80. in Regulators in Analysis, Geometryand Number Theory, Progress in Mathematics, Birkhäuser Verlag, 171 (2000),89-117.[16] Kato K., Milnor F'-theory and the Chow group of zero cycles, Applications ofalgebraic F'-theory to algebraic geometry and number theory, Proc. AMS-IMS-SIAM Joint Summer Research Conf. Boulder/Colo.(1983), Part 1, Contemp.Math., 55 (1986), 241-253.[17] Levelt G., Jordan decomposition for a class of singular differential operators,Ark. Math. 13 (1975), 1-27.[18] Rost M., Chow groups with coefficients, Doc. Math., J. DMV, 1 (1996), 209-214.[19] Reznikov A., All regulators of flat bundles are torsion, Ann. Math., (2) 141(1995), 373-386.[20] Saito T., e-factor of a tamely ramified sheaf on a variety, Inventiones Math.113 (1993), 389-417.[21] Simpson C, Higgs bundles and local systerms, Inst. Hautes Etudes Sci. Pubi.Math., 75 (1992), 5-95.[22] Tate J., Residues of differentials on curves, Ann. Sci. École Norm. Sup., sér. 4,1 (1968), 149-159.

ICM 2002 • Vol. II • 483-494Hilbert Schemes of Points on SurfacesL. Göttsehe*AbstractThe Hilbert scheme S*- n > of points on an algebraic surface S is a simpleexample of a moduli space and also a nice (crêpant) resolution of singularitiesof the symmetric power S-"''. For many phenomena expected for moduli spacesand nice resolutions of singular varieties it is a model case. Hilbert schemesof points have connections to several fields of mathematics, including modulispaces of sheaves, Donaldson invariants, enumerative geometry of curves,infinite dimensional Lie algebras and vertex algebras and also to theoreticalphysics. This talk will try to give an overview over these connections.2000 Mathematics Subject Classification: 14C05, 14J15, 14N35, 14J80.Keywords and Phrases: Hilbert scheme, Moduli spaces, Vertex algebras,Orbifolds.0. IntroductionThe Hilbert scheme S^ of points on a complex projective algebraic surface S isa a parameter variety for finite subschemes of length n on S. It is a nice (crêpant)resolution of singularities of the n-fold symmetric power S^ of S. If S is a K3surface or an abelian surface, then S^ is a compact, holomorphic symplectic (thushyperkähler) manifold. Thus S^ is at the same time a basic example of a modulispace and an example of a nice resolution of singularities of a singular variety. Thereare a number of conjectures and general phenomena, many of which originating fromtheoretical physics, both about moduli spaces for objects on surfaces and about niceresolutions of singularities. In all of these the Hilbert scheme of points can be viewedas a model case and sometimes as the main motivating example. Hilbert schemesof points on a surface have connections to many topics in mathematics, includingmoduli spaces of sheaves and vector bundles, Donaldson invariants, Gromov-Witteninvariants and enumerative geometry of curves, infinite dimensional Lie algebras andvertex algebras, noneommutative geometry and also theoretical physics.*Abdus Salarti International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste,Italy. E-mail: gottsche@ictp.trieste.it

484 L. GöttscheIt is usually best to look at the Hilbert schemes S^ for all n at the sametime, and to study their invariants in terms of generating functions, because newstructures emerge this way. For Euler numbers, Betti numbers and conjecturallyfor the elliptic genus these generating functions will be modular forms and Jacobiforms. This fits into general conjectures from physics about invariants of modulispaces. Also the cohomology rings of the S^ for different n are closely tied together.The direct sum over n of all the cohomologies is a representation for the Heisenbergalgebra modeled on the cohomology of S, and the cohomology rings of the S^ canbe described in terms of vertex operators. In the case that the canonical divisor ofthe surface S is trivial, this leads to an elementary description of the cohomologyrings of the S^n\which coincides with the orbifold cohomology ring of the symmetricpower, giving a nontrivial check of a conjecture relating the cohomology ring of anice resolution of an orbifold to the recently defined orbifold cohomology ring.The Hilbert schemes S^ are closely related to other moduli spaces of objectson S, including moduli of vector bundles and moduli of curves e.g. via the Serrecorrespondence and the Mukai Fourier transform. This leads to applications to thegeometry and topology of these moduli spaces, to Donaldson invariants, and alsoto formulas in the enumerative geometry of curves on surfaces and Gromov-Witteninvariants. We want to explain some of these results and connections. We will notattempt to give a complete overview, but rather give a glimpse of some of the morestriking results.1. The Hilbert scheme of pointsIn this article S will usually be a smooth projective surface over the complexnumbers. We will study the Hilbert scheme S^ = Hilb n (S) of subschemes of lengthn on S. The points of S^ correspond to finite subschemes W C S of length n,in particular a general point corresponds just to a set of n distinct points on S.S^ is projective and comes with a universal family Z n (S) C S^ x S, consisting ofthe (W, x) with x £ W. An important role in applications of S^ is played by thetautological vector bundles F^ := n*q*(L) of rank n on S^. Here n : Z n (S) —^ Sând q : Z n (S) —¥ S are the projections and F is a line bundle on S.Closely related to S^ is the symmetric power S^ = S n /G n , the quotientof S n by the action of the symmetric group G n . The points of S^ correspond toeffective 0-cycles ^ n»[xj], where the xi are distinct points of S and the sum of therii is n. The forgetful mapp : SW -• S {n) ,W ^ Y, len (°w,x) Mis a morphism. The symmetric power S^ is singular, as for instance the fix-locus ofany transposition in G n has codimension 2. On the other hand by [22] S^ is smoothand connected of dimension 2n and p : S^ —t S^ is a resolution of singularities.In fact this is a particularly nice resolution: If Y is a Gorenstein variety, i.e. thedualizing sheaf is a line bundle Ky, a resolution / : X —t Y of singularities is calledcrêpant if it preserves the canonical divisor, that is f*Ky = Kx- It is easy to see

Hilbert Schemes of Points on Surface 485that p : S^ —t S^ is crêpant. In the special case that S is an abelian surface or aK3 surface one can get a better result: A complex manifold X is called holomorphicsymplectic if there exists an everywhere non-degenerate holomorphic 2-form onX. If furthermore is unique up to scalar, X is called irreducible holomorphicsymplectic. A Kahler manifold X of real dimension in is called hyperkähler if itsholonomy group is Sp(n). Compact complex manifolds are holomorphic symplecticif and only of they admit a hyperkähler metric. In [7] it is shown that for a K3 surfaceS the Hilbert scheme S^ is irreducible holomorphic symplectic. There also, foran abelian surface A, the generalized Kummer varieties are constructed from AM.They form another series of irreducible holomorphic symplectic manifolds. The onlyotherexamples of compact hyperkähler manifolds, known not to be diffeomorphicto one in the above two series are the two isolated examples of resolutions of singularmoduli spaces of sheaves on K3 and abelian surfaces in [47],[48].2. Betti number, Euler numbers, elliptic genusFor many questions about the Hilbert schemes S^ one should look at alln at the same time. The first instance of this are the Betti numbers and Eulernumbers, for which we can find generating functions in terms of modular forms.Let % := {T £ C | S(r) > 0}. A modular form of weight k on Sl(2, Z) is a function/ : H -• C s.th.'(£ï!)='Furthermore, writing q = e 2 " T , we require that, in the Fourier development /(r) =12nez a nQ n ,a ll the the negative Fourier coefficients vanish. If also cto = 0, /is called a cusp form. The most well-known modular form is the discriminantA(T) := g[] n>0 (l — q n ) 24 , the unique cusp form of weight 12. The Dirichlet etafunction is n = A 1 / 24 .For a manifold X we denote by p(X, z) := '^2i(^l) t bi(X)z t the Poincaré polynomialand by e(X) = p(X, 1) the Euler number. The Betti numbers and Eulernumbers of the S^ have very nice generating functions [24]:j2p(s [n] ,z)t n =nn(i-^- 2+ v) ( - i),+iSi(s) . (2.1)n>0 k>\ i=0In particular Y, n >o e ( sln] )Q n ~

486 L. GöttscheA partition a = (ni,...,n r ) £ P(n) can also be written as a = (l ai ,... ,n a "),where a, is the number of occurences of i in (m,..., n r ). We put \a\ = r = ^ a^.Then (2.1) can be reformulated asp(S^,z)= Y, p(S {ai) x ...xS {a - ) ,z)z 2{n -^).(2.2)a£P(n)This result has been refined to Hodge numbers in [30], [11] and this was generalizedin [13] to the Douady space of a complex surface. It has been further refined todetermine the motive and the Chow groups [14] and the element in the Grothendieckgroup of varieties of S M [28].Partially motivated by (2.1) and using arguments from physics in [15] a conjecturalrefinement to the Krichever-Höhn elliptic genus is given. We restrict ourattention to the case that Kx = 0 when the elliptic genus is a Jacobi form. For acomplex vector bundle F on a complex manifold X and a variable t we putA t (F) :=0A'(E)t', St(E) := (£)S k (E)t k .k>0 k>0For the holomorphic Euler characteristic we write x(A^, A t (F)) := ^x(X,and similarly for St(E). Then the elliptic genus is defined byA k E)t k!Writing (S) := ^2 m>0 i c(m,l)q m y l ,the conjecture is£ 4>(S^)p N = u Ti n-^^-^ - LJ M yfiQniyl\c(nniN>0 n>0,m>0 - (1 — p n q m: l)3. Infinite dimensional Lie algebras and the cohomologyringWe saw that one gets nice generating functions in n for the Betti numbers of theS^. Now we shall see that the direct sum of all the cohomologies of the S M carriesa new structure which governs the ring structures of the Hilbert schemes. We onlyconsider cohomology with rational coefficients and thus write H*(X) for H*(X, Q).We write H := H*(S); for n > 0 let H„ := H*(S^). and H := ® n > 0 H„.We shall see that H is an irreducible module under a Heisenberg algebra. Thiswas conjectured in [54] and proven in [45],[32]. H contains a distingished element1 £ H 0 = Q. We denote by J s and J s[n] the evaluation on the fundamental classof S and S^. Define for n > 0 the incidence varietyZ hn := {(Z,x,W) £ SW xSx SV +n^ \ Z c W,p(W) - p(Z) = n[x]},

and use this to define operatorsHilbert Schemes of Points on Surface 487p n :H -> End(H); p n (a)(y) := pi^(pr* 2 (a) Upr{(y) D [Z,,„]).Let p- n (a) := ( —l) n p_ n (a)t, where t denotes the adjoint with respect to / 5 [ n] , andPo(a) := 0. By [45],[32] the p n (a) fulfill the commutation relations of a Heisenbergalgebra:[Pn(a),Pm(ß)] = (îT^nôn-n-J a-ßJidH, n, m £ Z, a, ß £ H. (3.1)J SWe can interpret this as follows. Let H = H + ® FA be the decomposition into evenand odd cohomology. Put S*(H) := @ i>0 S % (H + ) ® @ i>0 A*(FA). The Fock spaceassociated to H is F(H) := S*(H ® tQ[t]). Using the above theorems one readilyshows that there is an isomorphism of graded vector spaces F(H) —¥ H. With thisH becomes an irreducible module under the Heisenberg-Clifford algebra.The ring structure of the H*(S^) is connected to the Heisenberg algebraaction. Given an action of a Heisenberg algebra, a standard construction givesan action of the corresponding Virasoro algebra. The important fact however,proven in [37] is that the Virasoro algebra generators have a geometrical interpretationtying them to the ring structure of the cohomology of the S^. LetÖ : S —¥ S x S be the diagonal embedding, and let Ö* : H*(S) —¥ H*(S x S) bethe corresponding pushforward. Let p v p n - v 8(a) : H*(S) —¥ End(H) be definedas p v p n - v (ß x 7) := p„(/?)p n _„(7) applied to 5* (a) £ H x H. For B / 0 defineL n (a) := YaPvP-v8*(a). These operatorssatisfy the relations of the Virasoro algebra:Yi^ — fi / f \[L n (a),L m (ßj] = (n -m)L n+m (ab) + 6 n _ TO ——— f / c 2 (S , )a6jid H . (3.2)Let d : H —^ H be the operator which on each H*(S^) is the multiplication withci(€>M), where O^ = nJZ n (Sj) is the tautological vector bundle associated tothe trivial line bundle on S. The tie given in [37] to the ring structure is:[d,p n (a)] = nL n (a) + (f\p n (K s a), n>0,a£ H*(S). (3.3)In [42], for each a £ H*(S), classes a^ £ H*(S^) are defined as generatizationsof the Chern characters ch(F^) of tautological bundles, which are studied in [37].The homogeneous components of the a^ generate the ring H*(S^). [37],[42]relate the multiplication by the a M to the higher order commutators with d: Let«[*] : H -> H be the operator which on every H*(S^) is the multiplication with«["], then[a^,pi(ßj] =eMad(d))pi(aß), (3.4)where for an operator A : H —t H, ad(d)A = [d, A].(3.2),(3.3),(3.4) determine the cohomology rings of the S^. In case Kg = 0this is used in [38],[39] to give an elementary description of the cohomology ringsH*(S^) in terms of the symmetric group, which we will relate below to orbifoldcohomology rings.

488 L. Göttsche4. Orbifolds and orbifold cohomologyLet X be a compact complex manifold with an action of a finite group G andassume that for all 1 ^ g £ G the fixlocus X s has codimension > 2. The quotientX/G will usually be singular, but the stack quotient [X/G] is a smooth orbifold. Inphysics [16],[17] the following orbifold Euler characteristic has been introducede(X,G):= Y.gh=hg

Hilbert Schemes of Points on Surface 489If n :Y —¥ X/G is only a crêpant resolution but not hyperkähler, then usuallyH* rb ([X/G\) and H*(Y) are not isomorphic as rings. However in [51] a preciseconjecture is made relating the two: One has to correct H* rb ([X/G\) by Gromov-Witten invariants coming from classes of rational curves Y contracted by n. In thecase of the Hilbert scheme these curve classes are the multiples of a unique class.The conjecture was verified for S^.5. Moduli of vector bundlesWe denote by Mg(r,ci,c 2 ) the moduli space of Gieseker F-semistable coherentsheaves of rank ronS with Chern classes ci, c 2 . Here a sheaf T of rank r > 0 onS is called semistable, \fx(f3®H n )/r' < x(J 7 ®H n )/r for all sufficiently large n andfor all subsheaves Q C T of positive rank r'. As Mg(l,0,c 2 ) ~ Pic°(S) x S^- C2 \ theHilbert scheme of points is a special case. We will often restrict our attention to thecase of r = 2 and write Mg(ci,c 2 ). The Hilbert schemes of points are related in severalways to the Mg(ci,c 2 ). The most basic tie is the Serre correspondence whichsays that under mild assumptions rank two vector bundles on S can be constructedas extensions of ideal sheaves of finite subschemes by line bundles. Related to thisis the dependence of the Mg(ci,c 2 ) on the ample divisor H via a system of wallsand chambers. This has been studied by a number of authors (e.g. [49],[23],[18]).Assume for simplicity that S is simply connected. A class £ £ H 2 (S,Z) defines awall of type (01,02) if £ + ci £ 2H 2 (S, Z) and c\ — 4c2 < £ 2 < 0. The correspondingwall is W^ = {a £ H 2 (S,H) | a • £ = 0}. The connected components of thecomplement of the walls in H 2 (X, R) are called the chambers of type (01,02). If asheaf £ £ Mg(ci,c 2 ) is unstable with respect to F, then there is a wall W^ withF£ < 0 < F£ and an extension0^1 z ®A^£^l w xB^0,where A,B £ Pic(S) with A — B = £ and lz, lw are the ideal sheaves of zerodimensional schemes on S. It follows that Mg(ci,c 2 ) depends only on the chamberof H and the set theoretic change under wallcrossing is given in terms of Hilbertschemes of points on S. In the case e.g. of rational surfaces and K3-surfaces,the change can be described as an explicit sequence of blow ups along P*. bundlesover products S^ x S^ followed by blow downs in another direction [23],[18].The change of the Betti and Hodge numbers under wallcrossing can be explicitlydeterminedand this can be used e.g. to determine the Hodge numbers of Mg (01,02)for rational surfaces. For suitable choices of H one can find the generating functionsin terms of modular forms and Jacobi forms [27].The appearance of modular forms is in accord with the S-duality conjectures[54] from theoretical physics, which predict that under suitable assumptions thegenerating functions for the the Euler numbers of moduli spaces of sheaves onsurfaces should be given by modular forms. One of the motivating examples forthis conjecture is the case that S is a K3-surface. In this case the conjecture is that,if Mg(ci,c 2 ) is smooth, then it has the same Betti numbers as the Hilbert scheme ofpoints on S of the same dimension. Assuming this, the formula (2.1) for the Hilbert

490 L. Göttscheschemes of points implies that the generating function for the Euler numbers is amodular form. If Ci is primitive this was shown in [29]. The result was shown ingeneral for Mg(r, 01,02) with r > 0 in [57],[59], by relating the Hilbert scheme andthe moduli space via birational correspondences and deformations. One concludesthat Mg(r,ci,c 2 ) has the same Betti numbers as the Hilbert scheme of points ofthe same dimension, as both spaces are holomorphic symplectic [44] and birationalmanifolds with trivial canonical class have the same Betti numbers [5]. Similarresults are shown in [58] for abelian surfaces. Other motivating examples for theS-duality conjecture were the case of P2 [56] and the blowup formula relating thegenerating function for the Euler numbers of the moduli spaces of rank 2 sheaves ona surface S to that on the blowup of S in a point, which has since been established([40],[41], see also [27]).The moduli spaces Mg(ci,c 2 ) can be used to compute the Donaldson invariantsof S. In case p g = 0 these depend on a metric, corresponding to the dependenceof Mg(ci,c 2 ) on H. For rational surfaces one can use the above description of thewallcrossing for the Mg(ci,c 2 ) to determine the change of the Donaldson invariantsin terms of Chern numbers of generalizations of the tautological sheaves FMon products S^ x S^ of Hilbert schemes of points [18],[23]. The leading termsof these expressions can be explicitly evaluated. The wallcrossing of Donaldsoninvariants has also been studied in gauge theory (e.g.[35],[36]). There a conjectureabout the structure of the wallcrossing formulas is made. Assuming this conjectureone can determine the generating functions for the wallcrossing in terms of modularforms [25],[31].6. Enumerative geometry of curvesNow we want to see some striking relations between the Hilbert schemes Sând the enumerative geometry of curves on S. First let S be a K3 surface andF a primitive line bundle on S. Then F 2 = 2g — 2, where the linear system \L\has dimension g and a smooth curve in \L\ has geometric genus g. As a nodeimposes one linear condition, one expects a finite number of rational curves (i.e.curves of geometric genus 0) in \L\. Partially based on arguments from physics, aformula is given in [55] for the number of rational curves in \L\ and in [8] this mademathematically precise. Writing n g for the number of rational curves in \L\ withF 2 = 2g — 2 (counted with suitable multiplicities), the formula isE%^ = !> (6- 1 )9>0where A is again the discriminant. By (2.1) this implies the surprizing fact that n gis just the Euler number of S^. In fact the argument relates the number of curvesto S^: Yet C —t \L\ be the universal curve and let J —t \L\ be the correspondingrelative compactified Jacobian, whose fibre over the point corresponding to a curveC is the compactified Jacobian J(C) [3]. One can show that e(J(Cj) = 0 unlessg(C) = 0. It follows that e(J) is the sum over the e(J(Cj) for C £ \L\ withg(C) = 0. It is not difficult to show that S^ and J are birational. J is also smooth

Hilbert Schemes of Points on Surface 491and hyperkähler as a moduli space of sheaves on a K3 surface [44]. As already usedin the section on vector bundles, birational manifolds with trivial canonical bundlehave the same Betti numbers [5]. Thus J and S^ have the same Euler numbers.This shows (6.1), where the multiplicity of a rational curve C is e(J(Cj). By [20]this multiplicity is the multiplicity of the corresponding moduli space of stable maps,in particular it is always positive. In [26] a conjectural generalization of (6.1) toarbitrary surfaces S is given.Conjecture 6.11. For all ö > 0, there exists a universal polynomial Tg(x,y,z,w), such that forall projective surfaces S and all sufficiently ample line bundles L on S thenumber of ö-nodal curves in a general ö-dimensional linear subspace of \L\ isT s (x(L),x(Os),LKs,K 2 ).2. There are universal power series Bi,B 2 £ Z[[q]] whose coefficients can beexplicitely determined, such that^XE),x(Ps),LK s , Ki fi(DG2f= (^^Ûvf-Here D = q-fi and G2 = —2\^--The expectation that universal polynomials should exist is implicit in [53],[34]where the Tg are determined for Ö < 8. In [26] also another tie of the conjecture tothe Hilbert scheme of points is given: conjecturally the numbers Tg(x(L),x(Os),LKs, Kg) are suitable intersection numbers on the Hilbert scheme S^ of 3ó pointsof S. If S is a K3 surface or an abelian surface, then the conjecture predicts thatthe generating function can be written in terms of modular forms. In this casea modified version of Conjecture 6.1 was proven for primitive line bundles in [9]and [10], replacing the numbers of o"-nodal curves with the corresponding modifiedGromov-Witten invariants. In [43] a proof of the conjecture is published.References[1] D. Abramovich, T. Graber, A. Vistoli, Algebraic orbifold quantum products,preprint math.AG/0112004.[2] D. Abramovich, A. Vistoli, Compactifying the space of stable maps, J. Amer.Math. Soc, 15 (2002), 27^75.[3] A. Altman, S. Kleiman, Compactifying the Picard scheme. Adv. in Math., 35(1980), 50-112.[4] A. Altman, S. Kleiman, Compactifying the Picard scheme II, Amer. J. Math.,101 (1979), 10-41.[5] V. Batyrev, Birational Calabi-Yau n-folds have equal Betti numbers. Newtrends in algebraic geometry, 1-11, 1999.[6] V. Batyrev, L. Borisov, Mirror duality and string-theoretic Hodge numbers,Invent. Math., 126 (1996), 183^203.

492 L. Göttsche[7] A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle,J. Differential Geom,., 18 (1983), 755-782.[8] A. Beauville, Counting rational curves on K3 surfaces, Duke Math. J., 97(1999), 99-108.[9] J. Bryan, N. C. Leung, The enumerative geometry of K3 surfaces and modularforms. J. Amer. Math. Soc, 13 (2000), 371-110.[10] J. Bryan, N. C. Leung, Generating functions for the number of curves onabelian surfaces, Duke Math. J., 99 (1999), 311-328.[11] J. Cheah, On the cohomology of Hilbert schemes of points, J. Algebraic Geom.,5 (1996), 479-511.[12] W. Chen, Y. Ruan, A New Cohomology Theory for Orbifold, preprintmath.AG/0004129.[13] M. A. de Cataldo, L. Migliorini, The Douady space of a complex surface, Adv.Math., 151 (2000), 283-312.[14] M. A. de Cataldo, L. Migliorini, The Chow groups and the motive of the Hilbertscheme of points on a surface, preprint math. AG/0005249.[15] R. Dijkgraaf, G. Moore, E. Verlinde, H. Verlinde, Elliptic genera of symmetricproducts and second quantized strings, Comm. Math. Phys., 185 (1997), 197-209.[16] L. Dixon, J. Harvey, C. Vafa, E. Witten, Strings on orbifolds, Nuclear Phys.B, 261 (1985), 678-686.[17] L. Dixon, J. Harvey, C. Vafa, E. Witten, Strings on orbifolds II, Nuclear Phys.B 274 (1986), 285-314.[18] G. Ellingsrud, L. Göttsche, Variation of moduli spaces and Donaldson invariantsunder change of polarization, J. Reine Angew. Math., 467 (1995), 1-49.[19] G. Ellingsrud, S. A. Str0mme, On the homology of the Hilbert scheme of pointsin the plane, Invent. Math., 87 (1987), 343-352.[20] B. Fantechi, L. Göttsche, D. van Straten, Euler number of the compactifiedJacobian and multiplicity of rational curves J. Algebraic Geom., 8 (1999), 115-133.[21] B. Fantechi, L. Göttsche, Orbifold cohomology for global quotients, preprintmath.AG/0104207.[22] J. Fogarty, Algebraic families on an algebraic surface, Amer. J. Math., 90(1968), 511-521.[23] R. Friedman, Z. Qin, Flips of moduli spaces and transition formulas for Donaldsonpolynomial invariants of rational surfaces, Comm. Anal. Geom., 3 (1995),11-83.[24] L. Göttsche, The Betti numbers of the Hilbert scheme of points on a smoothprojective surface, Math. Ann. 286 (1990), 193-207.[25] L. Göttsche, Modular forms and Donaldson invariants for 4-manifolds with6+ = 1, J. Amer. Math. Soc, 9 (1996), 827-843.[26] L. Göttsche, A conjectural generating function for numbers of curves on surfaces,Comm. Math. Phys., 196 (1998), 523-533.[27] L. Göttsche, Theta functions and Hodge numbers of moduli spaces of sheaveson rational surfaces, Comm. Math. Phys., 206 (1999), 105-136.

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ICM 2002 • Vol. II • 495-502Vector Bundles on a K3 Surface*Shigeru Mukai 1 'AbstractA K3 surface is a quaternionic analogue of an elliptic curve from a viewpoint of moduli of vector bundles. We can prove the algebraicity of certainHodge cycles and a rigidity of curve of genus eleven and gives two kind ofdescriptions of Fano threefolds as applications. In the final section we discussa simplified construction of moduli spaces.2000 Mathematics Subject Classification: 14J10, 14J28, 14J60.1. IntroductionA locally free sheaf F of ox-modules is called a vector bundle on an algebraicvariety X. As a natural generalization of line bundles vector bundles have twoimportant roles in algebraic geometry. One is the linear system. If F is generatedby its global sections H°(X,E), then it gives rise to a morphism to a Grassmannvariety, which we denote by $E '• X —y G(H°(E),r), where r is the rank of F.This morphism is related with the classical linear system by the following diagram:X •^ G(H°(E),r)$L 1 I1 IPliicker(1)P*H°(L) ••••-•P*(A r F°(F)),where F is the determinant line bundle of F and $L is the morphism associated toit.The other role is the moduli. The moduli space of line bundles relates a(smooth complete) algebraic curve with an abelian variety called the Jacobian variety,which is crucial in the classical theory of algebraic functions in one variable.The moduli of vector bundles also gives connections among different types varieties,and often yields new varieties that are difficult to describe by other means.* Supported in part by the JSPS Grant-in-Aid for Scientific Research (A) (2) 10304001.1 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan. E-mail: mukai@kurims.kyoto-u.ac.jp

496 Shigeru MukaiIn higher rank case it is natural to consider the moduli problem of F underthe restriction that det F is unchanged. In view of the above diagram, vectorbundles and their moduli reflect the geometry of the morphism X —y P*H°(L)via Grassmannians and Pliicker relations. In this article we consider the case whereX is a K3 surface, which is one of two 2-dimensional analogues of an elliptic curveand seems an ideal place to see such reflection.2. Curves of genus oneThe moduli space of line bundles on an algebraic variety is called the Picardvariety. The Picard variety Pic C of an algebraic curve C is decomposed into thedisjoint union IJ d€Z Picd C by the degree d of line bundles. Here we consider thecase of genus 1. All components Pic^G are isomorphic to G if the ground field1is algebraically closed. But this is no more true otherwise. For example theJacobian Pico G has always a rational point but G itself does not. 2 We give otherexamples:Example 1 Let G4 be an intersection of two quadrics qi (x) = q 2 (x) = 0 in theprojective space P 3 and F the pencil of defining quadrics. Then the Picard varietyPÌC2 G4 is the double cover of F ~ P 1 and the branch locus consists of 4 singularquadrics in P. Precisely speaking, its equation is given by r 2 = disc (Xiqi + X2q2).Let G(2,5) C P 9 be the 6-dimensional Grassmann variety embedded into P 9by the Pliicker coordinate. Its projective dual is the dual Grassmannian G(5,2) cP 9 , where G(2,5) parameterizes 2-dimensional subspaces and G(5,2) quotient spaces.Example 2 A transversal linear section G = G(2,5) n Hi n • • • n H 5 is a curve genus1 and of degree 5. Its Picard variety PÌC2 G is isomorphic to the dual linear sectionG = G(5,2) n {Hi,... ,H 5 ), the intersection with the linear subspace spanned by 5points Hi,... ,H 5 £ P 9 .3. Moduli K3 surfacesA compact complex 2-dimensional manifold S is a K3 surface if the canonicalbundle is trivial and the irregularity vanishes, that is, Kg = F 1 (ös) = 0. Asmooth quartic surface S4 C P 3 is the most familiar example. Let us first look atthe 2-dimensional generalization of Example 1:Example 3 Let Ss be an intersection of three general quadrics in P 5 and N thenet of defining quadrics. Then the moduli space Ms(2,ös(l),2) is a double coverof N ~ P 2 and the branch locus, which is of degree 6, consists of singular quadricsin N.Here Ms(r, L,s), L being a line bundle, is the moduli space of stable sheavesF on a K3 surface S with rank r, det F ~ F and x(E) = r + s. Surprisingly two1 More precisely, this holds true if C has a rational point.2 Two components Pico C and Pic 9 _i C deserve the name Jacobian. They coincide in our case9=1-

Vector Bundles on a K3 Surface 497surfaces S$ = (2) n (2) n (2) c P 5 and Af s (2,O s (l),2) ^h P 2 in this exampleare both K3 surfaces. This is not an accident. In respect of moduli space, vectorbundles a K3 surface look like Picard varieties in the preceding section.Theorem 1 ([10],[11]) The moduli space M$(r, L, s) is smooth of dimension (L 2 ) —2rs + 2. Ms(r, L, s) is again a K3 surface if it is compact and and of dimension 2.A K3 surface S and a moduli K3 surface appearing as Ms(r,L,s) are notisomorphic in general 3 but their polarized Hodge structures, or periods, are isomorphicto each other over Q ([11]). The moduli is not always fine but there alwaysexists a universal P r_1 -bundle over the product S x Ms(r,L,s). Yet A be theassociated sheaf of Azumaya algebras, which is of rank r 2 and locally isomorphic tothe matrix algebra Mat r (OsxM)- A is isomorphic to End E if a universal family Eexists. The Hodge isometry between H 2 (S,Q) and H 2 (Ms(r,L,s),Q) is given byc 2 (-4)/2r G H 4 (S x Af,Q) ~ H 2 (S,Q) V ® H 2 (M,Q).Example 2 has also a K3 analogue. Let Si 2 C P 15 be a 10-dimensionalspinor variety SO(10)/U(5), that is, the orbit of a highest weight vector in theprojectivization of the 16-dimensional spinor representation. The anti-canonicalclass is 8 times the hyperplane section and a transversal linear section S = £12 nFi n • • • n H$ is a K3 surface (of degree 12). As is similar to G(2,5) the projectivedual Si2 C P 15 of Si2 is again a 10-dimensional spinor variety.Example 4 The moduli space Ms(2,Os(l),3)section S = £12 n {Hi,..., H$).is isomorphic to the dual linearIn this case, moduli is fine and the relation between S and the moduli K3 aredeeper. The universal vector bundle E on the product gives an equivalence betweenthe derived categories T)(S) and T)(S) of coherent sheaves, the duality S ~ S holds(cf. [17]) and moreover the Hilbert schemes HihV S and Hilb" S are isomorphic toeach other.Remark (1) Theorem 1 is generalized to the non-compact case by Abe [1].(2) If a universal family E exists, the derived functor with kernel E gives an equivalenceof derived categories of coherent sheaves on S and the moduli K3 (Bridgeland[4]). In even non-fine case the derived category D(S) is equivalent to that of themoduli K3 M twisted by a certain element a £ H 2 (M,Z/rZ) (Cäldäraru [5]).4. Shafarevieh conjectureLet S and T be algebraic K3 surfaces and / a Hodge isometry betweenH 2 (S, Z) and H 2 (T,Z). Then the associated cycle Z f £ H 4 (SxT,Z) ~H 2 (S,Z) V ®H 2 (T, Q) on the product S x T is a Hodge cycle. This is algebraic by virtue of theTorelli type theorem of Shafarevieh and Piatetskij-Shapiro. Shafarevieh conjecturedin [23] a generalization to Hodge isometries over Q. Our moduli theory is able toanswer this affirmatively.3 We take the complex number field C as ground field except for sections 2 and 7.

498 Shigeru MukaiTheorem 2 Let f : H 2 (S,Q) —• H 2 (T,Q) be a Hodge isometry. Then the associated(Hodge) cycle Zf £ H 4 (S x T, Q) is algebraic.In [11], we already proved this partially using Theorem 1 (cf. [21] also). Whatwe need further is the moduli space of projective bundles. Let F —• S be a P r_1 -bundle over S. The cohomology class [F] £ ^(SjPGL^jOsj) and the naturalexact sequence (in the classical topology)0 —•+ Z/rZ —•+ SL(r,O s ) —• PGL(r,O s ) —• 1define an element of H 2 (S,Z/rZ), which we denote by w(P).Fix a £ H 2 (S,Z) and r, we consider the moduli of P r_1 -bundles F over Swith w(P) = a mod r which are stable in a certain sense. If the self intersectionnumber (a 2 ) is divisible by 2r, then the moduli space contains a 2-dimensionalcomponent, which we denote by Ms(a/r). The following, a honest generalizationof computations in [11], is the key of our proof:Proposition 1 Assume that (a 2 ) is divisible by 2r 2 . Then H 2 (Ms(a/r),Z) isisomorphic to L 0 + Za/r c H 2 (S, Q) as polarized Hodge structure, where L 0 is thesubmodule of H 2 (S, Z) consisting of ß such that the intersection number (ß.a) isdivisible by r.For example let S2 be a double cover of P 2 with branch sextic. If a G H 2 (S, Z)satisfies (a. h) = 1 mod 2 and (a 2 ) = 0 mod 4, then Ms (a/2) is a K3 surface of degree8. This is the inverse correspondence of Example 1 (cf. [26], [20]). Details will bepublished elsewhere.5. Non-Abelian Brill-Noether locusLet G be a smooth complete algebraic curve. As a set a Brill-Noether locus ofG is a stratum of the Picard variety Pic G defined by h°(L), the number of globalsections of a line bundle F. The standard notation isW r d = {[L] I h°(L) > r + 1} c Pic rf G,for which we refer [2]. Non-Abelian analogues are defined in the moduli space lie (2)of stable 2-bundles on G similarly. The non-Abelian Brill-Noether locus of type IIIisSU C (2,K :n) = {F\ detF~O c (K c ), h°(F) > n} C U c (2)for a non-positive integer n, and type II isSU C (2,K : nG) = {F\ det F ~ det G ® O c (K c ), dimHom(G,F) > n} c U c (2)for a vector bundle G of rank 2 and n = deg G mod 2. By virtue of the (Serre)duality, these have very special determinantal descriptions. We give them schemestructures using these descriptions ([16]).

Vector Bundles on a K3 Surface 499Assume that G lies on a K3 surface S. If F belongs to Ms(r, L, s), then therestriction F|c is of canonical determinant and we have h°(E\c) > h°(E) > x(E) =r + s. So E\c belongs to Slic(2,K : r + s) if it is stable. This is one motivationof the above definition. The case of genus 11, the gap value of genera where Fano3-folds of the next section do not exist, is the moist interesting.Theorem 3 ([15]) If C is a general curve of genus 11, then the Brill-Noether locusT = Slic(2,K : 7) of type III is a K3 surface and the restriction L of thedeterminant line bundle is of degree 20.There exists a universal family £ on G x T. We moreover have the following:• the restriction £\ XX T is is stable and belongs to Mr(2, L, 5), for every x £ C,and• the classification morphism G —• T = MT(2,L,5) is an embedding.This embedding is a non-Abelian analogue of the Albanese morphism X —•Pico(PicoX) and we have the following:Corollary A general curve of genus eleven has a unique embedding to a K3 surface.In [9], we studied the forgetful map ip g from the moduli space V g of pairs ofa curve G of genus g and a K3 surface S with C C S to the moduli space M g ofcurves of genus g and the generically finiteness of ipn. The above correspondenceG H> T gives the inverse rational map of ipn. We recall the fact that ipio is notdominant in spite of the inequality dimoio = 29 > dim .Mio = 27 ([12]).6. Fano 3-foldsA smooth 3-dimensional projective variety is called a Fano 3-fold if the anticanonicalclass — Kx is ample. In this section we assume that the Picard groupPicX is generated by —Kx- The self intersection number (—Kx) 3 = 2g — 2 isalways even and the integer g > 2 is called the genus, by which the Fano 3-folds areclassified into 10 deformation types. The values of g is equal to 2,..., 10 and 12. AFano 3-folds of genus g < 5 is a complete intersection of hypersurfaces in a suitableweighted projective space.By Shokurov [25], the anticanonical linear system | — Kx\ always contain asmooth member S, which is a K3 surface. In [13] we classified Fano 3-folds X ofPicard number one using rigid bundles, that is, F £ Ms(r,L,s) with (F 2 ) — 2rs =—2. For example X is isomorphic to a linear section of the 10-dimensional spinorvariety, that is,x~Si 3 nffin"-nff 7) (2)in the case of genus 7 and a linear sectionx ~Zi 6 nHinH 2 nH 3 , (3)of the 6-dimensional symplectic, or Lagrangian, Grassmann variety Sie = SP(6)/U(3) C P 13 in the case of genus 9. The non-Abelian Brill-Noether loci shed newlight on this classification.

500 Shigeru MukaiTheorem 4 A Fano 3-fold X of genus 7 is isomorphic to the Brill-Noether locusSlic(2,K : 5) of Type III for a smooth curve C of genus 7.This description is dual to the description (2) in the following sense. First twoambient spaces of X, the moduli lie(2) and the Grassmannian G(5,10) D £12 areof the same dimension. Secondly let A r i and N 2 be the normal bundles of X inthese ambient spaces. Then we have A r i ~ A^7 ® Öx(-Kx), that is, two normalbundles are twisted dual to each other.Theorem 5 A Fano 3-fold of genus 9 is isomorphic to the Brill-Noether locusSUc(2,K:3G) of Type II for a nonsingular plane quartic curve C and G a rank 2stable vector bundle over C of odd degree.This descriptions is also dual to (3) in the above sense: The moduli lie (2) andthe Grassmannian G(3,6) D Eie are of the same dimension and the two normalbundles of X are twisted dual to each other. Each Fano 3-fold of genus 8, 12 andconjecturally 10 has also such a pair of descriptions.7. Elementary constructionThe four examples in sections 1 and 2 are very simple and invite us to asimplification of moduli construction. Let G4 be as in Example 1 and Mat2 theaffine space associated to the 16-dimensional vector space © i=0 (C 2 ® C 2 )ar,, where(xi) is the homogeneous coordinate of P 3 . Let Mat2,i be the closed subschemedefined by the condition thatA(x) = Xà=o AìXì £ Mat2 is of rank < 1 everywhere on G4and R its coordinat ring. Then the Picard variety PÌC2 G4 is the projective spectrumProjF SL(2^xSL(2^ of the invariant ring by construction. (See [18] for details.) Theabove condition is equivalent to that det A(x) is a linear combination of qi(x) andq 2 (x). The invariant ring is generated by three elements by Theorem 2.9.A of Weyl[28]. Two of them, say Bi,B 2 , are of degree 2 and correspond to qi(x) and q 2 (x),respectively. The rest, say T of degree 4, is the determinant of 4 by 4 matrixobtained from the four coefficients A 0 ,...,A 3 £ C 2 C 2 of A(x). There is onerelation T 2 = f±(Bi,B 2 ) . Hence ProjF SL(2^xSL(2^ is a double cover of P 1 asdesired.The moduli space Ms(2,ös(l),2) in Example 3 is constructed similarly. LetAlti be the affine space associated to the vector space © i=0 (A J C 4 )ar, and Alt^the subscheme defined by the condition that Xà=o -^* x * e -41*4 is of rank < 2everywhere on S$- Then the invariant ring of the action of SL(4) on Alt^2 isgenerated by four elements Bi,B 2 ,B$,T of degree 2, 2, 2, 6. There is one relationT 2 = fi(Bi,B 2 , B 3 ) and Ms(2,O s (l), 2), the projective spectrum Proj R SL{4) , is adouble cover of P 2 as described.The moduli space of vector bundles on a surface was first constructed byGieseker [6]. He took the Mumford's GIT quotient [19] of Grothendieck's Quotscheme [7] by PGL and used the Gieseker matrix to measure the stability of the

Vector Bundles on a K3 Surface 501action. In the above construction, we take the quotient of Alt^2, which is nothingbut the affine variety of Gieseker matrices of suitable rank 2 vector bundles, by ageneral linear group GF(4).The Jacobian, or the Picard variety, of a curve is more fundamental. Weil [27]constructed Pic s G as an algebraic variety using the symmetric product Sym s Gand showed its projectivity by Lefschetz' 30 theorem. Later Seshadri and Oda[24] constructed Pic^ G for arbitray d (over the same ground field as G) by alsotaking the GIT quotient of Quot schemes. The above constructions eliminate Quotschemes and the concept of linearization from those of Gieseker, Seshadri and Oda.References[i[2[io;[n[12;[13;[14;K. Abe: A remark on the 2-dimensional moduli spaces of vector bundles onK3 surfaces, Math. Res. Letters, 7(2000), 463^470.E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris : Geometry of algebraiccurves, I, Springer-Verlag, 1985.M. F. Atiyah : Vector bundles over an elliptic curve, Proc. London Math.Soc, 7(1957), 414-452.T. Bridgeland: Equivalences of triangulated categories and Fourier-Mukaitransformations, Bull. London Math. Soc, 31(1999), 25-34.A. Cäldäraru: Non-fine moduli spaces of sheaves on K3 surfaces, preprint.D. Gieseker: On the moduli of vector bundles on an algebraic surface, Ann.Math. 106(1977), 45-60.A. Grothendieck : Techniques de construction et theorem d'existence engéométrie algébrique, iV: Les schémas de Hilbert, Sem. Bourbaki, t. 13,1960/61, n° 221.V.A. Iskovskih : Fano 3-folds, II, Izv. Akad. Nauk SSSR,42(1978) : Englishtranslation, Math. USSR Izv. 12(1978), 469^505.S. Mori and S. Mukai: The uniruledness of the moduli space of curves of genus11, in 'Algebraic Geometry, Proceedings, Tokyo/Kyoto 1982', Series: LectureNotes in Mathematics, vol. 1016, (M, Raynaud and T. Shioda eds.), SpringerVerlag, 1983, 334^353.S. Mukai: Symplectic structure of the moduli space of sheaves on an abelianor K3 surface, Invent. Math., 77(1984), 10H16.—: On the moduli space of bundles on K3 surfaces, I, in 'Vector Bundles onAlgebraic Varieties ', Tata Institute of Fundamental Research, Bombay, 1987,341-413.— : Curves, K3 surfaces and Fano manifolds of genus < 10, in 'AlgebraicGeometry and Commutative Algebra in honor of Masayoshi NAGATA', (H.Hijikata and H. Hironaka et al eds.), Kinokuniya, Tokyo, 1988, 367^377.— : Biregular classification of Fano threefolds and Fano manifolds of coindex3, Proc. Nat. Acad. Sci., USA, 86 (1989), 3000^3002.— : New developments in the theory of Fano 3-folds: Vector bundle methodand moduli problem, Sugaku, 47(1995), 125^144.: English translation, SugakuExpositions, to appear.

502 Shigeru Mukai[15] —: Curves and K3 surfaces of genus eleven, in 'Moduli of Vector Bundles',Series: Pure and Applied Math., (Maruyama M. ed.), Mercel Dekker, NewYork, 1996, 189-197.[16] — : Non-Abelian Brill-Noether theory and Fano 3-folds, Sugaku, 49(1997),1-24.: English translation, Sugaku Expositions, 14(2001), 125-153.[17] —: Duality of polarized K3 surfaces, in Proceedings of Euroconference onAlgebraic Geometry, (K. Hulek and M. Reid ed.), Cambridge University Press,1998, 107-122.[18] —: Moduli theory, I, II, Iwanami Shoten, Tokyo, 1998, 2000: English translation,An introduction to invariants and moduli, to appear from CambridgeUniversity Press.[19] D. Mumford : Geometric invariant theory, Springer Verlag, 1965.[20] P. E. Newstead : Stable bundles of rank 2 and odd degree over a curve ofgenus 2, Topology, 7(1968), 205-215.[21] V. V. Nikulin : On correspondences between surfaces of K3 type, Izv. Akad.Nauk SSSR Ser. Mat., 51 (1987), 402-411. English translation, Math. USSRIzv., 30(1988), 375-383.[22] D. Orlov : Equivalences of derived categories and K3 surfaces, J. Math. Sci.(New York), 84(1997), 1361-1381.[23] I. R. Shafarevieh : Le théorème de Torelli pour les surfaces algébriques detype K3, Actes Congrès Intern. Math., Nice 1970, 413-417(1971).[24] C.S. Seshadri and T. Oda : Compactifications of the generalized JacobianVariety, Trans. Amer. Math. Soc, 253(1979), 1-90.[25] V.V. Shokurov : Smoothness of the general anticanonical divisor, Izv. Acad.Nauk SSSR 43(1979), 430-441 : English translation, Math. USSR Izv.14(1980), 395-405.[26] A. Tjurin : On intersection of quadrics, Russian Math. Survey 30(1975),51-105.[27] A. Weil : Variétés abéliennes et courbes algébriques, Hermann, Paris, 1948.[28] H. Weyl : The classical groups, Princeton Univ. Press, 1939.

ICM 2002 • Vol. II • 503-512Three Questionsin Gromov-Witten TheoryR. Pandharipande*AbstractThree conjectural directions in Gromov-Witten theory are discussed: Gorensteinproperties, BPS states, and Virasoro constraints. Each points to basicstructures in the subject which are not yet understood.2000 Mathematics Subject Classification: 14N35, 14H10.Keywords and Phrases: Gromov-Witten theory, Moduli of curves.1. IntroductionLet X be a nonsingular projective variety over C. Gromov-Witten theoryconcernsintegration over M Si „(X, fi), the moduli space of stable maps from genusg, n-pointed curves to X representing the class ß £ H 2 (X,Z). While substantialprogress in the mathematical study of Gromov-Witten theory has been made inthe past decade, several fundamental questions remain open. My goal here is todescribe three conjectural directions:(i) Gorenstein properties of tautological rings,(ii) BPS states for threefolds,(iii) Virasoro constraints.Each points to basic structures in Gromov-Witten theory which are not yet understood.New ideas in the subject will be required for answers to these questions.2. Gorenstein properties of tautological ringsThe study of the structure of the entire Chow ring of the moduli space ofpointed curves M g>n appears quite difficult at present. As the principal motiveis to understand cycle classes obtained from algebro-geometric constructions, we* Department of Mathematics, Princeton University, Princeton, NJ 08544, USA. E-mail:rahulp@math.princeton.edu

504 R. Pandharipandemay restrict inquiry to the system of tautological rings, R*(M g^n). The tautologicalsystem is defined to be the set of smallest Q-subalgebras of the Chow rings,F*(M S ,„)CA*(M S; „),satisfying the following three properties:(i) R*(M g: „) contains the cotangent line classes fii,... ,fi n wherefii=ci(Li),the first Chern class of the ith cotangent line bundle.(ii) The system is closed under push-forward via all maps forgetting markings:Tr,:R*(M g , n )^R*(M gî).(iii) The system is closed under push-forward via all gluing maps:7T» : F*(Af Slj „ lU {»}) ®Q F*(Af ff2j „ 2U {»}) —t R*(M gi+g2:ni+n2 ),7T» : F*(Af Slj „ lU {» j »}) —¥ R*(Mg 1+ i^ni ).Natural algebraic constructions typically yield Chow classes lying in the tautologicalring. See [7], [18] for further discussion.Consider the following basic filtration of the moduli space of pointed curves:Mg tn D M£ n D M r g' n D Cg^n.Here, M c denotes the moduli of pointed curves of compact type, M g * n denotesthe moduli of pointed curves with rational tails, and C g>n denotes the moduli ofpointed curves with a fixed stabilized complex structure C g . The choice of C g willplay a role below.The tautological rings for the elements of the filtration are defined by theimages of R*(M g: „) in the associated quotient sequence:R*(M g ,n) -> R*(Ml n ) -+ R*(M r g%) -+ R"(C g , n ) ^ 0. (2.1)Remarkably, the tautological rings of the strata are conjectured to resemble cohomologyrings of compact manifolds.A finite dimension graded algebra R is Gorenstein with socle in degree s ifthere exists an evaluation isomorphism,cj>:R s^Q,for which the bilinear pairings induced by the ring product,R r x R s - r-ìF4Q,are nondegenerate. The cohomology rings of compact manifolds are Gorensteinalgebras.

Three Questions in Gromov-Witten Theory 505Conjecture 1. The tautological rings of the filtration of M g>n are finite dimensionalGorenstein algebras.The Gorenstein structure of R*(M g ) with socle in degree g — 2 was discoveredby Faber in his study of the Chow rings of M g in low genus. The general conjectureis primarily motivated by Faber's original work and can be found in various stagesin [5], [19], and [7].The application of the conjecture to the stratum C g>n takes a special form dueto the choice of the underlying curve C g . The conjecture is stated for a nonsingularcurve C g defined over Q or, alternatively, for the tautological ring in F*(G Sj „,Q).The tautological ring of C g>n in Chow is not Gorenstein for all C g by recent resultsof Green and Griffiths.Two main questions immediately arise if the tautological rings are Gorensteinalgebras:(i) Can the ring structure be described explicitly?(ii) Are the tautological rings associated to embedded compact manifolds in themoduli space of pointed curves?The tautological ring structures are implicitly determined by the conjectural Gorensteinproperty and the Virasoro constraints [10].As the moduli space of curves may be viewed as a special case of the modulispace of maps, a development of these ideas may perhaps be pursued more fully inGromov-Witten theory. It is possible to define a tautological ring for Af Sj „(X, ß)in the context of the virtual class by assuming the Gorenstein property, but nostructure has been yet been conjectured. Again, the Virasoro constraints in principledetermine the tautological rings.3. BPS states for threefoldsLet X be a nonsingular projective variety over C of dimension 3. Let {7 Q } QGJ 4be a basis of H*(X,Z) modulo torsion. Let {7a}ae£>2 an( l {7a}ae£>>2 denote theclasses of degree 2 and degree greater than 2 respectively. The Gromov-Witteninvariants of X are defined by integration over the moduli space of stable maps(against the virtual fundamental class):(ja 1 ,---,Ja„) g ,ß= _ evJ(7 Ql )...e

506 R. PandharipandeThe constant map contribution F? Q may be further divided by genus:Eß =0 = Fj^=0 + Fj =0 + Y^ Fß=o-The genus 0 constant contribution records the classical intersection theory of X:7^0 \ —2 \ "* t a3 t a2 t ai /F ß=o = A Z^ 3J J Tai U 7a 2 U 7 0s .The genus 1 constant contribution is obtained from a virtual class calculation:^=0 = E *af=l,^=o = - E If / 7a UC 2 (X).oGD 2 a€D 2XSimilarly, the genus g > 2 contributions areF 3 ß=o = (l)lß=o = M )s^ I (MX) - ci(X) U c 2 (X)) • / Xg]_i.2 .IX JM„The Hodge integrals which arise here have been computed in [6]:a \B 2n \ \B 2n — 2 \ 1A 3 _ \*-> 2 g\ |-"2g-2|s-iMo - 2«? 2«?^ 2 (2p-2)!'where F2 S and B 2g -2 are Bernoulli numbers. The constant map contributions toF x are therefore completely understood.The second term in (3.2) is the nonconstant map contribution:^ = EE*f-S>0/3#0Since the virtual dimension of the moduli space M g (X, ß) is[c 1 (X) + 3g-3 + 3-3g = f ci(X),JßJßthe classes ß satisfying fi ci(X) < 0 do not contribute to the potential F x .Therefore,F xmay be divided into two sums:F x-- - ES>0+Es>oE nE n-ß?0, f ßCl (X) = 0ßÔ,f ßCl (X)>0In case ß ^ 0, we will write the series Fî(t, À) in the following form:nin>0 ai,...,a n £.D>2

Three Questions in Gromov-Witten Theory 507The degree 2 variables {t a }aeD 2 are formally suppressed in q via the divisor equation:8 TT h 7a t„ 2 for all i.(ii) n + j ß ci(X) = YJi=i deg(7 Qi )-The invariants will be defined to satisfy the divisor equation (which allows for theextraction of degree 2 classes 7 a ) and defined to vanish if degree 0 or 1 classes areinserted or if condition (ii) is violated. If L ci (X) = 0, then nî is well-definedwithout cohomology insertions.The new invariants nî (7 Ql ,..., 7 Q „ ) are defined via Gromov-Witten theory bythe following equation:F X = E E »s A*- 2 s 1 (sm(dX/2)y-" dßqd I A/2 Ig>0 ß^to, f ßCl (X)=0 d>0v'7E E^y E t*~ •••**!g>0 ßô, f ß ci(X)>0 »>0 ai,...,o„G-D>2g. , x2a_ 2 /sin(A/2)V g - 2+ W x)•n ß (la 1 ,--- ,la n )X 9 I I ^.The above equation uniquely determines the invariants n|(7 Ql ,... ,7 Q „).Conjecture 2. For all nonsingular projective threefolds X,(i) the invariants nî (7 Ql ,..., 7 Q „ ) are integers,(ii) for fixed ß, the invariants nî (7 Ql ,..., 7 Q „ ) vanish for all sufficiently largegenera g.If X is a Calabi-Yau threefold, the Gopakumar-Vafa conjecture is recovered[15], [16]. Here, the invariants nî arise as BPS state counts in a study of TypeIIA string theory on X via M-theory. The outcome is a physical deduction of theconjecture in the Calabi-Yau case.Gopakumar and Vafa further propose a mathematical construction of theCalabi-Yau invariants nî using moduli spaces of sheaves on X. The invariantsnî should arise as multiplicities of special representations of SI2 in the cohomologyof the moduli space of sheaves. The local Calabi-Yau threefold consisting of a curveG together with a rank 2 normal bundle N satisfying ci(N) = OJC should be themost basic case. Here the BPS states n s d should be found in the cohomology of anappropriate moduli space of rank d bundles on G. A mathematical development of

508 R. Pandharipandethe proposed connection between integrals over the moduli of stable maps and thecohomology of the moduli of sheaves has not been completed. However, evidencefor the program can be found both in local and global calculations in several cases[1], [20], [21].The conjecture for arbitrary threefolds is motivated by the Calabi-Yau casetogether with the degeneracy calculations of [29]. Evidence can be found, for example,in the low genus enumerative geometry of P 3 [9], [29]. If the conjecture istrue, the invariants nî(âi ,... ,7 Qn ) of P 3 may be viewed as defining an integralenumerative geometry of space curves for all g and ß. Classically the enumerativegeometry of space curves does not admit a uniform description.The conjecture does not determine the Gromov-Witten invariants of threefolds.A basic related question is to find some means to calculate higher genus invariants ofCalabi-Yau threefolds. The basic test case is the quintic hypersurface in P 4 . Thereare several approaches to the genus 0 invariants of the quintic: Mirror symmetry, localization,degeneration, and Grothendieck-Riemann-Roch [2],[3], [8],[11],[23]. But,the higher genus invariants of the quintic are still beyond current string theoreticand geometric techniques. The best tool for the higher genus Calabi-Yau case,the holomorphic anomaly equation, is not well understood in mathematics. Onthe other hand, all the invariants of P 3 may be in principle calculated by virtuallocalization [17].4. Virasoro constraintsLet X be a nonsingular projective variety over C of dimension r. Let {â} bea basis of H*(X,C) homogeneous with respect to the Hodge decomposition,1«éP"*(I,C).The descendent Gromov-Witten invariants of X are:(r kl ( 7oi )... -r*, (7a„))lß = _ '

Three Questions in Gromov-Witten Theory 509(i) the intersection pairing g a t, = J x 7 a U 7&,(ii) the Hodge decomposition 7 a £ H Pa ' Qa (X,C),(iii) the action of the anticanonical class ci(X).The formulas for the operators L k are:Ek =2.^ / A [bg + lrili (C % )gtmdb,m+k-iro=0 j=0+ ^(-l)'»+ 1 [J a -m-l]f(C«)« J Ö 0ira Ö M _ ra „ 1 „ 1 )+ Ç(C k+1 ) ab t a 0t b oà~ko48 JX {(3-r)cr(X)-2c 1(X)c r -i(X)),where the Einstein convention for summing over the repeated indices a,b £ A isfollowed.Several terms require definitions. For each class 7 a , a half integer b a is obtainedfrom the Hodge decomposition,b a =p a + (l-r)/2.The combinatorial factor [x]f is defined by:[x]i = eu+i-i(x,x + l,...,x + k),where e^ is the fcth elementary symmetric function. The matrix C b a is determinedby the action of the anticanonical class,G Q 6 76=ci(X)U7 Q ,The indices of G are lowered and raised by the metric g a t, and its inverse g ab . Theterms t^ and 9 QjTO are defined by:fa l _ 4-a _ x xm — l mu aO"rol)da,m = 9/dt a m,where both are understood to vanish if m < 0.Conjecture 3. For all nonsingular projective varieties X, L\,(Z X ) = 0.The conjecture for varieties X with only (p,p) cohomology was made byEguchi, Hori, and Xiong [4]. The full conjecture involves ideas of Katz. In caseX is a point, the constraints specialize to the known Virasoro formulation of Witten'sconjecture [22], [30] (see also [25]). After the point, the simplest varietiesoccur in two basic families: curves C g of genus g and projective spaces P" of dimensionn. A proof of the Virasoro constraints for target curves C g is presentedin a sequence of papers [26], [27], [28]. Gi ventai has recently proven the Virasoro

510 R. Pandharipandeconstraints for the projective spaces P" [12], [13], [14]. The two families of varietiesare quite different in flavor. Curves are of dimension 1, but have non-(p,p)cohomology, non-semisimple quantum cohomology, and do not always carry torusactions. Projective spaces cover all target dimensions, but have algebraic cohomology,semisimple quantum cohomology, and always carry torus actions.The Virasoro constraints are especially appealing from the point of view ofalgebraic geometry as all nonsingular projective varieties are covered. While manyaspectsof Gromov-Witten theory may be more naturally pursued in the symplecticcategory, the Virasoro constraints appear to require more than a symplecticstructure to define. For example, the bracket[Fi, F_i] = 2Fo,depends upon formulas expressing the Chern numbers,C r (X), / Ci(X)c r -l(X),x Jxin terms of the Hodge numbers h p,q of X (see [24]]).The Virasoro constraints may be a shadow of a deeper connection betweenthe Gromov-Witten theory of algebraic varieties and integrable systems. In casethe target is the point or the projective line, precise connections have been made tothe KdV and Toda hierarchies respectively. The connections are proven by explicitformulas for the descendent invariants in terms of matrix integrals (for the point)and vacuum expectation in ATTV (for the projective line) [22], [25], [27]. Theextent of the relationship between Gromov-Witten theory and integrable systemsis not known. In particular, an understanding of the surface case would be of greatinterest. Perhaps a link to integrable systems can be found in the circle of ideasinvolving Hilbert schemes of points, Heisenberg algebras, and Göttsche's conjecturesconcerning the enumerative geometry of linear series.Finally, one might expect Virasoro constraints to hold in the context of Gromov-Witten theory relative to divisors in the target X. For the relative theory of 1-dimensional targets X, Virasoro constraints have been found and play a crucial rolein the proof of the Virasoro constraints for the absolute theory of X [28].References[1] J. Bryan and R. Pandharipande, BPS states of curves in Calabi-Yau 3-folds,Geom. Topol. 5 (2001), 287^318.[2] P. Candelas, X. de la Ossa, P. Green and L. Parkes, A pair of Calabi-Yaumanifolds as an exactly soluble superconformai field theory, Nuclear PhysicsB359 (1991), 21^74.[3] T. Coates and A. Gi ventai, Quantum Riemann-Roch, Lefschetz, and Serre,math.AG/0110142.[4] T. Eguchi, K. Hori, and C.-S. Xiong, Quantum cohomology and Virasoro algebra,Phys. Lett. B402 (1997), 71^80.

Three Questions in Gromov-Witten Theory 511[5] C. Faber, A conjectural description of the tautological ring of the moduli spaceof curves, in Moduli of Curves and Abelian Varieties (The Dutch Intercity-Seminar on Moduli), C. Faber and E. Looijenga, eds., 109^129, Aspects ofMathematics E33, Vieweg: Wiesbaden, 1999.[6] C. Faber and R. Pandharipande, Hodge integrals and Gromov-Witten theory,Invent. Math. 139 (2000), 173^199.[7] C. Faber and R. Pandharipande, Logarithmic series and Hodge integrals inthe tautological ring, with an appendix by Don Zagier, Michigan Math. J. 48(2000), 215^252.[8] A. Gathmann, Relative Gromov-Witten invariants and the mirror formula,math.AG/0202002._[9] E. Getzler, Intersection theory on Afi j4 and elliptic Gromov-Witten invariants,J. Amer. Math. Soc. 10 (1997), 973^998.[10] E. Getzler and R. Pandharipande, Virasoro constraints and the Chern classesof the Hodge bundle, Nucl. Phys. B530 (1998), 701^714.[11] A. Givental, Equivariant Gromov-Witten invariants, Int. Math. Res. Notices13 (1996), 613^663.[12] A. Givental, Semisimple Frobenius structures at higher genus,math.AG/0008067.[13] A. Givental, Gromov-Witten invariants and quantization of quadratic hamiltonians,math.AG/0108100.[14] A. Givental, in preparation.[15] R. Gopakumar and C. Vafa, M-theory and topological strings I, hep-th/9809187.[16] R. Gopakumar and C. Vafa, M-theory and topological strings II, hepth/9812127.[17] T. Graber and R. Pandharipande, Localization of virtual classes, Invent. Math.135 (1999), 487^518. _[18] T. Graber and R. Pandharipande, A non-tautological algebraic class on M2,22,math.AG/0104057.[19] R. Hain and E. Looijenga, Mapping class groups and moduli spaces of curves,in Proceedings of Symposia in Pure Mathematics: Algebraic Geometry SantaCruz 1995, J. Kollâr, R. Lazarsfeld, D. Morrison, eds., Volume 62, Part 2,97^142.[20] S. Hosono, M.-H. Saito, and A. Takahashi, Holomorphic anomaly equation andBPS state counting of rational elliptic surface, Adv. Theor. Math. Phys. 1(1999), 177^208.[21] S. Katz, A. Klemm, and C. Vafa, M-theory, topological strings, and spinningblack holes, hep-th/9910181[22] M. Kontsevich, Intersection theory on the moduli space of curves and the matrixAiry function, Comm. Math. Phys. 147 (1992), 1-23.[23] B. Lian, K. Liu, and S.-T. Yau, Mirror principle I, Asian J. Math. 1 (1997),729^763.[24] A. Libgober and J. Wood, Uniqueness of the complex structure on Kahler manifoldsof certain homology types, J. Diff. Geom. 32 (1990), 139^154.[25] A. Okounkovand R. Pandharipande, Gromov-Witten theory, Hurwitz numbers,

512 R. Pandharipandeand matrix models, I, math. AG/0101147 .[26] A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz theory,and completed cycles, math.AG/0204305.[27] A. Okounkov and R. Pandharipande, The equivariant Gromov-Witten theoryo/P 1 , in preparation.[28] A. Okounkov and R. Pandharipande, The Gromov-Witten theory of targetcurves, in preparation.[29] R. Pandharipande, Hodge integrals and degenerate contributions, Comm. Math.Phys. 208 (1999), 489^506.[30] E. Witten, Two dimensional gravity and intersection theory on moduli space,Surveys in Diff. Geom. 1 (1991), 243^310.

ICM 2002 • Vol. II • 513^524Update on 3-foldsMiles Reid*AbstractThe familiar division of compact Riemann surfaces into 3 casesg = 0, 5 = 1 and g > 2corresponds to the well known trichotomy of spherical, Euclidean and hyperbolicnon-Euclidean plane geometry. Classification aims to treat all projectivealgebraic varieties in terms of this trichotomy; the model is Castelnuovo andEnriques' treatment of surfaces around 1900 (reworked by Kodaira in the1960s). The canonical class of a variety may not have a definite sign, so weusually have to beat it up before the trichotomy applies, by a minimal modelprogram (MMP) using contractions, flips and fibre space decompositions. Theclassification of 3-folds was achieved by Mori and others during the 1980s.New results over the last 5 years have added many layers of subtlety tohigher dimensional classification. The study of 3-folds also yields a rich crop ofapplications in several different branches of algebra, geometry and theoreticalphysics. My lecture surveys some of these topics.2000 Mathematics Subject Classification: 14E30, 14J30, 14J32, 14J35,14J45, 14J81.Keywords and Phrases: Mori theory, Minimal model program, Classificationof varieties, Fano 3-folds, Birational geometry.1. Popular introduction: the great trichotomyA trichotomy is a logical division into three cases, where we expect to winsomething in each case. The cases here are similar to the "much too small, justright, much too big" of Goldilocks and the Three Bears, or the geometric divisionof conic sections into ellipse, parabola and hyperbola due to Appollonius of Perga(200 BC), or the cosmological question of whether the universe contracts again intoa big crunch, tends to an asymptotic state or continues expanding exponentially.* Mathematics Institute, University of Warwick, Coventry CV4 7AL, England, UK. E-mail:miles@maths.warwick.ac.uk

514 Miles Reid1.1. Euclidean and non-Euclidean geometryEuclid's famous parallel postulate (c. 300 BC) states thatif a line falls on two lines, with interior angles on one side adding to< 180°, the two lines, if extended indefinitely, meet on the side onwhich the angles add to < 180°.We are in plane geometry, assumed homogeneous so that any construction involvinglines, distances, angles, triangles and so on can be carried out at any point and inany orientation with the same effect. In this context the great trichotomy is theobservation, probably due originally to Omar Khayyam (11th a), Nasir al-Din al-Tusi (13th c.) and Gerolamo Saccheri (1733), that two other cases besides Euclid'sare logically coherent (see Figure 1). In spherical geometry, the two lines meetspherical Euclidean hyperbolicFigure 1: The parallel postulateon both sides whatever the angles, whereas in hyperbolic non-Euclidean geometry,the two lines may diverge even though the angle sum is < 180°. Whether lineseventually meet is a long-range question, but it reflects the local curvature of thegeometry.1.2. Gauss and Riemann on differential geometryA local surface S in 3-space is positively curved if all its sections bend in thesame direction like the top of a sphere (see Figure 2). S is flat (or developable) ifpositiveflatFigure 2: Local curvaturenegativeit is straight in one direction like a cylinder, and negatively curved if its sectionsbend in opposite directions like a saddle or Pringle's chip. Gauss in his TheoremaEgregium (1828) and Riemann in his Habilitationsschrift (1854) found that curvatureis intrinsic to the local distance geometry of S, independent of how S sits in3-space: living on a sphere S of radius R, we can measure the perimeter of a disc ofradius r, which is 27r(sin j^)R, always less than the Euclidean value 27rr. If we lived

Update on 3-folds 515in the hyperbolic plane, the perimeter of a disc of radius r would be 27r(sinh jj)F,bigger than the Euclidean value, and growing exponentially with r.Riemann in particular generalised Gauss' ideas on surfaces to a space given locallyby an n-tuple (xi,..., x n ) of real parameters (a "many-fold extended quantity"or manifold), with distance arising from a local arc length ds given by a quadraticform ds 2 = ^gijdxidxj. The curvature is then a function of the second derivativesof the metric function #y. Riemann's differential geometry works with manifoldsthat are not homogeneous, e.g., having positive, zero, or negative curvature at differentpoints. It was a key ingredient in Einstein's general relativity (1915), whichtreats gravitation as curvature of space-time.1.3. Riemann surfacesThe story moves on from real manifolds (e.g., surfaces depending on 2 realvariables) to Riemann surfaces, parametrised instead by a single complex variable.The point here is Cauchy's discovery (c. 1815) that differentiable functions of acomplex variable are better behaved than real functions, and much more amenableto algebraic treatment. Riemann discovered that a compact Riemann surface C hasan embedding C ^y P^ into complex projective space whose image is defined by aset of homogeneous polynomial equations.A projective algebraic curve C C P^ is nonsingular if at every point P £ Cwe can choose N — 1 local equations fi,..., /jv-i so that the Jacobian matrix -?^fihas maximal rank N — 1. It follows from the implicit function theorem that oneof the linear coordinates z = zi of f N can be chosen as a local analytic coordinateon C. In other words, a compact Riemann surface is analytically isomorphic to anonsingular complex projective curve.1.4. The genus of an algebraic curveThe canonical class Kc = û^ = T c of a curve C is the holomorphic linebundle of 1-forms on C; it has transition functions on U n U' the Jacobian of thecoordinate change ^-, where z,z' are local analytic coordinates on U,U'. If z is arational function on C that is an analytic coordinate on an open set U C C then a1-form on U is f(z)dz with / a regular function on U. That is, ii c = Ö • dz, or dzis a basis of ii c on U.The genus g(C) can be defined in several ways: topologically, a compact Riemannsurface is a sphere with g handles (see Figure 3). It has Euler numberg = 0, sphere 9=1, torus g > 2, general typeFigure 3: The genus of a Riemann surfacee(C) = 2 — 2g, which equals deg To- The most useful formula for our purpose is

516 Miles Reiddeg Kc = 2g — 2. We see thatK C < 0 g = 0, K C = 0 g=l, K c > 0 g > 2.This trichotomy is basic for the study of a curve C from every point of view,including topology, differential geometry, complex function theory, moduli, all theway through to algebraic geometry and Diophantine number theory. To relatethis briefly to curvature as discussed in Section 1.2, for an arbitrary Riemannianmetric, the average value of curvature over C equals — deg Kc by the Gauss^Bonnettheorem; moreover, by the Riemann mapping theorem, there exists a metric on C inthe conformai class of the complex structure with constant positive, zero or negativecurvature in the three cases.2. Classification of 3-foldsThe great trichotomy also drives classification in higher dimensions. Themeaning of "higher dimensions" is time-dependent: dim 2 was worked out around1900 by Castelnuovo and Enriques, dim 3 during the 1980s by Mori and others, anddim 4 is just taking off with Shokurov's current work. I concentrate on dim 3, wherethese issues first arose systematically.2.1. Preliminaries: the canonical class K xAn n-dimensional projective variety X can be embedded X ^y P^, and isgiven there by homogeneous polynomial equations; nonsingular means that at everypoint F e X, we can choose N — n of the defining equations so that the Jacobianmatrix -jfifi has rank N — n, with n linear coordinates of P^ providing local analyticcoordinates on X.The canonical class of X is Kx = iì x = /\ n ii x . It has many interpretations:it is the line bundle obtained as the top exterior power of the holomorphic cotangentbundle; it has transition functions on U n U' the Jacobian determinant det jfi 1 ,where x,x' are systems of local analytic coordinates on open sets U,U' C X; itssections are holomorphic n-forms; at a nonsingular point P £ X, its sections aregenerated by the holomorphic volume form d#i A- • • Adx n , so that ii x = Ox -dxi A• • • A dx n .For MMP to work in dim > 3, we are eventually forced to allow certain mildsingularities. The theory in dim 3 is now standard and not very hard (see [YPG] andcompare the foreword to [CR]). We always insist that the first Chern class of Kxrestricted to the nonsingular locus X° c X comes from an element of H 2 (X, Q),that I continue to denote by Kx- This ensures that the pullback f*Kx by amorphism /: Y —t X is defined, together with the intersection number KxC withevery curve C C X (obtained by evaluating Kx £ H 2 against the class [C] £H 2 (X,Q)). Note that —KxC is an integral or average value of Ricci curvature(a 2-form) calculated over a 2-cycle [C] corresponding to a holomorphic curve; wehave taken several steps back from varieties of constant curvature suggested by thecolloquial pictures in Section 1.

Update on 3-folds 517In many contexts, the canonical class of a variety is closely related to thediscrepancy divisor. If /: Y —¥ X is a birational morphism, its discrepancy A/is defined by Ky = f*Kx + A/; if X and Y are nonsingular this is the divisorof zeros A/ = div (det ^fifi J of the Jacobian determinant of /, or its appropriategeneralisation if X and Y are singular. Since the components of A/ are exceptional,it follows that if A/ > 0, then there exists a component F of A/ such that KyC < 0for almost every curve C C F. It is known that every section s £ H°(Y,nKy)vanishes along A/ for every n > 0. A morphism / is crêpant if A/ = 0; thenKy = f*Kx, so that Ky is numerically zero relative to /.2.2. The trichotomy: K x < 0, K x = 0 or K x > 0?The naive section heading is misleading: Kx may have "different sign" atdifferent points of X and in different directions. The aim is not to apply thetrichotomy to X itself, but to modify it first to a variety X' by a MMP. We needto be more precise; we say that Kx is nef or numerically nonnegative if KxC > 0for every C C X (nef is an acronym for numerically eventually free - we hope that\nKx\ is a free linear system for some n > 0). As we saw at the end of Section 2.1,a discrepancy divisor A/ > 0 for a birational morphism /: Y —t X is a localobstruction to the nefdom of Ky. Mori theory (or the MMP) is concerned with thecase that Kx is not nef.2.3. Results of MMP for 3-foldsThe Mori category consists of (quasi-)projective n-folds X with Q-factorialterminal singularities; see [YPG] for details. For X in the Mori category, an elementarycontraction is a morphism ip : X —t Xi such that(i) Xi is a normal variety and ip has connected fibres.(ii) All curves C C X contracted by Ai —> • • • —> A n = Awhere (1) each step Xi —•* X i+i is an elementary divisorial contraction or flip ofthe Mori category, and (2) the final object X' either has Kx> nef or has a Mfsstructure X' —t S.Each birational step Xj —•* X i+i removes a subvariety of X on which Kx isnegative. A divisorial contraction contracts an irreducible surface in X to a curve

518 Miles Reidor a point. A flip is a surgery operation that cuts out a finite number of curves inXj on which K is negative, replacing them with curves on which K is positive. Atthe end of the MMP comes the dichotomy: either Kx> is nef, or —Kx 1 is ample ona global structure of X'.The main theorem on varieties with Kx nef is the existence of an Iitaka^Kodaira fibration X —t Y, with fibres the curves C C X with KxC = 0. Thisgives a natural case division according to dim Y. The extreme cases are Calabi-Yauvarieties (CY), where Kx = 0, and varieties of general type, where X —t Y isbirational to a canonical model Y having canonical singularities and ample Ky.This takes my story up to around 1990; for more details, see Kollâr and Mori[KM] or Matsuki [M].3. Lots of recent progress3.1. Extension of MMP to dimension 4Already from the mid 1980s, it was understood that the MMP could in largeparts be stated in all dimensions as a string of conjectures (or the log MMP, wherewe proceed in like manner, but directed by a log canonical class Kx + D). Thedifficult parts in dim > 3 are the existence of flips (or log flips), and the terminationof a chain of flips. Recent work of Shokurov [Sh] has established the existence of logflips in dim 4; the key idea is the reduction to prelimiting flips, already prominentin Shokurov's earlier work (see [FA], Chapter 18).3.2. Rationally connected varietiesA variety X is rational if it is birationally equivalent to P". That is, there aredense Zariski open sets X 0 C X and U C V n , and an isomorphism X 0 — U suchthat both ip and ip^1are given by rational maps. In other words, X has a one-to-oneparametrisation by rational functions. By analogy with curves and surfaces, onemight hope that rational varieties have nice characterisations, and that rationalitybehaveswell under passing to images or under deformation. Unfortunately, indim > 3, our experience is that this is not the case, and we are obliged to give upon the question of rationality. 1However, it turns out that the notion of rationally connected variety developedindependently by Campana and by Kollâr, Miyaoka and Mori is a good substitute.X is rationally connected if there is a rational curve through any two points P,Q £X. See [Ca], [KMM], [Ko] and [GHS] for developments of this notion.1 This is of course exaggerated. Rationality itself remains the major issue in many contexts, inparticular the rationality of GIT quotients. Iskovskikh's conjectured rationality criterion for conicbundles remains one of the driving forces of 3-fold birational geometry. Thanks to Slava Shokurovfor reminding me of this important point.

Update on 3-folds 5193.3. Explicit classification results for 3-foldsSection 2.3 discussed the Mori category and its elementary contractions. Theexplicit classification manifesto of the foreword of [CPR] calls for the abstract definitionsand existence results to be translated into practical lists of normal forms.The ideal result here is Mori's theorem [YPG], Theorem 6.1, that classifies 3-foldterminal singularities into a number of families; these relate closely to cyclic coversbetween Du Val singularities, and deform to varieties having only the terminalcyclic quotient singularities ^(l,a,—a).To complete our grasp of Mori theory, we hope for explicit classification resultsin this style for divisorial contractions, flips and Mfs. The last few years have seenremarkable progress by Kawakita [Kal], [Ka2] on divisorial contractions to points.A guiding problem in this area was Corti's 1994 conjecture ([Co2], p. 283) thatevery Mori divisorial contraction ip: X —t Y to a nonsingular point F £ Y is a(l,a,b) weighted blowup. Kawakita proved this, and went on to classify explicitlythe divisorial contractions to compound Du Val singularities of type A. There arealso results of Tziolas on contractions of surfaces to curves. For progress on Mfs seeSection 4.3.3.4. Calabi-Yau 3-folds and mirror symmetryA CY manifold X is a Kahler manifold with Kx = 0, usually assumed simplyconnected, or at least having H x (öx) = 0. A popular recipe for constructingCY 3-folds is due to Batyrev, based on resolving the singularities of toric completeintersections. This gives some 500,000,000 families of CY 3-folds, so much more impressivethan a mere infinity (see the website [KS]). There are certainly many more;I believe there are infinitely many families, but the contrary opinion is widespread,particularly among those with little experience of constructing surfaces of generaltype.Calabi-Yau 3-folds are the scene of exciting developments related to theStrominger-Yau-Zaslow special Lagrangian approach to mirror symmetry. For lackof space, I refer to Gross [Gr] for a recent discussion.3.5. Resolution of orbifolds and McKay correspondenceKlein around 1870 and Du Val in the 1930s studied quotient singularities

520 Miles Reid3.6. The derived category as an invariant of varietiesThe derived category D(A) of an Abelian category A was introduced byGrothendieck and Verdier in the 1960s as a technical tool for homological algebra.A new point of view emerged around 1990 inspired by results of Beilinson andMukai: for a projective nonsingular variety X over C, write D(X) for the boundedderived category of coherent sheaves on X; following Bondal and Orlov, we considerD(X) up to equivalence of C-linear triangulated category as an invariant ofX, somewhat like a homology theory; the Grothendieck group K 0 (X) is a naturalquotient of D(X).The derived category D(X) is an enormously complicated and subtle object(already for P 2 ); in this respect it is like the Chow groups, that are usually infinitedimensional, and contain much more information than anyone could ever use.Despite this, there are contexts, usually involving moduli constructions, in which"tautological" methods give equivalences of derived categories between D(X) andD(Y). An example is the method of [BKR] that establishes the McKay correspondenceon the level of derived categories by Fourier-Mukai transform. There is nosuch natural treatment for the McKay correspondence in ordinary (co-)homology(see [Cr]).The following conjectural discussion is based on ideas of Bondal, Orlov andothers, as explained by Bridgeland (and possibly only half-understood by me). As Isaid, classification divides up all varieties into FJ>0,F' = 0,F' 0, D(X) should be very infinite but rigid and indecomposable. Bondaland Orlov [BO] have proved that F(X) determines X uniquely if ±Kx is ample,but as far as I know, they have not established a qualitative difference between thetwo cases.Right up to Kodaira's work on surfaces in the 1960s, minimal models wereseen in terms of tidying away — 1-curves to make a really neat choice of model in abirational class, that eventually turns out to be unique. In contrast, starting fromaround 1980, the MMP in Mori theory sets itself the direct aim of making K nefif possible. Derived categories give us a revolutionary new aim: each step of theMMP chops off a little semi-orthogonal summand of F(X).4. Fano 3-folds: biregular and birational geometry4.1. The Sarkisov program

Update on 3-folds 521The modem view of MMP and classification of varieties is as a biregular theory:although we classify varieties up to birational equivalence, the aims and the methodsare stated in biregular terms. Beyond the MMP, the main birational problems arethe following:(1) If X is birational to a Mfs as in Theorem 1, then in how many different waysis it birational to a Mfs?(2) Can we decide when two Mfs are birationally equivalent?(3) Can we determine the group of birational selfmaps of a Mfs?The Sarkisov program gives general answers to these questions, at least in principle.It untwists any birational map ip: X —•* Y between the total spaces of two MfsX/S and Y/T as a chain of links, generalising Castelnuovo's famous treatment ofbirational maps of P 2 . A Sarkisov link of Type II consists of a Mori divisorialextraction, followed by a number of antiflips, flops and flips (in that order), then aMori divisorial contraction.More generally, the key idea is always to reduce to a 2-ray game in the Moricategory (see [Co2], 269^272). By definition of Mfs, we have p(X/S) = 1, but fora 2-ray game we need a contraction X' —t S' with p(X' /S') = 2. A Sarkisov linkstarts in one of two ways (depending on the nature of the map ip we are tryingto untwist): either blow X up by a Mori extremal extraction X' —t X and leaveS" = S; or find a contraction S —¥ S' of S so that p(X/S') = 2 and leave X = X'.In either case, the Mori cone of the new X'/S" is a wedge in R 2 with a marked Moriextremal ray, and we can play a 2-ray game that contracts the other ray, flippingit whenever it defines a small contraction. It is proved that, given ip: X —t Y,one or other of these games can be played, and the link ends as it began in a Moridivisorial contraction or a change of Mfs structure, making four types of links. Eachlink decreases a (rather complicated) invariant of ip, and it is proved that a chainof links terminates. See [Co] and Matsuki [M] for details.4.2. Birational rigidityWhile the Sarkisov program factors birational maps as a chain of links thatare elementary in some categorical sense, an explicit description of general links isstill a long way off. To obtain generators of the Cremona group of P 3 would involveclassifying every Mfs X/S that is rational, and every Sarkisov link between these;for the time being, this is an impossibly large problem. There is, however, a largeand interesting class of Mfs for which there are rather few Sarkisov links.A Mori fibre space X —t S is birationally rigid if for any other Mfs Y —t T, abirational map ip: X —•* Y can only exist if it lies over a birational map S —•* Tsuch that X/S and Y/T have isomorphic general fibres (but ip need not induce anisomorphism of the general fibres - this is a tricksy definition). If S = pt., so thatX is a Fano variety with p(X) = 1, the condition means that the only Mfs Y/Tbirational to X is Y = X itself. For example, P 2 is not rigid, since it is birational toall the scrolls F n . Following imaginative but largely non-rigorous work of Fano inthe 1930s, Iskovskikh and Manin proved in 1971 that a nonsingular quartic 3-foldX 4 c P 4 is birationally rigid. This proof has since been simplified and reworked

522 Miles Reidby many authors. The main result of [CPR] is that a general element X of anyof the famous 95 families of Fano hypersurface X^ c P(l,ai,... ,a 4 ) is likewisebirationally rigid.It is interesting to take a result of Corti and Mella [CM] as an example goingbeyond the framework of [CPR]. The codim 2 complete intersection X 3j4 cP 5 (l, 1,1,1,2,2) is a Fano 3-fold; write xi,...,X/i,yi,y 2 for homogeneous coordinatesand fi = £4 = 0 for the equations of X 3j4 . By a minor change of coordinates,I can assume that # 4 = yiy 2 + g'(xi,... ,£4). Then X 3j4 has 2 x |(1,1,1) quotientsingularities at the t/i, y 2 coordinate points. [CM] shows that blowing up either ofthese point leads to a Sarkisov linkA3,4n(1 4 ,2 2 )-> Y

Update on 3-folds 523is entertaining and nontrivial: there are two different families of Fano 3-folds incodim 4 with the same Hilbert series, obtained by unprojections that are numericallyidentical, and that differ only in the way that their unprojection planes embedII = P 2 < L -¥ wGr(2,5) in the weighted Grassmannian. These are the Tom and Jerryunprojections of [Ki], Section 8. The K3 surface sections of the two families forma single unobstructed family, but their extension to Fano 3-folds break up into twofamilies; this is reminiscent of the extension-deformation theory of the del Pezzosurface of degree S$,References[Al] S. Altinok, Graded rings corresponding to polarised K3 surfaces and Q-Fano 3-folds, Univ. of Warwick Ph.D. thesis, Sep. 1998, vii+93, get fromwww.maths.warwick.ac.uk/~miles/doctors/Selma.[ABR] S. Altinok, G. Brown and M. Reid, Fano 3-folds, K3 surfaces and gradedrings, in Singapore Internat. Symp. in Topology and Geometry (NUS,2001), Ed. A.J. Berrick and others, Contemp. Math. AMS, 2002; preprintmath.AG/0202092, 29.[BO] Alexei Bondal and Dmitri Orlov, Reconstruction of a variety from the derivedcategory and groups of autoequivalences, Comp. Math. 125 (2001)327^344.[BKR] Tom Bridgeland, Alastair King and Miles Reid, Mukai implies McKay, J.Amer. Math. Soc. 14 (2001) 535^554.[Ca] F. Campana, Connexité rationnelle des variétés de Fano, Ann. Sci. ÉcoleNorm. Sup. 25 (1992) 539^545.[Co] Alessio Corti, Factoring birational maps of threefolds after Sarkisov, J. Alg.Geom. 4 (1995) 223^254.[Co2] Alessio Corti, Singularities of linear systems and 3-fold birational geometry,in [CR], 259^312.[CM] Alessio Corti and Massimiliano Mella, Birational geometry of terminalquartic threefolds. I, can.dpmms.cam.ac.uk/~corti/cm.pdf, 41.[CPR] Alessio Corti, Sasha Pukhlikov and Miles Reid, Fano 3-fold hypersurfaces,in [CR], 175^258.[CR] Alessio Corti and Miles Reid, Explicit birational geometry of 3-folds, CUP,2000.[CR2] Alessio Corti and Miles Reid, Weighted Grassmannians, in memorial volumefor Paolo Francia (Genova, Sep 2001), M. Beltrametti Ed., de Gruyter2002, 22, preprint math.AG/0206011, 27.[Cr] Alastair Craw, An explicit construction of the McKay correspondence forA-HilbC 3 , math.AG/0010053, 30.[GHS] Tom Graber, Joe Harris and Jason Starr, Families of rationally connectedvarieties, math.AG/0109220, 21.[Gr] Mark Gross, Examples of special Lagrangian fibrations, math. AG/0012002,29.[K] Kawamata Yujiro, Francia's flip and derived categories math.AG/0111041,

524 Miles Reid23.[Kai] Kawakita Masayuki, Divisorial contractions in dimension three which contractdivisors to smooth points, Invent. Math. 145 (2001) 105-119.[Ka2] Kawakita Masayuki, Divisorial contractions in dimension three whichcontract divisors to compound Ai points, Comp. Math, to appear,math.AG/0010207, 23.[Ko] Kollâr Jânos, Rational curves on algebraic varieties, Springer 1996.[FA] Kollâr Jânos and others, Flips and abundance for algebraic threefolds,Astérisque 211 (1992), SMF 1992.[KM] Kollâr Jânos and Mori Shigefumi, Birational geometry of algebraic varieties,CUP 1998.[KMM] Kollâr Jânos, Miyaoka Yoichi and Mori Shigefumi, Rational connectednessand boundedness of Fano manifolds, J. Diff. Geom. 36 (1992) 765-779.[KS] Maximilian Kreuzer and Harald Skarke, Calabi-Yau data, websitetphl6.tuwien.ac.at/~kreuzer/CY.[M] Matsuki Kenji, Introduction to the Mori program, Springer, 2002.[O] Dmitri Orlov, Projective bundles, monoidal transformations, and derivedcategories of coherent sheaves, Izv. 56 (1992) 852-862 = Russian Acad.Sei. Izv. Math. 41 (1993) 133-141.[PR]S. Papadakis and M. Reid, Kustin-Miller unprojection without complexes,to appear in J. Algebraic Geometry, math.AG/0011094, 18.[YPG] Miles Reid, Young person's guide to canonical singularities, in Algebraicgeometry (Bowdoin, 1985), Proc. Sympos. Pure Math. 46, Part 1, AMS1987, 345-414.[Ki][Bou]Miles Reid, Graded rings and birational geometry, in Proc. of algebraicgeometry symposium (Kinosaki, 2000), K. Ohno Ed., 1-72, get fromwww.maths.warwick.ac.uk/~miles/3folds.Miles Reid, La correspondance de McKay, Séminaire Bourbaki, Astérisque276, SMF 2002, 53-72.[Sh] V.V. Shokurov, Prelimiting flips, to appear in Proc. SteklovInst., first draft Aug 1999, Mar 2002 draft 247, available fromwww.maths.warwick.ac.uk/~miles/3folds.

ICM 2002 • Vol. II • 525-532Sur les Algèbres Vertex Attachéesaux Variétés AlgébriquesVadim Schechtman*AbstractSganarelle: ... Mais encore faut-il croire quelque chose dans le monde:qu'est-ce donc que vous croyez?Dom Juan: Ce que je crois?Sganarelle: Oui.Dom Juan: Je crois que deux et deux sont quatre, Sganarelle, et quequatre et quatre sont huit.Molière, Dom, JuanOne dicusses sheaves of vertex algebras over smooth varieties and theirconnections with characteristic classes.2000 Mathematics Subject Classification: 17B69, 57R20.Keywords and Phrases: Vertex algebras, Characteristic classes.1. IntroductionLe but de cette note est de presenter une classification de certaines algèbresvertex, qui peuvent être associées à des variétés algébriques lisses; ceci est l'occasionde rencontrer des classes caractéristiques "style Pontryagin-Atiyah-Chern-Simons".Ceci a été obtenu dans [GMS] dont la presente note est un complément. On proposeici une définition plus simple d'une algébroïde vertex (infra, 2.4, 2.6), un énoncéplus précis et une démonstration courte du résultat principal de op. cit. (infra,3.4, 3.6, 3.7). À la fin on propose une construction directe des algèbres vertexassociées aux courbes, à l'aide des algèbres de Virasoro introduites par A.Beilinson.Le point de départ de cette note était une tentative de comprendre le complexe dede Rham chiral découvert par F.Malikov, [MSV]. Je remercie vivement mes amis etcollaborateurs Arkady Vaintrob, Fyodor Malikov et Vassily Gorbounov.*Laboratoire Emile Picard, Université Paul Sabatier, 118, Route de Narbonne, 31062 ToulouseCedex 4, France. E-mail: schechtman@picard.ups-tlse.fr

526 Vadim Schechtman2. Algébroïdes vertex2.1. On fixe un corps de base k de caractéristique 0. Rappelons qu'une algèbrevertex est un fc-espace vectoriel V muni d'un vecteur distingué 1 £ V (dit vacuum)et d'une famille d'applications fc-linéaires (^ : V ®k V —y V (i £ Z) telles quex (ì)y = 0 pour i assez grand. Si l'on pose dx := #(_ 2 )1, on obtient un opérateurd : V —• V. Les axiomes de [B] doivent être vérifiés. On n'est intéressé que parles algèbres vertex Z> 0 - graduées, ce qui signifie que l'espace V est muni d'une Z> 0 -graduation (dite poids conforme), V = ® n >o V n , 1 £ Vo et V n (j)V TO C V n +m-«-i-En particulier dV n C V n +i. Les morphismes de telles algèbres étant définis demanière évidente, on obtient la catégorie Vert des algèbres vertex Z > 0 -graduées.2.2. "Données classiques" associées à une algèbre vertex. Soit V £Vert. Pour être bref on écrira xy au lieu de X(_i)j/; c'est une opération non commutativeet non associative en général. On a V n V m C V n + TO . Posons A(V) = Vo;l'opération xy est commutative et associative sur Vo; donc A(V) devient une k-algèbre commutative avec unité 1. Posons A(V) = Y\. Soit O(F) le sous-fcvectorielde A(V) engendré par les éléments adb, a,b £ A(V). Alors O(F) devientun A(F)-module et d : A(V) —y 0(F) est une dérivation. En outre, si l'onpose T(V) := A(V)/Sï(V), l'opération ax induit une structure de A(F)-module surT(V). (Par contre, A(V) n'est pas un A(F)-module en général, à cause de la nonassociativité de l'opération ax.) L'opération ( 0 ) : A(V) x A(V) —y A(V) induitl'application [,] : T(A) x T(A) —• T(A) qui est un crochet de Lie; l'opération(o) : A(V) x A(V) —y A(V) induit une action de T(V) sur A(V) par dérivations;on a [r,ar'] = a[T,r'] + T(a)[T,r'], i.e. T(V) devient une A(V)-algébroïde deLie. Ya première opération induit aussi une action de l'algèbre de Lie T(V) surfi(V) telle que d est un morphisme de T(V)-modules, et T(OUJ) = r(a)oj + ar(oj).Enfin, l'opération ^ : A(V) x A(V) —y A est symétrique et induit un accouplementA(V)-bilinéaire (,) : T(V) x 0(V) —y A(V) telle que T((T',OJ)) =([T,T'],OJ) + (T',T(üJ)) et (ar)(uj) = ar(oj) + (r,oj)da.2.3. "Données quantiques." Les propriétés (Algl) — (Alg3) ci-dessoussont vérifiées, où a G A(V), x,y,z £ A(V), n : A(V) —y T(A) étant la projectioncanonique.(Algl) (ax) { i)y = a(x { i ) y) - n(x)n(y)(a).(Alg2) x {0) y + y {0 )X = d(x { i ) y); (dx) {0) y = 0.(Alg3) x { o)(y { i)z) = (x {0) y) {i) z + y {i) (x {0) z), i = 0,1.2.4. Soit A une fc-algèbre commutative de type fini, lisse sur k. PosonsQ(A) = £t\/ k , T(A) = Derk(A,A) (l'algèbre de Lie de fc-dérivations de A). Soitd = ÔDR '• A —y Q(A) la dérivation universelle. On a l'accouplement non dégénéréA-bilinéaire (,) : T(A) x Q(A) —y A; l'algèbre de Lie T(A) agit sur Q(A) par ladérivée de Lie. Ces données vérifient toutes les propriétés de 2.2.Une A-algébroïde vertex est un fc-espace vectoriel A muni d'un sous-espaceF 1 A C A avec les identifications de fc-vectoriels F 1 A = 0(A), A/F 1 A = T(A)et des opérations fc-bilinéaires (_i) : Ax A —y A, (a,x) H- ax = a(_i)X, (i) :

Sur les Algèbres Vertex Attachées aux Variétés Algébriques 527A x A —y A symétrique, ( 0 ) : Ax A —y A. On demande que (i) A(_i)Q(A) cO(A) et que l'action de A sur Q(A) et sur T(A) induite par (_i) coïncide avecl'action canonique; (ii) 0(A)(j)Q(A) = 0 (i = 0,1); O( 0 )-4 C Û(A), l'opérationT(A) x T(A) —• T(A) induite par ( 0 ) coïncide avec le crochet de Lie, et l'actioninduite T(A) x Q(A) —• Q(A) coïncide avec la dérivée de Lie; (iii) Pacccouplement(, ) : T(A) x Q(A) —y A induit par ^ coïncide avec l'accouplement canonique.Enfin, les propriétés (Algl) — (Alg3) doivent être vérifiées. Dans (Alg3) pour i = 1on interprète la partie de gauche comme n(x)(y(i)z).Soit T'(A) une algèbre de Lie dg concentrée en degrés —1,0, avec T^1(A)=T°(A) = T(A), d : T^1(A)—y T°(A) l'identité, le crochet [,]o,-i l'action adjointe.Soit Si'(A) : 0 —• A —• Q(A) —• Q(A)/9A —y 0 le complexe concentré endegrés —2,-1,0 avec les différentielles évidentes. Ce complexe est un module dgsur T'(A) (l'action de T°(A) est par la dérivée de Lie, la composante [,]_!,_! :T^1(A)x Q _1 (A) —• Q _2 (A) étant l'accouplement canonique, et la composante[,]-i,o étant définie par [T,UI] = i T (dw), où a) G Q(A)/9A est l'image de u; € Û(A),d : î\/ k —y Â/k est â différentielle de de Rham, i T : ii 2 A, k —> &\/ k est laconvolution avec r). On peut exprimer les axiomes (Alg2) et (Alg3) en disant quel'on a une algèbre de Lie dg A' : 0 —y A —y A —y A/dA —• 0, concentréeen degrés —2,-1,0, extension de T'(A) par Q'(A) (considérée comme une sousalgèbrede Lie abélienne), telle que l'action de T'(A) sur Q'(A) induite coïncideavec celle décrite ci-dessus. Un morphisme g : A —y A' est une applicationfc-linéaire respectant les opérations (^ et les filtrations, qui induit l'identité surO(A), T( A). D'où la catégorie Alg A des A-algébroïdes vertex, qui est un groupoïde(chaque morphisme est un isomorphisme).2.5. Soit A comme dans 2.4. On définit la catégorie VertA dont les objetssont V £ Vert munies d'un isomorphisme de fc-algèbres A(V) ^> A, cet isomorphismeidentifiant les données classiques (T(V),Sï(V),d, (,)) correspondantes avecles données standardes (T(A),Sï(A),dDR,{,)) décrites dans 2.4. Les morphismessont les morphismes des algèbres vertex induisants l'identité sur les données classiques.La construction 2.2, 2.3 donne lieu au foncteur Alg : VertA —> Alg A, V >-¥A(V). Ce foncteur admet l'adjoint à gauche U : Alg A —> VertA, l'algèbre vertexUA étant appelée l'algèbre enveloppante d'un algébroïde vertex A. Pour chaqueA £ Alg A le morphisme d'adjonction A —y Alg(UA) est un isomorphisme.2.6. Le langage suivant est un peu plus explicite et est parfois commode.Appelons A-algébroïde vertex scindée un couple B = ((,),c), où (, ) : T(A) xT(A) —• A (resp. c : T(A) x T(A) —• 0(A)) est une application fc-bilinéairesymétrique (resp. antisymétrique). On demande que les propriétés (AlgScindl)-(AlgScind3) ci-dessous soient vérifiées.(AlgScindl) (arfir 1 ) - a(r,6r') - 6(ar,r') + a6(r,r') = ^r'(a)r(6).(AlgScind2) (T",C(T,T')) + (T',C(T,T")) = )+r'((r,r"))/2 + r"((r,r'))/2.

528 Vadim Schechtman(AlgScind3) 3{T(C(T',T")) + T'(C(T",T)) + T"(C(T,T'J) - C([T,T'],T")-c([r',r"],r)-c([r",r],r')}=ö{-A u) i u) ') = & e iï 2 A/k\dn = u) -u)'}.Ya composition des morphismes est l'addition de 2-formes. L'addition des 3-formesinduit une structure d'un groupe abélien en catégories sur ce groupoïde.

Sur les Algèbres Vertex Attachées aux Variétés Algébriques 529On remarque que si A, A' sont deux A-algébroïdes vertex avec le même espacesous-jacent la même opération ^, alors X( Q )y — xôyy £ 0(A); cet élément nedépend que des n(x),n(y), où n : A —y T(A) est l'application canonique, d'oùl'application CA,A' '• T(A) x T(A) —• fi (A). De plus, cette application est A-bilinéaire, et OJA,A'( T , T ', T ") '•= ( T , C A,A'( T ', T ")) est antisymétrique en T,T',T",donc UJA,A' Peut être considérée comme une 3-forme différentielle, et cette formeest fermée.Réciproquement, étant donné A = AlgA et u £ fi^/jT, on définit A' = A +OJ £ AlgA ayant le même espace sous-jacent que A et la même opération ^, avec(0)' =(o) -w.Si g : A + OJ —y A fi OJ' est un morphisme, alors (g — id) (A) C fi (A), (g —id)\n(A) = 0, donc g — id induit une application h g : T(A) —• fi(A). La fonction,r] g (T,T r ) := (T,hg(r)) est antisymétrique en T,T' et A-bilinéaire, donc peutêtre considérée comme une 2-forme différentielle; on a dr\ = OJ — OJ'. Ceci induitune bijection HoniAi gA (A + OJ,A + OJ') = {n £ î\/ k \dn = OJ — OJ'}. On aHornAigA (A, A') = HomAi gA(A + OJ, A! -i- OJ). Cela définit une Action+: Mg A x Gr(tt [ l 3) ) —• Alg A - (3.1.1)3.2. Théorème. Si A est petite (voir 2.7), alors le groupoïde AlgA est unTorseur sous Gr(Q A ) par rapport à l'Action (3.1.1), c'est à dire, pour chaqueA £ AlgA le foncteur A +? : Gr(ii l fi ') —y AlgA est une équivalence.Par exemple, l'ensemble no(AlgA) des classes d'isomorphisme de AlgA est untorseur sous Hp R (A). Grace à 2.6 le Torseur AlgA est non-vide pour A petite.3.3. Soient A petite, et b = {r,},b' = {fi} deux bases abéliennes, d'où lesalgébroïdes scindées Bb, B^; on a T[ = ^TJ (la règle de Einstein est sous-entendue),4> = (# J ) € GL n (A) (pour être bref, on écrit b' = b). On définit une applicationhb',b '• T(A) —y fi(A), comme étant l'unique opérateur satisfaisant (Mori), telque (fi,hb',b(Tj)) = ^k( T i, T 'j)b- De plus, on définit une application C(,',b : T(A) xT(A) —• fi(A) comme étant l'unique opérateur tel que ßb',b := ((>)b)Cb',b) s °itune algébroïde vertex scindée, et hb>,b soit un morphisme d'algébroïdes scindéesBb< —y Bb',b- D'où la 3-forme av,b € fi^/jT telle que Bb = Bb',b + QV,b- Sib" = {T"} est la troisième base abélienne, avec r" = # J Vj, on définit la 2-forme/?b",b',b :=hb",b' + hb\b — hb",b € Â/k-3.4. Théorème. ß b »,w,b = kjri^fi^dfidcj)}, a b ',b = ^-{(^dfi) 3 }.Démonstration. Il resuite de (2.7) queCbW,rj) = iHr'r^r'dT'M -r'rmr'r'j^r'dcP- (i ++j)}, (3A) Cd'où, en utilisant (Mori),(jl^b = tr{-20- 1 rfr_î(0) + ^MW-T 1^)} (3.4) {J)fcb'.bOn = Hr'dfi'^) - ±- 1 T!'()- 1 d + - 1 il>- 1 Tl'(il>)d}. (3.4) fc

530 Vadim SchechtmanPar définition, Cb',b(7j,rj) = fi(h(Tjj) — TJ(/I(TJ')); ov,b = Cb — Cb',b, d'où;ab'.bOf.TJ) = - i ^ ^ r / W ^ r j ^ ^ ö ^ - (* ++j)}.(3.4) QEn outre, (3.4)^ entraîne/V.b'.bOn = ^rlf'r'rfWÖ^ - ^'(«fV 1^} (3-%d'où le théorème. Ici l'on identifie une 3-forme a avec une application antisymétriqueT(A) x T(A) —y fi(A) définie par a(r,r') = i T i T >a. A3.5. Classe de Pontryagin. Soit X une variété algébrique lisse sur fc,E un fibre vectoriel sur X. Choisissons une recouvrement affine il = {£/,} devX, et des bases b 1 des r(o/j,öx)-modules Y(Ui,E), d'où le cocycle de Cech =(4>ij), 4>ij £ Y(Uij,GL n (Oxj), b 1 = ijb 3 sur t/y, 4>ij4>jk = 4>ik sur U ijk . Convsidérons les cochaînes de Cech P2(4>) = ( 2^tr{7 1 k ïj 1 dijdj k }) £ C 2 (il,il 2 x),PÂ4>) = (ItrU^dîj) 3 }) £ CÛ,^); \>n a d v &() = 0, d DR p 2 (^) =JCechdv P3(4>),dDRP3(4>) = 0. Il en résulte que p() := (p 2 (4'),P3 ()) € Z 2 (ii,ii x ' ')Cechoù Ct x ' ' := (Yl 2 x —y Yl er x ), la différentielle totale dans le bicomplexe de Cechà coefficients dans ce complexe étant d = don + ( —l)' Dfi 'dv . De plus, si l'onCechchoisit des autres bases 'b 1 = gib 1 , d'ù g = (gi) £ C°(il, GLn(Oxj), le cocyclecorrespondant est ' = 9 , où 9 ij = giîjgj 1 • On définitPï(,9) •= (^tr{NMg) ••= LHig^dgi) 3 }); p(4>,g) = (iH^,g), P3 (g)) £ C^fixK &Alors pz( a 4>) = Ps(^) + d y pz(g) + d DR p 2 ((j),g),P2((j) s ) = M'P) + c h ^2(^,9),CechCechd'où p( 9 (j>) = p((j>) + dp(fi,g). Donc la classe p(E) de p() dans H 2 (X,Q x ') qu'onpeut appeler la classe de Pontryagin-Atyiah-Chern-Simons (pacs), ne depend quede E. On remarque que p() = p(fiû),donc p(E) = p(E*).3.6. Les groupoïdes Algr(u,o x )i U C X, forment un champ Stlflx sur latopologie de Zariski (même étale), parsque les opérations ^ sont des opérateursdifférentiels qui se localisent. D'après 3.2, VYLgx est une gerbe sous Yl x ' (localementnon-vide, mais pas localement connexe). Donc la classe caractéristiquec(2Ug x ) £ H 2 (X,Q x ') est définie, telle que C(%LIQ X ) = 0 ssi Y(X,W,gx) est nonvide.Rappelons sa définition. On choisit un recouvrement affine il = {£/,} de Xavec Ui petites; on choisit les objets Ai £ T(Ui,%Ltg x )- Sur les doubles intersections,il existe les isomorphismes fty : Aj\u ió ~^y Aifiji, + «y, «y £ ii 3 'êr (Uij).Si l'on pose ß ijk := hij\ Uijk - hik\u ijk + h jk \u ijk € ü 2 (Uij k ), on a c({^},{%}) :=

Sur les Algèbres Vertex Attachées aux Variétés Algébriques 531((aij),(ßijkj) £ Z 2 (ii,Yl x '). Pour une autre famille ({v4-},{/iy}) il existenthi : Ai ^> A'i + a t , a t £ fi^/jT; alors (hj + ay)o/iy : „4, -^> A'j + (a.j + ay) et(h'ij + ai) ohi '• Ai ~^y A'j + («y + «»), donc il existe l'unique /3y £ fi^; fc telle quedßij = a'ij - (Xij + ai - aj. Alors d((a t ), (/%)) = c({„4-},{>y}) - c({^},{%});par définition c(%Ltg x ) est la classe de c({„4j}, {^y}) dans la cohomologie.Soit fix le fibre tangent de X. Choisissons des bases bonnes b % de T(Ui,Tx),avec b 1 =

532 Vadim Schechtman[B][GMS][MSV]R. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster,Proc. Natl. Acad. Sci, USA, 83 (May 1986), 3068-3071.V. Gorbounov, F. Malikov & V. Schechtman, Gerbes of chiral differentialoperators. II, math.AG/0003170.F. Malikov, V. Schechtman & A. Vaintrob, Chiral de Rham complex,Comm. Math. Phys., 204 (1999), 439-473.

ICM 2002 • Vol. II • 533-541Topology of Singular Algebraic VarietiesB. Tötaro*AbstractI will discuss recent progress by many people in the program of extendingnatural topological invariants from manifolds to singular spaces. Intersectionhomology theory and mixed Hodge theory are model examples of such invariants.The past 20 years have seen a series of new invariants, partly inspired bystring theory, such as motivic integration and the elliptic genus of a singularvariety. These theories are not defined in a topological way, but there areintriguing hints of their topological significance.2000 Mathematics Subject Classification: 14F43, 32S35, 58J26.Keywords and Phrases: Intersection homology, Weight filtration, Ellipticgenus.1. IntroductionThe most useful fact about singular complex algebraic varieties is Hironaka'stheorem that there is always a resolution of singularities [20]. It has long beenclear that the non-uniqueness of resolutions poses a difficulty in many applications.Many different methods have been used to get around this difficulty so as to defineinvariants of singular varieties. One approach is to try to describe the relationbetween any two resolutions, leading to ideas such as cubical hyperresolutions [18]and the weak factorization theorem ([1], [31]). Another idea, coming from minimalmodel theory, is to insist on the special importance of crêpant resolutions, and moregenerally to emphasize the role of the canonical bundle. Recently the interplaybetween these two approaches has been very successful, as I will describe.The recent methods tend to be more roundabout than the direct topologicaldefinition of intersection homology groups. It is tempting to try to define suitablegeneralizations of intersection homology groups in order to "explain" various resultsbelow (3.2, 3.4, 4.1, 5.2).* Department of Mathematics and Mathematical Statistics, University of Cambridge, WilberforceRoad, Cambridge CB3 0WB, UK. E-mail: b.totaro@dpmms.cam.ac.uk

534 B. Totaro2. The weight filtrationDeligne discovered a remarkable structure on the rational cohomology of anycomplexalgebraic variety, not necessarily smooth or compact: the weight filtration[9]. This filtration expresses the way in which the cohomology of any variety isrelated to the cohomology of smooth compact varieties. It is a deep fact that theresulting filtration is well-defined. For example, an immediate consequence of thewell-definedness of the weight filtration on cohomology with compact support is thefollowing fact, originally conjectured by Serre ([11], [6], [12], p. 92).Theorem 2.1. For any complex algebraic variety X, not necessarily smoothor compact, one can define "virtual Betti numbers" a,X £ Z for i > 0 such that(1) if X is smooth and compact, then the numbers a,X are the Betti numbershX = dim Q H* (X,Q);(2) for any Zariski-closed subset Y c X, a,X = a,F + afiX — Y).Using resolution of singularities, it is clear that the numbers a,X are uniquelycharacterizedby these properties. What is less clear is the existence of such numbers.It follows, for example, that if two smooth compact varieties X and Y can bewritten as finite disjoint unions of locally closed subsets, X = JJ Xj and Y = JJ Y t ,with isomorphisms Xj = Y t for all i, then X and Y have the same Betti numbers.This is a topological property of algebraic varieties which has no obvious analoguein a purely topological context.The existence of the weight filtration, and consequently of the virtual Bettinumbers OjX, was originally suggested by Grothendieck's approach to the Weilconjectures on counting rational points on varieties over finite fields. Indeed, thenumber of F g -points of a variety clearly has an additive property analogous toproperty (2) above. One proof of the existence of the weight filtration for complexvarieties reduces the problem to the full Weil conjecture for varieties over finitefields, proved by Deligne [8]. Around the same time, Deligne gave a more directproof of the existence of the weight filtration for complex varieties, using Hodgetheory [7]. This is a classic example of the philosophy that the deepest propertiesof algebraic varieties can often be proved using either number theory or analysis,while they have no "purely geometric" proof.In 1995, however, Gillet and Soulé gave a new proof of the existence of theweight filtration [13]. They used "only" resolution of singularities and algebraicif-theory, specifically the Gersten resolution. As a result of their more geometricproof, they were able to define the weight filtration on the integral cohomology orFj-cohomology of a complex algebraic variety, not only on rational cohomology.To understand what this means, let me describe the weight filtration for asmooth complex variety U, not necessarily compact. Using resolution of singularities,we can write U as the complement of a divisor with normal crossings D insome smooth compact variety X. For i > 0, let XW be the disjoint union of thei-fold intersections of divisors. Then there is a spectral sequenceEi = H j (X {i) ,k)=> Hl +j (U,k)for any coefficient ring fc. The weight filtration on the compactly supported cohomologyof U is defined as the filtration associated to this spectral sequence. Gillet

Topology of Singular Algebraic Varieties 535and Soulé show that for any coefficient ring fc, this filtration is an invariant of U,independent of the choice of compactification U. This is not at all clear from theknown invariance of this filtration for fc = Q.In fact, Gillet and Soulé proved more: for any coefficient ring fc, the spectralsequence is an invariant of U from the E 2 term on. For fc = Q, the spectral sequencedegenerates at E 2 , but this is not true with coefficients in Z or Fj. As a result, forgeneral coefficients fc, the groups in the E 2 term are interesting new invariants of Uwhich are not simply the associated graded groups to the weight filtration. Theysatisfy Mayer-Vietoris sequences, and so can be considered as a cohomology theoryon algebraic varieties.I can now explain a new application of the geometric proof that the weightfiltration is well-defined. Namely, one can try to define the weight filtration notonly for algebraic varieties. The point is that resolution of singularities holds moregenerally, for complex analytic spaces, and even for real analytic spaces. Gilletand Soulé 's construction of the weight filtration uses algebraic if-theory as wellas resolution of singularities, and it is not clear how to adapt the argument toan analytic setting. But Guillen and Navarro Aznar improved Gillet and Soulé'sargument so as to construct the weight filtration using only resolution of singularities[17]. The details of their argument use their idea of "cubical hyperresolutions" [18].Using the method of Guillen and Navarro Aznar, I have been able to definethe weight filtration for complex and real analytic spaces. In more detail, let usdefine a compactification of a complex analytic space X to be a compact complexanalytic space X containing X as the complement of a closed analytic subset. Ofcourse, not every complex analytic space has a compactification in this sense. Wesay that two compactifications of X are equivalent if there is a third which lies overboth of them.Theorem 2.2. Let k be any commutative ring. Then the compactly supportedcohomology H*(X,k) has a well-defined weight filtration for every complex analyticspace X with an equivalence class of compactifications.Any algebraic variety comes with a natural equivalence class of compactifications,but in the analytic setting this has to be considered as an extra piece ofstructure. On the other hand, the theorem says that the weight filtration is welldefinedon all compact complex analytic spaces, with no extra structure needed.For real analytic spaces, one has the difficulty that there is no natural orientation,unlike the complex analytic situation. This is not a problem if one usesF 2 -coefficients, and therefore one can prove:Theorem 2.3. For every real analytic space X with an equivalence class ofcompactifications, the compactly supported cohomology of the space X(R) of realpoints with F 2 coefficients has a well-defined weight filtration.In particular, one can define virtual Betti numbers OjX for a real analyticspace X with an equivalence class of compactifications, the integers OjX being theusual F 2 -Betti numbers in the case of a closed real analytic manifold.Example. Let X be the compact real analytic space obtained by identifyingtwo copies at the circle at a point, and let Y be the compact real analytic spaceobtained by identifying two points on a single circle (the figure eight). It is imme-

536 B. Totarodiate to compute that aoX = 1 and aiX = 2, whereas OQY = 0 and ai = Y. Theinteresting point here is that the spaces X(R) and Y(R) of real points are homeomorphic.Thus the numbers a» for a compact real analytic space are not topologicalinvariants of the space of real points. In a similar vein, Steenbrink showed that theweight filtration on the rational cohomology of complex algebraic varieties is not atopological invariant, using 3-folds [27].Nonetheless, it seems fair to say that extending the weight filtration and thevirtual Betti numbers to complex and real analytic spaces helps to bring out moreof the topological meaning of these invariants of algebraic varieties. A real analyticspace has in some ways a weak structure; for example, the classification of closedreal analytic manifolds up to isomorphism is the same as the classification of closeddifferentiable manifolds up to diffeomorphism. From this point of view, it is surprisingthat compactified real analytic spaces have the extra structure of the weightfiltration on their F 2 -cohomology. It seems natural to ask for an F 2 -linear abeliancategory of "mixed motives" associated to compactified real analytic spaces X, suchthat the F 2 -cohomology groups of X with their weight filtration are determined bythe mixed motive of X. On Beilinson's conjectured abelian category of mixed motivesin algebraic geometry, see for example Jannsen [21], 11.3, and [22]; on variousapproximations to this category, see the triangulated categories defined by Hanamura[19], Levine [26], and Voevodsky [29], and the abelian category defined byNori.It should be much easier to define mixed motives for real analytic spaces thanto do so for algebraic varieties. In particular, one might speculate that the mixedmotive of a real analytic space should not involve much more information than theweight spectral sequence converging to its F 2 -cohomology (starting at E 2 ), perhapsconsidered together with an action of the Steenrod algebra. In low dimensions, onecould hope for precise classifications of mixed motives along these lines.3. Stringy Betti numbersThe following result of Batyrev's [4] is related to his famous result that twobirational Calabi-Yau manifolds have the same Betti numbers. The proof usesKontsevich's idea of motivic integration [24], as developed by Denef and Loeser[10]. To be precise, Batyrev's statement involves Hodge numbers, but I will onlystatewhat it gives about Betti numbers.Theorem 3.1. Let Y be a complex projective variety with log-terminal singularities.Then one can define the "stringy Poincaré function" p s t r (Y), which is arational function, such that for any crêpant resolution of singularities n : X —t Y,the stringy Poincaré function of Y is the usual Poincaré polynomial of X.We recall Reid's important definitions which are used here. First, let Y beany normal complex variety such that the canonical divisor Ky is Q-Cartier. ByHironaka, Y has a resolution of singularities n : X —t Y such that the exceptionaldivisors Ei, i £ I, are smooth with normal crossings. The discrepancies a» of E t aredefined byK x = n*K Y + ^2 aiEi -

Topology of Singular Algebraic Varieties 537The variety Y is defined to have log-terminal singularities if and only if a» > — 1 forall z. A resolution X —t Y is said to be crêpant if Kx = n*K Y .Batyrev defines the stringy Poincaré function of Y by the formula:JWW=5>(£$) nJCI j€Jg-1qaj+1 _Here Ej is the open stratum of Ej := Cij e jEj, and p(Ej) denotes the virtualPoincaré polynomial of Ej, written as a polynomial in q 1 / 2 . Thus Pst*{Y) is arational function in q 1 / 2 for Y Gorenstein, and in q x l n for some n in general.Batyrev's proof that the stringy Poincaré function of Y is independent of thechoice of resolution, using motivic integration, rests on the additivity properties ofthe virtual Poincaré polynomial. Using our extension of virtual Betti numbers tocomplex analytic spaces, we find:Theorem 3.2. The stringy Poincaré function can be defined as a rationalfunction for any compactified complex analytic space with log-terminal singularities.For any crêpant resolution X —t Y with Y compact, the stringy Poincaré functionof Y is the usual Poincaré polynomial of X.Likewise for real analytic spaces:Theorem 3.3. An F 2 -analogue of the stringy Poincaré function can be definedas a rational function for compactified real analytic spaces with log-terminal singularities.For any crêpant resolution X —t Y with Y compact, the stringy Poincaréfunction of Y is the usual Poincaré polynomial of the F 2-cohomology of X.In particular, this answers part of Goresky and MacPherson's Problem 7 in[15]:Corollary 3.4. Given a compact real algebraic variety Y, the ~F 2 -Betti numbersof any two projective IH-small resolutions of Y are the same.This uses the relation between IH-small resolutions and crêpant resolutions,which I worked out in [28] using results of Kawamata [23] and Wisniewski [30].In the complex situation, the corollary (for Betti numbers with any coefficients)has a more direct proof, since the Betti numbers of any small resolution of Y areequal to the dimensions of the intersection homology groups of Y. It is not yetknown whether one can define a new version of intersection homology groups withF 2 -coefficients which would be self-dual for all compact real analytic spaces. Apossible framework for defining such a theory has been set up by Banagl [2].4. The elliptic genus of a singular varietyI found that any characteristic number which can be extended from smoothcompact complex varieties to singular varieties, compatibly with small resolutions,must be a specialization of the elliptic genus [28]. It was then an important problemto define the elliptic genus for singular varieties. This was solved in a completelysatisfyingway by Borisov and Libgober [5]:Theorem 4.1. Let Y be a projective variety with log-terminal singularities.Then one can define the elliptic genus ofY, ip(Y), such that for any crêpant resolutionX —t Y, we have ip(Y) = ip(X).fi

538 B. TotaroHere is Borisov and Libgober's definition of ip(Y). Yet n : X —t Y be aresolution whose exceptional divisors Ek have simple normal crossings, and let a^be the discrepancies as in section 3. Formally, let yi denote the Chern roots of X sothat c(TX) = Y\i(l + yi), and let e^ be the cohomology classes on X of the divisorsE k . Then ip(Y) is the analytic function of variables z and r defined byW ) - y y dl9(-z)9(ii-)} ( 11 9(^ - z)9(-(a k + l)z) hwhere 0(Z,T) is the Jacobi theta function. The proof that ip(Y) is independent ofthe choice of resolution for log-terminal Y uses the weak factorization theorem ofAbramovich, Karu, Matsuki, and Wlodarczyk ([1], [31]).In the spirit of earlier sections, the singular elliptic genus extends to compactcomplex analytic spaces with log-terminal singularities. But it remains a mysteryhowto define the elliptic genus for some topologically defined class of singular spacesthat would include singular analytic spaces with log-terminal singularities.5. Possible characteristic numbers for real analyticspacesIn my paper [28], in trying to define characteristic numbers for singular complexvarieties, it was very helpful to require that these numbers are compatible withIH-small resolutions, as Goresky and MacPherson had suggested ([15], Problem 10).The problem thereby becomes more precise: it may be possible to show that somecharacteristic numbers extend to singular varieties and some do not. This can helpto suggest valuable invariants for singular varieties, such as Borisov and Libgober'selliptic genus for singular varieties, even if one is not a priori interested in IH-smallresolutions. (The same comments apply to crêpant resolutions.)With this in mind, we here begin to analyze which characteristic numberscan be defined for real analytic spaces, or for topological spaces with similar singularities,compatibly with IH-small resolutions. In the complex situation, thefundamental example of a singularity with two different IH-small resolutions is the3-fold node; one says that the two IH-small resolutions are related by the simplesttype of "flop." Likewise, in the real situation, the real 3-fold node has two differentIH-small resolutions. For convenience, let us say that two closed n-manifolds arerelated by a "real flop" if they are the two different IH-small resolutions Xi and X 2of a singular space with singular set of real codimension 3 that is locally isomorphicto the product of the 3-fold node with an (n — 3)-manifold.Let us first consider characteristic numbers for unoriented spaces. By Thom,the bordism ring MO* for unoriented manifolds is detected by Stiefel-Whitneynumbers.Therefore we can ask which Stiefel-Whitney numbers (meaning F 2 -linearcombinations of Stiefel-Whitney monomials) are unchanged under real flops. Or,more or less equivalently: what is the quotient of the bordism ring MO* by theideal of real flops Xi — X 2 , for Xi and X 2 as above? There is a good answer:

Topology of Singular Algebraic Varieties 539Theorem 5.1. The F 2 -vector space of Stiefel-Whitney numbers which areinvariant under real flops of n-manifolds is spanned by the numbers w\w n -i for0 of this space of invariant Stiefel-Whitneynumbers, modulo those which vanish for all n-manifolds, is Oforn odd and [n/2j + lfor n even. The quotient ring of MO* by the ideal of real flops is isomorphic to:F 2 [RP 2 ,RP 4 ,RP 8 ,.. .]/((RP 2 ") 2 = (RP 2 ) 2 " for all a > 2).This class of Stiefel-Whitney numbers has occurred before, in Goresky andPardon's calculation of the bordism ring of locally orientable F 2 -Witt spaces [16].To be precise, the latter ring coincides with the above ring in even dimensions butis also nonzero in odd dimensions. Goresky defined a Wu classWj in intersectionhomology for F 2 -Witt spaces [14], so that the square v 2 lives in ordinary homology,and the characteristic numbers for locally orientable F 2 -Witt spaces Y are obtainedby multiplying these homology classes by powers of the cohomology class Wi.This does not explain the invariance of these Stiefel-Whitney numbers forreal flops, however. The problem is that the 3-fold node is not an F 2 -Witt space.(Topologically, it is the cone over S 1 x S 1 , whereas the cone over an even-dimensionalmanifold is a Witt space if and only if the homology in the middle dimension iszero.) That is, the standard definition of intersection homology is not self-dual on aspace with 3-fold node singularities. This again points to the problem of defining anew version of intersection homology with F 2 coefficients which is self-dual on realanalytic spaces. That should yield an L-class in the F 2 -homology of such a space,which we can also identify with the square of the Wu class, and which thereforeshould allow the above characteristic numbers to be defined for a large class of realanalytic spaces. There are related results by Ban agi [3], for spaces which admit anextra "Lagrangian" structure.We now ask the analogous question for oriented singular spaces: what characteristicnumbers can be defined, compatibly with IH-small resolutions? We couldbegin by asking for the quotient ring of the oriented bordism ring M SO* by orientedreal flops Xi — X 2 , defined exactly as in the unoriented case (Xi and X 2 are thetwo small resolutions of a family of real 3-fold nodes), except that we require Xiand X 2 to be compatibly oriented. It turns out that this is not enough: all Pontrjaginnumbers are invariant under oriented real flops, whereas they can changeunder other changes from one IH-small resolution to another, such as complex flops(between the two small resolutions of a complex family of complex 3-fold nodes).By considering both real and complex flops, we get a reasonable answer:Theorem 5.2. The quotient ring of M SO* by the ideal generated by orientedreal flops and complex flops is:Z[(5,2 7 ,27 2 ,2 7 4 ,...],where CP 2 maps to ö and CP 4 maps to 2 7 + o" 2 . This quotient ring is exactly theimage of M SO* under the Ochanine elliptic genus ([25], p. 63).

540 B. TotaroThis result suggests that it should be possible to define the Ochanine genus fora large class of compact oriented real analytic spaces, or even more general singularspaces.References[1] D. Abramovich, K. Karu, K. Matsuki, and J. Wlodarczyk, Torification andfactorization of birational maps, math. AG/9904135.[2] M. Banagl, Extending intersection homology type invariants to non-Wittspaces, Memoirs of the AMS, to appear.[3] M. Banagl, The L-class of non-Witt spaces, to appear.[4] V. Batyrev, Stringy Hodge numbers of varieties with Gorenstein canonicalsingularities, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997),1-32, World Scientific, 1998.[5] L. Borisov and A. Libgober, Elliptic genera of singular varieties, Duke Math.J., to appear.[6] V. Danilov and A. Khovanskii, Newton polyhedra and an algorithm for computingHodge-Deligne numbers, Math. USSR Izv. 29 (1987), 279-298.[7] P. Deligne, Théorie de Hodge I, II, III, Proc. ICM 1970, v. 1, 425-430; Pubi.Math. IHES 40 (1972), 5-57; 44 (1974), 5-78.[8] P. Deligne, La conjecture de Weil I, Pubi. Math. IHES 43 (1974), 273-308.[9] P. Deligne, Poids dans la cohomologie des variétés algébriques, Actes ICMVancouver 1974,1, 79-85.[10] J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivicintegration, Invent. Math. 135 (1999), 201-232.[11] A. Durfee, Algebraic varieties which are a disjoint union of subvarieties, Geometryand topology: manifolds, varieties and knots, 99-102, Marcel Dekker,1987.[12] W. Fulton, Introduction to toric varieties, Princeton, 1993.[13] H. Gillet and C. Soulé, Descent, motives, and if-theory, J. reine angew. Math.478 (1996), 127-176.[14] M. Goresky, Intersection homology operations, Comment. Math. Helv. 59(1984), 485-505.[15] M. Goresky and R. MacPherson, Problems and bibliography on intersectionhomology, Intersection homology, ed. A. Borei, Birkhäuser, 1984, 221-233.[16] M. Goresky and W. Pardon, Wu numbers of singular spaces, Topology 28(1989), 325-367.[17] F. Guillen and V. Navarro Aznar, Un critère d'extension d'un foncteur définisur les schémas lisses, math.AG/9505008.[18] F. Guillén, V. Navarro Aznar, P. Pascual, and F. Puerta, Hyperrésolutionscubiques et descente cohomologique, Lecture Notes in Mathematics 1335,Springer, 1988.[19] M. Hanamura, Homological and cohomological motives of algebraic varieties,Invent. Math. 142 (2000), 319-349.[20] H. Hironaka, Resolution of singularities of an algebraic variety over a field of

Topology of Singular Algebraic Varieties 541characteristic zero, Ann. Math. 79 (1964), 109-326.[21] U. Jannsen, Mixed motives and algebraic K-theory, LNM 1400, Springer, 1990.[22] U. Jannsen, Motivic sheaves and filtrations on Chow groups, Motives (Seattle,1991), 245-302, AMS, 1994.[23] Y. Kawamata, K. Matsuda, and K. Matsuki, Introduction to the minimalmodel program, Algebraic Geometry (Sendai, 1985), ed. T. Oda, 283-360,Kinokuniya-North Holland, 1987.[24] M. Kontsevich, lecture at Orsay, 7 December 1995.[25] P. Landweber, Elliptic cohomology and modular forms, Elliptic curves andmodular forms in algebraic topology, 55-68, LNM 1326, Springer, 1988.[26] M. Levine, Mixed motives, AMS, 1998.[27] J. Steenbrink, Topological invariance of the weight filtration, Indag. Math. 46(1984), 63-76.[28] B. Totaro, Chern numbers for singular varieties and elliptic homology, Ann.Math. 151 (2000), 757-791.[29] V. Voevodsky, A. Suslin, and E. Friedlander, Cycles, transfers, and motivichomology theories, Princeton, 2000.[30] J. Wisniewski, On contractions of extremal rays of Fano manifolds, J. reineangew. Math. 417 (1991), 141-157.[31] J. Wlodarczyk, Combinatorial structures on toroidal varieties and a proof ofthe weak factorization theorem, math.AG/9904076.

Section 7. Lie Group and Representation TheoryPatrick Delorme: Harmonic Analysis on Real ReductiveSymmetricSpaces 545Pavel Etingof: On the Dynamical Yang-Baxter Equation 555D. Gaitsgory: Geometric Langlands Correspondence for GL n 571Michael Harris: On the Local Langlands Correspondence 583Alexander Klyachko: Vector Bundles, Linear Representations, and SpectralProblems 599Toshiyuki Kobayashi: Branching Problems of Unitary Representations 615Vikram Bhagvandas Mehta: Representations of Algebraic Groups andPrincipal Bundles on Algebraic Varieties 629E. Meinrenken: Clifford Algebras and the Duflo Isomorphism 637Maxim Nazarov: Representations of Yangians Associated with SkewYoung Diagrams 643Freydoon Shahidi: Automorphic L-Functions and Functoriality 655Marie-France Vignéras: Modular Representations of p-adic Groups andof Affine Hecke Algebras 667

ICM 2002 • Vol. II • 545-554Harmonic Analysis on Real ReductiveSymmetric SpacesPatrick Delorme*AbstractLet G be a reductive group in the Harish-Chandra class e.g. a connectedseniisiniple Lie group with finite center, or the group of real points of a connectedreductive algebraic group defined over R. Let a be an involution of theLie group G, H an open subgroup of the subgroup of fixed points of a. Onedecomposes the elements of L 2 (G/H) with the help of joint eigenfunctionsunder the algebra of left invariant differential operators under G on G/H.2000 Mathematics subject classification: 22E46, 22F30, 22E30, 22E50,33C67.Keywords and Phrases: Reductive symmetric space, Fiancherei formula,Meromorphic continuation of Eisenstein integrals, Temperedness, Truncation,Maass-Selberg relations.1. IntroductionLet G be a real reductive group in the Harish-Chandra class [H-Cl], e.g. aconnected semisimple Lie group with finite center, or the group of real points of aconnected reductive algebraic group defined over R. Let a be an involution of theLie group G, H an open subgroup of the subgroup of fixed points of a.Important problems of harmonic analysis on the so-called reductive symmetricspace G/H are :(a) to make the simultaneous spectral decomposition of the elements of thealgebra B(G/H) of left invariant differential operators under G on G/H. In otherwords, one wants to write the elements of L 2 (G/H) with the help of joint eigenfunctionsunder B(G/H).(b) to decompose the left regular representation of G in L 2 (G/H) into anHilbert integral of irreducible unitary representations of G : this is essentially thePlancherel formula.•"Institut de Mathématiques de Luminy, TJ.P.R. 9016 du C.N.R.S., Faculté des Sciencesde Luminy, 163 Avenue de Luminy, Case 930, 13288 Marseille Cedex 09, France. E-mail:delorme@iml.univ-mrs.fr

546 Patrick Delorme(e) to decompose the Dirac measure at eH, where e is the neutral element ofG, into an integral of ff-fixed distribution vectors : this is essentially the Fourierinversion formula.These problems were solved for the "group case" (i.e. the group viewed as asymmetric space : G = Gi x Gi, a(x,y) = (y,x), H is the diagonal of Gi x Gi) by Harish-Chandra in 1970s (see [H-C1,2, 3]), the Riemannian case (H maximalcompact) had been treated before (see [He]). Later, there were deep results by T.Oshima [01]. When G is a complex group and H is a real form, the Problems (a),(b), (c) were solved by P. Harinck, together with an inversion formula for orbitalintegrals ([Ha], see also [D3] for the link of her work with the work of A. Bouazizon real reductive groups).Then, E. van den Ban and H. Schlichtkrull, on the one hand, and I, on the otherhand, obtained different solutions to problems (a), (b), (c). Moreover, they obtaineda Paley-Wiener theorem (see [BS3] for a presentation of their work). I present heremy point of view, with an emphasize on problem (a), because it simplifies theformulations of the results (nevertherless, the important aspect of representationtheory is hidden). It includes several joint works, mainly with J. Carmona , andalso with E. van den Ban and J.L. Brylinski. Severals works of T. Oshima, linkedto the the Flensted-Jensen duality, alone and with T. Matsuki are very importantin my proof, as well as earlier results of E. van den Ban and H. Schlichtkrull.I have to acknowledge the deep influence of Harish-Chandra 's work. Thecrucial role played by the work [Be] of J. Bernstein on the support of the Plancherelmeasure, and some part of Arthur's article on the local trace formula [A] will beappearant in the main body of the article.2. Temperedness of the spectrumLet 9 be a Cartan involution of G commuting with a, let K be the fixed pointset of 9. Let g be the Lie algebra of G, etc. Let s (resp. q) be the space of elements ing which are antiinvariant under the differential of 9 (resp. a). Yet a$ be a maximalabelian subspace of s n q. If F is a a9-stable parabolic subgroup of G, containingA$ := exp

Harmonie Analysis on Real Reductive Symmetric Spaces 547Some spaces of r-spherical functions on G/H play a crucial role in the theory,namely:(a) G(G/H, T) : the Schwartz space of r-spherical functions on G/H which arerapidly decreasing as well as their derivatives by elements of the enveloping algebraU(g) of g (see [B2]).(b) A(G/H, T): the space of smooth r-spherical functions on G/H which areB(G/H) finite. Here A is used to evoke automorphic forms.(c) Atem,p(G/H, T): the space of elements of A(G/H, r) which have temperedgrowth as well as their derivatives by elements of U(g) ([D2]). Integration offunctions on G/H defines a pairing between At emp (G/H, r) and G(G/H, r).(d) A2(G/H, T): the space of square integrable elements of A(G/H, r). Thisis a subspace of the three proceeding spaces.One has:Theorem 1 ( [D2] ): The space ./^(G/ff, r) is finite dimensional.This is deduced from the theory of discrete series for G/H initiated by M.Flensted-Jensen [F-J] and achieved by T. Oshima and T. Matsuki, using the Flensted-Jensen duality [OM]. One has also to use the behaviour of the discrete series undercertain translation functors, studied by D. Vogan [V] and a result of H. Schlichtkrull[S] on the minimal if-types of certain discrete series.The next result follows from the work of J. Bernstein [Be] on the support ofthe Plancherel measure.Theorem 2 ([CD1], Appendice C): Every function in G(G/H, r) can be canonicallydesintegrated as an integral of elements of At emp (G/H, r).This information appeared to be crucial at the end of our proof.3. The continuous spectrum: Eisenstein integralsLet P = MAN the Langlands ^-decomposition of a -¥ E(P,fi,X) admits a meromorphic continuationin X £ o£- This meromorphic continuation, denoted in the same way,multiplied by a suitable product, p v , of functions of type X H> (a, A) + c, where a isa root of a and c £ C, is holomorphic around ia*.

548 Patrick DelormeThis meromorphic continuation is an interesting feature of the theory. For thethe "group case", it comes down to the meromorphic continuation of Knapp-Steinintertwining integrals. My proof with Brylinski uses D-modules arguments.The case where P is minimal had been treated separately by E. van den Ban[BI] and G. Olafsson [01]. One has also to mention the earlier work of T. Oshimaand J. Sekiguchi [OSe] on the spaces of type G/K e .The proof which gives the best results uses a method of tensoring by finitedimensional representations. It is a joint work with J. Carmona. It was initiated byD. Vogan and N. Wallach (see [W], chapter 10) for the meromorphic continuationof the Knapp-Stein intertwining integrals. For symmetric spaces and the mostcontinuous spectrum, the proof is due to E. van den Ban [B2]. This proof usesBruhat's thesis and tensoring by finite dimensional modules. This implies roughestimates for Eisenstein integrals, which generalize those obtained by E. van denBan when P is minimal [B2].Theorem 4 ([Dl]): If X £ ia* is such that E(P,fi,X) is well defined, then it istempered, i.e. is an element of At emp (G/H,T).This is a natural result but the proof is quite long. It uses the behaviourunder translation functors of ff-fixed distribution vectors of discrete series and ofgeneralized principal series, and also of the Poisson transform. Moreover the dualityof M. Flensted-Jensen, [F-J], and a criteria of temperedness due to Oshima [02] playa crucial role (apparently, J. Carmona has a way to avoid boundary values).With the help of this theorem and by using techniques due to E. van den Ban[B2], the rough estimates for Eisenstein integrals can be improved to get uniformsharp estimates for p v (X)E(P,fi,X), X £ ia* (cf. [Dl]).4. C-funetionsLet P be as above and let L be equal to MA. The theory of the constantterm, due to J. Carmona [CI] (Harish-Chandra for the group case, [H-Cl]), gives alinear map from At emp (G/H, r) into At emp (L/L n H, rfi), *fi >-*• *pp, characterizedby:1/2linit^+oo o~p•((exptX)l)ip((exptX)l) — (pp((exptX)l) = 0,where I £ L/L n H, X £ ap is F-dominant and öp is the modular function of P.Yet Q be a a9-stable parabolic subgroup of G with the same 0-stable Levisubgroup L other than P. Yet W(a) be the group of automorphisms of o inducedby an element of Ad(G). One defines meromorphic functions on oj, A H> CQ^P(S, X)with values in End(A 2 (M/M n H,TM)) such that :E(P,fi,X) Q (ma) = Y^ (C \p(s,Xj(p)(m)a^sX Q , m£ M, a£ A, X£ia*,seW(a)or rather for A in an open dense subset of ia*.The G-functions allow to normalize Eisenstein integrals as folllows:E°(P,fi,X) := E(P,Cp l p(l,X)- 1 fi,X).

Harmonie Analysis on Real Reductive Symmetric Spaces 5495. Truncation, Maass-Selberg relations and the regularityof normalized Eisenstein integralsLet P be as above and let P' = M'A'N' be the Langlands ^-decompositionof another a9-stable parabolic subgroup of G. Let fi (resp. ip') be an element ofA 2 (M/M fl H,TM) (resp. A 2 (M'/M' C\H,TM'))- One chooses p v as in Theorem 3,such that the product of p v with the G-functions are holomorphic in a neighbourhoodof ia* which is a product of ia* with a neighbourhood of 0 in o* . We do thesame for ip'. One defines F(X) := p v (X)E(P,fi,X). One defines similarly F'.One assumes, for the rest of the article, that G is semisimple. This is just tosimplify the exposition. One chooses T £ a$, regular with respect to the roots of00 in g. Yet C\. be the convex hull of W(a$)T and let Cj be equal to the subsetK(exp C^)H of G/H.Theorem 5 ([D2]):(i) One gets an explicit expression OJ T (X, X), involving the C-functions (see anexample below) and vanishing when A and A' are not conjugate under K, which isasymptotic to0 T (A,A'):= / (F(X)(x),F'(X')(x))dx,when T goes to infinity and X £ ia*, X £ ia'*. More precisely for ö > 0 there existsC > 0, k £ N and e > 0, such that for all T satisfying | a(T) |> Ö || T || for everyroot a of 00 in g, one has:n T (X,X)-oj T (X,X) |< G(l+ || A ||)*(1+ || A' ||) k — ^[[Tll(ii) Moreover OJ T is analytic in (A, A') £ ia* x ia'*.This generalizes a result of J. Arthur for the group case [A], Theorem 8.1. My proofis quite similar, but I was able to avoid his use of the Plancherel formula.(ii) is an easy consequence of (i). In fact, the explicit form of OJ T implies thatit is meromorphic around ia* x ia'*. Moreover Q T is holomorphic, hence locallybounded,around ia* x ia'*. From the inequality in (i), one deduces that OJ T islocally bounded, hence holomorphic, around ia* x ia'*.We will now show, by an example, how the explicit form of OJ T and its analyticityin (A, A') £ ia* x ia'* imply the Maass-Selberg relations.Let a be equal to 9, H = K, and r be the trivial representation. Let P,P'be minimal parabolic subgroups of G. Then dimA 2 (M/M C\H,T) = 1 and the C-functions are scalar valued. One assumes g to be semisimple and that the dimensionof A is one. Then W(a) has 2 elements, ±1, and one has the following explicitexpression of OJ T :oj T (X,X)=p v (X)p v i(X) YI e sXT - s ' x ' T C Pi p(s,X)C Pl p(s',X')(sX - s'A')" 1 .s=±l,s'=±lThus OJ T (X, A') is the sum of a product of (A — A') -1 by an analytic function witha product of (A + A') -1 by an analytic function. The analyticity at (A, A) implies

550 Patrick Delormeeasily that the factor in front of (A — A') -1 vanishes for A = A'. Hence we get| Gp[p(l,A) | 2 = | Cp|p( —1,A) | 2 , A £ ia*. This is one of the Maass-Selberg relations^.[D2], Theorem 2, and the work with J. Carmona [CD2], Theorem 2 for thegeneral case, see [BI], [B2] for the case where P is minimal). These relations implythatthe G-functions attached to normalized Eisenstein integrals are unitary, whendefined, for A purely imaginary. Hence they are locally bounded . This implies thatthey are holomorphic around the imaginary axis. This implies in particular someholomorphy property of the constant term of normalized Eisentein integrals. Fromthis, with the help of [BCD], one deduces:Theorem 6 (Regularity theorem for normalized Eisenstein integrals, [CD2], [BS1]for P minimal): The normalized Eisenstein integrals are holomorphic in a neighbourhoodof the imaginary axis.6. Fourier transform and wave packetsTheorem 7 ([CD2], [BS1] for F minimal): For f £ e(G/H,r), one has $ P f £§(ia*) ®A 2 (M/M fl H,TM), where Ipf is characterized by:((3%fi)(X),fi)= [ (f(x),E°(P,fi,X)(xj)dx, X£ia*, fi £ A 2 (M/M n H,T M ),JG/Hhere S (io*) is the usual Schwartz space.This theorem follows from the sharp estimates of Eisenstein integrals.Theorem 8 ([BCD]): If ^ is an element of$(ia*) A 2 (M/M n H,T M ), one has3° P £e(G/H,T), where :fp(x) :-- f E°(P,

Harmonie Analysis on Real Reductive Symmetric Spaces 551Theorem 9 ([CD2]): Let ¥ be a set of representative of a-association classes ofa9-stable parabolic subgroups. Here a-association means that the a are conjugatedunder K. Define:•j> T =PerY((w(^)r 1^°p-Then T T is an orthogonal projection operator in G(G/H,T) endowed with the L 2scalar product.7. The Plancherel formulaEssentially, the solution to problem (a) is contained in the following:Theorem 10 ([D4]): The projection T T is the identity operator on G(G/H,T).Actually, this gives an expression of every element in G(G/H,T) as a wavepacket of normalized Eisenstein integrals. The proof goes as follows. If T T was notthe identity, using Theorem 1 on the temperedness of the spectrum, one could finda non zero element of At emp (G/H, r) which is orthogonal to the image of T T . Then,generalizing Theorem 5 to the truncated inner product of an Eisenstein integral witha general element of At emp (G/H,r), this orthogonality can be explicitely described(cf. the evaluation of I before Theorem 9). As a result, this function has to be zero,a contradiction which proves the theorem.The theorem translates to if-finite functions, involving representations andF-fixed distribution vectors.The theorem can also be expressed with the unnormalized Eisenstein integrals.Then there are certain Plancherel factors involved. They are linked, as in thegroup case , to the intertwining integrals. Following the approach of A.Knapp andG.Zuckerman, [KZ], their computation is reduced to find an embedding of discreteseries into principal series attached to minimal parabolic subgroups. For connectedgroups, this has been done by J. Carmona [C2].8. Applications and open problemsSchwartz space for the hypergeometric Fourier transformThe image of a natural Schwartz space by the hypergeometric Fourier transformis characterized [D5]. The work uses the Plancherel formula of E. Opdam [Op],and the techniques mentioned above : theory of the constant term, G-functions,truncation ...Generalized Schur orthogonality relationsUsing the Plancherel formula for reductive symmetric spaces groups, K. Ankabout,[An], has proved generalized Schur orthogonality relations for generalizedcoefficients related to real reductive symmetric spaces. In particular, at least if weassume multiplicity one in the Plancherel formula, it implies the following:There exists an explicit positive function d on G/H, such that, for almost allrepresentations n occuring in the Plancherel formula, for £ an F-fixed distributionvector of n occuring in the decomposition of the Dirac measure, there exists an

552 Patrick Delormeexplicit non zero constant C* such that, for all v,v', FJ-finite vectors in the spaceof n:lim e"" f e- ed(x) < n'(g)Ç,v >dx = C*(v,v').6^0+ JG/HHere n* is the dimension of the support of the Plancherel decomposition,around n. This refines and generalizes a work of Mirodikawa. It suggests to lookfor such type of relations in other situations.D(G/iî)-finite r-spherical functions on reductive symmetric spacesS. Souaifi [So] showed how these functions appear as linear combinations ofderivatives along the complex parameter A, of Eisenstein integrals. For if-finitefunctions, filtrations are introduced, whose subquotients are described in terms ofinduced representations. The starting point is an adaptation of ideas used by J.Franke to study spaces of automorphic forms. The use of the spectral decompositionby Langlands is replaced here by the use of the Plancherel formula. For reductivep-adic groups, and for the group case, I got similar results.Invariant harmonic analysis on real reductive symmetric spacesThe goal is to study the F-invariant eigendistributions under B(G/H) on G/Hand to express invariant measures on certain F-orbits in terms of these distributions( cf [D3] for the work of A.Bouaziz and P.Harinck for the group case and G(C)/G(R),see also [OSe]).Harmonic analysis on p-adic reductive symmetric spacesFor the group case, the Problems (b) and (c) of the Introduction have beensolved by Harish-Chandra, up to the explicit description of the discrete series. Ingeneral, the problems are open (see [HH] for interesting structural results).References[An] K.Ankabout, Relations d'orthogonalité de Schur généralisées pour les espacessymétriques réductifs , J. Funct. Anal., 185 (2001), 63^110.[A] J. Arthur, A local trace formula, Pub. Math. I.H.E.S , 73 (1991), 5^96.[BI]E.van den Ban, The principal series for a reductive symmetric space I, Ann.Sc. Ec. Norm. Sup., 21 (1988), 359-112.[B2] E.van den Ban, The principal series for a reductive symmetric space II, J.of Funct. Analysis, 109 (1992), 331-441.[BCD] E.van den Ban, J.Carmona, P.Delorme, Paquets d'ondes dans l'espace deSchwartz d'un espace symétrique réductif, J. of Funct. Analysis, 139 (1996),225^243.[BS1][BS2]E.van den Ban , H.Schlictkrull, Fourier transform on a semisimple symmetricspace, Preprint Universiteit Utrecht, No 888, Nov. 1994.E. van den Ban , H. Schlictkrull, The most continuous part of the Planchereldecomposition for a reductive symmetric space , Ann. of Math., 145 (1997), 267^364.

[BS3][BS4][Be]Harmonie Analysis on Real Reductive Symmetric Spaces 553E.van den Ban , H.Schlictkrull, Harmonic analysis on reductive symmetricspaces , European Congress of Mathematics, Barcelona, 2000, Vol 1, 565-582, Birkhäuser Verlag, Basel, 2001.E.van den Ban , H.Schlictkrull, The Plancherel decomposition for a reductivesymmetric space I, II, Preprints 2001.J.N. Bernstein, On the support of the Plancherel measure, J. Geom. Phys.,5 (1988), 663-710.[BrD] J.L.Brylinski, P.Delorme, Vecteurs distributions H-invariants pour les sériesprincipales généralisées d'espaces symétriques réductifs et prolongementméromor--phe d'intégrales d'Eisenstein , Inv. Math., 109 (1992), 619-664.[Cl][C2]J.Carmona, it Terme constant des fonctions tempérées sur un espacesymétrique réductif,, J. fur Reine Angew. Math., 491 (1997), 17-63.J.Carmona, Plongement de séries discrètes pour un espace symétriqueréductif, J. of Funct. Analysis, 182 (2001), 16—51.[CD1] J.Carmona, P.Delorme, Base méromorphe de vecteurs distributions F-invariants pour les séries principales généralisées d'espaces symétriquesréductifs, J. of Funct. Analysis, 122 (1994), 152-221.[CD2] J.Carmona, P.Delorme, Transformation de Fourier pour les espaces symétriquesréductifs , Invent. Math., 134 (1998).[Dl][D2]P.Delorme, Intégrales d'Eisenstein pour les espaces symétriques réductifs.Tempérance. Majorations. Petite matrice B, J. of Funct. Analysis, 136(1996), 422-509.P.Delorme, Troncature pour les espaces symétriques réductifs , Acta Math.,179 (1997), 41-77.[D3] P.Delorme, Inversion des intégrales orbitales sur certains espacessymétriques réductifs, d'après A. Bouaziz et P. Harinck, Séminaire Bourbaki,Vol. 1995-1996, Astérisque 241, Exp. 810, 417-452.[D4] P.Delorme, Formule de Plancherel pour les espaces symétriques réductifs ,Ann. of Math., 147 (1998), 417-452.[D5][D6][F-J]P.Delorme, Espace de Schwartz pour la transformation de Fourier hypergéométrique, Appendice par M. TINFOU, J. Funct. Anal. , 168 (1999), 239-312.P.Delorme, The Plancherel formula on reductive symmetric spaces from thepoint of view of the Schwartz space. Lectures for the European SummerSchool of Group Theory, CIRM, Luminy 2001.M.Flensted-Jensen, Discrete series for semisimple symmetric spaces , Ann.of Math., 111(1980), 253-311.[Ha] P.Harinck, Fonctions orbitales sur GC/GR. Formule d'inversion desintégrales orbitales et formule de Plancherel , J. Funct. Anal., 153 (1998),52-107.[H-Cl] Harish-Chandra, Harmonie analysis on real reductive groups I , J. Funct.Anal, 36 (1976), 1-35.[H-C2] Harish-Chandra, Harmonic analysis on real reductive groups II, InventionesMath. , 36 (1976), 1-35.

554 Patrick Delorme[H-C3] Harish-Chandra, Harmonie analysis on real reductive groups III. The Maass-Selberg relations and the Plancherel formula , Ann. of Math., 104 (1976),117-201.[He] S. Helgason, Groups and geometric analysis, Academic Press, 1984.[HH] A.G.Helminck, G.F. Helminck , A class of parabolic fc-subgroups associatedwith symmetric fc-varieties , Trans. Amer. Math. Soc, 350 (1998), 4669-4684.[KZ] A.Knapp, G.Zuckerman, Classification of irreducible tempered representationsof semisimple groups I , Ann. of Math. 116 (1982), 389-455.[01] G.Olafsson, Fourier and Poisson transformation associated to a semisimplesymmetric space, Inv. Math., 90 (1987), 1-51.[Op] E.Opdam, Cuspidal hypergeometric functions , dedicated to R. Askey, Partl,Methods Appi. Anal. 6 (1999) 67-80.[Ol] T.Oshima, Fourier analysis on semisimple symmetric spaces, Non commutativeharmonic analysis and Lie groups (Marseille-Luminy, 1980), LectureNotes in Math., vol 880, Springer, 1991, 357-369.[02] T.Oshima, Asymptotic behaviour of spherical functions on semisimple symmetricspaces , Adv. Studies in Pure Math. 14 (1988), 561-601.[OM] T.Oshima, T.Matsuki, A description of discrete series for semisimple symmetricspaces, Adv. Studies in Pure Math., 4 (1984), 331-390.[OSe] T.Oshima, J.Sekiguchi, Eigenspaces of invariant differential operators on asemisimple symmetric space, Inv. Math. 57 (1980), 1-81.[S] H.Schlichtkrull, The Langlands parameter of Flensted-Jensen's discrete seriesfor semisimple symmetric spaces ,J. of Funct Anal, 50 (1983), 133-150.[So] S.Souaifi , Fonctions D(G/F)-finies sur un espace symétrique réductif,Preprint, 2001, to appear in J. of Funct Anal.[V] D. Vogan, Irreducibility of discrete series representation for semisimple symmetricspaces , Adv. Studies in Pure Math., 14 (1988), 191-221.[W] N. Wallach, Real Reductive Groups II, Academic Press, Inc., Boston, 1992.

ICM 2002 • Vol. II • 555-570On the Dynamical Yang-Baxter EquationPavel EtingoP( To Mira)AbstractThis talk is inspired by two previous ICM talks, by V. Drinfeld (1986),and G.Felder (1994). Namely, one of the main ideas of Drinfeld's talk is thatthe quantum Yang-Baxter equation, which is an important equation arisingin quantum field theory and statistical mechanics, is best understood withinthe framework of Hopf algebras, or quantum groups. On the other hand, inFelder's talk, it is explained that another important equation of mathematicalphysics, the star-triangle relation, may (and should) be viewed as a generalizationof the quantum Yang-Baxter equation, in which solutions dependon additional "dynamical" parameters. It is also explained there that to asolution of the quantum dynamical Yang-Baxter equation one may associatea kind of quantum group. These ideas gave rise to a vibrant new branch of"quantum algebra", which may be called the theory of dynamical quantumgroups. My goal in this talk is to give a bird's eye review of this theory andits applications.The quantum dynamical Yang-Baxter equation (QDYBE) is an equationwith respect to a function R : I)* —• Endf,(V ® V), where I) is a commutativefinite dimensional Lie algebra, and V is a semisimple t)-module. It readsÄ 12 (A - h 3 )R 13 (X)R 23 (X - h 1 ) = R 23 (X)R 13 (X - h 2 )R 12 (X)on V ® V ® V, where for instance R 12 (X — h 3 ) is defined by the formulaÄ 12 (A — h 3 )(vi ® v-2 ® Vî) := f Ä 12 (A — ß)(vi ® V2)) ® Vì if Vì is of weight ßunder I). If I) = 0, this equation turns into the usual quantum Yang-Baxterequation.I will start with explaining how rational solutions of QDYBE arise alreadyin the classical representation theory of finite dimensional simple Lie algebras,via the so called fusion construction ((the role of I) will be played by a Cartansubalgebra of the simple Lie algebra). Then I will explain generalizationsof this construction to quantum groups, affine Lie algebras, quantum affinealgebras, which yields trigonometric and elliptic solutions of QDYBE. I willthen define Felder's elliptic quantum group, and formulate a q-analog of theKazhdan-Lusztig equivalence between representations of affine Lie algebrasand quantum groups.* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139,USA. E-mail: etingof@math.mit.edu

556 Pavel EtingofAfter this I will define the classical dynamical Yang-Baxter equation (theclassical limit of QDYBE), and discuss the classification result (joint withVarchenko), that says, roughly, that all solutions are of the above three types- rational, trigonometric, and elliptic. This is a dynamical version of a resultof Belavin and Drinfeld.In the second part of the talk I will consider applications of the theoryof QDYBE to integrable systems. Namely, I will use the dynamical transfermatrix construction to define generalized Macdonald operators, and thenconstruct their common eigenfunctions as (renormalized) traces of the formTr( M g 2A ), where

On the Dynamical Yang-Baxter Equation 557The quantum dynamical Yang-Baxter equation (QDYBE), proposed by Gervais,Neveu, and Felder, is an equation with respect to a (meromorphic) functionR : Ì)* —¥ End[j(F V), where h is a commutative finite dimensional Lie algebraover C, and V is a semisimple h-module. It readsF 12 (A - h 3 )R 13 (X)R 23 (X - h 1 ) = R 23 (X)R 13 (X - h 2 )R 12 (X)on V ® V ® V. Here h % is the dynamical notation, to be extensively used below:for instance, R 12 (X — h 3 ) is defined by the formula R 12 (X — h 3 )(vi ® v 2 ® W3) :=(F 12 (A — p)(vi ® v 2 )) ®W3 ifW3 is of weight p under h.It is also useful to consider QDYBE with spectral parameter. In this case theunknown function R depends meromorphically on an additional complex variableu. Yet Uij = Ui — Uj. Then the equation readsF 12 («i2,A^/i 3 )F 13 («i3,A)F 23 («23,A^/i 1 ) = F 23 («23,A)F 13 («i3,A^/i 2 )F 12 («i2,A).Sometimes it is necessary to consider QDYBE with step 7 £ C*, which differsfrom the usual QDYBE by the replacement h % —¥ 7/1*. Clearly, R(X) satisfiesQDYBE iff F(A/7) satisfies QDYBE with step 7.Invertible solutions of QDYBE are called quantum dynamical R-matrices (withor without spectral parameter, and with step 7 if needed). If h = 0, QDYBE turnsinto the usual quantum Yang-Baxter equation R 12 R 13 R 23 = R 23 R 13 R 12 .2.2. Examples of solutions of QDYBELet V be the vector representation of sl(n), and h the Lie algebra of tracelessdiagonal matrices. In this case A £ h* can be written as A = (Ai,...,A„), whereA, £ C. Let v a , a = 1, ...,n be the standard basis of V. Yet E ab be the matrix unitsgiven by E ab v c = ö bc v a .We will now give a few examples of quantum dynamical R-matrices. Thegeneral form of the R-matrices will beR=YEaa® Eaa + YI a abEaa ®E bb + Y ßabE ab ® E ba , (1)a a=£b a=£bwhere a ab , ß ab are functions which will be given explicitly in each example.Example 1 The basic rational solution. Let ß ab = x Â , a ab = 1 + ß ab .Then R(X) is a dynamical R-matrix (with step 1).Example 2 The basic trigonometric solution. Let ß ab = x i ,l"x!_ 1) O-ab =q + ß ab . Then R(X) is a dynamical R-matrix (also with step 1).Example 3 The basic elliptic solution with spectral parameter (Felder's solution).Let 9(U,T) = ^^2j € z+i/2 e7 ' %^ T+2 A t *+ 1 / 2 )) be the standard theta-function(for brevity will write it as 9(uj). Let ß ab = gl^+^L^j, a ab = ggÎ^g^-These functions can be viewed as functions of z = e 27!tu . Then R(u, X) is a quantumdynamical R-matrix with spectral parameter and step 7. One may also define thebasic trigonmetric and basic rational solution with spectral parameter, which differfrom the elliptic solution by replacement of 9(x) by sin(x) and x, respectively.

558 Pavel EtingofRemark 1 In examples 1 and 2, the dynamical R-matrix satisfies the Heckecondition (PR — 1)(PR + q) = 0, with q = 1 in example 1 (where P is the permutationon V ® V), and in example 3 the unitarity condition R(u, X)R 21 (—u, A) = 1.Remark 2 The basic trigonometric solution degenerates into the basic rationalsolution as q —¥ 1. Also, the basic elliptic solution with spectral parametercan be degenerated into the basic trigonometric and rational solutions with spectralparameter by renormalizing variables and sending one, respectively both periods oftheta functions to infinity (see [EV2]). Furthermore, the basic trigonometric andrational solutions with spectral parameter can, in turn, be degenerated into thesolutions of Examples 1,2, by sending the spectral parameter to infinity. Thus, inessense, all the examples we gave are obtained from Felder's solution.2.3. The tensor category of representations associated to aquantum dynamical R-matrixLet R be a quantum dynamical R-matrix with spectral parameter. Accordingto [Fe], a representation of R is a semisimple h-module W and an invertiblemeromorphic function L = Lyy : C x h* —^ Endf,(V ® W), such thatR 12 (ui2,X-h 3 )L 13 (ui 3 ,X)L 23 (u23,X-h 1 ) = L 23 (u2 3 ,X)L 13 (ui 3 ,X-h 2 )R 12 (ui2,X).(2)(In the case of step 7, h % should be replaced by 7/1*).For example: (C, 1) (trivial representation) and (V,R) (vector representation).A morphism / : (W,Lw) —¥ (W',Lwi) is a meromorphic function f : Ì)* —¥End(,(IA~) such that (1 ® f(Xj)L w (u,X) = L w (u,X)(l ® /(A - 7ft 1 )). With thisdefinition, representations form an (additive) category Rep(72). Moreover, it is atensor category [Fe]: given (W,Lw) and (U,Ljj), one can form the tensor productrepresentation (W ® U,Lw®u), where L\vû(u, X) = L\^(u, X — , yh 3 )Lj 3 (u, A);tensor product of morphisms is defined by (/ ® g)(X) = /(A — 7ft 2 ) ® g(X).In absence of spectral parameter, one should use the same definitions without2.4. Gauge transformations and classificationThere exists a group of rather trivial transformations acting on quantum dynamicalR-matrices with step 7. They are called gauge transformations. If f) and Vare as in the previous section, then gauge transformations (for dynamical R-matriceswithout spectral parameter) are:1. Twist by a closed multiplicative 2-form : a ab —¥ a ab «)+v(*-'r«>«-'r«»>))a ab ,ß b —Ï e«('?( A )-'?( A -7Wa)-¥'(A-7W i ,)+¥)(A-7W a -7W i ,)) a4. Multiplication of u by a constant.

On the Dynamical Yang-Baxter Equation 559Theorem 2.1 [EV2] Any quantum dynamical R-matrix for h, V satisfying the Heckecondition with q = 1 (respectively, q ^ 1) is a gauge transformation of the basicrational (respectively, trigonometric) solution, or a limit of such R-matrices.In the spectral parameter case, a similar result is known only under ratherserious restrictions (see [EV2], Theorem 2.5). For more complicated configurations(h, V), classification is not available.Remark Gauge transformations 2-4 do not affect the representation categoryof the R-matrix. Gauge transformation 1 does not affect the category if the closedform 4> is exact: = d£, i.e. a b(X) = £o(A)£&(A - 7w a )£ a (A - 7W{,) _1 £{,(A) _1 ,where Ca (A) is a collection of meromorphic functions. We note that this is a verymildcondition: for example, if 7 is a formal paremeter and we work with analyticfunctions of A in a simply connected domain, then a multiplicative 2-form is closediff it is exact ("multiplicative Poincaré lemma").2.5. Dynamical quantum groupsEquation 2 may be regarded as a set of defining relations for an associativealgebra An (see [EV2] for precise definitions). This algebra is a dynamical analogueof the quantum group attached to an R-matrix defined in [FRT], and representationsof 12 are an appropriate class of representations of this algebra. The algebra An iscalled the dynamical quantum group attached to 12. If 12 is the basic elliptic solutionthen An is the elliptic quantum group of [Fe]. The structure and representations ofAn are studied in many papers (e.g. [Fe, EV2, FV1, TV]).To keep this paper within bounds, we will not discuss An in detail. However,let us mention ([EV2]) that An is a bialgebroid with base h*. This corresponds tothe fact that the category Rep(12) is a tensor category. Moreover, if 12 satisfies anadditional rigidity assumption (valid for example for the basic rational and trigonometricsolutions) then the category Rep(72) has duality, and An is a Hopf algebroid,or a quantum groupoid (i.e. it has an antipode).Remark For a general theory of bialgebroids and Hopf algebroids the readeris referred to [Lu]. However, let us mention that bialgebroids with base X correspondto pairs (tensor category, tensor functor to O(X)-bimodules), similarly tohow bialgebras correspond to pairs (tensor category, tensor functor to vector spaces)(i.e. via Tannakian formalism).2.6. The classical dynamical Yang-Baxter equationRecall that if 72 = 1 — hr + 0(h 2 ) is a solution QYBE, then r satisfies theclassical Yang-Baxter equation (CYBE), [ 12 r ,r 13 ] + [ 12 r ,r 23 ] + [ 13 r ,r 23 ] = 0, andthat r is called the classical limit of 72, while 72 is called a quantization of r. Similarly,let 72(A, h) be a family of solutions of QDYBE with step h given by a series 1 —hr(X) + 0(h 2 ). Then it is easy to show that r(X) satisfies the following differentialequation, called the classical dynamical Yang-Baxter equation (CDYBE):E X ' (1) 7£- - ^Sr" + X ' (8) S- 1 + [rl2 ' rl3] + [rl2 ' r23] + [rl3 ' r23] = °' (3)

560 Pavel Etingof(where xi is a basis of fi). The function r(X) is called the classical limit of R(X,h),and 72(A, h) is called a quantization of r(A).Define a classical dynamical r-matrix to be a meromorphic function r : \)* —tEnd(,(F ® V) satisfying CDYBE.Conjecture 2.2 Any classical dynamical r-matrix can be quantized.This conjecture is known in the non-dynamical case ([EK]), and was provedin [Xu] in the dynamical case for skew-symmetric solutions (r 21 = —r) satisfyingadditional technical assumptions. A general proof for the skew-symmetric case isrecently proposed in a preprint by T.Mochizuki. However, the most interestingnon-skew-symmetric case is still open.Similarly, the classical limit of QDYBE with spectral parameter is CDYBEwith spectral parameter. It is an equation with respect to r(u,X) and differs fromthe usual CDYBE by insertion of «y as an additional argument of r y .Remark 1 Similarly to CYBE, CDYBE makes sense for functions with valuesin g ® g, where g is a Lie algebra containing h.Remark 2 The classical limit of the notion of a quantum groupoid is thenotion of a Poisson groupoid, due to Weinstein. By definition, a Poisson groupoidis a groupoid G which is also a Poisson manifold, such that the graph of the multiplicationis coisotropic in G x G x G (where G is G with reversed sign of Poissonbracket). Such a groupoid can be attached ([EVI]) to a classical dynamical r-matrixr : h* —^ g ® g, such that r 21 + r is constant and invariant (i.e. r is a "dynamicalquasitriangular structure" on g). This is the classical limit of the assignment of aquantum groupoid to a quantum dynamical R-matrix ([EV2]).2.7. Examples of solutions of CDYBEWe will give examples of solutions of CDYBE in the case when g is a finitedimensional simple Lie algebra, and f) is its Cartan subalgebra. We fix an invariantinner product on g, It restricts to a nondegenerate inner product on h. Using thisinner product, we identify h* with f) (A € I)* 4 Ä 6 I)), which yields an innerproduct on h*. The normalization of the inner product is chosen so that short rootshave squared length 2. Let xi be an orthonormal basis of h,and let e a , e_ Q denotepositive (respectively, negative) root elements of g, such that (e Q ,e_ Q ) = 1.Example 1 The basic rational solution r(X) = £ e Q>0 ?A a")° •Example 2 The basic trigonometric solution r(X) = ^+Tâ>0 9 c °t aim ( C ),where 0 £ S 2 g is the inverse element to the inner product on g.Example 3 The basic elliptic solution with spectral parameter (Felder'ssolution) r(u, X) = j $ £* x t ® x t + £ Q ^fp^fy^pe Q ® e_ Q .Remark 1 One says that a classical dynamical r-matrix r has coupling constante if r + r 21 = eii. If r is with spectral parameter, one says that it has couplingconstant e if r(u, X) + r 21 (—u, X) = 0, and r(u, X) has a simple pole at u = 0 withresidue eO (these are analogs of the Hecke and unitarity conditions in the quantumcase). With these definitions, the basic rational solution has coupling constant 0,while the trigonometric and elliptic solutions have coupling constant 1.

On the Dynamical Yang-Baxter Equation 561Remark 2 As in the quantum case, Example 3 can be degenerated intoits trigonometric and rational versions where 9(x) is replaced by sin(x) and x,respectively, and Examples 1 and 2 can be obtained from Example 3 by a limit.Remark 3 The classical limit of the basic rational and trigonometric solutionsof QDYBE (modified by A —¥ X/K) is the basic rational, respectively trigoinometric,solutions of CDYBE for g = si n (in the trigonometric case we should set q = e^hl 2 ).The same is true for the basic elliptic solution (with 7 = h).Remark 4 These examples make sense for any reductive Lie algebra g.2.8. Gauge transformations and classification of solutions forCDYBEIt is clear from the above that it is interesting to classify solutions of CDYBE.As in the quantum case, it should be done up to gauge transformations. Thesetransformations are classical analogs of the gauge transformations in the quantumcase, and look as follows.1. r —¥ r + oj, where OJ = VA • CìJ(X)XìAXJ is a meromorphic closed differential2-form on if*.2. r(u, X) —¥ ar(aX — v); Weyl group action.In the case of spectral parameter, there are additional transformations:3. u —¥ bu.4. Yet r = £. . Syar, ® Xj + £ Q (p a e a ® e_ Q . The transformation is S'y —¥Sij + u () X .Q X . , a e u9aV , where tp is a function on h* with meromorphic dtp.Theorem 2.3 [EVI] (i) Any classical dynamical r-matrix with zero coupling constantis a gauge transformation of the basic rational solution for a reductive subalgebraof g containing if, or its limiting case.(ii) Any classical dynamical r-matrix with nonzero coupling constant is a gaugetransformation of the basic trigonometric solution for g, or its limiting case.(iii) Any classical dynamical r-matrix with spectral parameter and nonzerocoupling constant is a gauge transformation of the basic elliptic solution for g, orits limiting case.Remark One may also classify dynamical r-matrices with nonzero couplingconstant (without spectral parameter) defined on I* for a Lie subalgebra I C h, onwhich the inner product is nondegenerate ([Sch]). Up to gauge transformations theyare classified by generalized Belavin-Drinfeld triples, i.e. triples (Fi,F 2 ,T), whereFj are subdiagrams of the Dynkin diagram F of g, and T : Y\ —t F 2 is a bijectionperserving the inner product of simple roots (so this classification is a dynamicalanalog of the Belavin-Drinfeld classification of r-matrices on simple Lie algebras,and the classification of [EVI] is the special case Y\ =Y 2 = Y,T = 1). The formulafor a classical dynamical r-matrix corresponding to such a triple given in [Sch] worksfor any Kac-Moody algebra, and in the case of affine Lie algebras yields classicaldynamical r-matrices with spectral parameter (however, the classification is thiscase has not been worked out). Explicit quantization of the dynamical r-matricesfrom [Sch] (for any Kac-Moody algebra) is given in [ESS].

562 Pavel Etingof3. The fusion and exchange constructionUnlike QYBE, interesting solutions of QDYBE may be obtained already fromclassical representation theory of Lie algebras. This can be done through the fusionand exchange construction, see [Fa, EV3].3.1. Intertwining operatorsLet g be a simple finite dimensional Lie algebra over C, with polar decompositiong = n + ® fi ® n_. For any g-module V, we write V[v] for the weight subspaceof V of weight v £ if. Let M\ denote the Verma module over g with highest weightA G if*, x\ its highest weight vector, and x* x the lowest weight vector of the dualmodule. Let V be a finite dimensional representation of g. Consider an intertwiningoperator $ : M\ —¥ M ß ® V. The vector x* ß (§x\) £ V[A — p] is called theexpectation value of $, and denoted ($).Lemma 3.1 7/ M ß is irreducible (i.e. for generic p), the map $ —¥ ($) is anisomorphism Eom s (M ß+l/ , M ß ® V) —¥ V[v\.Lemma 3.1 allows one to define for any v £ V[v] (and generic A) the intertwiningoperator $ v x : M\ —¥ M\- v ® V, such that ($ x ) = v.3.2. The fusion and exchange operatorsNow let V, W be finite dimensional g-modules, and v £ V, tv £ W homogeneousvectors, of weights wt(w),wt(w). Consider the composition of two intertwiningoperators*£'" := (1>t- wt(v) ® mi • M x -+ Af A _ wt(w) _ wt(œ) W ® V.The expectation value of this composition, ($ x ' v ), is a bilinear function of tv andv. Therefore, there exists a linear operator Jwy(X) £ End(W V) (of weight zero,i.e. commuting with h), such that (& x ' v ) = Jwv(X)(w ® v). In other words, wehave (^_ wt{v) ® 1)*X = $^^( A H»®«). The operator J WV (X) is called the fusionoperator (because it tells us how to "fuse" two intertwining operators).The fusion operator has a number of interesting properties, which we discussbelow. In particular, it is lower triangular, i.e. has the form J = 1 + N, whereN is a sum of terms which have strictly positive weights in the second component.Consequently, N is nilpotent, and J is invertible.Define also the exchange operator Rvw(X) := JvwW^wvW '• V ® W ~*V ® W. This operator tells us how to exchange the order of two intertwiningoperators, in the sense that if Rwv(X)(w ® v) = £jWj ® w» (where w,,w, arehomogeneous), then $ x ' v = 7 3 £ i $^' u ' i (where P permutes V and W).Theorem 3.2 [EV3] R V v(X)is a solution of QDYBE.

On the Dynamical Yang-Baxter Equation 5633.3. Fusion and exchange operators for quantum groupsThe fusion and exchange constructions generalize without significant changesto the case when the Lie algebra g is replaced by the quantum group U q (g), whereq is not zero or a root of unity. The only change that needs to be made is in thedefinition of the exchange operator: namely, one sets 72(A) = Jy\ v (X)TZ 21 J 2 y V (X),where R is the universal 72-matrix of U q (g). This is because when changing theorder of intertwining operators, we must change the order of tensor product ofrepresentations V ® W, which in the quantum case is done by means of the R-matrix.Example 3.3 Let g = sl n , and V be the -vector representation of U q (g). Then theexchange operator has the form 72 = g 1_1// "72, where 72 is given by (1), with ß ab =„2(\ a -\ h -a+b)_i ! &ab — Q jor a < u, anaa ab — , 2(\ h -\ a +a-b)_^\2if a > b. The exchange operator for the vector representation of g is obtained bypassing to the limit q —¥ 1; i.e. it is given by (1), with ß ab = x _ x 1 _ b+a , a ab = 1for a

564 Pavel EtingofNamely, the universal fusion operator J(X) is the unique lower triangular solutionof the ABRR equation in a completion of U q (g)® 2 . This solution can be found inthe form of a series J = ^n>Q J n , Jo = 1, where J n £ U q (g)®U q (g) has zero weightand its second component has degree n in principal gradation; so J n are computedrecursively.This allows one to compute the universal fusion operator quite explicitly. Forexample, if q = 1 and g = sl 2 , then the universal fusion operator is given by theformula J(X) = ]T„> ( 0 ~f-f n ®(X-h + n+ l)- 1 ...(X-h + 2n)- 1 e n .3.6. The dynamical twist equationAnother important property of the fusion operator is the dynamical twistequation (which is a dynamical analog of the equation for a Drinfeld twist in a Hopfalgebra).Theorem 3.5 The universal fusion operator J(X) satisfies the dynamical twistequation J 12 ' 3 (X)J 1 ' 2 (X - h^) = J 1 ' 23 (X)J 2 ' 3 (X).Here the superscripts of J stand for components where the first and secondcomponent of J acts; for example J 1 ' 23 means (1 ® A)(J), and J 1 ' 2 means J ®1.3.7. The fusion operator for affine algebrasThe fusion and exchange construction can be generalized to the case when thesimple Lie algebra g is replaced by any Kac-Moody Lie algebra. This generalizationis especially interesting if g is replaced with the affine Lie algebra g, and V, W arefinite dimensional representations of U q (g) (where q is allowed to be 1). In thiscase, for each z £ C* we have an outer automorphism D z : U q (g) —¥ U q (g), whichpreserves the Chevalley generators q h and e,,/,, i > 0, of U q (g), while multiplyingeo by z and fi by z^1.Define the shifted representation V(z) by ny( z )(a) =n\/(D z (aj). Yet A = (A,fc) be a weight for g (fc is the level). Then similarlyto the finite dimensional case one can define the intertwining operator $ Xk (z) :M x —t M x _ wt , v) ®\~(z) (to a completed tensor product); it can be written as aninfinite series XêzÂ.fcN 0- "' where 3>^fc [n] : M~ x -t M x _ wt ,-. ® V are linearoperators. Furthermore, the expectation value of the composition ((^x_ wt , v^ k(zi)®l)$x k( Z2 )) ' 1S defined as a Taylor series in z 2 /zi. Thus, one can define the fusionoperator Jwv(z) £ Endf,(W ® V)[[z]] such that this expectation value is equal toJwv(z)(w ®v).3.8. Fusion operator and correlators in the WZW modelOne may show (see [FR, EFK]) that the series Jwv(z) is convergent in someneighborhood of 0 to a holomorphic function. This function has a physical interpretation.Namely, if q = 1, the operators $ x k (z) are, essentially, vertex operators(primary fields) for the Wess-Zumino-Witten conformai field theory, and the functionJw\/(z)(w ® v) is a 2-point correlation function of vertex operators. If q ^ 1,

On the Dynamical Yang-Baxter Equation 565this function is a 2-point correlation function of q-vertex operators, and has a similarinterpretation in terms of statistical mechanics.3.9. The ABRR equation in the affine case as the KZ (q-KZ)equationThe ABRR equation has a straightforward generalization to the affine case,which also has a physical interpretation. Namely, in the case q = 1 it coincideswith the (trigonometric) Knizhnik-Zamolodchikov (KZ) equation for the 2-pointcorrelation function, while for q ^ 1 it coincides with the quantized KZ equationfor the 2-point function of q-vertex operators, derived in [FR] (see also [EFK]).One may also define (for any Kac-Moody algebra) the multicomponent universalfusion operator J 1 - N (X) = J 1 > 2 - N (X)...J 2 > 3 - N (X)J N - 1 > N (X). It satisfiesa multicomponent version of the ABRR equation. In the affine case, J 1 '" JV (A) isinterpreted as the N-point correlation function for vertex (respectively, q-vertex)operators, while the multicomponent ABRR equation is interpreted as the KZ (respectively,qKZ) equation for this function. See [EV5] for details.3.10. The exchange operator for affine algebras and monodromyof KZ (q-KZ) equationsThe generalization of the exchange operator to the affine case is rather tricky,and there is a serious difference between the classical (q = 1), and quantum(q ^ 1) case. The naive definition would be Rvw(u,X) = Jvw(z,X)~ 1/ R? 1 \v 1 along a curve passing 1 from above (for +) and below (for -). Then thefunction R vw (u, X) := Jvw(z, A) _1 A ± ( J 2 y V (z~ 1 , A)) is of zero weight, and satisfiesthe QDYBE with spectral parameter (for V = W). The operator R vw (u,X) is theappropriate analog of the exchange operator (depending on a choice of sign).If q ^ 1, the functions Jvw(z,X) and R 21 \v®w(z)J'wv( z^1 ,^) are both solutionsof the quantized Knizhnik-Zamolodchikov equation, which is a differenceequation with multiplicative step p = q^2n% ( k +s) ; where m is the ratio of squarednorms of long and short roots, fc the level of A, and g the dual Coxeter number ofg. Therefore, if \p\ ^ 1, these functions admit a meromorphic continuation to thewhole C* (unlike the q = 1 case, they are now single-valued), and the naive formulafor Ryw(u, A) makes sense. As in the q = 1 case, this function is of zero weight and

566 Pavel Etingofsatisfies the QDYBE with spectral parameter (for V = W); it is the appropriategeneralization of the exchange operator (see [EFK] for details).The operators R vw (z,X), essentially, represent the monodromy of the KZ differentialequation. In particular, R vw is "almost constant" in u: its matrix elementsin a homogeneous basis under h are powers of e 2nm (in fact, Ry V (u,X) is gaugeequivalent, in an appropriate sense, to a solution of QDYBE without spectral parameter).Similarly, the operator Ry\y(z, A) for q ^ 1 represents the q-monodromy(Birkhoff's connection matrix) of the q-KZ equation. In particular, the matrix elementsof Rvw(u,X) are quasiperiodic in u with period r = \ogp/2ni. Since theyare also periodic with period 1 and meromorphic, they can be expressed rationallyvia elliptic theta-functions.Example 3.6 Let g = sl n , and V the -vector representation of U q (g). If q ^ 1,the exchange operator Ryy(u,X) is a solution of QDYBE, gauge equivalent to thebasic elliptic solution with spectral parameter (see [Mo] and references therein, andalso [FR, EFK]). The gauge transformation involves an exact multiplicative 2-form, which expresses via q-Gamma functions with q = p. Similarly, if q = 1,the exchange operators R vv (u,X) are gauge equivalent to the basic trigonometricsolution without spectral parameter, with q = e^/">-(k+g). the gauge transformationinvolves an exact multiplicative 2-form expressing via classical Gamma-functions.This is obtained by sending q to 1 in the result of [Mo].Remark Note that the limit q —¥ 1 in this setting is rather subtle. Indeed,for q ^ 1 the function Jy\y(u,X) has an infinite sequence of poles in the z-plane,which becomes denser as q approaches 1 and eventually degenerates into a branchcut; i.e. this single valued meromorphic function becomes multivalued in the limit.3.11. The quantum Kazhdan-Lusztig functorLet g = sl n , q ^ 1. Example 3.6 allows one to construct a tensor functorfrom the category fiepf(U q (gj) of finite dimensional C/ g (g)-modules, to the categoryRepj(72) of finite dimensional representations of the basic elliptic solution72 of QDYBE with spectral parameter (i.e. to the category of finite dimensionalrepresentations of Felder's elliptic quantum group). Namely, let V be the vectorrepresentation, and for any finite dimensional representation W of U q (g), setLw = Rvw- Then (W,Lw) is a representation of the dynamical R-matrix Ryy.This defines a functor F : Repf(U q (g)) —¥ Repf(Ryy). Moreover, this functor isa tensor functor: the tensor structure F(W) ® F(U) —¥ F(W ® U) is given by thefusion operator Jwu(X) (the axiom of a tensor structure follows from the dynamicaltwist equation for J). On the other hand, since Ryy and 72 are gauge equivalentby an exact form, their representation categories are equivalent, so one may assumethat F lands in Rep^(72).If the scalars for Repf(U q (g)) are taken to be the field of periodic functionsof A (in particular, fc is regarded as a variable), then F is fully faithful; it can beregarded as a q-analogue of the Kazhdan-Lusztig functor (see [EM] and referencestherein). It generalizes to infinite dimensional representations, and allows one to

On the Dynamical Yang-Baxter Equation 567construct elliptic deformations of all evaluation representations of U q (gj) (which wasdone for finite dimensional representations in [TV]). The versions of this functorwithout spectral parameter, from representations of g (U q (gj) to representations ofthe basic rational (trigonometric) solution of QDYBE can be found in [EV3].4. Traces of intertwining operators and MacdonaldfunctionsIn this section we discuss a connection between dynamical R-matrices and certainintegrable systems and special functions (in particular, Macdonald functions).4.1. Trace functionsLet V be a finite dimensional representation of U q (g) (q ^ 1), such thatV[0] 7^ 0. Recall that for any v £ V[0] and generic p, one can define an intertwiningoperator $ ß such that ($ ß ) = v. Following [EV4] set ^v(X,p)=Tr|M f ,( < £Jlg 2A )- This is an infinite series in the variables q^( x > ai i (where a, are thesimple roots) whose coefficients are rational functions of qb À > ai ï (times a commonfactor g 2( A^). For generic p this series converges near 0, and its matrix elementsbelong to q 2(x^(C(q (x ' a^) ® C(q (âi^)).Yet \Py(A, p) be the End(V[0])-valued function, such that ^y(X, pjv = ^V(X,p).The function \Py has remarkable properties and in a special case is closely relatedto Macdonald functions. To formulate the properties of vPy, we will consider arenormalized version of this function. Namely, let 5 q (X) be the Weyl denominatorria>o( < 3' (A ' a)^w acting on functions on h* with values in V[0]. Namely, we set (T>\yfi)(X) =Si/eb* Tr|i4/(72i4/y(Â — pj)f(X + v). These operators are dynamical analogs oftransfer matrices, and were introduced in [FV2]. It can be shown that ÎVi®w 2=T>W 1 'D\Y 2 ; in particular, V\y commute with each other, and the algebra generatedby them is the polynomial algebra in V\y i , where W, are the fundamental representationsof U q (g).4.3. Difference equations for the trace functionsIt turns out that trace functions Fy(X,p), regarded as functions of A, arecommon eigenfunctions of T>\y.

568 Pavel EtingofTheorem 4.1 [EV4] One hasV$ F v (X, p) = Xw(q- 2ß )Fv (X, p), (1)where Xw(x) = Tr|i4/(a:) is the character ofW.In fact, it is easy to deduce from this theorem that if w, is a basis of V[0]then Fy(X,p)vi is a basis of solutions of (1) in the power series space. Thus,trace functions allow us to integrate the quantum integrable system defined by thecommuting operators V\y i .Theorem 4.2 [EV4] The function Fy is symmetric in X and p in the followingsense: Fy*(p,X) = Fy(X,p)*.This symmetry property implies that Fy also satisfies "dual" difference equationswith respect to p: T>\£ Fy(X,p)* = xw((l^2X )Fy(X,p)*.4.4. Macdonald functionsAn important special case, worked out in [EKi], is g = sl n , and V = L^nwi ,where OJI is the first fundamental weight, and fc a nonnegative integer. In thiscase, dimF[0] = 1, and thus trace functions can be regarded as scalar functions.Furthermore, it turns out ([FV2]) that the operators V\y can be conjugated (by acertain explicit product) to Macdonald's difference operators of type A, and thus thefunctions Fy(X,p) are Macdonald functions (up to multiplication by this product).One can also obtain Macdonald's polynomials by replacing Verma modules Af^with irreducible finite dimensional modules Ly, see [EV4] for details. In this case,Theorem 4.2 is the well known Macdonald's symmetry identity, and the "dual"difference equations are the recurrence relations for Macdonald's functions.Remark 1 The dynamical transfer matrices V\y can be constructed not onlyfor the trigonometric but also for the elliptc dynamical R-matrix; in the case g = sl n ,V = Lknu! this yields the Ruijsenaars system, which is an elliptic deformation ofthe Macdonald system.Remark 2 If q = 1, the difference equations of Theorem 4.1 become differentialequations, which in the case g = sl n , V = L\, nwi reduce to the trigonometricCalogero-Moser system. In this limit, the symmetry property is destroyed, butthe "dual" difference equations remain valid, now with the exchange operator forg rather than U q (g). Thus, both for q = 1 and g / 1, common eigenfunctionssatisfy additional difference equations with respect to eigenvalues - the so calledbispectrality property.Remark 3 Apart from trace \P" of a single intertwining operator multipliedby q 2X , it is useful to consider the trace of a product of several such operators.After an appropriate renormalization, such multicomponent trace function (takingvalues in End((Vi ®...® VJV)[0]) satisfies multicomponent analogs of (1) and its dualversion, as well as the symmetry. Furthermore, it satisfies an additional quantumKnizhnik-Zamolodchikov-Bernard equation, and its dual version (see [EV4]).

On the Dynamical Yang-Baxter Equation 569Remark 4 The theory of trace functions can be generalized to the case ofaffine Lie algebras, with V being a finite dimensional representation of U q (g). Inthis case, trace functions will be interesting transcendental functions. In the caseg = sl n , V = Lkniü!, they are analogs of Macdonald functions for affine root systems.It is expected that for g = sl 2 they are the elliptic hypergeometric functions studiedin [FV3]. This is known in the trigonometric limit ([EV4]) and for q = 1.Remark 5 The theory of trace functions, both finite dimensional and affine,can be generalized to the case of any generalized Belavin-Drinfeld triple; see [ESI].Remark 6 Trace functions Fy(X,p) are not Weyl group invariant. Rather,the diagonal action of the Weyl group multiplies them by certain operators, calledthe dynamical Weyl group operators (see [EV5]). These operators play an importantrole in the theory of dynamical R-matrices and trace functions, but are beyond thescope of this paper.ReferencesABRR] Arnaudon, D., Buffenoir, E., Ragoucy, E. and Roche, Ph., Universal Solutionsof quantum dynamical Yang-Baxter equations, Lett. Math. Phys.44 (1998), no. 3, 201^214.Dr] Drinfeld, V. G., Quantum groups, Proceedings of the InternationalCongress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 798^820,Amer. Math. Soc, Providence, RI, 1987.EFK] Etingof, P., Frenkel, I., Kirillov, A., Jr. Lectures on representation theoryand Knizhnik-Zamolodchikov equations, AMS, (1998).EK] Etingof P., Kazhdan D., Quantization of Lie bialgebras I, Selecta Math.,2 (1996), 1-41.EKi] Etingof, P. I., Kirillov, A. A., Jr, Macdonald's polynomials and representationsof quantum groups, Math. Res. Let., 1 (3) (1994), 279^296.EM] Etingof, P., Moura, A., On the quantum Kazhdan-Lusztig functor,math.QA/0203003, 2002.ESI] Etingof, P., Schiffmann, O., Twisted traces of quantum intertwiners andquantum dynamical 72-matrices corresponding to generalized Belavin-Drinfeld triples, CMP 218 (2001), no. 3, 633^663.ES2] Etingof, P.; Schiffmann, O., Lectures on the dynamical Yang-baxter equations,math. QA/9908064.ESS] Etingof, P., Schedler, T., Schiffmann, O., Explicit quantization of dynamicalr-matrices, J. Amer. Math. Soc, 13 (2000), 595^609.EVI] Etingof, P., Varchenko, A., Geometry and classification of solutions ofthe classical dynamical Yang-Baxter equation, Commun. Math. Phys, 192(1998), 77^120 .EV2] Etingof, P., Varchenko, A., Solutions of the quantum dynamical Yang-Baxter equation and dynamical quantum groups, Commun. Math. Phys,196 (1998), 591^640 .EV3] Etingof, P., Varchenko, A., Exchange dynamical quantum groups, CMP205 (1999), no. 1, 19^52.

570 Pavel Etingof[EV4] Etingof, P., Varchenko, A., Traces of intertwiners for quantum groups anddifference equations, I, Duke Math. J., 104 (2000), no. 3, 391-432.[EV5] Etingof, P., Varchenko, A., Dynamical Weyl groups and applications,math.QA/0011001, to appear in Adv. Math.[Fa] Faddeev, L., On the exchange matrix of the WZNW model, CMP, 132(1990), 131-138.[Fe] Felder, G., Conformai field theory and integrable systems associated toelliptic curves, Proceedings of the International Congress of Mathematicians,Zürich 1994, 1247-1255, Birkhäuser, 1994; Elliptic quantum groups,preprint hep-th/9412207, Xlth International Congress of MathematicalPhysics (Paris, 1994), 211-218, Internat. Press, Cambridge, MA, 1995.[FR] Frenkel, I., Reshetikhin, N., Quantum affine algebras and holonomic differenceequations, Commun. Math. Phys. 146 (1992), 1-60.[FRT] Faddeev, L. D., Reshetikhin, N. Yu., Takhtajan, L. A., Quantization ofLie groups and Lie algebras. Algebraic analysis, Vol. I, 129-139, AcademicPress, Boston, MA, 1988.[FV1] Felder, G., Varchenko, A., On representations of the elliptic quantum groupElfish), Commun. Math. Phys., 181 (1996), 746-762.[FV2] Felder, G., Varchenko, A., Elliptic quantum groups and Ruijsenaars models,J. Statist. Phys., 89 (1997), no. 5-6, 963-980.[FV3] Felder, G., Varchenko, A., The g-deformed Knizhnik-Zamolodchikov-Bernard heat equation, CMP 221 (2001), no. 3, 549-571.[JKOS] Jimbo, M., Odake, S., Konno, H., Shiraishi, J., Quasi-Hopf twistors forelliptic quantum groups, Transform. Groups, 4 (1999), no. 4, 303-327.[Lu] Lu, J. H., Hopf algebroids and quantum groupoids, Inter. J. Math., 7 (1)(1996), 47-70.[Mo] Moura, A., Elliptic Dynamical R-Matrices from the Monodromy of theq-Knizhnik-Zamolodchikov Equations for the Standard Representation ofUq(sl(n+1)), math.RT/0112145.[Sch] Schiffmann, O, On classification of dynamical r-matrices, MRL, 5 (1998),13-30 .[TV] Tarasov, V.; Varchenko, A. Small elliptic quantum group e Tj7 (sljv), MoseMath. J., 1 (2001), no. 2, 243-286, 303-304.[Xu] Xu, P., Triangular dynamical r-matrices and quantization, Adv. Math.,166 (2002), no. 1, 1-49.

ICM 2002 • Vol. II • 571-582Geometrie LanglandsCorrespondence for GL nD. Gaitsgory*AbstractWe will review the geometric Langlands theory (mainly for the groupGLn), whose development was initiated in the works of V. Drinfeld and G.Laumon.Let A be a (smooth, complete) algebraic curve over a ground field fc, andlet E be an ^-adic n-dimensional irreducible local system on X.The geometric Langlands conjecture (for GL„) says that to E one canassociate an automorphic sheaf SE, which is a perverse sheaf on the modulistack Bun n (A) classifying vector bundles of rank noni.We will explain the motivation for this conjecture in terms of the classicaltheory of automorphic forms, and the methods involved in the constructionof SE-2000 Mathematics Subject Classification: 14H60.Introduction0.1. Let X be a (smooth, complete) curve over a ground field fc, and G a (split)reductive group. In the main body of the paper we will assume that G = GL n ,but now we would like to make a general overview of the theory, in which G can bearbitrary. Let Bunc denote the moduli stack of G-bundles on X.Our object of study is the category D(Bunc)-the derived category of sheaveson Bunc with constructible cohomology. By "sheaves" we will mean either Q r adicsheaves, which can be done over any fc, or D-modules, when char(fc) = 0.Finally, let G be the Langlands dual of G (thought of as an algebraic groupover Q f or fc, depending on the sheaf-theoretic context).0.2. It is believed that if o is a G-local system on X (i.e., a homomorphism fromthe appropriate version of ni(X) to G), which is sufficiently generic, then to o onecan attach an automotphic sheaf 3> £ D(Bunc), which is a Hecke eigen-sheaf withrespect to a. (See [BD], Section 5 for the definition of the Hecke eigen-property foran arbitrary G, or Section 2 below for the GL n case.)* Department of Mathematics, The University of Chicago, 5734 University Ave., Chicago, IL60637, USA. E-mail: gaitsgde@math.uchicago.edu

572 D. GaitsgoryUnfortunately, the geometric Langlands conjecture, i.e., the conjecture predictingthe existence of 3>, is not well formulated, because it is not known (at leastto the author) what "sufficiently generic" is for an arbitrary group G.The exception is the case when G = GL n , and "sufficiently generic" is understoodas "irreducible". In this situation, the geometric Langlands conjecture in theabove form was formulated by G. Laumon in [Lai], following the pioneering workof V. Drinfeld, [Dr], where the corresponding statement was proved for GL 2 .0.3. Although it is still not clear under what circumstances 3> should exist, at leastthree different constructions have been suggested by A. Beilinson and V. Drinfeld,in addition to the original Drinfeld's construction (the latter is, however, specificto the case G = GL n ). To the best of my knowledge, there are no theorems thatestablish the equivalence between any two of the four constructions described below.The first construction works in the D-module context (in particular, we have toassume that char(fc) = 0), under an additional assumption on a: one assumes thata is an oper, cf. [BD]. In this case, the corresponding 3> is a D-module (and notjust an object of the derived category). Moreover, 3> is holonomic. (Unfortunately,it is not clear from the construction whether 3> has regular singularities.)The second construction, via the so-called chiral Hecke algebra, also takes placein the D-module context. Here a can be arbitrary, but it is not clear under whatassumptions on a the object 3> thus constructed is non-zero, or when it is a singleD-module. It is not clear either whether the corresponding complex always hasholonomic (or even finitely generated) cohomologies.In the above two constructions, the fact that we work with D-modules is usedvery essentially, as the corresponding 3> is constructed by generators and relations.The other two constructions are more geometric in the sense that 3> is producedstarting from a, viewed as a sheaf on X, using the 6 functors.The third construction, uses the "spectral projector", and makes sense overany fc and for any a. It is again not clear under what assumption on a, the resulting3> is non-zero or when it lies in the bounded derived category.0.4. Finally, the fourth approach which, as was mentioned above, works for GL nonly and goes back to the original work of Drinfeld, is the subject of the presentpaper. In this case it can actually be shown that for an irreducible representation a,which can be thought of as an n-dimensional local system E on X, the correspondingautomorphic sheaf 3> (or JE) indeed exists and has all the desired properties.Let us add a few words about the history of this approach. After [Dr], wherethe case of n = 2 was solved, the (conjectural) generalization of the construction ofJE was suggested by G. Laumon in [Lai] and [La2]. Laumon's approach was furtherdeveloped by E. Frenkel, D. Kazhdan, K. Vilonen and the author (cf. [FGKV] and[FGV1]). The present paper can be regarded as a summary of these works. Finally,a certain vanishing result that was missing in order to complete the proof of theconjecture has been established in [Ga].Let us now briefly explain how the paper is organized. In Section 1 we reviewthe classical (i.e., function-theoretic) Langlands correspondence for GL n . In Section

Geometrie Langlands Correspondence for GL n 5732 we define Hecke eigen-sheaves and state the main theorem about the existence ofthe Hecke eigen-sheaf JE corresponding to an irreducible local system E. In Section3 we describe the construction of JE via a geometric analog of the Whittaker model.In Section 4 we explain how the main theorem about the existence of JE followsfrom a certain vanishing result. Finally, in Section 5 we indicate the steps involvedin the proof of the vanishing theorem.1. The classical theoryIn this section we will review the formulation of the classical Langlands conjecturefor function fields, and the technique of construction of automorphic formsvia Whittaker models.1.1. Let X be the global field corresponding to a (smooth, complete) curve X overa finite field ¥ q . We will denote by Â (resp., O) the corresponding ring of adeles(resp., the subring of integral adeles).Consider the quotient GL n (X)\GL n (K). The space Funct(GL n (X)\GL n (Âj)of (smooth) functions (with values in an arbitrary algebraically closed field of char.0, which we will take to be Q f ) is naturally a representation of the group GL n (A).Consider the subspace of functions that are invariant with respect to GL n (Q),i.e. the space of functions on the double quotient GL n (X)\GL n (K)/GL n (&). Thisis a module over the Hecke algebra GL n (K) with respect to GL n (

574 D. Gaitsgory1.3. Theorem. To every irreducible a as above there corresponds a (non-zero)function f a £ Yunct(GL n (X)\GL n (A)/GL n (G)), such thath-U = xAh) • U, Vft e H(GL„(A),GL„(O)).Moreover, such f„ is unique up to a scalar, and is cuspidal (see below).1.4. We will now sketch the construction oi /„• using the method oi Piatetski-Shapiro and Shalika, ci. [PS],[Sha].First, let us recall the notion oi cuspidality oi a iunction on GL„(A). Let / £Funct(G7/„(A)) be left-invariant with respect to a subgroup F (DC), where F c GL nis a subgroup that contains the unipotent radical N of the standard Borei subgroup.In particular, F contains also the unipotent radical N(Q) of any standard parabolic0 C GL n . The function / is called cuspidal if for every such parabolicy€N(Q)(X)\N(Q)(A)f(y -x) = 0,Vx£ GL n (A). (1.2)For fc = 1,..., n let Pu C GL n be the group of matrices, for which the ay entryis8ij if i > n — k,i > j. Note that P := Pi is the so-called mirabolic subgroup, i.e.the subgroup of matrices whose last row is (0, ...,0,1), and for any fc, Pj. D N.Yet N/, c GL n be the subgroup of (strictly) upper-triangular matrices, inwhich only the last fc — 1 columns may be non-zero. We shall fix a non-trivialcharacter tp : DC\A —t Q ( . It gives rise to a character \Pj. : A r j ! (A) —t Q ( , but takingthe sum of values of tp on the supdiagonal matrix entries.We define the space W* to consist of all functions/ G Funct(7*(DC)\GL„(A)/GL„(Q)),that satisfy f(n • x) = \Pfc(n) • f(x), Vn £ N/,. We define W cusp k as the subspaceof W* that corresponds to cuspidal functions. It is easy to see that for fc = n,W — W* v cusp n — * v n-1.5. Proposition. There are isomorphisms W cusp u — W cusp u+i f° r k = 1,..., n —1, which respect the H(GL„(A),GL n (

Geometrie Langlands Correspondence for GL n 5752. The geometric settingIn this section we will give a formulation of the geometric Langlands correspondence.We will assume that the ground field fc is of positive characteristic and isalgebraically closed. (The case of D-modules over a ground field of char. 0 iscompletely analogous.) We will denote by Bun„ be the moduli stack of rank nvector bundles on our curve X; D(Bun„) will denote the corresponding derivedcategory of sheaves on Bun„.2.1. First, we will introduce the Hecke functors that act on the category D(Bun„).For a point x £ X consider the stack JC X classifying the data of (M,M',ß), whereAt, At' are vector bundles on X, and ß is an embedding of coherent sheaves At' < L -¥ At,such that the quotient At/At' is isomorphic to the residue field k x , viewed as acoherent sheaf on X.We have two projections: Bun„

576 D. Gaitsgoryrespects the S^-equivariant structure for any d. 1Few remarks are in order:Remark 1. In addition to the "basic" Hecke functor H, one can introduce thefunctors H* : D(Bun„) —t D(X x Bun„) for i = 1, ...,n (classically, they correspondto the generators TJ of the Hecke algebra of E(GL n (X x ),GL n (O x jj). We haveH ~ H 1 , and H" is the pull-back functor with respect to the multiplication map(x, M) —¥ M(x). One can show (cf. [FGV1], Sect. 2) that if JE is a Hecke eigensheafwith respect to E, then ïV(7 E ) ~ A i (E) M J E -Remark 2. Note, that formally in the definition of eigen-sheaves, we did not usethe fact that the local system E was n-dimensional. However, one can show (usingRemark 1 above) that if JE £ D(Bun„) is a Hecke eigen-sheaf with respect to Eand the rank of E is different from n, then JE = 0.2.5. The following is the statement of the (unramified) geometric Langlands correspondencefor GL n , conjectured by G. Laumon in [Lai] and proved in [FGV1] and[Ga]:2.6. Theorem. Suppose that the local system E is irreducible. Then there existsa Hecke eigen-sheaf JE £ D(Bun„) with respect to E, which is, moreover, anirreducible perverse sheaf over every connected component of Bun„, and cuspidal. 2Remark 3. Of course, if JE is a Hecke eigen-sheaf, then so is JE ® K, whereK is any complex of vector spaces. An additional conjecture, which has not beenfully established yet, is that any Hecke eigen-sheaf with respect to E has this form,where JE is the eigen-sheaf constructed in [FGV1].3. Geometric Whittaker modelsFrom now on our goal will be to sketch the steps involved in the construction3.1. Let BunJ, be the stack classifying the data of (M,K), where M is a rank nvector bundle, and K is a non-zero map Q" _1 —t M. 3 Yet n denote the naturalprojection Bun^ —t Bun„.We will construct an object J' E £ D(Bun^), and then show that it descendsto the sought-for perverse sheaf JE on Bun„.For us, the category D(Bun^) is the geometric analog of the space of functionson the quotient P(K)\GL n (K)/GL n (

Geometrie Langlands Correspondence for GL n 5773.2. For an integer fc, 1 < fc < n, let Q„ : k be the stack classifying the dataof (M,Ki,...,Kk), where M is a rank n vector bundle, K,'S are non-zero mapsQra-i+...+ra-« _^ A*(M), satisfying the Pliicker relations. The latter means thatat the generic point of X, the collection of maps KI,...,K,U defines a flag of subbundles0 = At 0 C Mi C ... C M k C At,with identifications M»/Mj_i ~ Q" - * (cf. [BG], Sect. 1.3 for details). This flagwould be defined over the entire X if the maps K, were bundle maps, and not justinjective maps of sheaves.In addition, let Q n ,k,ex be a slightly larger stack, where the last map Ku isallowed to vanish. Let us denote by jk the open embedding jk : Q n^ —t Q n ,k,ex-In addition, we have the natural maps n k +i,k '• Q»,fc+i —^ Q n ,k (resp., nk+i,k,ex '•Qn,k+i,ex —* Q»,fc)j which "forget" the data of Kk+i-3.3. Next, we claim that for each fc = 1, ...,n there exist a certain full subcategoryD H (Q»,fc) C D(Q ni fc), which is a geometric analog of the space W*.The definition of D H ' (Q n ,k) involves an action on Q ni j. of a certain naturalgroupoid, which comes from the same unipotent group Nk as the one used in thedefinition of Whittaker functions on Pk(X)\GL n (A.)/GL n ( < 0). We refer the readerto [FGV] or [Ga], Sections 4 and 5 for a detailed discussion. Note that for fc = 1,Q n i is nothing but the stack Bun^ introduced above, and in this case D H ' (Q n i) =D(Q nil ) = D(Bm4).The relation between the categories D H ' (Q n ,k) for different fc's is given bythe next proposition. Before stating it, we should mention that in addition toD H (Q»,fc)j we have the corresponding full subcategory D H ' (Q n ,k,ex) € D(Q n ,k,ex)and the functors jki,jk*,jkma P the categories D H ' (Q ni j ie x) an d D H ' (Q n ,k) to oneanother.3.4. Proposition. The direct image functor nk+i,k,ex< '• D(Qn,fc+i,ex) —* D(Q»,fc)maps the subcategory D ' (Q n ,k+i,ex) to D ' (Q n ,k) and induces an equivalence betweenthe latter categories. Moreover, nk+i,k,exi restricted to D ' (Q n ,«+i,ex) coincideswith nk+i,k,ex*! hence it preserves the subcategory of perverse sheaves andcommutes with the Verdier duality.We introduce the Whittaker functor Wk,k+i '• D H (Q»,fc) —^ D H ' (Q n ,fc+i) as acomposition of the quasi-inverse of nk+i,k,ex< followed by the restriction j%. Theleft (resp., right) adjoint of Wk,k+i, is nothing but ^+1^! (resp., 7^+1^»).3.5. The object J' E £ D(Bun^), that we promised to construct in this section isobtained as follows: J' E is by definition7T2,1! °ÎT3,2! ° -7r n ,„_l! ( ^ ), (3.1)where J E is a certain canonical perverse sheaf in D H ' (Q n , n ), that we will presentlydescribe.Remark 4. The above definition of J' E can be rewritten as J' E ~ ft (7^), wheref is the natural map Q„ : „ —¥ Bun^. However, we had to break fi into the above

578 D. Gaitsgoryelementary steps (i.e. nk+i,k0symmetric power of X. This map sends the data of (At,Ki,...,K n ) to the divisor ofzeroes of the map K„, which, we recall, is a map between line bundles.The second map is denoted by ev : Q„ : „ —t G a , and it is defined as follows. Bythe definition of Q n , n , a point of this stack defines a complete flag of vector bundles0 = Mo C Mi C ... C M n = M, and identifications Mj/Mj_i ~ ft n-i for i < n, anda map G —¥ M n /M n _i for i = n. Therefore, we have n — 1 short exact sequences0 —¥ iV +1 —¥ M J —¥ iV —t 0, where M J is the corresponding rank 2 subquotient ofM, and each such extension defines a class in 77 1 (X, Q) ~ G a . The value of ev onthe above point of Q„ : „ is the sum of the above extension n — 1 classes in G a .oWe define the perverse sheaf J E on Q n^n as a tensor product of r* (U E^ ) (heredE^ denotes the symmetric power of E which lives on X^) and ev*(A-Sch), whereA-Sch is the Artin-Schreier sheaf on G a . We apply an appropriate cohomologicalshift to the above tensor product to make it a perverse sheaf.oFinally, we define J E £ D(Q„ : j) a s Goresky-MacPherson extension of J l E , i.e.o o rJ E := j\*(J x E). The fact that J E belongs to the Whittaker category D (Q„,„)follows from the construction.3.6. In the next section we will explain how the irreducibility assumption on Eimplies that the complex J' E on Bun^, constructed in this way, descends to Bun„.Here we will comment on the action of the Hecke functors.In [Ga], Section 6, it was shown that one can lift the Hecke functors E, x andH from Bun„ on the stacks Q n ,k-More precisely, for every fc we have the functors H^"'* : D(Q ni fc) —^ D(Q ni fc),which preserve the subcategory D H ' (Q n ,k)- Moreover, for fc = 2, ...,n, the functorsWk-i,k>^k,k-iW^k,k-i* intertwine in the natural sense the functors E n x - k andjj^n.fc-i_ p or n — i ; t Vj e pull-back functor (with an appropriate cohomological shift)n* intertwines H^"' 1 acting on D(Bun^) and E x acting on D(Bun„).Analogusly, we have the global Hecke functors H n - k : D(Q ni fc) —^ D(X x Q n ,k),where the assertions similar to the above ones hold.The basic fact about the perverse sheaf J E is its eigen-property, cf. [FGV1],Appendix:3.7. Proposition. There is a canonical isomorphism E n ' n (J E ) ~ E[l] M J E ,compatible with iterations (cf. Section 2).

Geometrie Langlands Correspondence for GL n 579From this proposition it follows that J' E satisfies the Hecke eigen-propertywith respect to H "• 1 . This implies that the complex JE £ D(Bun„), to which J' Edescends, has the required Hecke eigen-property with respect to H.4. Irreducibility and descentThe main step in the proof of Theorem 2.6. is the following theorem:Recall that J' E was obtained from the perverse sheaf J E by applying to it thefunctor 7T 2 ,i! o 7r 3j 2! ° ...7r„ : „_i!. Set J E := J E and J E := nk+i,ki ° --^n,n-v.(Ê)-Yet also Bun^* c Bun„ be the open substack corresponding to vector bundlesM, for which Hom(£°,M) = 0 for some fixed line bundle £. For every fc we thenobtain an open substack Q n k := Bun^* x Q n^ c Q n ,k-Bun„4.1. Theorem. Assume that E is irreducible. Then for fc = n, ...,2 the naturalma P Jkt(-?E) —* Jì*(-?E) * S an isomorphism over Q n k .From this theorem, we obtain that (over Q n k ) J E is a perverse sheaf, whichis irreducible on every connected component. Indeed, this is true for fc = n byconstruction,and by induction we can assume that this is true also for fc' > fc. Butthen we obtain that J E ~ Trk+i,k,exi(jki*(Ê +1 ))'an^ we know from Proposition 3.4.that the functor nk+i,k,ext preserves perversity and irreducibility on the Whittakersubcategory. In particular, we obtain that J' E is a perverse sheaf, irreducible onevery connected component of Bun^* '.4.2. Theorem 4.1. has been established in [FGV1], in a slightly different form. Themain ingredient in the proof of Theorem 4.1. is the following vanishing result:For any local system E', and positive integers n' and d, let us consider thefollowing functor Av^, E, : D(Bun„/) —^ D(Bun„/):Let Mod n ; be the stack of "upper modifications of length d", i.e. Mod n ;classifies the data of triples (M,M',/3), where M and M' are rank-n' vector bundleson X, and ß is an embedding M' n'. Supposealso that d > (2g — 2) • rk(E') • n', where g is the genus of X. Then the functorAv^, E' '• D(Bun„/) —^ D(Bun„/) vanishes identically.The idea of the proof of Theorem 4.1. is that the cone of the map jk\ (JE) —*j*!*(3 r^) can be expressed in terms of the functors Av^ E for various d applied tovarious sheaves on Bun n _fc + i.

580 D. Gaitsgory4.4. The next step is to show that perversity and irreducibility of J' E on Bun^* 'implies that J' E descends to a perverse sheaf on Bun„, cf. [FGV1], Section 5.First, one shows that Îßun»" descends to a perverse sheaf on Bun^*. Thisis done using a trick with Euler characteristics. It turns out, that since we alreadyknow that Îßun»* ' is perverse and irreducible, it is sufficient to show that theEuler-Poincaré characteristics of the stalks of J' E are constant along the fibers ofthe map n : Bun^ —t Bun„. Secondly, one observes that the above Euler-Poincarécharacteristics are actually independent of the local system E (and depend only onits rank).Thus, it is sufficient to find just one local system of a given rank n, for whichthe above constancy of Euler-Poincaré characteristics takes place, and one easilyfinds one like that.Finally, using the Hecke eigen-property of J' E , one shows that there exists aperverse sheaf JE defined on the entire Bun„, whose (cohomologically normalized)pull-back to Bun^ is isomorphic to J' E - The fact that the resulting sheaf JE iscuspidal follows from the construction.5. The vanishing resultIn this section we will indicate the main ideas involved in the proof of Theorem4.3., following [Ga]. We will change the notation slightly, and replace n' by nand E' by E.5.1. First, let us rewrite the definition of the functor Av n E . Consider first thefunctor A\- n>E : D(Bun„) —t D(Bun„) given byJ^p,(q*(E l )®H(Jj),where p,q are the two projections from X x Bun„ to Bun„ and X, respectively.From Proposition 2.3. it follows that the d-fold iteration ItAv^ E := Av„^ 0...0k\' n} E' : D(Bun„) —^ D(Bun„) maps to the equivariant derived category D Sd (Bun„),where the E^-action on the base Bun„ is, of course, trivial.Moreover, it follows from the definition of Laumon's sheaf H E , that there is afunctorial isomorphismAVIE(J) CZ (ltAvl E (Jjffi (5.1)where the superscript E^ designates the functor of Ed-invariants.5.2. The first step in the proof of Theorem 4.3. is the observation that instead ofproving that the functor AY„ : E vanishes, it is in fact enough to show that it is exactin the sense of the perverse i-structure.The fact that the seemingly weaker exactness assertion is equivalent to vanishingis proved using the Euler characteristics trick, similar to what we did in theprevious section.Since, as we have seen above, the functor Av^ E can be expressed throughmore elementary functors Av n: E, we will analyze the exactness properties of the

Geometrie Langlands Correspondence for GL n 581latter. Unfortunately, it is not true that the "elementary" functor Av„^ is exact.However, it will be exact when regarded as a functor acting on a certain quotientcategory.5.3. We introduce the category D(Bun„) as the quotient of D(Bun„) by a triangulatedsubcategory Dd egen (Bun n ). (An object J £ D(Bun„) belongs to Drf ese „(Bun„)essentially when it is degenerate, i.e., when it vanishes in the Whittaker model.)The quotient D(Bun„) possesses the following three crucial properties:1) The perverse i-structure on D(Bun„) induces a well-defined i-structure on D(Bun^2) The Hecke functors E, x : D(Bun„) —t D(Bun„) gives rise to well-defined functorsD(Bun„) —t D(Bun„), and the latter functors are exact in the sense of the t-structure on D(Bun„).3) The subcategory Drf ese „(Bun„) is orthogonal to the subcategory of cuspidalsheaves. I.e., if 3^1,3^2 £ D(Bun„) are such that 7i is cuspidal and the imageof 3^ in D(Bun„) vanishes, then Hom D ( B un„)(3 r i ) 3^) — Hom D ( B un„)(3 r 2 ) 3 r i) = 0.5.4. The main step in the proof that the functor Av n E is the following:5.5. Theorem. The functor AY„ : E : D(Bun„) —t D(Bun„) descends to a welldefinedfunctor D(Bun„) —t D(Bun„), and the latter functor is exact.The main idea behind the proof of Theorem 5.5. is the same phenomenon asthe one that forbids the existence of Hecke eigen-sheaves on Bun„ with respect tolocal systems of a wrong rank. Namely, if J is a perverse sheaf on Bun„ such thatthe corresponding object of D(Bun„) violates Theorem 5.5., then by looking at thebehavior of H d (J) £ Y)(X d x Bun„) around the various diagonals in X d , we arriveto a contradiction.Using Theorem 5.5. the proof of the exactness of Av^ E proceeds as follows:From (5.1) (and using the fact that taking invariants is an exact functor), we obtainthat the functor Av^ E is well-defined and exact on the quotient category D(Bun„).Moreover, by induction on n we can assume that for any J £ D(Bun„), Av n E (J)is cuspidal.Hence, if J is a perverse sheaf on Bun„, in the cohomological truncationsarrowsA°(Av^(3-)) and r

582 D. Gaitsgory[BG] A. Braverman, D. Gaitsgory, Geometric Eisenstein series, Preprintmath.AG/9912097, to appear in Invent. Math.[CS] W. Casselman, J. Shalika, The unramified principal series of p-adic groupsII. The Whittaker function, Comp. Math., 41 (1980), 207-231.[Dr] V.G. Drinfeld, Two-dimensional iadic representations of the fundamentalgroup of a curve over a finite field and automorphic forms on GL(2), Amer.J. Math., 105 (1983), 85-114.[FGKV] E. Frenkel, D. Gaitsgory, D. Kazhdan, K. Vilonen, Geometric realization ofWhittaker functions and the Langlands correspondence, Jour. Amer. Math.Soc, 11 (1998), 451-484.[FGV] E. Frenkel, D. Gaitsgory, K. Vilonen, Whittaker patterns in the geometry ofmoduli spaces of bundles on curves, Annals of Math., 153 (2001), 699-748.[FGV1] E. Frenkel, D. Gaitsgory, K. Vilonen, On the Geometric Langlands conjecture,Jour. Amer. Math. Soc, 15 (2002), 367-417.[Ga] D. Gaitsgory, On a vanishing conjecture appearing in the geometric Langlandscorrespondence, Preprint math.AG/0204081 (2002).[Laf] L. Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent.Math., 147 (2002), 1-241.[Lai] G. Laumon, Correspondance de Langlands géométrique pour les corps defonctions, Duke Math. J., 54 (1987), 309-359.[La2] G. Laumon, Faisceaux automorphes pour GL n : la première constructionde Drinfeld, Preprint alg-geom/9511004 (1995).[PS] LI. Piatetski-Shapiro, Euler subgroups, in: Lie Groups and Their Representations,ed. I.M. Gelfand, Adam Hilder Pubi., (1975), 597-620.[Sha] J.A. Shalika, The multiplicity one theorem for GL n , Ann. Math., 100(1974), 171-193.[Shi] T. Shintani, On an explicit formula for class 1 Whittaker functions onGL n over 9ß-adic fields, Proc. Japan Acad., 52 (1976), 180-182.

ICM 2002 • Vol. II • 583-597On the Local Langlands CorrespondenceMichael Harris*AbstractThe local Langlands correspondence for GL{n) of a non-Archimedean localfield F parametrizes irreducible admissible representations of GL(n, F) interms of representations of the Weil-Deligne group WD F of F. The correspondence,whose existence for p-adic fields was proved in joint work of the authorwith R. Taylor, and then more simply by G. Henniart, is characterized by itspreservation of salient properties of the two classes of representations.The article reviews the strategies of the two proofs. Both the author'sproof with Taylor and Henniart's proof are global and rely ultimately on anunderstanding of the i?-adic cohomology of a family of Shimura varieties closelyrelated to GL(n). The author's proof with Taylor provides models of the correspondencein the cohomology of deformation spaces, introduced by Drinfeld,of certain p-divisible groups with level structure.The general local Langlands correspondence replaces GL(n, F) by an arbitraryreductive group G over F, whose representations are conjecturallygrouped in packets parametrized by homomorphisms from WDp to the Langlandsdual group L G. The article describes partial results in this direction forcertain classical groups G, due to Jiang-Soudry and Fargues.The bulk of the article is devoted to motivating problems that remainopen even for GL{n). Foremost among them is the search for a purely localproof of the correspondence, especially the relation between the Galoistheoreticparametrization of representations of GL(n, F) and the group-theoreticparametrization in terms of Bushnell-Kutzko types. Other open questions includethe fine structure of the cohomological realization of the local Langlandscorrespondence: does the modular local Langlands correspondence of Vignerasadmit a cohomological realization?2000 Mathematics Subject Classification: 11, 14, 22.IntroductionCompared to the absolute Galois group of a number field, e.g. Gal(Q/Q),the Galois group Yp of a non-archimedean local field F has a ridiculously simplestructure. Modulo the inertia group IF, there is a natural isomorphismYp/Ip -^r Gal(kp/kp),*Institut de Mathématiques de Jussieu-UMR CNRS 7586, Université Paris 7. Membre,Institut Universitaire de France, France. E-mail: harris@math.jussieu.fr

584 Michael Harriswhere kp is the residue field of F. Then Gal(kp/kp) is topologically generated bythe geometric Frobenius Frob(x) = x~i, where q = \kp\ = pf for p prime. Theinertia group has a two step filtration,1 -^ Pp -+ I F -+ JJZ.^1,where the wild ramification group Pp is a pro-p group.Thus if a : Yp^yGL(n,C) is a continuous homomorphism, n > 1, then theimage of a is solvable, and a(Pp) is nilpotent. This is still true when a is a finitedimensionalcomplex representation of the Weil group, the subgroup Wp C Ypof elements whose image in Gal(kp/kp) is an integral power of Frob. Despitethis simplicity, our understanding of the set of equivalence classes of n-dimensionalrepresentations of Wp is far from complete, at least when p divides n.The reciprocity map of local class field theory:F x-^rW rabF ,identifies the set Q(1,F) of one-dimensional representations of Wp with the setA(1,F) of irreducible representations of F x = GL(1,F). More than a simplebijection, this identification respects a number of salient structures, and its behaviorwith respect to field extensions F'/F is well understood. Moreover, it is compatible,in a straightforward way, with global class field theory, and was historically firstderived as a consequence of the latter.A simple special case of Langlands' functoriality principle is the so-calledstrong Artin conjecture, which identifies the Artin L-function attached to an irreduciblen-dimensional representation of Gal(Q/K), for a number field K, as theL-function of a cuspidal automorphic representation of GL(U)K- AS a local counterpart,Langlands proposed a parametrization of irreducible admissible representationsof reductive groups over the local field F in terms of representations of Wp.The prototypical example is the local Langlands conjecture for GL(n). By analogywith the case n = 1, the set of equivalence classes of irreducible admissible representationsof G n = GL(n, F) is denoted A(n, F). By Q(n, F) we denote the set ofequivalence classes of n-dimensional representations, not of Wp but rather of theWeil-Deligne group WDp, and only consider representations for which any liftingof Frob acts semisimply. Then the general local Langlands conjecture for GL(n),in its crudest form, asserts the existence of a family of bijections, as F and n vary:a = a n ,p : A(n, F) -^ Ç(n,F). (0.1)A normalization condition is that the central character ^ of n £ A(n, F) correspondto det a(n) via local class field theory.The first general result of this type was proved by Henniart [Hel]. Early workof Bernstein and Zelevinsky reduced (0.1) to the existence of bijectionsa = a n , F : A 0 (n,F) Â- £„(n,F), (0.2)

On the Local Langlands Correspondence 585where Q Q (n,F) are the irreducible representations of Wp and Ao(n,F) is the supercuspidalsubset of A(n,F). Both sides of (0.2) are homogeneous spaces underA(1,F), and thus under its subset A unr (l, F) of unramified characters x of F x : ifn £ Ao(n,F) (resp. a £ Go(n,Fj), we denote by n ® x (resp. a ® x) the tensorproduct of n (resp. a) with the one-dimensional representation x ° det of G n (resp.with the character

586 Michael Harriscorresponding to r is then the unique subquotient n(r) = ,E(fl'));£(s,0-n,E(n) ®0-n',E(fl'j) = JJ £ v (s, (T n>E (YY) ® CT„',£;(II'), tp v )is the product of local Deligne-Langlands e factors.VHere " denotes contragredient. The local additive characters tp v are assumedto be the local components of a continuous character of AE/E.1.6. The map a = a n: F'• Ao(n,F)^-Q(n,F)(i) takes values in Qo(n,F);(ii) defines a bijection Ao(n,F) ^> Qo(n,F);(iii) satisfies the remaining requirements of a local Langlands correspondence,especially (0.4)-The main burden of [LRS] is the construction of a class A sood (n, E) largeenough to satisfy (1.4): now a moot point, since Lafforgue has proved that all cuspidalautomorphic representations of GL(n) of a function field are "good" in thissense. The A sood (n, E) in [LRS] are the automorphic representations that contributeto the cohomology of an appropriate Drinfeld modular variety, constructedfrom scratch for the occasion, attached to the multiplicative group of a divisionalgebra of dimension n 2 over E, unramified at the chosen tv. Property (1.5) in this

On the Local Langlands Correspondence 587case follows from general results of Deligne in [D], valid only in equal characteristic.Now by (1.2), for a sufficiently large set S of places of E we haveJJ L(s,E v x n;) = JJ L(s,o n , Ev (Yl v ) ®a nÊv (K)); (1-7)v

588 Michael Harristhen its local component n^ is tempered at every finite prime v. Clozel in [Cll]already showed this to be true for almost all unramified v. Generalizing a methoddeveloped by Lubotzky, Phillips, and Sarnak for the 2-sphere, Clozel [C12] uses theversion of the Generalized Ramanujan Conjecture proved in [HT] to obtain effectiveconstructions of families of equidistributed points on odd-dimensional spheres.With (0.1) out of the way, we can propose the following improvement of (1.3):Problem 1. Show that0-TI,E(YY)WE V —+ On,E v (Yl v ). (1-10)For n = 2 this was established by Carayol assuming standard conjectures onthe semisimplicity of Frobenius. Theorem 1.9 shows that it holds up to semisimplification.The techniques of [HT], like the earlier work of Kottwitz treating unramifiedplaces, is based on a comparison of trace formulas, and cannot detect thedifference between two representations with the same semisimplification. Assumingsemisimplicity of Frobenius, the equality (1.10) follows easily from Theorem 1.9and Deligne's conjecture, apparently inaccessible, on the purity of the monodromyweight filtration.Compatibility with functoriality. Given cuspidal automorphic representationsuj of GL(Hì)E, for i = 1,2, ...,r, and a homomorphism p : GL(n{) x ••• xGL(n r )—¥GL(N) of algebraic groups, Langlands functoriality predicts the existenceof an automorphic representation p»(IIi ®... E r ), not necessarily cuspidal, ofGL(N)E, such that, for almost all unramified places v of E,ON,E V (P*(Y1I ...n r )„) = po (®l = ia ni: E v (flì,vj)- (1A1)In recent years this has been proved for general number fields E in several importantspecial cases: the tensor products GL(2) x GL(2)^GL(4) (Ramakrishnan)and GL(2) x GL(3)^GL(6) (Kim-Shahidi), and the symmetric powers Sym 3 :GL(2)^GL(4) (Kim-Shahidi) and Sym 4 : GL(2)^GL(5) (Kim). It has been verifiedin all four cases that (1.11) holds for all v.Construction of supercuspidal representations by "backwards lifting".The unitary representation n £ Ao(n,F) is isomorphic to its contragredientif and only if the local factor L(s,n x n) has a pole at s = 0, which is necessarilysimple.The local factor can be decomposed as a product:L(s,n x n) = L(s,n,Sym 2 )L(s,n, A 2 ), (1A2)where the two terms on the right are defined for unramified n by Langlands and ingeneral by Shahidi. Only one of the factors on the right has a pole. Using the classA good (n,E) of automorphic representations, Henniart has shown that it is the firstfactor (resp. the second factor) if and only if a(n) is orthogonal (resp. symplectic);

On the Local Langlands Correspondence 589the symplectic case only arises for n even. One thus expects that n is obtained byfunctorial transfer from an L-packet of a classical group G over F, via the map ofL-groups L G^yGL(n,C), where L G = SO(n,C), resp. Sp(n,C), if the first, resp.the second factor in (1.12) has a pole at s = 0.In particular, when n = 2m and L(s, n, A 2 ) has a pole at s = 0, n should comefrom an L-packet on the split group SO(2m + 1, F). Using a local analogue of themethod of "backwards lifting," or automorphic descent, due to Ginzburg, Rallis, andSoudry, Jiang and Soudry have constructed a generic supercuspidal representationn' of SO(2m +1, F) for every n £ Ao(n, F) with the indicated pole. More generally,they have obtained a complete parametrization of generic representations of splitG = SO(2m+l, F) in terms of Langlands parameters WDp^y L G [JS]. These resultsshould certainly extend to other classical groups.2. Cohomological realizations of the localcorrespondenceThe theory of the new vector implies easily that any irreducible admissiblerepresentation n £ A(n, F) has a rational model over the field of definition of itsisomorphism class: the Brauer obstruction is trivial for G n . The analogous assertionfails for representations in G(n, F). Thus one cannot expect the existence of a spaceM, with a natural action of G n x Wp, whose cohomology of whatever sort realizesthe local Langlands correspondence, as an identity of virtual representationsO»,FM = ±[77oro G „(77 c (Af),7r)] := ±^2(-l) z Hom Gn (H t c(M),n). (2.1)We add a third group to the picture by taking J to be an inner form of G n ,the multiplicative group of a central simple algebra D over F of dimension n 2 ,with Hasse invariant r^-. The set A(n,F) contains a subset A> 2 ) (n, F) of discreteseries representations, character twists of those realized in the regular representationon L 2 (G n ) (modulo center). The set A(J) of equivalence classes of irreducibleadmissible representations contains an analogous subset A( 2 ) (J), equal to A(J) if Dis a division algebra. The J acquêt-Langlands correspondence [R,DKV] is a bijectionJL : A(2)(G n ) -^-¥ „4(2) (J) determined by the identity of distribution charactersìxAg) = e(J)xjL(*)(j), K e A< 2) (G) (2.2)if e (J) = ±1 is the Kottwitz sign and g and j are elliptic regular elements with thesame eigenvalues. When rp, = 1 there are two spaces 0 F and M LTF with naturalG n x J-actions. The former is a countable union, indexed by Z, of copies of theprofinite étale cover Q F ' of the rigid-analytic upper half space 0 F = P" _1 (Cj,) —W n^1(F), defined by Drinfeld in [D2]. The latter is the rigid generic fiber of theformal deformation space M FT F of the one-dimensional height n formal o^-modulewith Drinfeld level structures of all degrees [DI]. A relation analogous to (2.1) wasconjectured by Carayol in [CI], with ± = (^1)" _1 :

590 Michael HarrisTheorem 2.3. For n supercuspidala*(n) ® JL(n) =a*(n) ® n =±[Hom Gn (H c (Û F ),n)]±[Hom,j(H c (Ml T: p),JL(n))].The notation a#(n) indicates that a(n) has been twisted by an elementary factor.We use the rigid-analytic étale cohomology introduced by Berkovich [B] withcoefficients in Q f , tfi^p. For M LT F this can be interpreted as a space of vanishingcycles for the formal deformation space, viewed as a formal scheme over Spf(Op).The case of 0 F was proved in [HI], using the existence of Shimura varieties admittingrigid-analytic uniformizations by 0 F . This has recently been extended toF of equal characteristic by Hausberger [Hau]. The case of M LTF , again for nsupercuspidal, was initially treated by Boyer [Bo] in the equal-characteristic case.The analogous statement for F p-adic, and for any n, is the logical starting pointof the proof of Theorem 1.9 in [HT].Theorem 2.3 is extended in [HT] to general n £ „4(2) (G). The explicit formulafor the alternating sum of the Homj(H l c(M LT F ), JL(nj) is awkward but yields asimple expression forJ2(^y +J Exfi G (Hom,j(Hl(M n LT:p),JL(n)),n)= ^(-l) i +J+ k ExtUExt k J(HUMlT,p),JL(K)),K) (2-4)«JAin terms of the semisimplification of a(n). An analogous conjectural expression forindividual 77*(OF) has been circulating for several years and should appear in aforthcoming joint paper with Labesse. Faltings has proved [F2] that the spaces OFand M LT F become isomorphic after p-adic completion of the latter. Thus the twoquestions in the following problem reduce to a single question:Problem 2. Determine the individual representations H l c(M.), and the spacesExt J Gn(Hi(Üp),n) and Hom,j(Hi(M LTF ), JL(nj) for all i,j, all n £ A(n,F). Inparticular, show that Ext J G (77*(OF), n) vanishes unless there exists n' £ „4(2) (n, F)such that n and n' induce the same character of the Bernstein center.The results of [HT] imply that, for any n £ A> 2 )(n,F), with Bernstein characterßx, the Bernstein center acts on î( — Y) % Homj(Hl(M LT F), JL(nj) via ß„.For n supercuspidal it is known in all cases that the spaces in Problem 2vanish for i ^ n — 1 (and for j ^ 0). This vanishing property should characterizesupercuspidal n among representations in A( 2 )(n,F). When n is the Steinbergrepresentation, the 77*(OF), as well as the corresponding Ext groups, are calculatedexplicitly in [SS]. The calculation in [SS] is purely local, whereas the vanishingoutside the middle degree for n £ Ao(n,F) is based on properties of automorphicforms.

On the Local Langlands Correspondence 591Problem 3. Find a purely local proof of the vanishing property for n £ Ao(n, F).The covering group of Ö F over 0 F can be identified with the maximal compactsubgroup J° C J. Thus 77*(OF) can be written as a sum ® T 77*(OF)[T] ofits r-isotypic components, where r runs over irreducible representations of J° or,equivalently, over inertial equivalence classes of representations of J. Closely relatedto Problem 3 is the followingProblem 4. Characterize r £ A(J) such that JL -1^) £ Ao(n,F). Equivalently,calculate the Jacquet functors of the G„-spaces 77*(OF)[T] geometrically, in termsof r.When n is prime JL -1^) £ Ao(n, F) if and only if dim r > 1; when dimr = 1JL -1^) is a twist of the Steinberg representation. For general n practically nothingis known.Results of L. Fargues [Fa]. For certain classical F-groups G, Rapoport and Zink,using the deformation theory of p-divisible (Barsotti-Tate) groups, have definedpro-rigid analytic spaces M admitting continuous G x J x WE actions on their£-adic cohomology, where J is an inner form of G and E, the reflex field of M,is a finite extension of F [RZ]. In [R] Rapoport proposes a conjectural formula,which he attributes to Kottwitz, for the discrete series contribution to the virtualTneG x J x IA~F-module [H(Mj] = Yrii-lYH^M^fi).pairs (G,J) consideredin [RZ] include (G n ,D x ) with general Hasse invariant ^fi, G = J = GU(n), thequasi-split unitary similitude group attached to the unramified quadratic extensionof F, and the symplectic similitude group G = GSp(2n,F).Theorem 2.5 (Fargues). Suppose F/Q p unramified, (G,J) = (G n ,D x ), with(rpi,n) = 1. For any n £ Ao(n,F) we have^(^l)*77omj(77*(,M,Q f ), JL(nj) 0= ±n ® AV(TT)up to a simple twist. Here the subscript o denotes the G-supercuspidal part and A ? isa certain tensor product of exterior powers of a(n) with total weight rp,, dependingon auxiliary data defining M.This confirms the Kottwitz-Rapoport conjectures in the case in question. ForG = J = GU(3) Rogawski has defined a local Langlands correspondence via basechange to GL (3). In that case the supercuspidal representations of G are groupedinto L-packets. Fargues's techniques apply to this case as well, and he obtains aversion of the Kottwitz-Rapoport conjectures, more difficult to state than Theorem2.5 (higher Ext's are involved, and the formula is averaged over L-packets). 1 More1 The statement of the general Kottwitz-Rapoport conjectures in [H3] for general discreteseries representations is based on a misreading of Rapoport's use of the term "discrete L parameter". The correct conjecture should involve the analogue of the alternating sum on the right-handside of (2.4), with JL(%) replaced by %' in the L-packet associated to %.

592 Michael Harrisgenerally, Fargues' methods apply to classical groups attached to Shimura varieties,whenever the trace formula is known to be stable and functorial transfer from G toGL(n) has been established.In contrast to [HT], Fargues' methods are essentially rigid-analytic, and makeno use of equivariant regular integral models of Shimura varieties in wildly ramifiedlevel - fortunately so, since such models are not known to exist. Heuristically,the characters of the representations of G and J on [77(M)] can be related byapplyinga Lefschetz trace formula to £-adic cohomology of the rigid space M. Thisapproach, which in principle provides no information about the Wp action, has beensuccessfully applied to 0 F by Faltings in [Fl], and to Af£ TF by Strauch [S] whenn = 2. For higher M FTF , and for the Rapoport-Zink spaces studied by Fargues,one does not yet know how to deal with wild boundary terms in Huber's Lefschetzformula [Hu] and its higher-dimensional generalizations.Using work of Oort and Zink on stratification of families of abelian varietiesand the slope filtration for p-divisible groups, Mantovan [M] has developed anotherapproach to the cohomology of Shimura varieties of PEL type. Closer in spirit to[HT] than to [F], [M] obtains finer results on the geometry of the special fiber anda description of the cohomology in ramified level similar to that of [F].Cohomological realizations with torsion coefficients.It would be convenient if the following question had an affirmative answer:Question 5. Is 77*(0 F ,Zf) a torsion-free Zf-module?The global trace formula methods used in [HI] and [HT] to derive Theorem2.3 from an analysis of the cohomology of the "simple" Shimura varieties of the titleof [K] are insensitive to torsion in cohomology. When £ > n it may be possible, asin recent work of Mokrane and Tilouine, to combine £-adic Hodge theory with thegeneralized Eichler-Shimura congruence formula, for the same "simple" Shimuravarieties, to answer Question 5. For £ < n completely new ideas are needed.When fc is an algebraically closed field of characteristic £ ^ p, Vignéras hasdefined a class of smooth supercuspidal representations Ao,k(n, F) of G n with coefficientsin fc, and has proved that they are in bijection with the set Go,k(n,F) ofirreducible n-dimensional representations of Wp over fc (see article in these Proceedings).It is natural to expect that this modular local Langlands correspondenceis realized on the spaces H*(M,k), with M = 0 F or M LTF .Problem 6. Define a modular Jacquet-Langlands map n H> JL(n) from Ao,k(n, F)to fc-representations of J, and formulate the last sentence precisely. Does the virtuallA'F-module(^l) n - 1^2(^l) i +i+ k Exi{ } (Ext k j(Hl(M,k),JL(nj),n)realize the modular local Langlands correspondence?

On the Local Langlands Correspondence 593Implicit in the second question is the assumption that the modular Jacquet-Langlands map can be extended to a wider class of fc-representations of G n , perhapsincluding reduction (mod £) of supercuspidal representations in characteristic zero.One can of course ask the same questions when £ = p. In this case we can considerrigid (de Rham) cohomology, in the sense of Berthelot, as well as p-adic étale cohomology.All three groups G n , J, and Wp have large analytic families of p-adicrepresentations. It is not at all clear whether the p-adic cohomology of 0 F is sufficientlyrich to account for all p-adic deformations - in categories yet to be defined- of a given representation occurring in cohomology with coefficients in F p .3. Explicit parametrization of supercuspidalrepresentationsDistribution characters.The distribution character X-K, a locally integrable function on the set of regularsemisimple elements of G n = GL(n, F), is the fundamental analytic invariant of n £A(n,F). For n £ A(2)(n,F), XJL(-K), related to XT by (2.2), extends continuouslyto an invariant function on J = D x provided (rp,,n) = 1, which we assume. Underthis hypothesis every element of J is elliptic and every elliptic regular elementj is contained in a unique maximal torus T(j), isomorphic to the multiplicativegroup of an extension K of F of degree n. Since JL(n) is finite-dimensional, itsrestriction to T(j) equals J2e a^(C)C where £ runs over characters of K x and thecoefficients a^(£) = a n (K, £) are non-negative integers, almost all zero. In this wayn £ A' 2 )(n,F) is determined by the integer-valued function a T (Ä",£) where K runsover degree n extensions of F and £ over characters of K x . Invariance entails thesymmetry condition a^(7i"',' T £) = a n (K,£) where a : K ^fi K' is an isomorphismover F; in particular, if a € Autp(K).Problem 7. Express a T (Ä",£) in terms of numerical invariants of a(n).Of course a T (Ä",£) = 0 unless £|F* coincides with the central character £„of n. When n = 2 a n (K, £) £ {0,1}, and a theorem of Tunnell, completed by H.Saito, relates the nonvanishing of a n (K, £) to the local constant £(|,).For n prime to p a conjecture of Reimann, following an earlier conjecture of Moy,expresses x-n m terms of a(n); work in progress of Bushnell and Henniart shows thatthis conjecture is almost right (probably up to an unramified character of degree attwo).Parametrization via types.A fundamental theorem of Bushnell and Kutzko asserts that every supercuspidaln can be obtained by compactly supported induction from a finite-dimensionalrepresentation r of a subgroup 77 c G n which is compact modulo the center Z n ofG n . The pair (77,p), called an extended type, is unique up to conjugation by G n .The character x* can be obtained from (77, p) by a simple integral formula [BH,(A.14)].

594 Michael HarrisThe outstanding open problem concerning the local Langlands correspondenceis undoubtedly-Problem 8. (a) Define a(n) directly in terms of (77,p) (and vice versa).(b) Show directly that the definition of a in (a) has the properties of a localLanglands correspondence.Note that (b) presupposes a direct construction of the local Galois constants.Problem 8 formulates the hope, often expressed, for a purely local constructionof the local Langlands correspondence. Bushnell, Henniart, and Kutzko have madeconsiderable progress toward this goal. Among other results, they have obtained:• A formula for the conductor a(n x n'), n £ Ao(n, F), n' £ Ao(n', F) [BHK];• A purely local candidate for the base change map A(n, F)—¥A(n, K) whenK/F is a tame, not necessarily Galois extension [BH, I], agreeing withArthur-Clozel base change for K/F cyclic;• A bijection between wildly ramified supercuspidal representations of G p mand wildly ramified 2 representations in Go(p m , F), preserving local constants[BH, II].In each instance, the constructions and proofs are based primarily on the theory oftypes. A complete solution of Problem 8 remains elusive, however, absent a betterunderstanding of the local Galois constants.Question 9. Can the types (77, p) be realized in the cohomology (£-adic or p-adic)of appropriate analytic subspaces of 0 F or M LT F ?Positive results for certain (77, p) have been announced by Genestier andStrauch, at least when n = 2.Acknowledgments. I thank R. Taylor, G. Henniart, and L. Fargues for theircomments on earlier versions of this report.ReferencesMore or less detailed accounts of the history of the local Langlands conjecture,and of its proofs, can already be found in the literature: [Rd] and [Ku] describe theproblem and the work of Bernstein and Zelevinsky, while the proofs are outlined in[C2,C3], [He4], [W], as well as the introduction to [HT].[AC] Arthur, J. and L. Clozel,, Simple algebras, base change, and the advancedtheory of the trace formula, Annals of Math. Studies, 120, Princeton:Princeton University Press (1989).[B] Berkovich, V.G., Étale cohomology for non-archimedean analytic spaces,Pubi. Math. I.H.E.S., 78, 5-161 (1993).2 A wildly ramified irreducible representation of Wp is one that remains irreducible uponrestriction to Pp; a wildly ramified supercuspidal is one not isomorphic to its twist by any nontrivialunramified character.

On the Local Langlands Correspondence 595[Bo] Boyer, P., Mauvaise réduction de variétés de Drinfeld et correspondance deLanglands locale, Invent. Math., 138, 573^629 (1999).[BHK] Bushnell, C, G. Henniart, and P. Kutzko, Local Rankin-Selberg convolutionsfor GL n : explicit conductor formula, J. Amer. Math. Soc, 11,703^730 (1998).[BK] Bushnell, C. and P. Kutzko, The admissible dual of GL(N) via compactopen subgroups, Annals of Math. Studies, 129 (1993).[BH] Bushnell, C. and G. Henniart, Local tame lifting for GL(n), I. Pubi. Math.IHES, (1996); II: wildly ramified supercuspidals, Astérisque, 254 (1999).[Cl] Carayol, H., Non-abelian Lubin-Tate theory, in L. Clozel and J.S. Milne,eds., Automorphic Forms, Shimura varieties, and L-functions, New York:Academic Press, vol II, 15^39 (1990).[C2] Carayol, H., Variétés de Drinfeld compactes, d'après Laumon, Rapoport, etStuhler, Séminaire Bourbaki exp. 756 (1991-1992), Astérisque 206 (1992),369-109.[C3] Carayol, H., Preuve de la conjecture de Langlands locale pour GL n : Travauxde Harris-Taylor et Henniart, Séminaire Bourbaki exp. 857. (1998-1999).[CU] Clozel, L., Représentations Galoisiennes associées aux représentations automorphesautoduales de GL(n), Pubi. Math. I.H.E.S., 73, 97^145 (1991).[C12] Clozel, L., Automorphic forms and the distribution of points on odddimensionalspheres, (manuscript, 2001).[CL] Clozel, L. and J.-P. Labesse, Changement de base pour les représentationscohomologiques de certains groupes unitaires, appendix to J.-P. Labesse,Cohomologie, stabilisation, et changement de base, Astérisque, 257 (1999).[D] Deligne, P., Les constantes des équations fonctionnelles des fonctions L,Modular Functions of one variable II, Lect. Notes Math., 349, 501^595(1973).[DKV] Deligne, P., D.Kazhdan, and M.-F.Vigneras, Représentations des algèbrescentrales simples p-adiques, in J.-N.Bernstein, P.Deligne, D.Kazhdan, M.-F.Vigneras, Représentations des groupes réductifs sur un corps local, Paris:Hermann (1984).[Dl] Drinfeld, V., Elliptic modules, Math. USSR Sbornik, 23, 561^592 (1974).[D2] Drinfeld, V., Coverings of p-adic symmetric domains, Fun. Anal. Appi.,10, 107-115 (1976).[Fl] Faltings, G., The trace formula and Drinfeld's upper halfplane, Duke Math.J., 76, 467-181 (1994).[F2] Faltings, G., A relation between two moduli spaces studied by V. G. Drinfeld,preprint, 2001.[Fa] Fargues, L., Correspondances de Langlands locales dans la cohomologie desespaces de Rapoport-Zink, thèse de doctorat, Université Paris 7 (2001).[Fu] Fujiwara, K., Rigid geometry, Lefschetz-Verdier trace formula and Deligne'sconjecture, Invent. Math., 127, 489^533 (1997).

596 Michael Harris[HI] Harris, M., Supercuspidal representations in the cohomology of Drinfel'dupper half spaces; elaboration of Carayol's program, Invent. Math., 129,75-119 (1997).[H2] Harris, M., The local Langlands conjecture for GL(n) over a p-adic field,n

On the Local Langlands Correspondence 597[RZ][Ro][Sh][S][SS][W][Z]Rapoport, M. et T. Zink, Period Spaces for p-divisible Groups, Princeton:Annals of Mathematics Studies 141 (1996).Rogawski, J., Representations of GL(n) and division algebras over a p-adicfield, Duke Math. J., 50, 161-196 (1983).Shahidi, F., Local coefficients and normalization of intertwining operatorsfor GL(n), Comp. Math., 48, 271-295 (1983).Strauch, M., On the Jacquet-Langlands correspondence in the cohomologyof the Lubin-Tate deformation tower, Preprintreihe SFB 478 (Münster), 72,(1999).Schneider, P. and U. Stuhler, The cohomology of p-adic symmetric spaces,Invent. Math., 105, 47-122 (1991).Wedhorn, T., The local Langlands correspondence for GL(n) over p-adicfields, lecture at the summer school on Automorphic Forms on GL(n) atthe ICTP Trieste, ICTP Lecture Notes Series (to appear).Zelevinsky, A. V., Induced representations of reductive p-adic groups II:on irreducible representations of GL(n), Ann. Sci. E.N.S., 13, 165-210(1980).

ICM 2002 • Vol. II • 599-613Vector Bundles, LinearRepresentations, and Spectral ProblemsAlexander Klyachko*AbstractThis paper is based on my talk at ICM on recent progress in a number ofclassical problems of linear algebra and representation theory, based on new approach,originated from geometry of stable bundles and geometric invariant theory.2000 Mathematics Subject Classification: 14F05, 14M15, 14M17, 14M25, 15A42.Keywords and Phrases: Bundles, Linear representations, Spectral problems.1. IntroductionTheory of vector bundles brings a new meaning and adds a delicate geometricflavour to classical spectral problems of linear algebra, relating them to geometricinvariant theory, representation theory, Schubert calculus, quantum cohomology,and various moduli spaces. The talk may be considered as a supplement to that ofHermann Weyl [35] from which I borrow the following quotation"In preparing this lecture, the speaker has assumed that he is expected to talkon a subject in which he had some first-hand experience through his own work. Andglancing back over the years he found that the one topic to which he has returnedagain and again is the problem of eigenvalues and eigenfunctions in its variousramifications. "2. Spectra and representationsLet's start with two classical and apparently independent problems.Hermitian spectral problem. Find all possible spectra X(A + B) of sumof Hermitian operators A, B with given spectraX(A):X(B) : Xi(B) >X 2 (B)>--->Xi(A)>X 2 (A)>--->X n (A),X n (B).*Department of Mathematics, Bilkent University, Bilkent 06533, Ankara, Turkey. E-mail:klyachko@fen.bilkent.edu.tr

600 A. KlyachkoAmong commonly known restrictions on spectra are trace identityY^X i (A + B) = Y^X j (A) + Yl^(B)ì j kand a number of classical inequalities, like that of Weyl [34]A i+i _!(A + B)< Xi(A) + Xj(B). (2.0)Tensor product problem. Find all components \fi C V a ® Vß of tensorproduct of two irreducible representations of GL„ with highest weights (=Youngdiagrams)a : cti > 02 > • • • > a nß: h > b 2 > • • • > b n .In contrast to the spectral problem (2.1) the coefficients of tensor productdecompositionV a ®V ß =^2cl ß V 7 (2.1)can be evaluated algorithmically by Littlewood Richardson rule, which may be describedas follows. Fill z-th row of diagram ß by symbol i. Then cÂ is equal tonumber of ways to produce diagram 7 by adding cells from ß to a in such a waythat the symbolsi) weakly increase in rows,ii) strictly increase in columns,iii) reading all the symbols from right to left, and from top to bottom producesa lattice permutation, i.e. in every initial interval symbol i appears at leastas many times as i + 1.It turns out that these two problems are essentially equivalent and have thesame answer. To give it, let's associate with a subset Id {1,2,... , n} of cardinalityp = \I\ Young diagram 07 in a rectangular of format p x q, p+ q = n, cut out bypolygonalline Y1, connecting SW and NE corners of the rectangular, with i-thunit edge running to the North, for i £ I, and to the East otherwise. One canformally multiply the diagrams by L-R rule7oi(Jj = Y,cfjOK (2.2)where afj := c^'fij are L-R coefficients. Geometrically (2.2) is decomposition ofproduct of two Schubert cycles in cohomology ring of Grassmannian G^ of linearsubspaces of dimension p and codimension q.k

Spectral Problems 601Theorem 2.1. The following conditions are equivalenti) There exist Hermitian operators A, B, C = A + B with spectra X(A), X(B),X(C).ii) InequalityXK(C) 0, (2-3')and its equivalence to (2.3), known as saturation conjecture, was later proved byA. Knutson and T. Tao [22], and in more general quiver context by H. Derksen andJ. Weyman [6].Note that inequalities (UK), although complete, are too numerous to be practicalfor large n. That is why L-R rule, in its different incarnations [22, 11], oftenprovides a more intuitive way to see possible spectra for sum of Hermitian operators.Example 2.3. Let A be Hermitian matrix with integer spectrum A(A) : cti >a2 > ... > a n and B > 0 be a nonnegative matrix of rank one with spectrumX(B) : b > 0 > • • • > 0. Viewing the spectra as Young diagrams, and applying L-Rrule we find out that A(A) ® X(B) is a sum of diagrams 7 : ci > C2 > • • • > c nsatisfying the following intrlacing inequalitiesci > ai > C2 > a2 • • • > c n > a Tl

602 A. KlyachkoBy Theorem 2.3 this implies Cauchy interlacing theorem for spectraXi(A) < Xi(A + B)< Xi-i(A), rkB=l, B>0,known in mechanics as Rayleigh-Courant-Fisher principle: Let mechanical systemS" is obtained from another one S, by imposing a linear constraint, e.g. by fixing apoint of a drum. Then spectrum of S separates spectrum of S".3. Toric bundlesHistorically Theorem 2.3 first appears as a byproduct of theory of toric vectorbundles and sheaves, originated in [15, 17]. See other expositions of the theory in[21, 30], and further applications in [16, 33]. Vector bundles form a cross point atwhich the diverse subjects of this paper meet together.3.1. FiltrationsTo avoid technicalities let's consider the simplest case of projective planeon which diagonal torusP 2 = {(x a : x ß : x r )\x £ C}T={(t a :t ß :t 1 )\t£C*} (3.1)acts by the formula4- , ry ! 4- ry 1 -*- • 4~ -, ryh^ - -4- ry Ï \Orbits of this action are vertices, sides and complement of the coordinate triangle. Inparticular there is unique dense orbit, consisting of points with nonzero coordinates.The objects of our interest are T-equivariant (or toric for short) vector bundles£ over P 2 . This means that £ is endowed with an action T : £ which is linear onfibers and makes the following diagram commutative£ —i-^r £tLet us fix a generic point po £ P 2 not in a coordinate line, and denote byE := £(po)the corresponding generic fiber. There is no action of torus T on the fiber E.Instead the equivariant structure produces some distinguished subspaces in E bythe following construction. Let us choose a generic point p Q £ X a in coordinate

Spectral Problems 603line X a : x a = 0. Since T-orbit of po is dense in P 2 , we can vary t £ T so that tpotends to p a . Then for any vector e £ E = £(p 0 ), we have te £ £(tpo) and can trythe limitlim (te)which either exists or not. Let us denote by E a (0) the set of vectors e £ E forwhich the limit exists:L; a (0) := {e G E\ lim (te) exists}.Evidently E a (0) is a vector subspace of E, independent of po and p Q .An easy modification of the previous construction allows to define for integerm £ Z, the subspaceE a (m):=\e£E\ lim ( —) -(te) exists > .(^ tpo^p a \tß) JRoughly speaking E a (m) consists of vectors e £ E for which te vanishes up to orderm as tpo tends to coordinate line X a . The subspaces E a (m) form a non-increasingexhaustive Z-filtration:E a :••• D E a (m - 1) D B a (m) D E a (m + 1) D • • • ,E a (m) = 0, for ro>0, (3.2)E a (m) = E, for m«0.Applying this construction to other coordinate lines, we get a triple of filtrationsE a , E ß , E 1 in generic fiber E = £(po), associated with toric bundle £.Theorem 3.1. The correspondence£^(E a ,E ß ,E r ) (3.3)establishes an equivalence between category of toric vector bundles on P 2 and categoryof triply filtered vector spaces.We'll use notation £(E a ,E ßfiltrations E a ,E ß ,E~>.3.2. Stability,E~>) for toric bundle corresponding to triplet ofThe previous theorem tells that every property or invariant of a vector bundlehas its counterpart on the level of filtrations. For application to spectral problemsthe notion of stability of a vector bundle £ is crucial. Recall that £ —¥ P 2 is said tobe Alumford-Takemoto stable iffÇi(T)_ ci(£)_ ( ,(àA)rk^ < rk£for every proper subsheaf T C £, and semistable if weak inequalities hold. Hereci(£) = deg det £ is the first Chern class. Donaldson theorem [7] brings a deepgeometrical meaning to this seemingly artificial definition: Every stable bundlecarries unique Hermit-Einstein metric (with Ricci curvature proportional to metric).

604 A. Klyachkois stable iff for every proper sub-Theorem 3.2. Toric bundle £ = £(E a ,E ß ,E~>)space F e E the following inequality holds— l — Y i dim pM(i)

3.4. Components of tensor productSpectral Problems 605In the previous section we explain that stability inequalities (3.5) (44> (UK))via toric Donaldson-Yau theorem solve Hermitian spectral problem. To relate thiswith tensor product part of Theorem 2.1 we need another interpretation of thestability inequalities via Geometric Invariant Theory [26].Recall, that point x £ P(V) is said to be GIT stable with respect to linearaction G : V if G-orbit of the corresponding vector x £ V is closed and its stabilizeris finite. LetX = T a xT ß x T 1be product of three flag varieties of the same types as flags of the filtrationsE a ,E ß ,E~>, and £ a be line bundle on the flag variety T a induced by characteriv a : diagli,x 2 ,... ,x n ) >-¥ x^1 x^2 • • • x\ n ,where a : cti > 02 > • • • > a n is the spectrum of filtration E a , i.e. spectrum of thecorresponding operator H a .Observation 3.4. Vector bundle £ = £(E a ,E ß ,triplet of flagsE 1 ) is stable iff the correspondingi = F a xP 3 xF 1 ÉFxf' î xP = l 4 P(F(X, Cj)is a GIT stable point w.r. to group SY(E) and polarization £ = £ a M £ ß M C 1 .This observation is essentially due to Alumford [25]. Notice that by Borei-Weil-Bott theorem [5] the space of global sections F(J 7a ,£ a ) = V a is just an irreduciblerepresentation of SL(£') with highest weight a. Hence Y(X,£j) = V a ® \fi ® \fi.Every stable vector x can be separated from zero by a G-invariant section of £ N .Therefore triplet of flags in generic position is stable iff [VJV Q ® Yfiß ® VJV 7 ] SL(^ 7^ 0for some N > 1. This proves the last part of Theorem 2.1, modulo the saturationconjecture.4. Unitary operators and parabolic bundlesWe have seen in the previous section that solution of the Hermitian spectralproblem amounts to stability condition for toric bundles. A remarkable ramificationof this idea was discovered by S. Angihotri and Ch. Woodward [2] for unitaryspectralproblem.Let U £ SU(n) be unitary matrix with unitary spectrumLet's normalize exponents A, as followse(U) = (e 2 " Al ,e 2 " A V-- ,e 2 " A ").' Ai > A 2 > • • ^ X n ,+ x n =X(U) := t Ai + A 2 + --- • • + A„ = 0, (4.1)^ Ai — A„ A„ < < 1, 1,and, admitting an abuse of language, call X(U) spectrum of U.

606 A. KlyachkoUnitary spectral problem. Find possible spectra of product X(UV), when spectraof the factors X(U), X(V) are given.To state the result we need in quantum cohomology H*(Gp of GrassmannianGp of linear subspaces of dimension p and codimension r. This is an algebra overpolynomial ring C[q] generated by Schubert cycles 07, J C {1,2,... ,n}, |J| = p,n = p + r with multiplication given by the formulaO-I * O-J = ^2K,dC fj( d ) ( Ì d(T Kwhere structure constants cfj(d) are defined as follows. Let G| < L -¥ W(f\ p C n )Pliicker imbedding andbetp : P 1 -• G r pbe a rational curve of degree d in Gressmanian G| C W(f\ p C n ). One can checkthat Lp depends on dimG| + nd parameters. For fixed point x £ P 1 the conditionifi(x) £ a 1 imposes codim 07 constraints on ip. Hence forthe numberscodim a 1 + codim a j + codim OK = dim G r p + nd(a 1, a j, a K )d = #{

Spectral Problems 607i.e. £ be topologically trivial. Narasimhan-Seshadri theorem [27] claims that everystable bundle carries unique flat metric, and hence defines unitary monodromyrepresentationpe : m(X,x 0 ) -^SY(E), E = £(x 0 ).This gives rise to equivalence_ /stable bundles A _ /irreducible uitary represent9 ' \of degree zero J Stations p : ni ^-SY(E) JThis theorem is an ancestor of the Donaldson-Yau generalization [7] to higher dimensions,and may be seen as a geometric version of Langlands correspondence.In algebraic terms the theorem describes stable bundles in terms of solutionof equation[Ui,V 1 ][U 2 ,V 2 ]---[U g ,V g ] = lin unitary matrices Ui,Vj £ SU(£'). This is not the matrix problem we arecurrently interested in. To modify it let's consider punctured Riemann surfaceX = X\{pi,p 2 ,... ,pi). It has distinguished classes7 Q = (small circle around p Q )in fundamental group m(X), and we can readily define an analogue of RHS of (4.2):M g (\W,\W,--- ,X W ) = {P •• MX) -+ SUCE) | A(p( 7 a)) = \ (a) }, (4.3)where Aâ^ is a given spectrum of monodromy around puncture p Q . C. S. Seshadri[31] manages to find an analogue of more subtle holomorphic LHS of (4.2) in termsof so called parabolic bundles.Parabolic bundle £ on X is actually a bundle on compactification X togetherwith R-filtration in every special fiber E a = £(p a ) with support in an interval oflength < 1. The filtration is a substitution for spectral decomposition of p(-y a ), cf.(4.1). Seshadri also defines (semi)stability of parabolic bundle £ by inequalitiesPar deg T Par deg £^ k ^ - ^ ^ k ^ 'V^ C £ '(44)where the parabolic degree is given by equation Par deg £ = deg£ + ^2 ai X\ a .Aletha-Seshadri theorem [24] claims that every stable parabolic bundle £ on Xcarries unique flat metric with given spectra of monodromies A(7 Q ) = Aâ^. Thisgives a holomorphic interpretation of the space (4.3)M r\(!) \( 2 ) \W\ — fiable parabolic bundles of degree zeroA , .9' ' ' y with given types of the filtrations J 'In the simplest case of projective line with three punctures (4.3) amounts to spaceof solutions of equation UVW = 1 in unitary matrices U, V, W £ SU(n) with given

608 A. Klyachkospectra. By Metha-Seshadry theorem solvability of this equation is equivalent tostability inequalities (4.4). In the case under consideration holomorphic vectorbundle £ on P 1 is trivial, £ = E x P 1 , and hence its subbundle T C £ of rank pis nothing but a rational curve ip : P 1 —¥ G P (E) in Grassmannian. This allows towrite down stability condition (4.4) in terms of quantum cohomology, and eventuallyarrive at Theorem 4.1.5. Further ramificationsThe progress in Hermitian and unitary spectral problems open way for solutionof a variety of others classical, and not so classical, problems. Alost of them,however, have no holomorphic interpretation, and require different methods, borrowedfrom harmonic analysis on homogeneous spaces, symplectic geometry, andgeometric invariant theory.5.1. Multiplicative singular value problemThe problem in question is about possible singular spectrum a (AB) of productof complex matrices with given singular spectra a (A) and a(B). Recall, that singularspectrum of complex matrix A is spectrum of its radial part a (A) := A(v / A*A).For a long time it was observed that every inequality for Hermitian problem hasa multiplicative counterpart for the singular one. For example multiplicative versionof Weyl's inequality A i+i _i(A + B) < A,(A) + Xj(B) is a i+j -i(AB) < afiA)aj(B).The equivalence between these two problems was conjectured by R. C. Thompson,and first proved by the author [20] using harmonic analysis on symmetric spaces.Later on A. Alekseev, E. Alenreken, and Ch. Woodward [1] gave an elegant conceptualsolution based on Drinfeld's Poisson-Lie groups [8]. Here is a precise statementfor classical groups.Theorem 5.1. LetG be one of the classical groups SY(n,C), SO(n,C), orSp(2n,C)and L be the corresponding compact Lie algebra of traceless skew Hermitian complex,real, or quaternionic nxn matrices respectively. Then the following conditionsare equivalent(1) There exist A t £ G with given singular spectra a (Ai) = ai andAiA 2 • • • Apf = 1.(2) There exist Hi £ L with spectra A(ffj) = y^ldogCTJ andHi + H 2 + --- + H N = 0.Note, however, that neither of the above approaches solve the singular problemper se, but reduces it to Hermitian one. Both of them suggest that all three problems

Spectral Problems 609must be treated in one package. Alore precisely, every compact simply connectedgroup G give birth to three symmetric spaces• The group G itself,• Its Lie algebra LQ,• The dual symmetric space Hg = G c /G,of positive, zero, and negative curvature, and to three "spectral problems" concernedwith support of convolution of G orbits in these spaces, see [20] for details. ForG = SU(n) we return to the package of unitary, Hermitian, and singular problems.The first two problems may be effectively treated in framework of vector bundleswith structure group G, as explained in sections 2-4. Many flat, i.e. additive"spectral problem" has been solved by A. Berenstein and R. Sjammar in a verygeneralsetting [4].5.2. Other symmetric spacesAs an example of unresolved problem let's consider symmetric spaces associatedwith different incarnations of Grassmannian• Compact U(p + q)/U(p) x U(q),• Flat Mat(p,q) = coomplex px q matrices,• Hyperbolic U(p,q)/U(p) x U(q).In compact case the corresponding spectral problem is about possible angles betweenthree p-subspaces U, V, W C IP in Hermitian space IP of dimension n = p + q,p < q. The Jordan anglesUV = (ip>i,ip>2,-.. ,tp P ),0

610 A. KlyachkoAgain our experience with the unitary triplet suggests that the exponential mapestablishes a Thompson's type correspondence between O'Shea-Sjamaar inequalitiesfor additive singular problem and that of for hyperbolic angles.5.3. P-adic spectral problemsThere is also a nonarchimedian counterpart of this theory, which deals withclassical Chevalley groups G p = SL(n,Q p ), SO(n,Q p ), or Sp(2n,Q p ) over p-adicfield Qp and their maximal compact subgroups Kp = SL(n,Z p ), SO(n,Z p ), orSp(2n, Z p ) respectively. Double coset Kp#Kp may be treated as a complete invariantof lattice L = gL 0 , L 0 = Z®" with respect to Kp. We call lattice L = gL 0unimodular, orthogonal or symplectic if respectively g £ SL(n,Q p ), g £ SO(n,Q p )or g £ Sp(2n,Q p ).It is commonly known that in the unimodular case there exists a basis e, ofL 0 such that ê, = p Qi e, form a basis of L for some a, £ Z. We define index (L : L 0 )by(L:L 0 ) = (p a \p a V..,p a »), ai>a 2 > •••> a„_3 > ... > az- n > Gi-n> an d a~i = —a«.Similarly, for symplectic lattice L we can choose symplectic basis e,, fj of L 0such that è, = p Qi e, and fj = pâj fj form a basis of L. In this case we have(L:L 0 ) = (p a »,p a »- 1 ,...,p ai ,p- Ql ,...,p- a »- 1 ,p- a »), (5.3)with a n > a n -i >,... , > cti > 0.Notice that the spectra (5.1)-(5.3) have the same symmetry, as singular spectruma (A) of a matrix A e G in the corresponding classical complex group.Theorem 5.2. The following conditions are equivalent(1) There exists a sequence of (unimodular, orthogonal, symplectic) latticesLQ, LI, ... , Ljv-i, LM = LQof given indices ai = (L t : L,_i).(2) The indices ai satisfy the equivalent conditions of Theorem 5.1 for the correspondingcomplex group G.We'll give proof elsewhere. The theorem is known for the unimodular lattices,see [10].

Spectral Problems 6115.4. Final remarksIn the talk I try to trace the flaw of ideas from the theory of vector bundlesto spectral problems. It seems C. Simpson [32] was the first to note that vectorbundles technic has nontrivial implications in linear algebra. He proved that productCiC 2 --- CM of conjugacy classes G, C SL(n, C) is dense in SL(n, C) iffdim Gi + dimG2 + • • • + dimGjv > (n + l)(n — 2),ri +r 2 + h r N > n,(5.4)where r t is maximal codimension of root space of a matrix A t £ Ci. This problemwas suggested by P. Deligne, who noted that under conditiondim Gi + dim G2 + h dim CN = 2n 2 — 2an irreducible solution of equation AiA 2 • • • AM = 1, a» £ Ci is unique up to conjugacy,see book of N. Katz [14] on this rigidity phenomenon.I think that inverse applications to moduli spaces of vector bundles are sillahead. One may consider polygon spaces [18, 12] as a toy example of this feedback,corresponding to toric 2-bundles. A similar space of spherical polygons in S 3 withgiven sides is a model for moduli space of flat connections in punctured Riemannsphere. Its description is a challenge problem.There are many interesting results, e.g. infinite dimensional spectral problems,which fall out of this survey. I refer to Fulton's paper [10] for missing details.References[1] A. Alekseev, E. Aleinrenken, & C. Woodward, Linearization of Poisson actionsand singular values of matrix product, Ann. Inst. Fourier (Grenoble), 51(2001), no. 6, 1691-1717.[2] S. Angihotri & C. Woodward, Eigenvalues of products of unitary matricesand quantum Schubert calculus, Math. Res. Letters, 5 (1998), 817-836.[3] P. Belkale, Local systems on P 1 — S for S a finite set, Compositio Math., 129(2001), no. 1, 67-86.[4] A. Berenstein & R. Sjamaar, Coadjoint orbits, moment polytopes, and theHilbert-Mumford criterion, J. Amer. Math. Soc, 13 (2000), no. 2, 433-466.[5] R. Bott, Homogeneous vector bundles, Ann. of Math., 66 (1957), 203-248.[6] H. Derksen & J. Weyman, Semi-invariants of quivers and saturation forLittlewood-Richardson theorem, J. Amer. Math. Soc, 13 (2000), no. 3,467-479.[7] S. K. Donaldson, Infinite determinants, stable bundles and curvature, DukeMath. J., 54 (1987), 231-247.[8] V. G. Drinfeld, Quantum groups, Proceedings of the International Congressof Mathematicians, vol. 1,2 (Berkeley, 1986), Amer. Alath. Soc, Providence,RI, 1987, 798-820.

612 A. Klyachko[9] G. Faltings, Mumford-Stabilität in der algebraischen Geometrie, Proceedingsof the International Congress of Mathématiciens, vol. 1,2, (Zürich, 1994),Birkhäuser, Basel, 1995, 648-655.[10] W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus,Bull. Amer. Math. Soc, 37 (2000), no. 3, 209-249.[11] O. Gleizer & A. Postnikov, Littlewood-Richardson coefficients via Yang-Baxterequation, Internat. Math. Res. Notices (2000), no. 14, 741-774.[12] J.-C. Hausmann & A. Knutson, The cohomology ring of polygon spaces, Ann.Inst. Fourier (Grenoble), 48 (1998), no. 1, 281-321.[13] A. Horn, Eigenvalues of sum of Hemitian matrices, Pacific J. Math., 12(1962), 225-241.[14] N. Al. Katz, Rigid local systems, Princeton University Press, Princeton, 1996.[15] A. A. Klyachko, Equivariant bundles on toric varieties, Izv. Akad. NaukSSSR Ser. Mat, 53 (1989), no. 5, 1001-1039 (Russian); Math. USSR-Izv.,35 (1990), no. 2, 63-64.[16] A. A. Klyachko, Aloduli of vector bundles and class numbers, Functional, iPriozhen., 25 (1991), 81-83 (Russian); Funct. Anal. Appi., 25 (1991), no. 1,67-69.[17] A. A. Klyachko, Vector bundles and torsion free sheaves on the projectiveplane, Preprint Max-Planck-Institute fur Mathematik A1PI/91-59, (1991).[18] A. A. Klyachko, Spatial polygons and stable configurations of points in theprojective line, Algebraic geometry and its applications (Yaroslavl, 1992),Vieweg, Braunschweig, 1994, 67-84.[19] A. A. Kyachko, Stable bundles, repesentation theory and Hermitian operators,Selecta Mathematica, 4 (1998), 419-445.[20] A. A. Klyachko, Random walks on symmetric spaces and and inequalities formatrix spectra, Linear Algebra Appi., 319 (2000), no. 2-3, 37-59.[21] A. Knutson & E. Sharp, Sheaves on toric varieties for physics, Adv. Theor.Math. Phys., 2 (1998), no. 4, 873-961.[22] A. Knutson & T. Tao, The honeycomb model of GL(n,C) tensor products.I. Proof of the saturation conjecture, J. Amer. Math. Soc, 12 (1999), no. 2,1055-1090.[23] A. Knutson, T. Tao & Ch. Woodward, The honecomb model for GL(n,C)tensor products II: Facets of Littlewood-Richardson cone, Preprint (2001).[24] V. B. Aletha & C. S. Seshadri, Aloduli of vector bundles on curves withparabolic sructure, Math. Ann., 258 (1980), 205-239.[25] D. Alumford, Projective invariants of projective structures, Proc. Int. Congressof Alath. Sockholm, 1963, Almquist & Wiksells, Uppsala, 1963, 526-530.[26] D. Alumford, J. Fogarty, & F. Kirwan, Geometric invariant theory, Springer,Berlin, 1994.[27] Al. S. Narasimhan & C. S. Seshadri, Stable and unitary vector bundles on acompact Riemann surface, Ann. Math., 82 (1965), 540-567.

Spectral Problems 613[28] Yu. Neretin, On Jordan angles and triangle inequality in Grassmannian,Geom,. Dedicata, 86 (2001), no. 1-3, 81-92.[29] L. O'Shea & R. Sjamaar, Aloments maps and Riemannian symmetric pairs,Math. Ann., 317 (2000), no. 3, 415-457.[30] M. Perling, Graded rings and equivariant sheaves on toric varieties, PreprintUniv. Kaiserslauten, (2001).[31] C. S. Seshadri, Aloduli of vectir bundles on curves with parabolic structures,Bull. Amer. Math. Soc, (1977), 124-126.[32] C. T. Sympson, Product of Alatrices, Differential geometry, global analysis,and topology, Canadian Alath. Soc. Conf. Proc, vol. 12, AA1S, ProvidenceRI, 1992, 157-185.[33] C. Vafa & E. Witten, A strong coupling test of S-duality, Nuclear Phys. B,431 (1994), no. 1-2, 3-77.[34] H. Weyl, Das asymptotischer Verteilungsgesetz der Eigenwerte lineare partiallerDifferentialgleichungen, Math. Ann., 71 (1912), 441-479.[35] H. Weyl, Ramifications, old and new, of the eigenvalue problem, Bull. Amer.Math. Soc, 56 (1950), 115-139.

ICAl 2002 • Vol. II • 615-627Branching Problems ofUnitary RepresentationsToshiyuki Kobayashi*AbstractThe irreducible decomposition of a unitary representation often contains continuousspectrum when restricted to a non-compact subgroup. The author singles outa nice class of branching problems where each irreducible summand occurs discretelywith finite multiplicity (admissible restrictions). Basic theory and new perspectivesof admissible restrictions are presented from both analytic and algebraic view points.We also discuss some applications of admissible restrictions to modular varieties andL p -harmonic analysis.2000 Mathematics Subject Classification: 22E46, 43A85, 11F67, 53C50, 53D20.Keywords and Phrases: Unitary representation, Branching law, Reductive Liegroup.1. IntroductionLet n be an irreducible unitary representation of a group G. A branchinglaw is the irreducible decomposition of n when restricted to a subgroup G':r®TT\QI ~ / m n (T)T dp(r) (a direct integral). (1.1)JCÎ'Such a decomposition is unique, for example, if G' is a reductive Lie group, andthe multiplicity m* : G' —¥ N U {oo} makes sense as a measurable function on theunitary dual G'.Special cases of branching problems include (or reduce to) the fallowings:Clebsch-Gordan coefficients, Littlewood-Richardson rules, decomposition of tensorproduct representations, character formulas, Blattner formulas, Plancherel theoremsfor homogeneous spaces, description of breaking symmetries in quantum mechanics,theta-lifting in automorphic forms, etc. The restriction of unitary representationsserves also as a method to study discontinuous groups for non-Riemannian homogeneousspaces (e.g. [Alg, Oh]).*RIMS, Kyoto University, Kyoto 606-8502, Japan. E-mail: toshi@kurims.kyoto-u.ac.jp

616 T. KobayashiOur interest is in the branching problems for (non-compact) reductive Liegroups G D G'. In this generality, there is no known algorithm to find branchinglaws. Even worse, branching laws usually contain both discrete and continuous spectrumwith possibly infinite multiplicities (the multiplicity is infinite, for example,in the decomposition of the tensor product of two principal series representationsof SL(n,C) for n > 3, [Ge-Gr]).The author introduced the notion of admissible restrictions and infinitesimaldiscrete decomposability in [K05] and [K09], respectively, seeking fora good framework of branching problems, in which we could expect especially asimple and detailed study of branching laws, which in turn might become powerfulmethods in other fields as well where restrictions of representations naturally arise.The criterion in Theorem B indicates that there is a fairly rich examples ofadmissible restrictions; some are known and the others are new. In this framework, anumber of explicit branching laws have been newly found (e.g. [D-Vs, Gr-Wi^, Hu-P-S, Koi^^g, Ko-01,2, Li 2 , Loi^, X]). The point here is that branching problemsbecome accessible by algebraic techniques if there is no continuous spectrum.The first half of this article surveys briefly a general theory of admissiblerestrictions both from analytic and algebraic view points (§2, §3). For the simplicityof exposition, we restrict ourselves to unitary representations, although a part of thetheory can be generalized to non-unitary representations. The second half discussessome applications of discretely decomposable restrictions. The topics range fromrepresentation theory itself (§4) to some other fields such as L p -analysis on nonsymmetrichomogeneous spaces (§5) and topology of modular varieties (§6).2. Admissible restrictions to subgroupsLet G' be a subgroup of G, and n £ G. In light of (1.1), we introduce:Definition 2.1. We say the restriction 7r|c is G'-admissible if it decomposesdiscretely and the multiplicity ro^r) is finite for any r £ G'.One can easily prove the following assertion:Theorem A ([K05, Theorem 1.2]). Let G D G' D G" be a chain of groups, andn £ G. If the restriction n\a" is G"-admissible, then n\a' is G'-admissible.Throughout this article, we shall treat the setting as below:Definition 2.2. We say (G,G r ) is a pair of reductive Lie groups if1) G is a real reductive linear Lie group or its finite cover, and2) G' is a closed subgroup, and is reductive in G.Then, we shall fix maximal compact subgroups K D K' of G D G', respectively.A typical example is a reductive symmetric pai (G, G'), by which we meanthat G is as above and that G' is an open subgroup of the set G" of the fixed points ofan involutive automorphism a of G. For example, (G,G r ) = (GL(n,C),GL(n,Rj),(SL(n,R), SO(p, n — pj) are the cases.

Branching Problems of Unitary Representations 617Let (G,G r ) be a pair of reductive Lie groups. Here are previously knownexamples of admissible restrictions:Example 2.3. The restriction 7r|c is G'-admissible in the following cases:1) (Harish-Chandra's admissibility theorem) n £ G is arbitrary and G' = K.2) (Howe, [Hoi]) n is the Segal-Shale-Weil representation of the metaplectic groupG, and its subgroup G' = G'iG 2 forms a dual pair with Gi compact.In these examples, either the subgroup G' or the representation n is veryspecial,namely, G' is compact or n has a highest weight. Surprisingly, without suchassumptions, it can happen that the restriction n\a' is G'-admissible. The followingcriterion asserts that the "balance" of G' and n is crucial to the G'-admissibility.Theorem B (criterion for admissible restrictions, [K07]). Let G D G' be apair of reductive Lie groups, and n £ G. IfCone(G')nASif(7r) = {0}, (2.1)then the restriction n\K> is K'-admissible. In particular, the restriction 7r|c isG'-admissible, namely, decomposes discretely with finite multiplicity.A main tool of the proof of Theorem B is the microlocal study of charactersby using the singularity spectrum of hyperfunctions. The idea goes back to Atiyah,Howe, Kashiwara and Vergne [A, H02, Ks-Vr] in the late '70s. The novelty ofTheorem B is to establish a framework of admissible restrictions with a numberof new examples of interest, which rely on a deeper understanding of the unitarydualdeveloped largely in the '80s (see [Kn-Vo] and references therein).Let us briefly explain the notation used in Theorem B. We write i' 0 C 60 forthe Lie algebras of K' c K, respectively. Take a Cartan subalgebra t 0 of 6o- Then,ASif(7r) is the asymptotic if-support of n ([Ks-Vr]), and Cone(G') is defined asCone(G') := V^ì(t* 0 n Ad*(if)(e^)). (2.2)By definition, both ASK(TT) and Cone(G') are closed cones in ^/^Ttg.Example 2.4. If G' = K, then the assumption (2.1) is automatically fulfilledbecause Cone(G') = {0}. The conclusion of Theorem B in this special case isnothing but Harish-Chandra's admissibility theorem (Example 2.3 (1)).To apply Theorem B for non-compact G', we rewrite the assumption (2.1)more explicitly in specific settings. On the part Cone(G'), we mention:Example 2.5. Cone(G') is a linear subspace ^/^T(tg) -

618 T. Kobayashito Kirillov-Kostant. This representation is a unitarization of a Zuckerman-Voganmodule A q (A) after some p-shift, and can be realized in the Dolbeault cohomologygroup on Ox by the results of Schmid and Wong. (Here, we adopt the samepolarization and normalization as in a survey [K04, §2], for the geometric quantizationOx =$• n\.) We note that nx £ G for "most" A. Let g = 6 + p be thecomplexification of a Cartan decomposition of the Lie algebra go of G. We setA+(p) — {a £ A(p,t) : (A, a) > 0}, for A G V^ïl* 0 -The original proof (see [K05]) of the next theorem was based on an algebraic methodwithout using microlocal analysis. Theorem B gives a simple and alternative proof.Theorem C ([K05]). Letnx £ G be attached to an elliptic coadjoint orbit Ox. Ifthen the restriction nx\a' is G'-admissible.R-span A+(p) n Cone(G') = {0}, (2.2)Let us illustrate Theorem C in Examples 2.6 and 2.7 for non-compact G'.For this, we note that a maximal compact subgroup K is sometimes of the formKi x K 2 (locally). This is the case if G/K is a Hermitian symmetric space (e.g.G = Sp(n,R),SO* (2n), SU(p, qj). It is also the case if G = 0(p,q), Sp(p,q), etc.Example 2.6 (K ~ Ki x K 2 ). Suppose K is (locally) isomorphic to the directproduct group Ki x K 2 . Then, tie restriction nx\a' is G'-admissible if A| tn e 2 = 0and G' D Ki. So does the restriction n\a' if n is any subquotient of a coherentcontinuation of nx- This case was a prototype of G'-admissible restrictions n\a'(where G' is non-compact and n is a non-highest weight module) proved in 1989 bythe author ([K01; K02, Proposition 4.1.3]), and was later generalized to Theorems Band C. Special cases include:(1) Ki ~ T, then n is a unitary highest weight module. The admissibility ofthe restrictions n\a' in this case had been already known in '70s (see Alartens[Alt], Jakobson-Vergne [J-Vr]).(2) Ki ~ SU(2), then nx is a quaternionic discrete series. Admissible restrictionsn\(}i in this case are especially studied by Gross and Wallach [Gr-Wi] in '90s.(3) Ki ~ 0(q),U(q),Sp(q). Explicit branching laws of the restriction nx\a' forsingular A are given in [K03, Part I] with respect to the vertical inclusions ofthe diagram below (see also [Koi,Kos] for those to horizontal inclusions).0(4p,4q) D U(2p,2q) D Sp(p,q)U U U0(4r) x 0(4p - Ar, Aq) D U(2r) x U(2p - 2r, 2q) D Sp(r) x Sp(p - r, q)

Branching Problems of Unitary Representations 619Example 2.7 (conformai group). There are 18 series of irreducible unitary representationsof G := U(2, 2) with regular integral infinitesimal characters. Amongthem, 12 series (about "67% " !) are G'-admissible when restricted to G' := Sp(l, 1).The assumption in Theorem B is in fact necessary. By using the technique ofsymplectic geometry, the author proved the converse statement of Theorem B:Theorem D ([K013]). Let G D G' be a pair of reductive Lie groups, and n £ G.If the restriction n\K> is K'-admissible, then Cone(G') n ASK(TT) = {0}.3. Infinitesimal discrete decomposabilityThe definition of admissible restrictions (Definition 2.1) is "analytic", namely,based on the direct integral decomposition (1.1) of unitary representations. Next,we consider discrete decomposable restrictions by a purely algebraic approach.Definition 3.1 ([K09, Definition 1.1]). Let g be a Lie algebra. We say a g-moduleX is discretely decomposable if there is an increasing sequence of g-submodulesof finite length:00X = [J X m , X 0 c Xi c X 2 C • • • . (3.1)ro=0We note that dim X m = 00 in most cases below.Next, consider the restriction of group representations.Definition 3.2. Let G D G' be a pair of reductive Lie groups, and n £ G. Wesay that the restriction n\a> is inflnitesimally discretely decomposable if theunderlying (g,K)-module nK is discretely decomposable as a g'-module.The terminology "discretely decomposable" is named after the following fact:Theorem E ([K09]). Let (G,G r ) be a pair of reductive Lie groups, and nK theunderlying (g,K)-module ofn£G. Then (i) and (ii) are equivalent:i) The restriction 7r|c is inflnitesimally discretely decomposable.ii) The (g,K)-module nK has a discrete branching law in the sense that nKis isomorphic to an algebraic direct sum of irreducible (g',K r )-modules.Aloreover, the following theorem holds:Theorem F (infinitesimal =^> Hilbert space decomposition; [Kon]). Letn £ G. If the restriction 7r|c is inflnitesimally discretely decomposable, then therestriction 7r|c decomposes without continuous spectrum:TT\QI ~ yj m n (T)T (a discrete direct sum of Hilbert spaces). (3.2)TGG 7At this stage, the multiplicity ro^r) := dim Home (r, 7T|G') can be infinite.However, for a reductive symmetric pair (G,G r ), it is likely that the multiplicityof discrete spectrum is finite under the following assumptions, respectively.(3.3) n is a discrete series representation for G.(3.4) The restriction 7r|c is inflnitesimally discretely decomposable.

620 T. KobayashiConjecture 3.3 (Wallach, [X]). ro^r) < oo for any T £ G' if (3.3) holds.Conjecture 3.4 ([Kon, Conjecture C]). ro^r) < oo for any T £ G' if (3.4) holds.We note that Conjecture 3.4 for compact G' corresponds to Harish-Chandra'sadmissibility theorem. A first affirmative result for general non-compact G' wasgiven in [K09], which asserts that Conjecture 3.4 holds if n is attached to an ellipticcoadjoint orbit. A special case of this assertion is:Theorem G ([Ko 9 ]). m n (r) < 00 for any r £&if both (3.3) and (3.fi) hold.In particular, Wallach's Conjecture 3.3 holds in the discretely decomposablecase. We note that an analogous finite-multiplicity statement fails if continuousspectrum occurs in the restriction n\a' for a reductive symmetric pair (G,G r ):Counter Example 3.5 ([Kon]). ro^r) can be 00 if neither (3.3) nor (3.4) holds.Recently, I was informed by Huang and Vogan that they proved Conjecture 3.4for any n [Hu-Vo].A key step of Theorem G is to deduce the if'-admissibility of the restrictionn\K> from the discreteness assumption (3.4), for which we employ Theorem H below.Let us explain it briefly. We write V B (îT) for the associated variety of theunderlying (g, K)-module of n (see [Vo]), which is an algebraic variety contained inthe nilpotent cone of g*. Yet pr B _j, B / : g* —¥ (g')* be the projection correspondingto g' C g. Here is a necessary condition for infinitesimal discrete decomposability:Theorem H (criterion for discrete decomposability [K09, Corollary 3.4]). Letn £ G. If the restriction 7r|c is inflnitesimally discretely decomposable, thenpr B _j, B /(V s (nj) is contained in the nilpotent cone of (g')*•We end this section with a useful information on irreducible summands.Theorem I (size of irreducible summands, [K09]). Let n £ G. If the restrictionn\(}i is inflnitesimally discretely decomposable, then any irreducible summandhas the same associated variety, especially, the same G elf and-Kirillov dimension.Here is a special case of Theorem I:Example 3.6 (highest weight modules, [N-Oc-T]). Let G be the metaplectic group,and G' = G'iG' 2 is a dual pair with Gi compact. Let 9(a) be an irreducible unitaryhighestweight module of G 2 obtained as the theta-correspondence of a £ G[.Then the associated variety of 9(a) does not depend on a, but only on G\.An analogous statement to Theorem I fails if there exists continuous spectrumin the branching law 7r|c (see [Kon] for counter examples).4. Applications to representation theorySo far, we have explained basic theory of discretely decomposable restrictionsof unitary representations for reductive Lie groups G D G'. Now, we ask what

Branching Problems of Unitary Representations 621discrete decomposability can do for representation theory. Let us clarify advantagesof admissible restrictions, from which the following applications (and some more)have been brought out and seem to be promising furthermore.1) Study of G' as irreducible summands of n\a'-2) Study of G by means of the restrictions to subgroups G'.3) Branching laws of their own right.4.1. From the view point of the study of G' (smaller group), one of advantages ofadmissible restrictions is that each irreducible summand of the branching law n\a'gives an explicit construction of an element of G'.Historically, an early success of this idea (in '70s and '80s) was the constructionof irreducible highest weight modules (Howe, Kashiwara-Vergne, Adams, •••). Alarge part of these modules can be constructed as irreducible summands of discretebranching laws of the Weil representation (see Examples 2.3 (2) and 3.6).This idea works also for non-highest weight modules. As one can observe fromthe criterion in Theorem B, the restriction n\a' tends to be discretely decomposable,if ASK(T) is "small". In particular, if n is a minimal representation in the sensethat its annihilator is the Joseph ideal, then a result of Vogan implies that ASK (TT) isone dimensional. Thus, there is a good possibility of finding subgroups G' such thatn\(}i is G'-admissible. This idea was used to construct "small" representations ofsubgroups G' by Gross-Wallach [Gr-Wi]. In the same line, discretely decomposablebranching laws for non-compact G' are used also in the theory of automorphic formsfor exceptional groups by J-S. Li [Li 2 ].4.2. From the view point of the study of G (larger group), one of advantages ofadmissible restrictions is to give a clue to a detailed study of representations of Gby means of discrete branching laws.Needless to say, an early success in this direction is the theory of (g,K)-modules (Lepowsky, Harish-Chandra, •••). The theory relies heavily on Harish-Chandra's admissibility theorem (Example 2.3 (1)) on the restriction of n to K.Instead of a maximal compact subgroup K, this idea applied to a non-compactsubgroup G' still works, especially in the study of "small" representations of G. Inparticular, this approach makes sense if the if-type structure is complicated but theG'-type structure is less complicated. Successful examples in this direction include:1) To determine an explicit condition on A such that a Zuckerman-Vogan moduleA q (A) is non-zero, where we concern with the parameter A outside thegood range. In the setting of Example 2.6 (3), the author found in [K02] acombinatorial formula on Ki -types of A q (A) and determined explicitly whenA q (A) 7^ 0. The point here is that the computation of if-types of A q (A) istoo complicated to carry out because a lot of cancellation occurs in the generalizedBlattner formula, while ifi-type formula (or G'-type formula for somenon-compact subgroup G') behaves much simpler in this case.2) To study a fine structure of standard representations. For example, Leeand Loke [Le-Lo] determined the Jordan-Holder series and the unitarizability

622 T. Kobayashiof subquotients of certain degenerate non-unitary principal series representationsn, by using G'-admissible restrictions for some non-compact reductivesubgroup G'. Their method works successfully even in the case where if-typemultiplicity of n is not one.4.3. From the view point of finding explicit branching law, an advantage of admissiblerestrictions is that one can employ algebraic techniques because of the lack ofcontinuous spectrum. A number of explicit branching laws are newly found (e.g.[D-Vs, Gr-Wi^, Hu-P-S, Koi^^s, Ko-01,2, Li 2 , Loi^, X]) in the context of admissiblerestrictions to non-compact reductive subgroups. A mysterious feature isthat "different series" of irreducible representations may appear in discretely decomopsablebranching laws (see [K05, p. 184] for a precise meaning), although all ofthem have the same Gelfand-Kirillov dimensions (Theorem I).5. New discrete series for homogeneous spacesLet G D H be a pair of reductive Lie groups. Then, there is a G-invariantBorei measure on the homogeneous space G/H, and one can define naturally aunitary representation of G on the Hilbert space L 2 (G/H).Definition 5.1. We say n is a discrete series representation for G/H,is realized as a subrepresentation of L 2 (G/H).ifn£GA discrete series representation corresponds to a discrete spectrum in thePlancherel formula for the homogeneous space G/H. One of basic problems innon-commutative harmonic analysis is:Problem 5.2. 1) Find a condition on the pair of groups (G,H) such that thereexists a discrete series representation for the homogeneous space G/H.2) If exist, construct discrete series representations.Even the first question has not found a final answer in the generality that(G, H) is a pair of reductive Lie groups. Here are some known cases:Example 5.3. Flensted-Jensen, Alatsuki and Oshima proved in '80s that discreteseries representations for a reductive symmetric space G/H exist if and only ifrankG/H = rankK/(HnK). (5.1)This is a generalization of Harish-Chandra's condition, rank G = rank if, for agroup manifold G x G/diag(G) ~ G ([FJ, Alk-Os]).Our strategy to attack Problem 5.2 for more general (non-symmetric) homogeneousspaces G/H consists of two steps:1) To embed G/H into a larger homogeneous space G/H, on which harmonicanalysis is well-understood (e.g. symmetric spaces).2) To take functions belonging to a discrete series representation % (^y L 2 (G/Hj),and to restrict them with respect to a submanifold G/H (^y G/H).

Branching Problems of Unitary Representations 623If G/H is "generic", namely, a principal orbit in G/H in the sense of Richardson,then it is readily seen that discrete spectrum of the branching law n\a gives adiscrete series for G/H ([Koio, §8]; see also [Hu, Ko^s, Lii] for concrete examples).However, some other interesting homogeneous spaces G/H occur as nonprincipalorbits on G/H, where the above strategy does not work in general. Aremedy for this is to impose the admissibility of the restriction of n, whichjustifies the restriction of L p -functions to submanifolds, and then gives rise to manynon-symmetric homogeneous spaces that admit discrete series representations. Forexample, let us consider the case where G = G T and H = G" for commuting involutiveautomorphisms r and a of G such that G/H satisfies (5.1). Then by usingTheorem C and an asymptotic estimate of invariant measures [Kog], we have:Theorem J (discrete series for non-symmetric spaces, [Koio]). Assume thatthere is tv £ W„ such thatR+ -spanA+(p) frjtt ,n y/-l($)~ T = {0}. (5.2)Then there exist infinitely many discrete series representations for any homogeneousspace of G that goes through xH £ G/H for any x £ K.We refer to [Koio, Theorem 5.1] for definitions of a finite group W„ andA + (p)rr,w- The point here is that the condition (5.2) can be easily checked.For instance, if G ~ Sp(2n,R) ~ G/H (a group manifold), then Theorem Jimplies that there exist discrete series on all homogeneous spaces of the form:G/H = Sp(2n,R)/(Sp(n Q ,C) x GL(m,C) x • • • x GL(n k ,Gj), (5^n 4 = n).The choice of x in Theorem J corresponds to the partition (no, ni,..., nu)- We notethat the above G/H is a symmetric space if and only if m = n 2 = ••• = nk = 0.The restriction of unitary representations gives new methods even for symmetricspaces where harmonic analysis has a long history of research. Let us statetwo results that are proved by the theory of discretely decomposable restrictions.Theorem K (holomorphic discrete series for symmetric spaces). SupposeG/H is a non-compact irreducible symmetric space. Then (i) and (ii) are equivalent:i) There exist unitary highest weight representations of G that can be realized assubrepresentations of L 2 (G/H).ii) G/K is Hermitian symmetric and H/(H n K) is its totally real submanifold.This theorem in the group manifold case is a restatement of Harish-Chandra'swell-known result. The implication (ii) =$• (i) was previously obtained by a differentgeometric approach ('Olafsson-Orsted [01-0]). Our proof uses a general theory ofdiscretely decomposable restrictions, especially, Theorems B, H and J.

624 T. KobayashiTheorem L (exclusive law of discrete spectrum for restriction and induction).Let G/G' be a non-compact irreducible symmetric space, and n £ G. Thenboth (1) and (2) cannot occur simultaneously.1) The restriction n\a' is inflnitesimally discretely decomposable.2) n is a discrete series representation for the homogeneous space G/G'.We illustrate Theorems K and L by G = SL(2,R). The examples below arewell-known results on harmonic analysis, however, the point is that they can beproved by a simple idea coming from restrictions of unitary representations.Example 5.4. 1) Holomorphic discrete series exist for G/H = SL(2,R)/SO(l, 1)(a hyperboloid of one sheet). This is explained by Theorem K because the geodesicH/(H n K) is obviously totally real in the Poincaré disk G/K = SL(2,R)/SO(2).2) There is no discrete series for the Poincaré disk G/K = SL(2,R)/SO(2). Thisfact is explained by Theorem L because any representation of G is obviously discretelydecomposable when restricted to a compact K.6. Modular varieties, vanishing theoremRetain the setting as in Definition 2.2. Let F' c F be cocompact torsion-freediscrete subgroups of G' C G, respectively. For simplicity, let G' be a semisimpleLie group without compact factors. Then, both of the double cosets X :=Y\G/K and Y := Y'\G'/K' are compact, orientable, locally Riemannian symmetricspaces. Then, the inclusion G' < L -¥ G induces a natural map i : Y —¥ X. Theimage i(Y) defines a totally geodesic submanifold in X. Consider the induced homomorphismof the homology groups of degree m := dim F,i.:H m (Y;Z)->H m (X;Z).The modular symbol is defined to be the image i*[Y] £ H m (X;Z) of the fundamentalclass [Y] £ H m (Y;Z). Though its definition is simple, the understanding ofmodular symbols is highly non-trivial.Let us first recall some results of Matsushima-Murakami and Borei-Wallachon the de Rham cohomology group H*(X; C) summarized as:H*(X;C) = ($H*(X) 7r , H*(Xfi :=Eom G (n,L 2 (Y\Gj)®H*(g,K;n K ). (6.1)TreeThe above result describes the topology of a single X by means of representationtheory. For the topology of the pair (Y,X), we need restrictions of representations:Theorem M (vanishing theorem for modular symbols, [Ko-Od]). IfASK(TT) n Cone(G') = {0}, n # 1,

Branching Problems of Unitary Representations 625then the modular symbol i*[Y] is annihilated by the n-component H m (X) 7rperfect paring H m (X;C) x H m (X;C) -• C.in theTheorem M determines, for example, the middle Hodge components of totallyrealmodular symbols of compact Clifford-Klein forms of type IV domains.The discreteness of irreducible decompositions plays a crucial role both inMatsushima-Murakami's formula (6.1) and in a vanishing theorem for modular varieties(Theorem Al). In the former L 2 (F\G) is G-admissible (Gelfand and Piateski-Shapiro), while the restriction 7r|c is G'-admissible (cf. Theorem B) in the latter.References[A] M. F. Atiyah, The Harish-Chandra character, London Alath. Soc. LectureNote Series 34 (1979), 176-181.[D-Vs] M. Duflo and J. Vargas, in preparation.[FJ] M. Flensted-Jensen, Discrete series for semisimple symmetric spaces,Annals of Alath. Ill (1980), 253-311.[Ge-Gv] I. M. Gelfand and M. I. Graev, Geometry of homogeneous spaces, representationsof groups in homogeneous spaces, and related questions ofintegral geometry, Transi. II. Ser., A. M. S. 37 (1964), 351-429.[Gr-Wi] B. Gross and N. Wallach, A distinguished family of unitary representationsfor the exceptional groups of real rank = A, Progress in Alath. 123(1994), Birkhäuser, 289^304.[Gr-W2] B. Gross and N. Wallach, Restriction of small discrete series representationsto symmetric subgroups, Proc. Sympos. Pure Alath. 68 (2000),A.ALS., 255^272.[Hoi] R. Howe, 9-series and invariant theory, Proc. Sympos. Pure Alath. 33(1979), A.M.S., 275^285.[H02] R. Howe, Wave front sets of representations of Lie groups, Automorphicforms, representation theory, and arithmetic (1981), Tata, 117-140.[Hu] J-S. Huang, Harmonic analysis on compact polar homogeneous spaces,Pacific J. Alath. 175 (1996), 553^569.[Hu-P-S] J-S. Huang, P. Pandzic, and G. Savin, New dual pair correspondences,Duke Alath. 82 (1996), 447-171.[Hu-Vo] J-S. Huang and D. Vogan, personal communications (2001).[J-Vr] H. P. Jakobson and Al. Vergne, Restrictions and expansions of holomorphicrepresentations, J. Funct. Anal. 34 (1979), 29^53.[Ks-Vr] Al. Kashiwara and Al. Vergne, K-types and singular spectrum, Lect.Notes in Alath., vol. 728, Springer, 1979, 177^200.[Kn-Vo] A. Knapp and D. Vogan, Jr., Cohomological Induction and Unitary Representations,Princeton U.P., 1995.[K01] T. Kobayashi, Unitary representations realized in L 2 -sections of vectorbundles over semi-simple symmetric spaces, Proc. of the 27-28th Symp. ofFunct. Anal, and Real Anal. (1989), Alath. Soc. Japan, 39^54. (Japanese)

626 T. Kobayashi[K02] T. Kobayashi, Singular Unitary Representations and Discrete Series forIndefinite Stiefel Manifolds U(p, q; V)/U(p — m, q; F), Alemoirs of A.M.S.,vol. 462, 1992.[K03] T. Kobayashi, The Restriction of A q (A) to reductive subgroups, Part I,Proc. Japan Acad. 69 (1993), 262-267; Part II, ibid. 71 1995, 24-26.[K04] T. Kobayashi, Harmonic analysis on homogeneous manifolds of reductivetype and unitary representation theory, Transi., Series II, Selected Paperson Harmonic Analysis, Groups, and Invariants 183 (1998), A.ALS., 1-31.[K05] T. Kobayashi, Discrete decomposability of the restriction of A q (A) withrespect to reductive subgroups and its applications, Invent. Alath. 117(1994), 181-205.[Kog] T. Kobayashi, Invariant measures on homogeneous manifolds of reductivetype, J. reine und angew. Alath. 490 (1997), 37-53.[K07] T. Kobayashi, Discrete decomposability of the restriction of A q (A) withrespect to reductive subgroups II — micro-local analysis and asymptoticK-support, Annals of Alath. 147 (1998), 709-729.[Kog] T. Kobayashi, Multiplicity free branching laws for unitary highest weightmodules, Proceedings of the Symposium on Representation Theory heldat Saga, Kyushu (K. Alimachi, ed.), 1997, 7-13.[K09] T. Kobayashi, Discrete decomposability of the restriction of A q (A) withrespect to reductive subgroups III — restriction of Harish-Chandra modulesand associated varieties, Invent. Alath. 131 (1998), 229-256.[Koio] T. Kobayashi, Discrete series representations for the orbit spaces arisingfrom two involutions of real reductive Lie groups, J. Funct. Anal. 152(1998), 100-135.[Kon] T. Kobayashi, Discretely decomposable restrictions of unitary representationsof reductive Lie groups —examples and conjectures, Adv. Stud.Pure Alath. 26 (2000), 99-127.[K012] T. Kobayashi, Theory of discrete decomposable branching laws of unitaryrepresentations of semisimple Lie groups and some applications, SugakuExposition, Transi. Ser., A.ALS. (to appear).[K013] T. Kobayashi, in preparation.[Ko-Od] T. Kobayashi and T. Oda, Vanishing theorem of modular symbols onlocally symmetric spaces, Comment. Alath. Helvetic 73 (1998), 45-70.[Ko-0i] T. Kobayashi and B. Orsted, Conformai geometry and branching lawsfor unitary representations attached to minimal nilpotent orbits, C. R.Acad. Sci. Paris 326 (1998), 925-930.[Ko-02] T. Kobayashi and B. Orsted, Analysis on the minimal representation of0(p,q), I, II, III, preprint.[Le-Lo] S-T. Lee and H-Y. Loke, Degenerate principal series of U(p,q) andSpin(p,q), preprint.[Lii] J-S. Li, On the discrete series of generalized Stiefel manifolds, Trans.A.ALS. 340 (1993), 753-766.

Branching Problems of Unitary Representations 627[Li 2 ] J-S. Li, Two reductive dual pairs in groups of type E, Alanuscripta Alath.91 (1996), 163-177.[Loi] H-Y. Loke, Restrictions of quaternionic representations, J. Funct. Anal.172 (2000), 377-403.[L02] H-Y. Loke, Howe quotients of unitary characters and unitary lowestweight modules, preprint.[Alg] G. Alargulis, Existence of compact quotients of homogeneous spaces, measurablyproper actions, and decay of matrix coefficients, Bui. Soc. Alath.France 125 (1997), 1-10.[Alt] S. Alartens, The characters of the holomorphic discrete series, Proc. Nat.Acad. Sci. USA 72 (1975), 3275-3276.[Alk-Os] T. Alatsuki and T. Oshima, A description of discrete series for semisimplesymmetric spaces, Adv. Stud. Pure Alath. 4 (1984), 331-390.[N-Oc-T] K. Nishiyama, H. Ochiai, and K. Taniguchi, Bernstein degree and associatedcycles of Harish-Chandra modules — Hermitian symmetric case,Astérisque 273 (2001), 13-80.[Oh] H. Oh, Tempered subgroups and representations with minimal decay ofmatrix coefficients, Bull. Soc. Alath. France 126 (1998), 355-380.[O1-0] G. Olafsson and B. Orsted, The holomorphic discrete series of an affinesymmetric space, I, J. Funct. Anal. 81 (1988), 126-159.[0-Vs] B. Orsted and J. Vargas, Restriction of square integrable representations:discrete spectrum, preprint.[Vo] D. Vogan, Jr., Associated varieties and unipotent representations, Progressin Alath. 101 (1991), Birkhäuser, 315-388.[Vo-Z] D. Vogan, Jr. and G. Zuckerman, Unitary representations with non-zerocohomology, Compositio Alath. 53 (1984), 51-90.[X] J. Xie, Restriction of discrete series of SU(2,1) to S(U(1) x U(l, 1)), J.Funct. Anal. 122 (1994), 478-518, ph.D. dissertation, Rutgers University.

ICAl 2002 • Vol. II • 629-635Representations of Algebraic Groups andPrincipal Bundles on Algebraic VarietiesVikram Bhagvandas Mehta*AbstractIn this talk we discuss the relations between representations of algebraicgroups and principal bundles on algebraic varieties, especially in characteristicp. We quickly review the notions of stable and semistable vector bundles andprincipal G-bundles , where G is any semisimple group. We define the notionof a low height representation in characteristic p and outline a proof of thetheorem that a bundle induced from a semistable bundle by a low heightrepresentation is again semistable. We include applications of this result tothe following questions in characteristic p:1) Existence of the moduli spaces of semistable G-bundles on curves.2) Rationality of the canonical parabolic for nonsemistable principal bundleson curves.3) Luna's etale slice theorem.We outline an application of a recent result of Hashimoto to study thesingularities of the moduli spaces in (1) above, as well as when these spacesspecialize correctly from characteristic 0 to characteristic p. We also discussthe results of Laszlo-Beauville-Sorger and Kumar-Narasimhan on the Picardgroup of these spaces. This is combined with the work of Hara and Srinivas-Mehta to show that these moduli spaces are F-split for p very large. Weconclude by listing some open problems, in particular the problem of refiningthe bounds on the primes involved.2000 Mathematics Subject Classification: 22E46, 14D20.Keywords and Phrases: Semistable bundles, Low-height representations.1. Some DefinitionsWe begin with some basic definitions:Let V be a vector bundle on a smooth projective curve X of genus g over analgebraically closed field (in any characteristic).*Tata Institute of Fundamental Research, Mumbai, India. E-mail: vikram@math.tifr.res.in

630 Vikram Bhagvandas AlehtaDefinition 1.1: V is stable ( respectively semi-stable J if for all subbundles W ofV, we havep(W) fef deg W/rk W < ( 0, one constructs the moduli spaces U s (r, d)(U(r, dj) ofstable (semistable) vector bundles of rank r and degree d, using Geometric InvariantTheory (G.I.T.).If the ground field is C, the complex numbers, one has the basic (genus X > 2):Theorem 1.2: Let V have degree 0. Then V is stable 44> V ~ V„, for someirreducible representation a : ni(X) —¥ U(n).This is due to Narasimhan-Seshadri. Note that H —¥ X is a principal m (X)fibration, where H is the upper-half plane. Any a : m (X) —t GL(n, C) gives avector bundle of rank n on X, V a = H x^1^) C".Remark 1.3: It follows from Theorem 1.2 that if V is a semistable bundle on acurve X over C, then ® n (V),S n (V), in fact any bundle induced from V is againsemistable. By Lefschetz, this holds for any algebraically closed field of characteristic0.Remark 1.4: In general, a subbundle W of a vector bundle V is a reductionof the structure group of the principal bundle of V to a maximal parabolic ofGL(n),n = rank V. This is in turn equivalent to a section a of the associated fibrespace:ExGUn) G£,( n )/p.Now let X be a smooth curve and E —¥ X a principal G-bundle on X, where G isa semisimple (or even a reductive) group in any characteristic.Definition 1.5: E is stable (semistable) 44> V maximal parabolics P of G, V sectionsa of E(G/P), we have degree CT # T^ > 0(> 0), where TV is the relative tangentbundle of E(G/P) 4 X.Over C, we have the following [18]:Theorem 1.6: E —¥ X is stable 44> E ~ E„ for some irreducible representationa : m (X) —t K, the maximal compact of G.The analogue of Remark 1.3 is valid in this general situation.Remark 1.7: One can analogously define stable and semistable vector bundlesand principal bundles on normal projective varieties of dimension > 1. Again, incharacteristic 0, bundles induced from semistable bundles continue to be semistable.Remark 1.8: In characteristic p, bundles induced from semistable bundles neednot be semistable, in general[7]. In this lecture we shall examine some conditionswhen this does hold, and also discuss some applications to the moduli spaces ofprincipal G-bundles on curves.2. Low height representations

Representations of Algebraic Groups and Principal Bundles 631Here we introduce the basic notion of a low height representation in characteristicp. Yet f : G —¥ SL(n) = SL(V) be a representation of G in char p, Gbeing reductive. Fix a Borei B and a Torus T in G. Let L(A»), 1 of V. Write eachA, as yjgyOij, where {aj} is the system of simple roots corresponding to B andjqij £ Q Vi, j. Define htXi = /Jftj. Then one has the basic [9,20]:jDefinition 2.1: f is a low-height representation ofG, or V is a low-height moduleover G, if 2ht(Xi) of G, V is completely reducibleover F 44> F itself is completely reducible in G. By definition, an abstract subgroupF of G is completely reducible in G 44> for any parabolic P of G, if F is containedin P then F is contained in a Levi component L of P. These results were provedby Serre[20] using the notion of a saturated subgroup of G.In general, denote sup (2ht A,) by htaV. If V is the standard SL(n) module,then htsL(n)^l(V)= i(n — i),l of induced bundles [9]:Theorem 2.3: Let E —t X be a semistable G-bundle, where G is semisimple andthe base X is a normal projective variety. Let f : G —¥ SL(n) be a low-heightrepresentation. Then the induced bundle E(SL(nj) is again semistable.The proof is an interplay between the results of Bogomolov, Kempf, Rousseauand Kirwan in G.I.T. on one hand and the results of Serre mentioned earlier on theother. The group scheme E(G) over X acts on E(SL(n)/P) and assume that a isa section of the latter. Consider the generic point if of X and its algebraic closureK. Then E(G)-% acts on E(SL(n)/P)-^-, and a is a if-rational point of the latter.There are 2 possibilities:1) a is G.I.T semistable. In this case, one can easily prove that deg CT # T^ > 0.2) a is G.I.T. unstable, i.e., not semistable. Let P(a) be the Kempf-Rousseauparabolic for a, which is defined over K. For deg CT # T^ to be > 0 it is sufficientthat P(a) is defined over K. Note that since V is a low-height representationof G, one has p> h. One then has ([20]).Proposition 2.4: If p > h, there is a unique G-invariant isomorphism log:G u —¥ g , where G u is the unipotent variety of G and g is the nilpotent varietyof g = Lie G.Proposition 2.4 is used inProposition 2.5: Let H be any semisimple group and W a low-height representationof H. Let Wi C W and assume that 3X £ Lie H, X nilpotent such that X £Lie (Stab (Wij). Then in fact one has X £ Lie [Stab (Wi) re d]-Along with some facts from G.I.T, Proposition 2.5 enables us to prove thatP(a) is in fact defined over K, thus finishing the sketch of the proof of Theorem 2.3.See also Ramanathan-Ramanan [19]. One application of low-height representations

632 Vikram Bhagvandas Alehtais in the proof of a conjecture of Behrend on the rationality of the canonical parabolicor the instability parabolic. If V is a nonsemistable bundle on a variety X, thenone can show that there exists a flag V",of subbundles of V with the properties:0 = Vo C Vi C V 2 • • • C V n = V(1) Each Vi/Vi-i is semistable and p (Vj/Vj_i) > p (Vj+i/V»), 1 of definition of X and V. Such a flag corresponds to a reductionto a parabolic P of GL(n) and properties (1) and (2) may be expressed asfollows: the elementary vector bundles on X associated to P all have positivedegree and H°(X, E(g)/E(pj) = 0, where g = Lie GL(n) and p = Lie P.One may ask whether there is a such a canonical reduction for a nonsemistableprincipal G bundle E —¥ X. Such a reduction was first asserted first by Ramanathan[18], and then by Atiyah-Bott[l] ,both over C and both without proofs. It wasBehrend [ 5 ], who first proved the existence and uniqueness of the canonical reductionto the instability parabolic in all characteristics. Further, Behrend conjecturedthat H°(X,E(g)/E(p)) = 0.In characteristic zero, one can check that all three definitions of the instabilityparaboliccoincide and that Behrend's conjecture is valid. In characteristic p, oneuses low-height representations to show the equality of the three definitions andprove Behrend's conjecture [14].Theorem 2.6: Let E —t X be a nonsemistable principal G-bundle in char p. Assumethat p > 2dimG. Then all the 3 definitions coincide and further we haveH°(X,E(g)/E(pj) = 0, where p = Lie P and P is the instability parabolic.Theorem 2.6 is useful, among other things, for classifying principal G-bundleson P 1 and P 2 in characteristic p.If V is a finite-dimensional representation of a semisimple group G (in anycharacteristic),then the G.I.T. quotient V//G parametrizes the closed orbits in V.Now, let the characteristic be zero and let VQ £ V have a closed orbit. Then Luna'sétale slice theorem says that 3 a locally closed non-singular subvariety S of V suchthat VQ £ S and S//G Vg is isomorphic to V//G, locally at VQ, in the étale topology.Here G Vo is the stabilizer of VQ- The proof uses the fact that G Vo is a reductivesubgroup of G (not necessarily connected!), hence V is a completely reducible Gmodule. In characteristic p, one has to assume that V is a low-ht representationof G. Then the conclusion of Luna's étale slice theorem is still valid: to be moreprecise, let V be a low-ht representation of G and let VQ £ V have a closed orbit.Put H = Stab (vo)- The essential point, as in characteristic 0, is to prove thecomplete reducibility of V over H. Using the low-ht assumption, one shows thatevery X £ Lie H with X nilpotent can be integrated to a homomorphism G a —¥ Hwith tangent vector X. Now, under the hypothesis of low-ht, one shows that -ff rec [is a saturated subgroup of G and (-ff rec [ : -ff 0 J) is prime to p. This shows that V

Representations of Algebraic Groups and Principal Bundles 633is a completely reducible -ff rec [ module. Further, one shows that # re( j is a normalsubgroup of H with H/H ïe(^ a finite group of multiplicative type, i.e. a finitesubgroup of a torus. Now the complete reducibility of V over H follows easily [11].Just as in characteristic zero, one deduces the existence of a smooth ff-invariantsubvariety S of V such that VQ £ S and S//H is locally isomorphic to V//G at VQ-This result is used in the construction of the moduli space MQ to be described inthe next section.3. Construction of the moduli spacesThe moduli spaces of semistable G-bundles on curves were first constructed byRamanathan over C [16,17], then by Faltings and Balaji-Seshadri in characteristic0 [3,6]. There are 3 main points in Ramanathan's construction:1. If E —t X is semistable, then the adjoint bundle E(g) is semistable.2. If E —¥ X is polystable, then E(g) is also polystable.3. A semisimple Lie Algebra in char 0 is rigid.The construction of MQ in char p was carried out in [2,15]. We describethe method of [15] first : points (1) and (2) are handled by Theorem 2.3 and thefollowing [11] :Theorem 3.1: Let E —t X be a polystable G-bundle over a curve in char p. Leta : G —¥ SL(n) = SL(V) be a representation such that all the exterior powersA*V, 1 of uses Luna's étale slice theorem in char p and Theorem 2.3.Now one takes a total family T of semistable G bundle on X and takes thegood quotient of T to obtain MQ in char p. Theorem 3.1 is used to identify theclosed points of MQ as the isomorphism classes of polystable G-bundles, just as inchar 0. The semistable reduction theorem is proved by lifting to characteristic 0and then applying Ramanathan's proof (in which (3) above plays a crucial role).This construction follows Ramanathan very closely and, as is clear, one has to makelow-height assumptions as in Theorem 3.1.The method of [2] follows the one in [3] with some technical and conceptualchanges. One chooses an embedding G —¥ SL(n) and a representation W for SL(n)such that (1) G is the stabilizer of some WQ £ W. (2) W is a "low separable indexrepresentation" of SL(n), i.e., all stabilizers are reduced and W is low-height overSL(n). The semistable reduction theorem is proved using the theory of Bruhat-Tits.Here also suitable low-height assumptions have to be made.4. Singularities and specialization of the modulispacesWe first discuss the singularities of MQ, assuming throughout that G is simplyconnected.In char 0, MQ has rational singularities, this follows from Boutot'stheorem. In char p, the following theorem due to Hashimoto [8] is relevant:

634 Vikram Bhagvandas AlehtaTheorem 4.1: Let V be a representation of G such that all the symmetric powersS n (Y r ) have a good filtration. Then the ring of invariant [S'(V)] G is strongly F-regular.Strongly F-regularity is a notion in the theory of tight closure in commutativealgebra. We just note that if a geometric domain is strongly F-regular then itis normal,Cohen-Alacaulay, F-split and has "rational-like" singularities. Now lett £ MQ be the "worst point", i.e., the trivial G-bundle on X.The local ring (OM G ,*) A is isomorphic to (S'(W)//G) A , where W = directsum of g copies of g, with G acting diagonally. If p is a good prime for G , thenHashimoto's theorem implies that OM G ,t is strongly F-regular. The other points ofMQ are not so well understood. This would require a detailed study of the automorphismgroups of polystable bundles, both in char 0 and p, and of their invariantsof the slice representations. This is necessary also to study the specialization problem,i.e., when MQ in char 0 specializes to MQ in char p. One has to show thatthe invariants of the slice representations in char 0 specialize to the invariants inchar p. However for G=SL(n),the situation is much simpler. One can write downthe automorphism group of a polystable bundle and its representation on the localmoduli space. Consequently, one expects the moduli spaces to specialize correctlyand that the local rings of MQ are strongly F-regular in all positive characteristics.We briefly discuss Pic MQ in char 0. It follows from [4,10] that MQ has thefollowing properties in char 0:1. Pic MQ ~Z.2. MQ is a normal projective, Gorenstein variety with rational singularities andwith K negative ample.Now let X be a normal,Cohen-Alacaulay variety in char 0. It is proved in[13],in response to a conjecture of Karen Smith, that if X has rational singularities,then the reduction of X mod p is F-rational for all large p. This result togetherwith 1 and 2 above imply that MQ reduced mod p is F-split for all large p. Wecannot give effective bounds on the primes involved. One partial result is known inthis direction [12].Acknowledgement: I would like to thank my colleagues S. Ilangovan, A.J. Parameswaranand S. Subramanian for their help in preparing this report and T.T. Nayyaand H for constant help and encouragement.References[1] ALF. Atiyah, R. Bott, The Yang-Alills equations over Riemann Surfaces, Phil.Trans. R. Soc. London A 308 (1982), 523^615.[2] V. Balaji, A.J. Parameswaran Semistable Principal Bundles-II (in positivecharacteristics) to appear in Transformation Groups.[3] V. Balaji, C.S. Seshadri Semistable Principal Bundles-I (in characteristic zero),to appear in Journal of Algebra.[A] A. Beauville, Y. Laszlo, C. Sorger, The Picard Group of the moduli of G-bundles over curves, Compositio Math. 112, (1998), No.2, 183^216.

Representations of Algebraic Groups and Principal Bundles 635[5] K. Behrend, Semistability of reductive group schemes over curves, Math. Ann.301 (1995), 281^305.[6] G. Faltings, Stable G-bundles and projective connections, J. Algebraic Geom.2 (1993) No.3, 507^568.[7] D. Gieseker, Stable Vector Bundles and the Frobenius morphism, Ann. Sci.Ecol. Nor. Sup. 6, (1973).[8] Al. Hashimoto, Good filtrations of symmetric algebras and strong F-regularityof invariant subrings, Math. Z. 236 (2001), No.3, 605^623.[9] S. Ilangovan, V.B. Alehta, A.J. Parameswaran, Semistability and Semisimplicityin representations of low-height in positive characteristics, preprint.[10] S. Kumar, ALS. Narasimhan, Picard group of the moduli spaces of G-bundles,Math. Ann. 308, (1997), No.l, 155^173.[11] V.B. Alehta, A.J. Parameswaran, Geometry of low-height representations, Proceedingsof the International Colloquium on Algebra, Arithmetic and Geometry,(ed. R. Parimala), TIFR Alumbai 2000.[12] V.B. Alehta, T.R. Ramadas, Aloduli of vector bundles, Frobenius splitting andinvariant theory, Ann. of Math. (2) 144, (1996), 269^313.[13] V.B. Alehta, V. Srinivas, A characterization of rational singularities, Asian J.Math., Vol.1, (1997), No.2, 249-271,.[14] V.B. Alehta, S. Subramanian, On the Harder-Narasimhan Filtration of PrincipalBundles, Proceedings of the International Colloquium on Algebra, Arithmeticand Geometry, (ed. R. Parimala), TIFR Alumbai 2000.[15] V.B. Alehta, S. Subramanian, Aloduli of Principal G-bundles on curves in positivecharacteristic, in preparation.[16] A. Ramanathan, Aloduli for principal bundles over algebraic curves I, Proc.Indian Acad. Sei. Math. Sci, 106, (1996), No.3, 301^328.[17] A. Ramanathan, Aloduli for principal bundles over algebraic curves II, Proc.Indian Acad. Sei. Math. Sci, 106 (1996), No.4, 421-149.[18] A. Ramanathan, Aloduli for principal bundles, in: Algebraic Geometry, Proceedings,Copenhagen 1978, 527^533, Lecture Notes in Alathematics vol. 732,Springer.[19] S. Ramanan, A. Ramanathan, Some remarks on the instability flag, TohokuMath. Journal 36, (1984), 269-291.[20] J-P. Serre, Moursund Lectures, University of Oregon Alathematics Department,notes by W.E. Duckworth (1998).

ICAl 2002 • Vol. II • 637^642Clifford Algebras and theDuflo IsomorphismE. Meinrenken*AbstractThis article summarizes joint work with A. Alekseev (Geneva) on the Dufloisomorphism for quadratic Lie algebras. We describe a certain quantizationmap for Weil algebras, generalizing both the Duflo map and the quantizationmap for Clifford algebras. In this context, Duflo's theorem generalizes to astatement in equivariant cohomology.2000 Mathematics Subject Classification: 17B, 22E60, 15A66, 55N91.Keywords and Phrases: Clifford algebras, Quadratic Lie algebras, Duflomap, Equivariant cohomology.1. IntroductionThe universal enveloping algebra U(g) of a Lie algebra (g, [-, -] 0 ) is the quotientof the tensor algebra T(g) by the relations, ££' — £'£ = [£,£'] 0 . The inclusion ofthe symmetric algebra S(g) into T(g) as totally symmetric tensors, followed by thequotient map, gives an isomorphism of {(-modulessym : S(g) -> U(g) (1.1)called the symmetrization map. The restriction of sym to {(-invariants is a vectorspace isomorphism, but not an algebra isomorphism, from invariant polynomials tothe center of the enveloping algebra. Let J £ C°° (g) be the functionJ(C) = det(j(ad c )),j( z) = S^M^,and J 1 / 2 its square root (defined in a neighborhood of £ = 0). Denote by J 1 / 2the infinite order differential operator on S g C C°°(g*), obtained by replacing the* Department of Mathematics, University of Toronto, 100 St. George Street, Toronto, ONM6R1G7, Canada. E-mail: mein@math.toronto.edu

638 E. Aleinrenkenvariable £ £ g with a directional derivative -g-, where p is the dual variable on g*.Duflo's celebrated theorem says that the compositionsymojVà .SgÛ(g)restricts to an algebra isomorphism, (Sg) e —¥ Cent(U(gj). In more geometric language,Duflo's theorem gives an isomorphism between the algebra of invariant constantcoefficient differential operators on g and bi-invariant differential operators onthe corresponding Lie group G.The purpose of this note is to give a quick overview of joint work with A.Alekseev [1, 2], in which we obtained a new proof and a generalization of Duflo'stheorem for the special case of a quadratic Lie algebra. That is, we assume thatg comes equipped with an invariant, non-degenerate, symmetric bilinear form B.Examples of quadratic Lie algebras include semi-simple Lie algebras, or the semidirectproduct g = s x s* of a Lie algebra s with its dual. Using B we can definethe Clifford algebra 01(g). Duflo's factor J 1//2 (£) arises as the Berezin integralof exp(g(A(£))) £ Cl({(), where q : A(g) —¥ Cl(g) is the quantization map, andÀ : g —¥ A 2 g is the map dual to the Lie bracket.2. Clifford algebrasLet V be a finite-dimensional real vector space, equipped with a non-degeneratesymmetric bilinear form B. Fix a basis e a £ V and let e a £ V be the dual basis.We denote by o(V) C End(V) the space of endomorphisms A of V that are skewsymmetricwith respect to B. For any A £ o(V) we denote its components byA a b = B(e a ,Aeb). Consider the function S : o(V) —¥ A(V) given byS(A) = det 1 / 2 (j(A)) exp A(l/) ßf(A) ab e a A e(using summation convention), wheref( z ) = (li i j)'(z) = lcoth(^)-l (2.1)In turns out that, despite the singularities of the exponential, S is a global analyticfunction on all of o(V). It has the following nice property. Let C1(V) denote theClifford algebra of V, defined as a quotient of the tensor algebra T(V) by therelations vv' + v'v = B(v,v'). The inclusion of A(V) into T(V) as totally antisymmetrictensors, followed by the quotient map to C1(V), gives a vector spaceisomorphismq: /\(V)->Cl(V)known as the quantization map. Then S (A) relates the exponentials of quadraticelements l/2A a {,e a A e b in the exterior algebra with the exponentials of the correspondingelements l/2A a i,e a e b in the Clifford algebra:exp cl(l/) (l/2A Q6 e a e 6 ) = q (i(S(A)) exp A(l/) (l/2A a6 e a A e 6 )) . (2.2)

Clifford Algebras and the Duflo Isomorphism 639Here i : A(V) —¥ End(V) is the contraction operator. In fact, one may add linearterms to the exponent: Let E be some vector space of "parameters", and

640 E. AleinrenkenIn Etingof-Schiffmann [6], it is shown that t a (, is in fact the unique solution ofthis particular CDYBE, up to gauge transformation. Alore general CDYBE's areassociated to a pair h c ß of Lie algebras, here h = g. The proof sketched abovecan be modified to produce some of these more general solutions.4. The non-commutative Weil algebraUsing B to identify the Lie algebra g with its dual g*, the Weil algebra of g isthe Z -graded g-module given as a tensor productWg = Sg® Ag,where generators of S g are assigned degree 2. Let L for £ £ g denote the generatorsfor the g-action on Wg, and if' = 1 ® i$ the contraction operators. The Weildifferential d ' is a derivation of degree 1, uniquely characterized by its propertiesd H/ od H/ = 0 and d H/ (l ® £) = £ ® 1 for £ £ g. The Weil algebra Wg withthese three types of derivations is an example of a g-differential algebra: That is,Lf', i)fi, d 14 satisfy relations similar to contraction operators, Lie derivatives, andde Rham differential for a manifold with group action.In [1], we introduced the following non-commutative version of the Weil algebra,YVg = Ug®Cl(g).It carries a Z-filtration, where generators of U(g) are assigned filtration degree 2,with associated graded algebra gr(Wg) = Wg. Moreover, it carries a Z 2 -grading,compatible with the Z-filtration in the sense of [8]. Define contraction operators asZ 2 -graded commutators iY^ = [1 ® £, •], let fi^fi be the generators for the naturalg-module structure, and set d w = [D, •] whereis the cubic Dirac operator [7]. Its squareV = e 0 . ® e a - 1 ® C £ WgV 2 = \e a e a ® 1 - ^fabef abcis in the center of Wg, hence d is a differential. As it turns out, Wg is againa g-differential algebra. The derivations d ,t^,L^ respect the Z-filtration, andthe induced derivations on the associated graded algebra are just the standardderivations for the Weil algebra Wg.The vector space isomorphism sym ®q : Wg —¥ Wg intertwines the contractionoperators and Lie derivatives, but not the differentials. There does exist, however,a better "quantization map" Q : Wg —¥ Wg that is also a chain map. Using ourfunction S e £ C°°(g) ® Ag, let i(S s ) denote the operator on Wg, where the Agfactoracts by contraction on Ag and the G°° (g)-factor as an infinite order differentialoperator.

Clifford Algebras and the Duflo Isomorphism 641Theorem. [1] The quantization mapQ := (sym®q) o i(Si) : Wg -• Wgintertwines the contraction operators, Lie derivatives, and differentials on Wg andon Wg.The fact that Q intertwines the two differentials d 14 , d w relies on a numberof special properties of the function S e , including the CDYBE.Put differently, the quantization map Q defines a new, graded non-commutativering structure on the Weil algebra Wg, in such a way that the derivations i)fi, Lf , dare still derivations for the new ring structure, and in fact become inner derivations.Notice that Q restricts to the quantization map for Clifford algebras q : Ag —¥ Cl(g)on the second factor and to the Duflo map on the first factor, but is not just theproduct of these two maps.5. Equivariant cohomologyH. Cartan in [3] introduced the Weil algebra Wg as an algebraic model for thealgebra of differential forms on the classifying bundle EG, at least in the case Gcompact.In particular, it can be used to compute the equivariant cohomology HQ(M)(with real coefficients) for any G-manifold M. Yet if h ,Lf h ,d Rh denote the contractionoperators, Lie derivatives, and differential on the de Rham complex 0(Af )of differential forms. LetH B (M) = H((Wg ® tt(M)) basic , d w + d m )where (Wg ® 0(Af ))t> as ic is the subspace annihilated by all Lie derivatives Lf' +L Rh and all contraction operators if' + i Rh . Cartan's result says that H S (M) =HQ(M,R) provided G is compact.Alore generally, we can define H B (A) for any g-differential algebra A. YetR S (A) be defined by replacing Wg with Wg. The quantization map Q : Wg —¥ Wginduces a map Q : H S (A) —¥ R S (A).Theorem. [1] For any g-differential algebra A, the vector space isomorphism Q :H s (Â) —¥ R s (A) is in fact an algebra isomorphism.Our proof is by construction of an explicit chain homotopy between the twomaps Wg ® Wg —¥ Wg given by "quantization followed by multiplication" and"multiplication followed by quantization", respectively. Taking A to be the trivial g-differential algebra (i.e. A = Q(point)), the statement specializes to Duflo's theoremfor quadratic g.References[1] A. Alekseev & E. Aleinrenken, The non-commutative Weil algebra, Invent.Math. 139 (2000), 135^172.

642 E. Aleinrenken[2] A. Alekseev & E. Aleinrenken, Clifford algebras and the classical dynamicalYang-Baxter equation (in preparation).[3] H. Cartan, Notions d'algèbre différentielle; application aux groupes de Lie etaux variétés où opère un groupe de Lie., Colloque de topologie (espaces fibres),Bruxelles, (1950).[4] Al. Duflo, Opérateurs différentiels bi-invariants sur un groupe de Lie, Ann. Sci.École Norm. Sup. 10 (1977), 265^288.[5] P. Etingof & A. Varchenko, Geometry and classification of solutions of theclassical dynamical Yang-Baxter equation, Comm. Math. Phys., 192 (1988),77-120.[6] P. Etingof & O. Schiffmann, On the moduli space of classical dynamical r-matrices, Math. Res. Lett. 8 (2001), 157^170.[7] B. Kostant, A cubic Dirac operator and the emergence of Euler number multipletsof representations for equal rank subgroups, Duke Math. J. 100 (1999),no. 3, 447-501.[8] B. Kostant & S. Sternberg, Symplectic reduction, BRS cohomology, andinfinite-dimensional Clifford algebras, Ann. Physics 176 (1987), no. 1, 49-113.

ICAl 2002 • Vol. II • 643^654Representations of YangiansAssociated with Skew Young DiagramsMaxim Nazarov*AbstractThe Yangian of the Lie algebra gl N has a distinguished family of irreduciblefinite-dimensional representations, called elementary representations. Theyare parametrized by pairs, consisting of a skew Young diagram and a complexnumber. Each of these representations has an explicit realization, it extendsthe classical realization of the irreducible polynomial representations of gl N bymeans of the Young symmetrizers. We explicitly construct analogues of theseelementary representations for the twisted Yangian, which corresponds to theLie algebra SON • Our construction provides solutions to several open problemsin the classical representation theory. In particular, we obtain analogues ofthe Young symmetrizers for the Brauer centralizer algebra.2000 Mathematics Subject Classification: 17B35, 17B37, 20C30, 22E46.Keywords and Phrases: Branching rules, Brauer algebra, Classical groups,Intertwining operators, Reflection equation, Yangians, Young symmetrizers.1. Yangian of the general linear Lie algebra1.1. For each simple finite-dimensional Lie algebra g over the field C, Drinfeld[4] introduced a canonical deformation of the universal enveloping algebra of thepolynomial current Lie algebra g[x]. This deformation is a certain Hopf algebraover C, denoted by Y(g) and called the Yangian of the simple Lie algebra g. Nowconsider the general linear Lie algebra gl N , it contains the special linear Lie algebrasljv as a subalgebra. The Hopf algebra which is called the Yangian of the reductiveLie algebra gl^r and is denoted by Y(gl N ), was considered in the earlier works ofmathematical physicists from St.-Petersburg, see for instance [6]. The Hopf algebraY (g I N ) is a deformation of the universal enveloping algebra of the Lie algebra g I N [x],and the Yangian Y(sljv) of the simple Lie algebra sljv is a Hopf subalgebra of Y(gl N ).Throughout this article, we assume that N is a positive integer.* Department of Mathematics, University of York, York YO10 5DD, England. E-mail:mlnl@vork.ac.uk

644 Alaxim NazarovThe unital associative algebra Y(gl JV ) over C has a family of generators Tywhere a = 1,2,... and i, j = 1, ..., N. The defining relations for these generatorscan be written in terms of the formal power seriesTij(x) = öij • 1 + T^x- 1 + T^x- 2 + ... £ Y(gl N ) [[x' 1 ]]. (1.1)Here x is the formal parameter. Let y be another formal parameter, then thedefining relations in the associative algebra Y(gl JV ) can be written as(x- y )-[Tij(x),Tk l (y)]= T kj (x)T a (y)-T kj (y)T a (x), (1.2)where i,j,k,l = 1, ...,N. The square brackets in (1.2) denote usual commutator.In terms of the formal series (1.1), the coproduct A : Y(gl JV ) —t Y(gt N ) ® Y(gt N )is defined byA(Tij(x)) = J2 T i*(x) ® Tkj(x) ; (1.3)k=ithe tensor product on the right hand side of the equality (1.3) is taken over thesubalgebra C[[a: _1 ]] C Y(gl JV ) [[i -1 ]]- The counit homomorphism e : Y(gl JV ) —t Cis determined by the assignment e : Tj(u) >-¥ % • 1.For each i and j one can determine a formal power series Tj(x) in x^1withthe coefficients in Y(gl JV ) and the leading term o"y, by the system of equationsJVY^ Tik (x) T k j (x) = öij where i, j = 1, ..., N.k=iThe antipode S on Y(gl JV ) is the anti-automorphism of the algebra Y(gl N ), definedby the assignment S : Ty (x) >-¥ Tj (x). We also use the involutive automorphism£iv of the algebra Y(gt N ), defined by the assignment £JV : Tij(x) >-¥ Tij(-x).Take any formal power series f(x) £ C[[a: _1 ]] with the leading term 1. TheassignmentTj(x)^ f(x)-Tj(x) (1.4)defines an automorphism of the algebra Y(gl N ), this follows from (1.1) and (1.2).The Yangian Y(sljv) is the subalgebra in Y(gl JV ) consisting of all elements, whichare invariant under every automorphism (1.4).It also follows from (1.1) and (1.2) that for any z £ C, the assignmentT Z : Tij(x) H> Tj(xdefines an automorphism T Z of the algebra Y(gl N ). Here the formal power seriesin (x — z)^1should be re-expanded in x _1 . Regard the matrix units E t j £ gl N asgenerators of the universal enveloping algebra U(gl JV ). The assignment-z)«iv : Tij(x) >-¥ öij • 1 — Eji x^1defines a homomorphism ctN '• Y(gl N ) —¥ U(gl JV ). By definition, the homomorphism«iv is surjective. For more details and references on the definition of the YangianY(gl N ), see [7].

Representations of Yangians 6451.2. Let v = (vi,v 2 , ...) be any partition. As usual, the parts of v are arrangedin the non-increasing order: vi ^ v 2 ^ ... ^ 0. Let v' = (v[,v 2 , •••) be thepartition conjugate to v. In particular, v[ is the number of non-zero parts of thepartition v. An irreducible module over the Lie algebra gl^r is called polynomial,if it is equivalent to a submodule in the tensor product of n copies of the defininggl N -module C^, for some integer n ^ 0. The irreducible polynomial gl^r -modulesare parametrized by partitions v such that v[ ^ N. Here n = vi + v 2 + ... . Let V vbe the irreducible module corresponding to v. This gl^r -module is of highest weight(vi, ..., VM). Here we choose the Borei subalgebra in gl^r consisting of the uppertriangular matrices, and fix the basis of the diagonal matrix units En , ..., ENN inthe corresponding Cartan subalgebra of gl N .Take any non-negative integer M. Yet the indices i and j range over the set{1, ..., N + M}. Fix the basis of the matrix units E t j in the Lie algebra gijv+M •We suppose that the subalgebras gl N and gt M in gljv+M are spanned by elementsEij where respectively i,j = 1, ... ,N and i,j = N + 1, ...,N + M. Yet X and pbe two partitions, such that A{ ^ N + M and p[ ^ M. Consider the irreduciblemodules Y\ and V ß over the Lie algebras gijv+M an d B'M- The vector spaceHom 0[ M (^>V\) ( L5 )comes with a natural action of the Lie algebra gl^r • This action of gl N may bereducible. The vector space (1.5) is non-zero, if and only if X k ^ Pk and X' k —p' k ^ Nfor each k = 1,2,...; see for instance [8].Denote by Ajv(Af) the centralizer of the subalgebra U(gl M ) C U(gljv+M)- Thecentralizer Ajv(Af ) c U(gljv +Af ) contains U(gl JV ) as a subalgebra, and acts naturallyin the vector space (1.5). This action is irreducible. For every AT, Olshanski [16]defined a homomorphism of associative algebras Y(gl JV ) —t Ajv(Af). Along withthe centre of the algebra U(gljv+M)) the image of this homomorphism generates thealgebra Ajv(Af). We use a version of this homomorphism, it is denoted by UMM-The subalgebra in Y(gl JV+M ) generated by Ty where i,j = 1, ...,N, bydefinitioncoincides with the Yangian Y(gl N ). Denote by 1, Xi > j > Pi} .

646 Alaxim NazarovWhen p = (0,0,...), this is the usual Young diagram of the partition A. Considerthe Y(gl JV )-module obtained from the Ajv(Af )-module (1.5) by pulling back throughthe homomorphism O.NM ° T Z : Y(gl N ) —t Ajv(Af). Since the central elements ofU(gl JV+M ) act in (1.5) as scalar operators, this Y(gl JV )-module is irreducible. It isdenoted by V u (z), and is called an elementary module. Its equivalence class doesnot depend on the choice of the integer M, such that A{ ^ N + M and p[ ^ M.The elementary modules are distinguished amongst all irreducible Y(gl N )-modules by the following theorem. Consider the chain of algebrasY(gl 1 )cY(gl 2 )c...cY(gl JV ). (1.7)Here for every k = 1, ... ,N — 1 we use the embedding (pi : Y(gt k ) —¥ Y(gl fc+1 ).Consider the subalgebra of Y(gl JV ) generated by the centres of all algebras in thechain (1.7), it is called the Gelfand-Zetlin subalgebra. This subalgebra is maximalcommutative in Y(gl N ); see [3] and [13]. Take any finite-dimensional module Wover the Yangian Y(gl N ).Theorem 1. Two conditions on the Y(gl N )-module W are equivalent:a) W is irreducible, and the action of the Gelfand-Zetlin subalgebra of Y(gl N )in W is semi-simple;b) W is obtained by pulling back through some automorphism (1.4) from thetensor productV U!l (zi)®...®V U!m (z m ) (1.8)of elementary Y (gl N)-modules, for some skew Young diagrams OJI, ... ,oj m and forsome complex numbers zi, ...,z m such that Zk — z% $. Z for all k fi^l.This characterization of irreducible finite-dimensional Y(gl JV )-modules withsemi-simple action of the Gelfand-Zetlin subalgebra was conjectured by Cherednik,and was proved by him [3] under certain extra conditions on the module W. In fullgenerality, Theorem 1 was proved in [14]. An irreducibility criterion for the Y(gl JV )-module (1.8) with arbitrary parameters Zi, ... ,z m was given in [15].The classification of all irreducible finite-dimensional Y(gl JV )-modules has beengiven by Drinfeld [5]. However, the general structure of these modules needs a betterunderstanding. For instance, the dimensions of these modules are not explicitlyknownin general. The tensor products (1.8) provide a wide class of irreducibleY(gljv)~ m odules, which can be constructed explicitly.1.4. The Y(gl JV )-module V u (z) has an explicit realization. It extends the classicalrealization of irreducible g I N -module V v by means of the Young symmetrizers [20].Let us use the standard graphic representation of Young diagrams on the planeR 2 with two matrix style coordinates. The first coordinate increases from top tobottom, the second coordinate increases from left to right. The element (i,j) £ UJis represented by the unit box with the bottom right corner at the point (i,j) £ R 2 •Suppose the set UJ consists of n elements. Consider the column tableau of shapeUJ. It is obtained by filling the boxes of UJ with numbers 1, ..., n consecutively bycolumnsfrom left to right, downwards in every column. Denote this tableau by 0.

Representations of Yangians 6478 9 3 41234567-3-4-2-30-1-2For each k = 1, ..., n put Ck = j — i if the box (i, j) e a; is filled with thenumber k in the tableau 0. The difference j — i is called the content of the box(i,j) of the diagram UJ. Our choice of the tableau 0 provides an ordering of thecollection of all contents of UJ. In the above figure, on the left we show the columntableau 0 for the partitions À = (5,3,3,3,3,0,0,...) and p = (3,3,2,0,0,...). Onthe right we indicate the contents of all boxes of UJ.Introduce n complex variables ti, ... ,t n with the constraints tk = U for allk and I occuring in the same column of 0. The number of independent variablesamong ti, ... ,t n equals the number of non-empty columns in the diagram UJ. Orderlexicographically the set of all pairs (k,l) with 1 ^ k of U(gl JV ) in the space C^ back through thehomomorphisma N o T Z : Y(gl N ) -+ V(gl N ),we obtain a module over the algebra Y(gl N ), which is denoted by V(z) and calledan evaluation module. We have V(z) = V u (z) for À = (1,0, ...) and p = (0,0,...).For any partitions À and p, the operator EQ has the following interpretation, interms of the tensor products of evaluation modules over the Hopf algebra Y(gl N ).Yet PQ be the operator in (C N )® n reversing the order of the tensor factors.Proposition 2. The operator EQ P 0 is an intertwiner of the Y (gl N)-modulesV(c n + z)®...® V(ci +z) —y V(ci +z)®...® V(c n + z).By Proposition 2, the image of the operator EQ is a submodule in the tensorproduct of evaluation Y(gl JV )-modules V(ci + z) ® ... ® V(c n + z). Denote thisY(gl JV )-submodule by VQ(Z). For any À and p, we have the following theorem. Put-rj (x-pk + k)(x + k-l)~ M (x-fM k + k-l)(x + k) •This rational function of x expands as a power series in x^1(iAi)with the leading term 1.Theorem 2. The Y(gl N )-module VQ(Z) is equivalent to the elementary moduleV u (z), pulled back through the automorphism of the algebra Y(gl N ) defined by (1.4),where f(x) = f ß (x — z).This theorem is due to Cherednik [3], see also [12]. It provides an explicitrealization of the elementary Y(gl JV )-module V u (z) as a subspace in (C N )® n . Italso shows that the Y(gl JV )-module Vn(z) is irreducible, cf. [15]. The isomorphismbetween the Y(gl JV )-module VQ(Z), and the pull-back of the Y(gl JV )-module V u (z)as in Theorem 1, is unique up to a scalar multiplier.In Section 2 we give an analogue of Theorem 2 for the orthogonal Lie algebraSON, instead of gl N . The case of the symplectic Lie algebra sp N is similar to thatof SON, and is considered in the detailed version [12] of the present article.For any simple Lie algebra g the Yangian Y(g) as defined in [4], contains theuniversal enveloping algebra U(g) as a subalgebra. An embedding U(gl JV ) —¥ Y(gt N )can be defined by j^.. ,_. ^yW Q 12)The image of U(sljv) C U(gl JV ) under this emdedding belongs to Y(sljv) C Y(gl N ).The homomorphism O.N '• Y(gl N ) —¥ U(gl JV ) is identical on the subalgebra U(gl JV ).The restriction of O.N to Y(sljv) provides a homomorphism Y(sljv) —¥ U(sljv), whichis identical on the subalgebra U(sljv). For g ^ SIN a homomorphism Y(g) —t U(g)identical on the subalgebra U(g) C Y(g), does not exist [4]. For this reason, insteadof the Yangian Y(sojv) from [4], we will consider the twisted Yangian Y(gl N , a) from[17]. Here a is the involutive automorphism of the Lie algebra gl N , such that —a isthe matrix transposition. Then SON is the subalgebra of a-fixed points in gl^r.

Representations of Yangians 6492. Twisted Yangian of the orthogonal Lie algebra2.1. The associative algebra Y(gl N ,a) is a deformation of the universal envelopingalgebra of the twisted polynomial current Lie algebra{A(x) £ gl N [x] : a(A(xj) = A(-x)} .The deformation Y(gl N ,a) is not a Hopf algebra, but a coideai subalgebra in theHopf algebra Y(gljy). The definition of the twisted Yangian Y(gl N , a) was motivatedby the works of Cherednik [2] and Sklyanin [19] on quantum integrable systems withboundary conditions. This definition was given by Olshanski in [17].As in Subsection 1.1, let the indices i and j range over the set {1, ..., N}. Bydefinition,Y(gl N , a) is the subalgebra in Y(gl JV ) generated by the coefficients of allformal power seriesin x _1 . Due to (1.3), the subalgebra Y(gl N ,a)NJ2Tki(-x)Tkj(x) (2.1)k=iA(Y(gl JV ,a))cY(gl JV ,a)®Y(gl JV ).in Y(gl JV ) is a right coideal:To give the defining relations for the generators of Y(gl N ,a), introduce theextended twisted Yangian X(gl N ,a). The unital associative algebra X(gl N ,a) hasa family of generators Sy where a = 1,2,... . and i, j = 1, ..., N. PutSij(x) = öij • 1 + S^x- 1 + S^x- 2 + ... £ X(gl N ,a) [[x' 1 ]]. (2.2)Defining relations for the generators Sy of the algebra X (gljy, a) can be written as(x 2 - y 2 ) • [Sij(x),S M (y)] = (x + y)- (S kj (x)S a (y) - S kj (y)S a (x))- (x - y) • (Sik(x)Sji(y) - S ki (y)Sij(x)) + S ki (x)Sji(y) - S ki (y)Sji(x) -All these relations can be written as a single reflection equation, see [7]. One candefine a homomorphism TTN '• X(gl N ,a) —¥ Y(gl N ,a) by mapping the series Sij(x)to (2.1). The homomorpism TTN is surjective. As a two-sided ideal of X (gl N , a), thekernel of the homomorphism TTN is generated by the coefficients of all seriesSij(x) + (2x - l)Sij(-x) - 2xSji(x) (2.3)in x^1.This ideal is also generated by certain central elements of X (gl N , a), see [7].The algebra X(gl N ,a) admits an analogue of the automorphism £JV of Y(gl N ).Determine a formal power series S'y (x) in x^1with the coefficients in X (gl N , a) andthe leading term o"y, by the system of equationsJVY^ sik (x) S k j (x) = öij where i, j = 1, ..., N.k=i

650 Alaxim NazarovThen one can define an involutive automorphism TJN of the algebra X(gl N ,a)the assignment_n N : Sij(x) ^ Sij(-x - f ).byHowever, TJN does not determine an automorphism of Y(gl N ,a), because TJN doesnot preserve the ideal ofX(gl N ,a) generated by the coefficients of all series (2.3).For any formal power series f(x) £ C[[x -1 ]] with the leading term 1, theassignment Sy(x) H> f(x) • Sy(x) (2.4)defines an automorphism of the algebra X(gl N ,a). The defining relations of thealgebra X(gl N ,a) imply that the assignmentßN : Sy (x) H> öij • 1 + E ij -E Jidefines a homomorphism of associative algebras /3JV : X(gl N ,a) —¥ U(sojv). Bydefinition,the homomorphism /3JV is surjective. Moreover, /3JV factors through TTN •Note that the homomorphism Y(gl N ,a) —¥ U(sojv) corresponding to /3JV , cannot beobtained from O.N '• Y(gl N ) —¥ U(gl JV ) by restricting to the subalgebra Y(gl N ,a),because the image ofY(gl N ,a) relative to O.N is not contained in the subalgebraU(SOJV) C U(gl JV ); see [11]. An embedding U(sojv) —t Y(gl N ,a) can be defined byE t j-Ej t^T^-T^,cf. (1.12). The homomorphism Y(gl N ,a)identical on the subalgebra U(sojv) C—¥ U(sojv) corresponding to /3JV, is thenY(gl N ,a).2.2. For any partition v with v[ ^ N, the irreducible polynomial gl^-module V v canalso be regarded as a representation of the complex general linear Lie group GLN-Consider the subgroup ON C GLN preserving the standard symmetric bilinear form( , ) on C^. The subalgebra SON C gljv corresponds to this subgroup. Note thatthe complex Lie group ON has two connected components. In [20] the irreduciblefinite-dimensional representations of the group ON are labeled by the partitions v ofn = 0,1,2, ... such that v[ + v 2 ' ^ N. Denote by W v the irreducible representationof ON corresponding to v. As sojv-module, W v is irreducible unless 2v[ = N, inwhich case W v is a direct sum of two irreducible sojv-modules.Choose any embedding of the irreducible representation V v of the group GLNinto the space (C N )® n . Take any two distinct numbers k,l £ {1, ...,n). Byapplyingthe bilinear form ( , ) to a tensor w £ (C N )® n in the fcth and Ith tensorfactors, we obtain a certain tensor w £ (C N )® (" _2^. The tensor w is called traceless,if w = 0 for all distinct k and I. Denote by (C N )f n the subspace in (C N )® nconsisting of all traceless tensors, this subspace is ON -invariant. Then W v can beembedded into (C N )® n as the intersection V v n (C N )®. n , see [20].Let the indices i and j range over {1, ..., N + M}. Choose the embeddingof the Lie algebras gl^r and gl M into gijv+M as in Subsection 1.2. It determinesembeddings of groups GLN X GL M —^ GLN+M and ON X OM —^ OJV+M- Take anytwo partitions À and p such that A{ + X' 2 ^ N + M and p[ + p' 2 ^ M. Consider the

Representations of Yangians 651irreducible representations W\ and W ß of the groups OJV+M and OM respectively.The vector space Horn 0M ( W ß , W\ ) (2.5)comes with a natural action of the group ON • This action of ON may be reducible.The vector space (2.5) is non-zero, if and only if X k ^ Pk and X' k — p' k ^ N foreach k = 1,2,... ; see [18]. Thus for a given N, the vector spaces (1.5) and (2.5)are zero or non-zero simultaneously. Further, for a given N, the dimension of (2.5)does not exceed that of (1.5). Our results provide an embedding of (2.5) into (1.5),compatible with the action of the orthogonal group ON in these two vector spaces.Denote by BJV(AT) the subalgebra of OM -invariants in the universal envelopingalgebra Y(SON+M)- Then Bjv(Af) contains the subalgebra U(sojv) C Y(SON+M),and is contained in the centralizer of the subalgebra U(SOM) C Y(SON+M)- Thealgebra Bjv(Af) naturally acts in the vector space (2.5). The Bjv(Af)-module (2.5)is either irreducible, or splits into a direct sum of two irreducible Bjv(Af)-modules.In the latter case, (2.5) is irreducible under the joint action of the algebra Bjv(M)and the subgroup ON C OJV+M-For every non-negative integer M, Olshanski [17] defined a homomorphismY(gljvj

652 Alaxim Nazarovunder the joint action of the algebra X (gl N , a) and the subgroup Ojv C OJV+M- Ourmain result is an explicit realization of the X(gl JV ,o-)-module W U (M), similar tothe realization of the elementary Y(gl JV )-module given by Theorem 2. Our explicitrealization is compatible with the action of the group OJV in W U (M).Take the standard orthonormal basis ei, ..., ejv in C^, so that (e,,ej ) = o"y.The linear operatorNu®v^r (u,v) • ^2e t ® e ì (2-7)«=iin C N ® C N commutes with the action of OJV- Take the complex variables ti, • • • ,t nwith the same constraints as in Proposition 1. Consider the ordered product overthe pairs (k,l), —• , „ vTT M< iM ) (2 8)i J ^ V c k + ci + t k + ti + N + MJwhere Qki is the linear operator in (C N )® n , acting as (2.7) in the fcth and Ith tensorfactors, and acting as the identity in the remaining n — 2 tensor factors. Here thepairs (k,l) are ordered lexicographically, as in (1.9). Let us now multiply (2.8) by(1.9) on the right, and consider the result as an operator-valued rational functionof the constrained variables ti, • • • ,t n .Proposition 3. At ti = ... = t n = —\ the ordered product of (2.8) and (1.9)has the valueIT(M)iQklCk + ci + N + M -1E n-T\ ( 1Q ? T », , ) ; (2-9)c111 k + ci + N(M)r.4. r ,4.N4-M-\ì , + M - 1X 'the ordered products in (2.9) are taken over all pairs (k,l) such that the numbers kand I appear in different columns of the tableau 0.Denote the operator (2.9) by FQ(M). If k and I appear in different columnsof 0, thenCk + ci > 3 - Ai - À2 > 3 - N - M.Hence each of the denominators in (2.9) is non-zero for any choice of p. The algebraof operators in (C N )® n generated by all Pu and Qki with 1 ^ k of the operator FQ(M) is contained in the image of EQ .Suppose that M = 0, then p = (0,0,...). In this special case, the image ofthe operator EQ in (C N )® n is equivalent to Y\ as a representation of the groupGLN, see (1.10). It turns out that the image of the operator FQ(0) consists of alltraceless tensors from the image of EQ . In particular, the image of Fn (0) in (C N )® nis equivalent to W\ as a representation of the group OJV- Even in the special caseM = 0, the formulas (2.9) for the operator FQ(M) seem to be new; cf. [20].

Representations of Yangians 6532.4. Let us extend a to an automorphism of the associative algebra \J(gt N ). Forany z £ C, define the twisted evaluation module V(z) over the algebra Y(gl JV ) bypullingthe standard action of the algebra \J(gt N ) in the vector space C^ backthrough the composition of homomorphismsa o a N ° T- z : Y(gl N ) -+ Y(gl N ).The evaluation module V(z) and the twisted evaluation module V(z) over Y(gl N ),have the same restriction to the subalgebra Y(gl N ,a) C Y(gl N ); see (2.1).For any A and p, the operator FQ(M) has the following interpretation in termsof the restrictions to Y(gl N ,a) of tensor products of evaluation modules over theHopf algebra Y(gl N ); cf. Proposition 2. For each k = 1, ..., n put dk = Ck + ^ -\-We assume that A{ + X' 2 ^ N + M and p[ + p' 2 ^ M.Proposition 4. The operator FQ(M) is an intertwiner ofY(gl N ,a)-modulesV(di) ®...® V(d n ) —y V(di) ®...® V(d n ).By Proposition 4, the image of FQ(M) is a submodule in the restriction of thetensor product of evaluation Y(gl JV )-modules V(di)®... ® V(d n ) to the subalgebraY(gljvj

654 Alaxim NazarovReferenees[i[2[3;[4;[6;[9;[io;[n[12:[is;[w;[15[16[17;[18[19;[2o;R. Brauer, On algebras which are connected with the semisimple continuousgroups, Ann. Math., 38 (1937), 857-872.I. Cherednik, Factorized particles on the half-line and root systems, Theor.Math. Phys., 61 (1984), 977-983.I. Cherednik, A new interpretation of Gelfand-Zetlin bases, Duke Math. J., 54(1987), 563-577.V. Drinfeld, Hopf algebras and the quantum Yang-Baxter equation, SovietMath. Doklady, 32 (1985), 254-258.V. Drinfeld, A new realization of Yangians and quantized affine algebras, SovietMath. Doklady, 36 (1988), 212-216.P. Kulish and E. Sklyanin, Quantum spectral transform method: recent developments,Lecture Notes in Phys., 151 (1982), 61-119.A. Alolev, Al.Nazarov and G.Olshanski, Yangians and classical Lie algebras,Russian Math. Surveys, 51 (1996), 205-282.I. Alacdonald, Symmetric Functions and Hall Polynomials, Clarendon Press,1995.A. Alolev and G. Olshanski, Centralizer construction for twisted Yangians, SelectaMath., 6 (2000), 269-317.Al.Nazarov, Yangians and Capelli identities, Amer. Math. Soc. Translations,181 (1998), 139-163.Al. Nazarov, Capelli elements in the classical universal enveloping algebras,Adv. Stud. Pure Math., 28 (2000), 261-285.Al. Nazarov, Representations of twisted Yangians associated with skew Youngdiagrams, in preparation.Al. Nazarov and G. Olshanski, Bethe subalgebras in twisted Yangians, Comm.Math. Phys., 178 (1996), 483-506.Al. Nazarov and V. Tarasov, Representations of Yangians with Gelfand-Zetlinbases, J. Reine Angew. Math., 496 (1998), 181-212.Al. Nazarov and V. Tarasov, On irreducibility of tensor products of Yangianmodules associated with skew Young diagrams, Duke Math. J., 112 (2002),342-378.G. Olshanski, Extension of the algebra U(g) for infinite-dimensional classicalLie algebras g, and the Yangians Y(gl(mj), Soviet Math. Doklady, 36 (1988),569-573.G. Olshanski, Twisted Yangians and infinite-dimensional classical Lie algebras,Lecture Notes in Math., 1510 (1992), 103-120.R. Proctor, Young tableaux, Gelfand patterns, and branching rules for classicalgroups, J. Algebra 164 (1994), 299-360.E. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys.,Series A, 21 (1988), 2375-2389.H. Weyl, Classical Groups, their Invariants and Representations, PrincetonUniversity Press, 1946.

ICAl 2002 • Vol. II • 655-666Automorphic ^-Functionsand FunctorialityFreydoon Shahidi*AbstractThis is a report on the global aspects of the Langlands-Shahidi methodwhich in conjunction with converse theorems of Cogdell and Piatetski-Shapirohas recently been instrumental in establishing a significant number of new andsurprising cases of Langlands Functoriality Conjecture over number fields.They have led to striking new estimates towards Ramanujan and Selbergconjectures.2000 Mathematics Subject Classification: 11F70, 11R39, 11R42, 11S37,22E55.Keywords and Phrases: Automorphic L-function, Functoriality.1. PreliminariesLet F be a number field. For each place v of F, let F v be its completion atv. Assume v is a finite place and let O v denote the ring of integers of F v . Denoteby P v its maximal ideal and fix a uniformizing parameter w v generating P v . Yet[O v : P v ] = q v and fix and absolute value | |„ for which \w v \ v = q^1.Yet G be a quasisplit connected reductive algebraic group over F. Fix an F-Borel subgroup B = TU, where T is a maximal torus of B and U is its unipotentradical. Let A 0 C T be the maximal split subtorus of T. Throughout this article,P is a maximal parabolic subgroup of G, defined over F, with a Levi decompositionP = MN, where M is a Levi subgroup of P and N is its unipotent radical. We willassume P is standard in the sense that N c U. We fix M by assuming T c M.We finally use W to denote the Weyl group of A 0 in G.Let Ap denote the ring of adeles of F and for every algebraic group H over F,let H = H(Ap). Considering H as a group over each F v , we then set H v = H(F V ).Yet A denote the split component of M, i.e., the maximal split subtorus ofthe connected component of the center of M. For every group H defined over F, let*Purdue University, Department of Mathematics, West Lefayette, Indiana 47907, USA. E-mail:shahidi@math.purdue.edu

656 F. ShahidiX(H)p be the group of F-rational characters of H. We set o = Hom(X(M)F,R).Then o* = X(M)p®zR = X(A)p®zR and oj = o* ®R C is the complex dual of o.When G is unramified over a place v, we let K v = G(0„). Otherwise, we shallfix a special maximal compact subgroup K v C G u for which G v = P V K V = B V K V .Yet K = ® V K V Then G = PK = BK. Yet K M = K n M.For each v, the embedding X(M.)p ^y X(M.)p v induces a mapa v = Hom(X(M) Fi ,,R) -• o.There exists a homomorphism HM : M —¥ a defined byexp(x,H M (m))= JJ_\x(m v )\vfor every \ € X(M.)p and m = (m v ). We extend H M to Hp on G by making ittrivial on N and K.Yet a denote the unique simple root of A in N. It can be identified by aunique simple root of A 0 in U. If pp is half the sum of F-roots in N, we set5 = (pp,ct)^1pp£ a*, where for each pair of non-restricted roots a and ß of T,(a,ß) = 2(a,ß)/(ß,ß) is the Killing form.Given a connected reductive algebraic group H over F, let L H be its L-group.Considering H as a group over F v , we then denote by L H V its L-group over F v .Yet L H° = L H® be the corresponding connected component of 1. We then have anatural homomorphism from L H V into L H. We let r) v : L M V —t L M be this mapfor M (cf. [4]).Let L N be the L-group of N defined naturally in [4]. Let L n be its ( complex )mLie algebra, and let r denote the adjoint action of L M on L n. Decompose r = 0 r ti=1~to its irreducible subrepresentations, indexed according to the values (a, ß) = i asß ranges among the positive roots of T. Alore precisely, Xßy £ L n lies in the spaceof i'i if and only if (a,ß) = i. Here Xßy is a root vector attached to the coroot ß v ,considered as a root of the L-group. The integer m is equal to the nilpotence classof L n. We let r i:V = r t • r/ v for each i (cf. [34,40,41]).If A denotes the set of simple roots of A 0 in U, we use 9 C A to denote thesubset generating Al. Then A = 6 U {a}. There exists a unique element WQ £ Wsuch that WQ(9) C A, while wo(ct) < 0. We will always choose a representative WQfor wo in G(F) and use WQ to denote each of its components.V2. Eisenstein series and .L-funetionsLet n = ® v n v be a cusp form on M. Given a F^M-finite function ip in thespace of ir, we extend (p to a function fi> on G as in Section 2 of [39] as well as in[17], and for s £ C, sets(9) = tp(g)exp{sà + pp,Hp(g)). (2.1)

Automorphic L-functions and Functoriality 657The corresponding Eisenstein series is then defined byE(s,^,g,P)= J2 > 0, define the global intertwiningoperator M(s,ir) byM(s,n)f(g)= f f(ufi 1 n'g)dn' (g£G). (2.3)JN'Observe that if / = ® v f v , then for almost all v, f v is the unique K v -fixed functionsnormalized by f v (e v ) = 1. Finally, if at each v we define a local intertwiningoperator bythenA(s,n v , w 0 ) fv(g) = / f v (wô 1 n'g)dn', (2.4)M(S,TT) = ® V A(S,TT V ,WQ). (2.5)It follows form the general theory of Eisenstein series that the poles ofE(s,ip,g,P), as fi> and g vary, are the same as those of M(s,ir), and for Re(s) > 0,they are all simple and finite in number, with none on the line Re(s) = 0 (cf.[17,33,35]).By construction each

658 F. ShahidiThe main result of [34, also see 40] is thatM(s,n)f = ®vesA(s,n v ,w 0 )fv®®v(sfvmx JJ L s (is, n, n)/L s (l + is, n, n), (2.9)«=iwhere / = ® v f v is such that for each v $ S, f v is the unique K v -fixed functionin I(s,n v ) normalized by f v (e v ) = 1 and for each i, r t denotes the contragredientof ì'ì,ìJ= 1, • • • ,m, the irreducible components of the adjoint action of L M or L N.Here /„ is the F^-fixed function in the space of I(-s,wo(ir v j), normalized the sameway. Moreover /„ and /„ are identified as elements in spherical principal series.3. Generic representations and the non-eonstanttermSuppose F is a field, either local or global, and G is as before, with a Boreisubgroup B = TU over F. Fix an F-splitting {X a >}, i.e., a collection of rootvectors as a' ranges over simple roots of T in U which is invariant under the actionof FF = Gal(F/F). This then determines a map form U to EG a , ip(u) = (x a i) a i,where x a i is the «'-coordinate of« with respect to {X a >}. Yet {K Q /} be a collectionof elements in F such that

Automorphic L-functions and Functoriality 659a canonical Whittaker functional for I(s, n v ). Changing the splitting we now assumeK a i = 1. It now follows from Rodier's theorem that there exists a complex function(of«), C Xv (s,ir v ), depending on n v ,Xv an d wo such that (cf. [41,42,43])X x A s^v) = C Xv (s,n v )X Xv (s,w 0 (n v )) • A(s,n v ,w 0 ). (3.3)This is what we call the Local Coefficient attached to s,n v ,Xvof wo is now specified by our fixed splitting as in [43].Finally, ifan d wo- The choiceE x (s, s ,g,P) = E(s, s ,ug,P)x(u)du (3.4)J\J(F)\Uis the x-nonconstant term of the Eisenstein series, then ([7,41,42])mE x (s,„e,P)= YlW^JlLsil + is^ri)- 1 , (3.5)ves ì=iwhere now S is assumed to have the property that if v $ S, then Xv is also unramified.Applying Definition (3.4) to both sides of (2.6), using (3.5) now implies thecrude functional equation ([40,41])mmJJ_Ls(is,ir,ri) = JJ C Xv (s,n v ) JJ L s (l -is,n,ri). (3.6)4. The main induction, functional equations andmultiplicativityTo prove the functional equation for each r t with precise root numbers andL-function, we use (cf. [42]):Proposition 4.1. Given 1 of M t = M.i(Ap), unramified for every v ^ S,m'such that if the adjoint action r' of L M t on L n, decomposes as r' = Q) r'-, thenLs(s,n,ri) = L s (s,n',r[).Moreover m' < m.Remark 4.2. As was observed by Arthur [1], each Mj can be taken equalto M and n' = n. In fact each G, can be taken to be an endoscopic group for G,sharing M as a Levi subgroup. We shall record this asProposition 4.3. Given i, 1 < i < m, there exist a quasisplit connectedreductive F-group with M as a Levi subgroup and m' < m for which r[ = r t .3=1

660 F. ShahidiUsing this induction and local-global arguments (cf. Proposition 5.1 of [42]),it was proved in [42] thatTheorem 4.4. (Theorems 3.5 and 7.7 of [42]) a) For each i, 1 of apolynomial in q^swhose constant term is 1, if v < oo, and is the Artin L-functionattached to r t • ip' v , where tp' v : W' F —¥ L M V is the homomorphism of the Deligne-Weil group into L M V parametrizing n v , if either v = oo or n v has an Iwahori-fixedvector; and a root number e(s, / K v ,r i v ,ip v ) satisfying the same provisions, such thatifL(s,n,ri) = JJ_L(s,Tr v ,r i:V ) (4.1)andthen6) Lete(s,n,ri) = JJ_e(s,Tr v ,r i:V ,fi v ), (4.2)VL(s, / K,r i ) = e(s, / K,r i )L(l — s, 7r,Fj). (4.3)-y(s,ir v ,r i:V ,fi v ) = e(s,Tr v ,r i:V ,fi v )L(l - s,n v ,r iiV )/L(s,n v ,r iiV ). (4.4)Then each / y(s,ir v ,ri V ,fi v ) is multiplicative in the sense of equation (3.13) in Theorem3.5 of [42]. (See below.) Ifn v is tempered, then-f(s,ir v ,ri :V ,fi v ) determines thecorresponding root number and L-function uniquely and in fact that is how they aredefined. Suppose n v is non-tempered, then each L(s, Tr v ,r i:V ) is determined by meansof the analytic continuation of its quasi-tempered Langlands parameter and multiplicativityof corresponding 7 -functions. More precisely, if a v is the quasitemperedLanglands parameter that gives n v as a subrepresentation, thenL(s,TT v ,r i}V ) = JJ L(s,Wj(a v ),r' i{jhv ), (4.5)where the notation is as in part 3) of Theorem 3.5 of [42], provided that everyL-function on the right hand side is holomorphic for Re(s) > 0, whenever a v is(unitary) tempered (Conjecture 7.1 of [42], proved in many cases [3.6.42]). Theset Si,Wj and ifi.-. are defined as follows in which we drop the index v. Assumen C Lrad Me) (jv e) nM)tM (T ® 1J where M§(N^nM) is a parabolic subgroup o/M definedby a subset 9 C A, the set of simple roots of A 0 . Let 9' = WQ(9) C A and fix areduced decomposition WQ = w„-i • • • wi of WQ (Lemma 2.1.1 of [41])- For each j,there exists a unique root aj £ A such that Wj(ctj) < 0. For each j, 2 < j < n — 1,let Wj = Wj-i • • • wi- Set wi = 1. Let iij = 9j U {aj}, where 9i =9, 9 n = 9',and 9j + i = Wj(9j), 1 < j < n — 1. Then Mf^. contains M^.(N^. n MQ.) as amaximal parabolic subgroup andwj(a) is a representation of M$ i . The L-group L M$acts on the space of ri, but no longer necessarily irreducibly. Given an irreducibleconstituent of this action, there exists a unique j, 1 < j < n — 1, which under Wjis equivalent to an irreducible constituent of the action of L M$ j on the Lie algebraof the L-group of N^. n Mn r Let i(j) be the index of this subspace and denote by

Automorphic L-functions and Functoriality 661ìfi,-. the action of L M$ j on it. Finally, let Si denote the set of all such j's for agiven i. (See Theorem 3.5 and Section 7 of [42]. Also see the discussion just beforeProposition 5.2 of [2.8].)Remark 4.5. If G = GL t+ „,M = GL t x GL n and IT = ® v n v and n' =® V TT' V are cuspidal representations of GL t (Ap and GL n (Ap), then m = 1 andL(s,n ® n',ri) is precisely the Rankin-Selberg product L-function L(s,n x n') attachedto (7T, 7r') (cf. [21,43,44]). In this case each of the local L-functions androot numbers are precisely those of Artin through parametrization which is nowavailable for GLN(F V ) for any N due to the work Harris-Taylor [18] and Henniart[19]. As we explain later, this will also be the case for many of our local factors asa result of our new cases of functoriality which we shall soon explain. This is quiteremarkable, since our factors are defined using harmonic analysis, as opposed to thevery arithmetic nature of the definition given for Artin factors. This is a perfectexample of how deep Langlands' conjectures are.Remark 4.6. The multiplicativity of local factors, in the sense of Theorem3.4, are absolutely crucial in establishing our new cases of functoriality throughoutour proofs [12,23,28]. In fact, not only do we need them to prove our strong transfers,they are also absolutely necessary in establishing our weak ones.5. Twists by highly ramified characters, holomorphyand boundednessWhile the functional equations developed from our method are in perfect shapeand completely general, nothing that general can be said about the holomorphyand possible poles of these L-functions. On the other hand, there has recently beensome remarkable new progress on the question of holomorphy of these L-function,mainly due to Kim [24,25,31]. They rely on reducing the existence of the polesto that of existence of certain unitary automorphic forms, which in turn points tothe existence of certain local unitary representations. One then disposes of theserepresentations, and therefore the pole, by checking the corresponding unitary dualof the local group. In view of the functional equation, this needs to be checked onlyfor Re(s) > 1/2. In fact, to carry this out, one needs to verify that:Certain local normalized (as in [41]) intertwining operatorsare holomorphic and non-zero for Re(s) > 1/2, (5.1)in each case [24,25,31]. The main issue is that one cannot always get such a contradictionand rule out the pole. In fact, there are many unitary duals whosecomplementary series extend all the way to Re(s)=l.On the other hand, if one considers a highly ramified twist -n n (see Theorem5.1 below) of ir, then it can be shown quite generally that every L(s,ir rj ,ri) is entire(cf. [45] for its local analogue). In fact, if n is highly ramified, then wo(7r^) ^ -n n ,whose negation is a necessary condition for M(s,ir rj ) to have a pole, a basic factfrom Langlands spectral theory of Eisenstein series (Lemma 7.5 of [33]). This wasused by Kim [24], and in view of the present powerful converse theorems [8,9], that

662 F. Shahidiis all one needs to prove our cases of functoriality [12,23,28,30]. To formalize this,we borrow the following proposition (Proposition 2.1) from [28], in order to statethe result. It is a consequence of our general induction (Propositions 4.1 and 4.3)and [24].Theorem 5.1. Assume (5.1) is valid. Then there exists a rational character£ £ X(M.p) with the following property: Let S be a non-empty finite set of finiteplaces of F. For every globally generic cuspidal representation n of M = M.(Ap),there exist non-negative integers f v , v £ S, depending only on the local centralcharacters of n v for all v £ S, such that for every grössencharacter n = ® v r/ v of Ffor which conductor of r) v , v £ S, is larger than or equal to f v , every L-functionL(s,TT„,i'i), i = 1, • • • ,rn, is entire, where -n n = n® (n • £). The rational character £can be simply taken to be £(m) = det(Ad(ro)|n), m £ M, where n is the Lie algebrao/N.The last ingredient in applying converse theorems is that of boundedness ofeach L(s, / K,r i ) in every vertical strip of finite width, away from its poles, whichare finite in number, again using the functional equation and under Assumption(5.1). This was proved in full generality by Gelbart-Shahidi [15], using the theoryof Eisenstein series via [33] and [36]. The main theorem of [15] (Theorem 4.1) is infull generality, allowing poles for L-functions. Here we will state the version whichapplies to our -n n .Theorem 5.2. Under Assumption (5.1), let £ and n be as in Theorem 5.1.Assume n is ramified enough so that each L(s,ir rj ,ri) is entire. Then, given a finitereal interval I, each L(s,ir rj ,ri) remains bounded for all s with Re(s) £ I.The main difficulty in proving Theorem 5.2 is having to deal with reciprocalsof each L(s, / K,r i ), 2 of the criticalstrip, whenever m > 2, which is unfortunately the case for each of our cases offunctoriality. We handle this by appealing to equations (3.5) and estimating thenon-constant term (3.4) by means of [33,36].6. New cases of functorialityLanglands functoriality predicts that every homomorphism between L-groupsof two reductive groups over a number field, leads to a canonical correspondencebetween automorphic representations of the two groups. The following instances offunctoriality are quite striking and are consequences of applying recent ingeniousconverse theorems of Cogdell and Piatetski-Shapiro [8,9] to certain classes of L-functions whose necessary properties are obtained mainly from our method. (See[20] for an insightful survey.) We refer to [11] for more discussion of these results andthe transfer from GL2(Ap)xGL2(Ap) to GL 4 (AF), using Rankin-Selberg methodby Ramakrishnan [37]. (See [23] for a proof using our method.)6.a. Let ni = ® V TTI V and 7T2 = ® v n 2v be cuspidal representations of GL 2 (Ap)and GLs(Ap), respectively. For each v, let pi v be the homomorphism of Deligne-Weil group into GLj + i(C), parametrizing m v , i = 1,2. Let Trit, M TT2 V be theirreducible admissible representation of GL 6 (F V ) attached to pi v ® p2 V via [18,19].Set 7Ti M 7T2 = ® V (TTI V M n 2v ), an irreducible admissible representation of GL 6 (Ap).

Automorphic L-functions and Functoriality 663Next, let n = m, n v = m v and p v = pi v . Yet Sym 3 (iT v ) be the irreducibleadmissible representation of GLfiF v ) attached to Sym 3 (p v ) and set Sym 3 (7r) =® v Sym 3 (TT V ), an irreducible admissible representation of GLfiAp). We have:Theorem 6.1 [28,30]. a) The representations ni M 7r 2 and Sym 3 (n) areautomorphic.b) Sym 3 (n) is cuspidal, unless n is either of dihedral or of tetrahedral type.In view of [9], one needs to show that L(s, (ni M n 2 ) x (a ® nj) is nice in thesense that it satisfies the contentions of Theorems 4.4.a, 5.1 and 5.2 for a highlyramifiedgrössencharacter n, where a is a cuspidal representation of GL n (Ap), n =1,2,3,4, which is unramified in every place v where either ni v or n 2v is ramified.In particular for each v, one of TTi v ,ir2 V or a v is in the principal series. It thenfollows from multiplicativity (cf. Theorem 4.4) and the main results of [43,44],that these L-functions are equal to certain L-functions defined from our method.Alore precisely, we can take (G,M) to be: a) G = SL 5 , M^ = SL 2 x SL 3 ; b)G = Spin(lO), M D = SL 3 x SL 2 x SL 2 ; c) G = E S 6 C , M D = SL 3 x SL 2 x SL 3 ;d) G = Ff c , Mß = SL 3 x SL 2 x SL4, according as n = 1,2,3,4, respectively.This leads to a proof that ni M 7r 2 is weakly automorphic. The strong transferrequires a lot more work, involving base change, both normal [2] and non-normal[22], and finally a local result [5]. Automorphy of Sym 3 (7r) is a consequence ofapplying the first part to (7r,Ad(7r)), where Ad(7r) is the adjoint of n, establishedby Gelbart-Jacquet [14]. It does not require the use of [5].Observe that we have in fact proved that the homomorphisms GL 2 (C) ®GLfiC) C GL 6 (C) and Sym 3 : GL 2 (C) —¥ GLfiC) are functorial. Neither areendoscopic.6.b. Let II = „n„ be a cuspidal representation of GLfiAp) and let A 2 :GL 4 (C) —¥ GL 6 (C) be the exterior square map. Also with n as in 6.a, let Sym 4 (7r) =® v Sym 4 (TT V ), where Sym 4 (7^) is attached to Sym 4 (p v ). ThenTheorem 6.2 (cf. [23]). a) The map A 2 is weakly functorial, in the sensethat there exists an automorphic form on GL 6 (Ap) whose local components areequal to A^n^) for all v, except if v\2 or v\3. Here A^n^) is defined by the localLanglands conjecture [18,19].b) Sym (TT) is an automorphic representation ofGL 5 ( i F,We point out that b) is obtained by applying a) to Sym (TT). a) is proved byapplyingour method to Spin groups (Case D n — 1 of [40], n = k + A, k = 0,1,2,3).Proposition 6.3 (cf. [29]). Sym 4 (Tr) is cuspidal unless TT is either of dihedral,tetrahedral or octahedral type.Let n = ® v n v be a cusp form on GL 2 (Ap). For each unramified v, let a v andß v be the Hecke eigenvalues of n v . Then as corollary to Proposition 6.3 we havethe following striking improvements towards Ramanujan and Selberg conjectures.Corollary 6.4. a) (cf. [29]) Assume F is an arbitrary number field. Thenqv 1/9 < K| and \ß v \ < ql /9 . b) (cf. [27]). Assume F = Q. Then p- 7 / M < \a p \and \ß p \ of the Laplace operator on L 2 (F n)975for every congruence subgroup Y satisfies Xi(Y) > = 0.2380 • • •40966.C. Let i : Sp2 n (C) ^ GL 2 „(C) be the natural embedding. Let TT = ® V TT V

664 F. Shahidibe a generic cuspidal representation of S0 2n +i(Ap). For each unramified v, let{A v } C Sp2n(C) be the Hecke-Frobenius conjugacy class parametrizing TT V . Yetì(TT V ) be the unramified representation of GL 2n (F v ) attached to {i(A v )}. Then themain theorem of [12] proves:Theorem 6.5 [12]. The embedding i is weakly functorial, i.e., there exist anautomorphic representation of GL 2n (Ap) whose components are equal to ì(TT V ) foralmost all v.This is proved by applying our method to maximal parabolics of appropriateodd special orthogonal groups (Case B n of [40]). The strong transfer is now alsoestablished by Ginzburg-Rallis-Soudry [16] as well as Kim [26] by building uponTheorem 6.5.Final Comments. Many other cases are in progress. Among them are a proofof the existence of Asai transfer [32] using our method, which was originally provedby Ramakrishnan [38], using the Rankin-Selberg method. This is the first case whereone needs to use quasisplit groups. Since the issue of stability of root numbers [10](cf. [11]) seems to be close to being settled by means of our method [46], many otherstransfers should now be available. A similar approach for nongeneric representationswas initiated in [13].References[i[2[io;J. Arthur, Endoscopic L-functions and a combinatorial indentity, Dedicated toH.S.A1. Coxeter, Canad. J. Alath., 51 (1999), 1135-1148.J. Arthur and L. Clozel, Simple algebras, Base change, and the Advanced Theoryof the Trace Formula, Annals of Alath. Studies, no. 120, Princeton UniversityPress, 1989.Al. Asgari, Local L-functions for split spinor groups, Canad. J. Alath., (toappear).A. Borei., Automorphic L-functions, Automorphic Forms and AutomorphicRepresentations, Proc. Sympos. Pure Alath., vol. 33; II, Amer. Alath. Soc,Providence, RI, 1979, 27^61.C. J. Bushnell and G. Henniart, On certain dyadic representations, Annals ofAlath., Appendix to [28], (to appear).W. Casselman and F. Shahidi, On irreducibility of standard modules for genericrepresentations, Ann. Scient. Ec.Norm.Sup., 31 (1998), 561^589.W. Casselman and J. A. Shalika, The unramified principal series of p-adicgroups II; The Whittaker function, Comp. Alath., 41 (1980), 207-231.J. W. Cogdell and I. I. Piatetski-Shapiro, Converse theorems for GL n , Pubi.Alath. IHES, 79 (1994), 157^214., Converse theorems for GL n II, J. Reine Angew. Alath., 507 (1999),165^188., Stability of gamma factors for SO(2n + 1), Alanuscripta Alath., 95(1998), 437-461., Converse Theorems, Functoriality and Applications to Number Theory,These Proceedings.

Automorphic L-functions and Functoriality 665[12] J. W. Cogdell, H. Kim, I. I. Piatetski-Shapiro and F. Shahidi, On lifting fromclassical groups to GL N , Pubi. Alath. IHES, 93 (2001), 5^30.[13] S. Friedberg and D. Goldberg, On local coefficients for nongeneric representationsof some classical groups, Comp. Alath., 116 (1999), 133^166.[14] S. Gelbart and H. Jacquet, A relation between automorphic representations ofGL(2) and,GL(3), Ann. Scient. Éc. Norm. Sup., 11( 1978), 471^552.[15] S. Gelbart and F. Shahidi, Boundedness of automorphic L-functions in verticalstrips, Journal of AA1S, 14 (2001), 79^107.[16] D. Ginzburg, S. Rallis and D. Soudry, Generic automorphic forms onSO(2n+l): functorial lift to GL(2n), endoscopy and base change, IMRN 14(2001), 729^764.[17] Harish-Chandra, Automorphic forms on semisimple Lie groups, SLN 62 (1968),Berlin-Heidelberg-New York.[18] Al. Harris and R. Taylor, On the geometry and cohomology of some simpleShimura varieties, Annals of Alath. Studies, no. 151, Princeton University-Press, 2001.[19] G. Henniart, Une preuve simple des conjectures de Langlands pour GL(n) surun corps p-adique, Invefnt. Alath., 139 (2000), 439—155.[20] G. Henniart, Progrès récents en fonctorialité de Langlands, Seminaire Bourbaki,Juin 2001, Exposé 890, 890-1 to 890-21.[21] H. Jacquet, I. Piatetski-Shapiro and J. Shalika, Rankin-selberg convolutions,Amer. J. Alath., 105 (1983), 367-164.[22] , Relvement cubique non normal, C. R. Acad. Sci. Paris Sr. I Alath.,292, no. 12 (1981), 567-571.[23] H. Kim, Functoriality for the exterior square of GL4 and symmetric fourth ofGL 2 , preprint (2000).[24] , Langlands-Shahidi method and poles of automorphic L-functions: Applicationto exterior square L-functions, Can. J. Alath., 51 (1999), 835-849.[25] , Langlands-Shahidi method and poles of automorphic L-functions II,Israel J. Alath., 117 (2000), 261^284.[26] , Residual spectrum of odd orthogonal groups, IMRN, 17 (2000), 873^906.[27] H. Kim and P. Sarnak, Refined estimates towards the Ramanujan and Selbergconjectures, Appendix 2 to [23].[28] H. Kim and F. Shahidi, Functorial products for GL 2 x GL ?J and the symmetriccube for GL 2 , Annals of Alath., (to appear).[29] , Cuspidality of symmetric powers with applications, Duke Alath. J., 112(2002), 177^197.[30] , Functorial products for GL 2 x GL ?J and functorial symmetric cube forGL 2 , CR. Acad. Sci. Paris, 331 (2000), 599^604.[31] , Symmetric cube L-functions for GL 2 are entire, Ann. of Alath., 150(1999), 645^662.[32] Al. Krishnamurthy, The weak Asai transfer to GL(4) via Langlands-Shahidimethod, Thesis, Purdue University (2002).[33] R. P. Langlands, On the Functional Equations Satisfied by Eisenstein Series,

666 F. ShahidiLecture Notes in Alath., Vol 544, Springer-Verlag, 1976.[34] , Euler Products, Yale University Press, 1971.[35] C. Moeglin and J.-L. Waldspurger, Spectral decomposition and Eisenstein series,Cambridge Tracts in Alath., vol. 113, Cambridge University Press, 1995.[36] W. Müller, The trace class conjecture in the theory of automorphic forms, Ann.of Alath., 130 (1989), 473^529.[37] D. Ramakrishnan, Modularity of the Rankin-Selberg L-series, and multiplicityone for SL(2), Ann. of Alath., 152 (2000), 45-111.[38] , Modularity of solvable Artin representations of GO(4)-type, IMRN, 1(2002), 1-54.[39] F. Shahidi, Functional equation satisfied by certain L-functions, Comp. Alath.,37(1978), 171^207.[40] , On the Ramanujan conjecture and finiteness of poles for certain L-functions, Ann. of Alath., 127 (1988), 547^584.[41] , On certain L-functions, Amer, J. Alath., 103(1981), 297^355.[42] , A proof of Langlands conjecture on Plancherel measures; Complementaryseries for p-adic groups, Annals of Alath., 132 (1990), 273^330.[43] , Local coefficients as Artin factors for real groups, Duke Alath. J., 52(1985), 973^1007.[44] , Fourier transforms of intertwining operators and Plancherel measuresforGL(n), Amer. J. of Alath., 106 (1984), 67-111.[45] , Twists of a general class of L-functions by highly ramified characters,Canad. Alath. Bull., 43 (2000), 380^384.[46] , Local coefficients as Mellin transforms of Bessel functions; Towards ageneral stability, preprint (2002).

ICAl 2002 • Vol. II • 667^677Modular Representations of p-adic Groupsand of Affine Hecke AlgebrasMarie-France Vignéras*AbstractI will survey some results in the theory of modular representations of areductive p-adic group, in positive characteristic tfip and I = p.2000 Mathematics Subject Classification: 11S37, 11F70, 20C08, 20G05,22E50.Keywords and Phrases: Modular representation, Reductive p-adic group,Affine Hecke algebra.Introduction The congruences between automorphic forms and their applicationsto number theory are a motivation to study the smooth representations of areductive p-adic group G over an algebraically closed field R of any characteristic.The purpose of the talk is to give a survey of some aspects of the theory of R-representations of G. In positive characteristic, most results are due to the author;when proofs are available in the littérature (some of them are not !), references willbe given.A prominent role is played by the unipotent block which contains the trivialrepresentation. There is a finite list of types, such that the irreducible representationsof the unipotent block are characterized by the property that they contain aunique type of the list. The types define functors from the ^-representations of Gto the right modules over generalized affine Hecke algebras over R with differentparameters; in positive characteristic £, the parameters are 0 when £ = p, and rootsof unity when tfip.In characteristic 0 or tfip, for a p-adic linear group, there is a Deligne-Langlandscorrespondence for irreducible representations; the irreducible in the unipotent blockare annihilated by a canonical ideal J; the category of representations annihilatedby J is Alorita equivalent to the affine Schur algebra, and the unipotent block isannihilated by a finite power J k .* Institut de Mathématiques de Jussieu, Université de Paris 7, France. E-mail:vigneras@math.jussieu.fr

668 M.-F. VignérasNew phenomena appear when £ = p, as the supersingular representations discoveredby Barthel-Livne and classified by Ch. Breuil for GL(2,Q P ). The modulesfor the affine Hecke algebras of parameter 0 and over R of characteristic p, aremore tractable than the ^-representations of the group, using that the center Z ofa Z[q]-affine Hecke algebra H of parameter q is a finitely algebra and H is a generated^-module. The classification of the simple modules of the pro-p-Iwahori Heckealgebra of GL(2, F) suggests the possibility of a Deligne-Langlands correspondencein characteristic p.Complex caseNotation. C is the field of complex numbers, G = G (F) is the group of rationalpoints of a reductive connected group G over a local non archimedean field F withresidual field of characteristic p and of finite order q, and Mode G is the categoryof complex smooth representations of G. All representations of G will be smooth,the stabilizer of any vector is open in G. An abelian category C is right (left)Alorita equivalent to a ring A when C is equivalent to the category of right (left)A-modules.The modules over complex affine Hecke algebras with parameter q are relatedby the Borei theorem to the complex representations of reductive p-adic groups.Borei Theorem The unipotent block of Mode G is (left and right) Moritaequivalent to the complex Hecke algebra of the affine Weyl group of G with parameterq.The proof has three main steps, in reverse chronological order, Bernstein a)[B] [BK], Borei b) [Bo], [C], Iwahori-Alatusmoto c) [IM], [Al].a) (l.a.l) Mode G is a product of indecomposable abelian subcategories "theblocks".The unipotent block contains the trivial representation. The representationsin the unipotent block will be called unipotent, although this term is already usedby Lusztig in a different sense.(l.a.2) The irreducible unipotent representations are the irreducible subquotientsof the representations parabolically induced from the unramifìed charactersof a minimal parabolic subgroup ofG.b) Let J be an Iwahori subgroup of G (unique modulo conjugation).(l.b.l) The category of complex representations of G generated by their I-invaxiant vectors is abelian, equivalent by the functorV ^ V 1 = Homc G (C[J\G],F)to the category Alod He (G, I) of right modules of the Iwahori Hecke algebraH c (G,I) = End CG C[I\G].

Modular Representations 669(l.b.2) This abelian category is the unipotent block.c) (l.c) The Iwahori Hecke algebra HQ(G, I) is the complex Hecke algebra ofthe affine Weyl group of G with parameter q.The algebra has a very useful description called the Bernstein decomposition[LI] [BK], basic for the geometric description of Kazhdan-Lusztig [KL].From (l.b.l), the irreducible unipotent complex representations of G are innatural bijection with the simple modules of the complex Hecke algebra HQ(G,I).By the "unipotent" Deligne-Langlands correspondence, the simple HQ(G, J)-modules"correspond" to the G'-conjugacy classes of pairs (s,N), where s £ G' is semisimple,N £ Lie G' and Ad(s)N = qN, where G' is the complex dual group of G withLie algebra LieG'. This is known to be a bijection when G = GL(n,F) [Z] [R].When G is adjoint and unramified (quasi-split and split over a finite unramifiedextension), it is also known to be a bijection if one adds a third ingredient, a certainirreducible geometric representation p of the component group of the simultaneouscentralizer of both s and N in G'; this is was done by Chriss [C], starting from thebasic case where G is split of connected center treated by Kazhdan Lusztig [KL]and by Ginsburg [CG] *. The adjoint and unramified case is sufficient for manyapplicationsto automorphic forms; to my knowledge the general case has not beendone.According to R. Howe, the complex blocks should be parametrized by types.The basic type, the trivial representation of an Iwahori subgroup, is the type ofthe unipotent block. An arbitrary block should be right Alorita equivalent to theHecke algebra of the corresponding type. The Hecke algebra of the type shouldbe a generalized affine Hecke complex algebra with different parameters equal topositive powers of p. This long program started in 1976 is expected to be completedsoon. The most important results are those of Bushnell-Kutzko for GL(n, F) [BK],of Alorris for the description of the Hecke algebra of a type [Al], of Moy and Prasadfor the definition of unrefined types [MP].Conjecturally, the classification of simple modules over complex generalizedaffine Hecke algebras and the theory of types will give the classification of thecomplex irreducible representations of the reductive p-adic groups.We consider now the basic example, the general linear p-adic group GL(n, F).The the complex irreducible representations of GL(n,F) over R are related by the"semi-simple" Deligne-Langlands correspondence (proved by Harris-Taylor [HT1]Introduction page 18. Complex representations of the absolute Weil-Deligne group withsemi-simple part trivial on the inertia subgroup (6.1) are in natural bijection with the £-adicrepresentations of the absolute Weil group trivial on the wild ramifìcation subgroup for anyprime number ifip[T] [D]. In the Deligne-Langlands correspondence, one considers only therepresentations which are Frobenius semi-simple.

670 M.-F. Vignérasof a sepand Henniart [He]), to the representations of the Galois group Gal(F/F)arable algebraic closure F of F.Deligne-Langlands correspondence(l.d) The blocks of Mode GL(n, F) are parametrized by the conjugacy classesof the semi-simple n-dimensional complex representations r of the inertia groupI(F/F) which extend to the Galois group Gal(F/F).(l.e) The block parametrized by r is equivalent to the unipotent block ofa product of linear groups G T = GL(di,F{) x ... x GL(d r ,F r ) over unramifìedextensions F t of F where ^d»[.Fj : F] = n.(l.f) The irreducible unipotent representations ofGL(n, F) are parametrizedby the G'L(n,R)-conjugacy classes of pairs (s,N) where s £ GL(n,C) is semisimple,N £ M(n, C) is nilpotent, and sN = qNs.Modular case Let R be an algebraically closed field of any characteristic.When the characteristic of R is 0, the theory of representations of G is essentiallylikethe complex theory, and the above results remain true although some proofsneed to be modified and this is not always in the littérature. From now on, wewill consider "modular or modi" representations, i.e. representations over R ofcharacteristic £ > 0.Banal primes Although a reductive p-adic group G is infinite, it behavesoften as a finite group. Given a property of complex representations of G whichhas formally a meaning for mod £ representations of G, one can usually prove thatoutside a finite set of primes £, the property remains valid. This set of primes iscalled "banal" for the given property.For mod £ representations the Borei theorem is false, because the mod £ unipotentblock of GL(2, F) contains representations without Iwahori invariant vectorswhen q = — lmodl [VI].(2) The Borei theorem is valid for modi representations when £ does notdivide the pro-order of any open compact subgroup ofG.These primes are banal for the three main steps in the proof of the complextheorem.a) (2.a) Any prime is banal for the decomposition of Modp G in blocks.The complex proof of (l.a.l) does not extend. There is a new proof relying ofthe theory of unrefined types [V5 III.6] when £fip.b) (l.a.2), (l.b.l), (l.b.2) remain true because £ does not does not divide thepro-order of the Iwahori subgroup J and [V2] [V4]:(2.b) Any irreducible cuspidal mod ^representation of G is injective and projectivein the category of mod ^representations of G with a given central characterwhen £ is as in theorem 2.

Modular Representations 671e) Any prime £ is banal for the Iwahori-Alatsumoto step because the proofs ofIwahori-Alatusmoto and of Alorris are valid over Z, and for any commutative ringA, the Iwahori Hecke A-algebraH A (G, I) = End AG A[I\G] ~ H Z (G, I) ® z Ais isomorphic to the Hecke A-algebra of the affine Weyl group of G with parameterqA where qA is the natural image of q in A.The primes £ of theorem 2 are often called tie banal primes of G becausesuch primes are banal for many properties. For example, the category of mod £-representations of G with a given central character has finite cohomological dimension[V4]. In the basic example GL(n, F), £ is banal when £fip and the multiplicativeorder of q modulo £ is > n.Limit primes The set of primes banal for (l.a.2), (l.b.l) is usually largerthan the set of banal primes of G. The primes of this set which are not banal willbe called, following Harris, the limit primes of G. In the basic example GL(n, F),the limit primes £ satisfy q = 1 mod £ and £ > n [V3]. For number theoretic reasons,the limit primes are quite important [DT] [Be] [HT2]. They satisfy almost all theproperties of the banal primes. For linear groups, the limit primes are banal for theproperty that no cuspidal representation is a subquotient of a proper parabolicallyinduced representation. This is may be true for G general.Let Qf be an algebraic closure of the field Q( of l-adic numbers, Z,( its ring ofintegers and F,( its residue field. The following statements follow from the theory oftypes, or from the description of the center of the category of mod £ representations(the Bernstein center).(3.1) The reduction gives a surjective map from the isomorphism classes ofthe irreducible cuspidal integral Q f -representations of G to the irreducible cuspidalF (-representations of G, when £ is a banal or a limit prime for G.Natural characteristic The interesting case where the characteristic of Ris p is not yet understood. There is a simplification: ^-representations of G havenon zero vectors invariant by the pro-p-radical I p of I. The irreducible are quotientsof R[I P \G].Some calculations have been made for GL(2,F) [BY] [Br] [V9]. A direct classificationof the irreducible ^-representations of G = GL(2, Q p ) [BL] [Br] and ofthe pro-p-Iwahori Hecke A-algebra Hp(G,I p ) = Endp(} R[I P \G] (called a modppro-p-Iwahori Hecke algebra) shows:(4.1) Suppose R of characteristic p. The pro-p-Iwahori functor gives a bijectionbetween the irreducible R-representations of GL(2,Q P ) and the simple rightHp(G, Ip)- modules.This is the "modp simple Borei theorem" for the pro-p-Iwahori group of

672 M.-F. VignérasGL(2,Q P ). In particular p is banal for the simple version of (l.b.l) when G =GL(2,Q P ). Irreducible modp representations of GL(2, F) which are non subquotientsof parabolically induced representations from a character of the diagonal torusare called supersingular [BL]. There is a similar definition for the modp simple supersingularmodules of the pro-p-Iwahori Hecke algebra of GL(2, F).(4.2) There is a natural bijection between the modp simple supersingularmodules of the modp pro-p-Iwahori Hecke algebra of GL(2, F) and the modpirreducible dimension 2 representations of the absolute Weil group of F.This suggests the existence of a modp Deligne-Langlands correspondence.Some computations are beeing made by R. Ollivier for GL(3, F).We end this section with a new result on affine Hecke algebras as in [L3], whichis important for the theory of representations modulo p.(4.3) Let H be an affine Hecke Z [q]-algebra of parameter q associated to ageneralized affine Weyl group W. Then the center Z of H is a finitely generatedZ[q]-algebra and H is a finitely generated Z-module.The key is to prove that H has a Z[g]-basis (q kÊ w ) we w where (E w ) is aBernstein Z[g _1 ]-basis of H[q^]. The assertion (4.3) was known when the parameterq is invertible.Non natural characteristic R an algebraically closed field of positive characteristic£fip. Any prime £fip is banal for the "simple Borei theorem". The "simpleBorei theorem" is true mod £fip.(5.1) Suppose £fip. The Iwahori-invariant functor gives a bijection betweenthe irreducible R-representations of G with Y rI fiO and the simple right Hp(G,I)-modules.The existence of an Haar measure on G with values in R implies that Alod/j G isleft Alorita equivalent to the convolution algebra Hp(G) of locally constant, compactdistributions on G with values in R. When the pro-order of J is invertible in R, theHaar measure on G over R normalized by J is an idempotent of Hp(G), and (5.1)could have been already proved by I. Schur [V3]. In general (5.1) follows from thefact that R[I\G] is "almost projective" [V5].Alore generally, one expects that the Howe philosophy of types remains truefor modular irreducible representations. Their classification should reduce to theclassification of the simple modules for generalized affine Hecke iï-algebras of parametersequal to 0 if £ = p, and to roots of unity if £fip. This is known for lineargroups if £fip [V5] or in characteristic £ = p for GL(2, F) [V9].The unipotent block is described by a finite set S of modular types, the "unipotenttypes" [V7]. The set S contains the class of the basic type (J, id). In the banalor limit case, this is the only element of S. A unipotent type (P,T) is the G-

Modular Representations 673conjugacy class of an irreducible ^-representation of a parahoric subgroup P of G,trivial on the pro-p-radical P p , cuspidal as a representation of P/P p (the group ofrational points of a finite reductive group over the residual field of F). The isomorphismclass of the compactly induced representation ind P r of G determines theG-conjugacy class of r, and conversely. We have ind 7 id = R[I\G].(5.2) Theorem Suppose £fip. There exists a unite set S of types, suchthat- indp T is unipotent for any (P, r) £ S,- an irreducible unipotent R-representation V of G is a quotient of ind P T fora unique (P,T) £ S, called the type ofY r ,- the map V H> HonißG(i n dp r, V) between the irreducible quotients of ind P rand the right HR(G,T) = Endpo ind P r modules is a bijection.The set S has been explicitely described only when G is a linear group [V5].In the example of GL(2, F) and q = —1 modulo £, the set S has two elements, thebasic class and the class of (GL(2, Op), r) where r is the cuspidal representation ofdimension q — 1 contained in the reduction modulo £ of the Steinberg representationof the finite group GL(2,F g ).The Hecke algebra Hp(G,r) of the type (P,T) could probably be describeda generalized affine Hecke li-algebra with different parameters (complex case [Al][L2], modular case for a finite group [GHM]).The linear group in the non natural characteristic We consider thebasic example G = GL(n, F) and R an algebraically closed field of positive characteristic£fip.(6.1) Any prime £fip is banal for the Deligne-Langlands correspondence.This means that (l.d) (l.e) (l.f) remain true when C is replaced by R. Theproof is done by constructing congruences between automorphic representations forunitary groups of compact type [V6].The unipotent block is partially described by the affine Schur algebraS R (G, I) = EndfiG V,V = ®p D i indp id,which is the ring of endomorphisms of the direct sum of the representations ofG compactly induced from the trivial representation of the parahoric subgroups Pcontaining the Iwahori subgroup J. The functor of /-invariants gives an isomorphismfrom the endomorphism ring of the i?G-module V to the endomorphism ring of theright Hft(G,I)-module V 1 and the (Sp(G,L),Hp(G,Lj) module V 1 satisfies thedouble centralizer property [V8].(6.2) End fl - H ( Gj/ ) V =SR(G,I), Ends R (G,i)V =HR(G,I).In the complex case, the affine Schur algebra Sc(G,I) is isomorphic to analgebra already defined R.M. Green [Gr]: A complex affine quantum linear group

674 M.-F. VignérasU(gl(n,qj) has a remarkable representation W of countable dimension such thatthe tensor space W® n satisfies the double centralizer propertyEnd 5(n, 9 ) W0n= #("> ?)'End H(n, q) W® n = S(n, q)where S(n, q) is the image of the action of U(gl(n, qj) in W® n . The algebras S(n, q)and H(n,q) are respectively isomorphic to Sc(G,I) and HQ(G,I); the bimodulesW® n and V 1 are isomorphic.Yet J be the annihilator of R[I\G] in the global Hecke algebra Hp(G).(6.3) Theorem Suppose £fip.There exists an integer k > 0 such that the unipotent block of Mod« G is theset of R-representations of G annihilated by J k .An irreducible representation of G is unipotent if and only if it is a subquotientof R[I\G], if and only if it is annihilated by J.The abelian subcategory of representations of G annihilated by J is Moritaequivalent to the affine Schur algebra Sp(G,I).This generalizes the Borei theorem to mod £ representations when G is a lineargroup. The affine Schur algebra exists and the double centralizer property (6.2) istrue for a general reductive p-adic group G; in the banal case, the affine Schuralgebra is Alorita equivalent to the affine Hecke algebra.Integral structures Let £ be any prime number. There are two notionsof integrality for an admissible (^-representation V of G, dim V K < oo for allopen compact subgroups K of G, which coincide when £fip [V3]. One says thatV is integral if V contains a G-stable Z,(-submodule generated by a Q ( -basis of V,and V is locally integral if the HQ (G, JQ-module V K is integral, i.e. contains aiî^ (G, JQ-submodule Z,i-generated by a Q r basis of V K , for all K.When V is irreducible and integral, the action of the center Z of G on V, thecentral character, is integral, i.e. takes values in Zf. The situation is similar fora simple integral HQ (G, J)-module W. The central character is integral, i.e. itsrestriction to the center of H% (G, I) takes values in Z,(.(7.1) Théorèmea) An irreducible cuspidal Q ( -representation V of G is integral if and only ifits central character is integral.b) A simple HQ- (G, I)-module is integral if and only if its central character isintegral.c) An irreducible representation V of G with Y rI fiO is locally integral if andonly ifV 1 is an integral HQ (G,I)-module.The assertion b) results from (4.3). For a) [V3]. For £ = p, c) is due to J.-F.Dat, using its theory of l-adic analysis [D].

Modular Representations 675A general irreducible (^-representation V of G is contained in a parabolicallyinduced representation of an irreducible cuspidal representation W of a Levi subgroupof G. If W is integral then V is integral, but the converse is false when£ = p. When £fip, the converse is proved for classical groups by Dat using resultsof Moeglin (there is a gap in the "proof" of the converse in [V3]).(7.2) Brauer-Nesbitt principle [V3][V11] When £fip, the integral structuresL of an irreducible Q(-representation of G are T^G-finitely generated (hencecommensurable) and their reduction L ® F( are unite length F (-representations ofG with the same semi-simplication (modulo isomorphism).When £ = p, this is false. An integral cuspidal irreducible Q p -representationV of G embeds in Q p [F\G], for any discrete co-compact-mod-center subgroup F ofG, and has a natural integral structure with an admissible reduction [V10]. Whenthe theory of types is known, V is induced from an open compact-mod-center subgroup,hence has an integral structure with a non admissible reduction, which isnot commensurable with the first one.References[BL] Barthel L., Livne R., Alodular representations of GL 2 of a local field: theordinary unramified case, J. of Number Theory 55, 1995, 1-27. Irreduciblemodular representations of GL 2 of a local field, Duke Alath. J. 75, 1994,261-292.[Be] Bellaiche Joël, Congruences endoscopiques et représentations galoisiennes,Thèse Orsay 2002.[B] Bernstein J.N., Le "centre" de Bernstein. Dans J.N. Bernstein, P. Deligne,D. Kazhdan, Al.-F.Vignéras, Représentations des groupes réductifs sur uncorps local, Travaux en cours. Hermann Paris 1984.[Bo] Borei Armand, Admissible representations of a semisimple group over alocal field with vectors fixed under an Iwahori subgroup, Invent. Alath. 35,(1976), 233-259.[Br] Breuil Christophe, Sur quelques représentations modulaires et p-adiques deGL(2,Q p ) I, II, Preprints 2001.[BK] Bushnell Colin, Kutzko Phillip, The admissible dual of GL(N) via compactopen subgroups, Annals of Alath. Studies, Princeton Univeristy Press,129 (1993). Smooth representations of reductive p-adic groups: Structuretheory via types, Proc. London Alath. Soc. 77, (1988), 582-634.[Ca] Cartier Pierre, Representations of p-adic groups: a survey, Proc. of Symp.in pure math. AA1S XXXIII, part 1, 1979, 111-156.[C] Chriss, Neil A., The classification of representations of unramified Heckealgebras, Alath. Nachr. 191 (1998), 19-58.[CG] Chriss N., Ginzburg V., Representation theory and complex geometry,Birkhäuser 1997.

676 M.-F. Vignéras[D] Dat Jean-Francois, Generalized tempered representations of p-adic groups,Preprint 2002.[DT] Diamond Fred, Taylor Richard, Non-optimal levels of mod £ representations,Invent, math. 115, (1994), 435-462.[GHA1] Geek, Meinolf; Hiss, Gerhard; Alalie, Gunter, Towards a classification ofthe irreducible representations in non-describing characteristic of a finitegroup of Lie type. Alath. Z. 221 (1996), no. 3, 353-386.[Gr] Green R.M., The affine g-Schur algebra, Journal of Algebra 215 (1999)379-411.[IM] Iwahori N., Alatsumoto H., On some Bruhat decompositions and the structureof the Hecke rings of p-adic Chevalley groups, Pubi. Alath. I.H.E.S.25 (1965), 5-48.[HT1] Harris Michael, Taylor Richard, The geometry and cohomology of somesimple Shimura varieties, Annals of mathematics studies 151 (2001).[HT2] Harris Michael, Taylor Richard, Notes on p-adic uniformization and congruences,2002.[H] Henniart Guy, Une preuve simple des conjectures de Langlands pour GL nsur un corps p-adique, Invent. mat. 139 (2000), 339-350.[KL] Kazhdan D., Lusztig G., Proof of the Deligne-Langlands conjecture forHecke algebras, Invent. Alath. 87, (1987), 153-215 .[LI] Lusztig G., Some examples of square integrable representations of p-adicgroups, Trans. Amer. Alath. Soc. 277, (1983), 623-653.[L2] Lusztig G., Classification of unipotent representations of simple p-adicgroups, International Alathematics Research Notices Noll, (1995), 517-589[L3] Lusztig G., Representations of affine Hecke algebras, Soc. Alath. de France,Astérisque 171-171 (1989), 73-84.[Al] Alorris Lawrence, Tamely ramified intertwining algebras, Invent, math.114, (1993),1-54. Tamely ramified supercuspidal representations, Ann. Sci.cole Norm. Sup. (4) 29 no. 5, (1996), 639-667.[AIP] Moy, Allen; Prasad, Gopal. Jacquet functors and unrefined minimal K-types, Comment. Alath. Helv. 71 (1996), no. 1, 98-121. Unrefined minimalif-types for p-adic groups. Invent. Alath. 116 (1994), no. 1-3,393-408.[R] Rogawski John, On modules over the Hecke algebra of a p-adic group,Invent, math. 79, (1985), 443-465.[VI] Vignéras M.-F., Représentations modulaires de GL(2, F) en caractéristiquel, F corps p-adique, pfil, Compositio Alathematica 72 (1989), 33-66. Erratum,Compositio Alathematica 101,(1996), 109-113.[V2] Vignéras M.-F., Banal Characteristic for Reductive p-adic Groups, J. ofNumber Theory Vol.47, Number 3, 1994, 378-397.[V3] Vignéras M.-F., Représentations l-modulaires d'un groupes réductif p-adique avec Ifip, Birkhäuser Progress in Alath. 137 (1996).[V4] Vignéras M.-F., Cohomology of sheaves on the building and

Modular Representations 677^-representations, Inventiones Alathematicae 127, 1997, 349-373.[V5] Vignéras M.-F., Induced representations of reductive p-adic groups in characteristicIfip, Selecta Alathematica New Series 4 (1998) 549-623.[V6] Vignéras M.-F., Correspondance locale de Langlands semi-simple pourGL(n,F) modulo £fip, Inventiones 144, 2001, 197-223.[V7] Vignéras M.-F., Irreducible modular representations of a reductive p-adicgroup and simple modules for Hecke algebras, International EuropeanCongress Barcelone 2000. Birkhäuser Progress in Alath. 201, 117-133.[V8] Vignéras M.-F., Schur algebra of reductive p-adic groups I, Institut deAlathématiques de Jussieu, prépublication 289, Alai 2001, To appear inDuke Alath. Journal[V9] Vignéras M.-F., Representations modulo p of the p-adic group GL(2,F),Institut de Alathématiques de Jussieu, prépublication 30, septembre 2001.[V10] Vignéras M.-F., Formal degree and existence of stable arithmetic latticesof cuspidal representations of p-adic reductive groups, Invent. Alath. 98 no.3, (1989), 549-563.[VII] Vignéras M.-F., On highest Whittaker models and integral structures, Institutde Alathématiques de Jussieu, prépublication 308, septembre 2001.[Z] Zelevinski A., Induced representations of reductive p-adic groups II, Ann.scient. Ecole Norm. Sup. tome 13, (1980), 165-210.

Section 8. Real and Complex AnalysisA. Eremenko: Value Distribution and Potential Theory 681Juha Heinonen: The Branch Set of a Quasiregular Mapping 691Carlos E. Kenig: Harmonic Measure and "Locally Flat" Domains 701Nicolas Lerner: Solving Pseudo-Differential Equations 711C. Thiele: Singular Integrals Meet Modulation Invariance 721S. Zelditch: Asymptotics of Polynomials and Eigenfunctions 733Xiangyu Zhou: Some Results Related to Group Actions in SeveralComplex Variables 743

ICAl 2002 • Vol. II • 681-690Value Distribution and Potential Theory*A. EremenkoÂbstractWe describe some results of value distribution theory of holomorphic curvesand quasiregular maps, which are obtained using potential theory. Among theresults discussed are: extensions of Picard's theorems to quasiregular mapsbetween Riemannian manifolds, a version of the Second Main Theorem ofNevanlinna for curves in projective space and non-linear divisors, descriptionof extremal functions in Nevanlinna theory and results related to Cartan's1928 conjecture on holomorphic curves in the unit disc omitting hyperplanes.2000 Mathematics Subject Classification: 30D35, 30C65.Keywords and Phrases: Holomorphic curves, Quasiregular maps, Meromorphicfunctions.1. IntroductionClassical value distribution theory studies the following question: Let / bea meromorphic function in the plane. What can one say about solutions of theequation f(z) = a as a varies? The subject was originated in 1880-s with twotheorems of Picard (Theorems 1 and 4 below). An important contribution wasmade by E. Borei in 1897, who gave an "elementary proof" of Theorem 1, whichopened a way to many generalizations. Borel's result (Theorem 12 below) alsogives an extension of Picard's theorem to holomorphic curves C —¥ P". In 1925, R.Nevanlinna (partially in cooperation with F. Nevanlinna) created what is called nowthe Nevanlinna Theory of meromorphic functions, which was subject of intensiveresearch [5]. A good elementary introduction to the subject is [18]. Griffiths andKing [16] extended Nevanlinna theory to non-degenerate holomorphic maps / :C" —¥ Y, where Y is a compact complex manifold of dimension n. In modern timesthe emphasis has shifted to two multi-dimensional generalizations: holomorphiccurves in complex manifolds and quasiregular mappings between real Riemannianmanifolds. This survey is restricted to a rather narrow topic: generalizations of* Supported by NSF grant DMS 0100512 and by the Humboldt Foundation.1 Department of Mathematics, Purdue University, West Lafayette IN 47907, USA. E-mail:eremenko@math.purdue.edu

682 A. EremenkoPicard's theorem that are obtained with potential-theoretic methods. Some otherapplications of potential theory to value distribution can be found in [14, 20, 27].Recent accounts of other methods in the theory of holomorphic curves are [21, 29].We begin with Picard's Little Theorem:Theorem 1 Every entire function which omits two values in C is constant.To prove this by contradiction, we suppose that / is a non-constant entirefunction which omits 0 and 1. Then « 0 = log|/| and «i = log|/ — 1| are nonconstantharmonic functions in the plane satisfying\UQ—U{\—C, (1.1)where V stands for the pointwise sup, u + = u V 0, and c is a constant. Thereare several ways to obtain a contradiction from (1.1). They are based on rescalingarguments that permit to remove the c terms in (1.1). To be specific, one can findsequences z^ £ C, rj, > 0 and Aj, —^ +oo such that A^1v,j(zk+ rj,z) —^ vfiz), k —ôo, |z|0, Vj(0) = 0, (1.2)and Vj ^ 0. This gives a contradiction with the uniqueness theorem for harmonicfunctions. The idea to base a proof of Picard's theorem on (1.2) comes from thepaper [13] (the main result of this paper is described in Section 3 below). Twoversions of the rescaling argument (existence of appropriate z%,r^ and A\A are givenin [7, 12] and [19], respectively. The second version has an advantage that it usesonly one result from potential theory, Harnack's inequality. Thus Picard's theoremcan be derived from two facts: Harnack's inequality and the uniqueness theoremfor harmonic functions. This makes the argument suitable for generalizations.2. Quasiregular maps of Riemannian manifoldsWe recall that a non-constant continuous map / between regions in R n is calledif-quasiregular if it belongs to the Sobolev class W lo ' c n (first generalized derivativesare locally L"-summable), and in additionll/'IP < KJf almost everywhere, (2.1)where J is the Jacobian determinant and K > 1 is a constant. The standardreferences are [24, 25]. If n = 2, every quasiregular map can be factored as g o ,where g is analytic and a quasiconformal homeomorphism. It follows that Picard'sTheorems 1 and 4 (below) extend without any changes to quasiregular maps ofsurfaces. For the rest of this section we assume that n > 3, and that all manifoldsare connected. The weak smoothness assumption / £ Wfi" is important: if werequire more smoothness, the maps satisfying (2.1) will be local homeomorphisms(and even global homeomorphisms if the domain is R n ). A fundamental theoremof Reshetnyak says that all quasiregular maps are open and discrete, that is they

Value Distribution and Potential Theory 683have topological properties similar to those of analytic functions of one complexvariable. Several other results about analytic functions have non-trivial extensionto quasiregular mappings. One of the striking results in this area is Rickman'sgeneralization of Picard's theorem [25]:Theorem 2 A K-quasiregular map R n —¥ R n can omit only a finite set of pointswhose cardinality has an upper bound in terms on n and K.Even more surprising is that when n = 3, the number of omitted values canindeed be arbitrarily large, as Rickman's example in [26] shows.It turns out that the method of proving Picard's theorem outlined in Section 1,extends to the case of quasiregular maps. One has to use a non-linear version ofpotential theory in R n which is related to quasiregular maps in the same way aslogarithmic potential theory to analytic functions. This relation between quasiregularmaps and potential theory was discovered by Reshetnyak. He singled out a classof functions (which are called now A-harmonic functions), that share many basicproperties (such as the maximum principle and Harnack's inequality) with ordinaryharmonicfunctions, and such that u o / is A-harmonic whenever u is A-harmonicand / quasiregular. In particular, log \x — a\ is A-harmonic on R"\{a}, so if / omitsthe value a, then log |/ —ct| satisfies Harnack's inequality (with constants dependingon K and n). If m values are omitted by / we can obtain relations, similar to (1.2),vt = ... = v+, ViV Vj >0, «i(0)=0, (2.2)for certain A-harmonic functions Vj ^ 0, j = l,...,ro. Rickman's example mentionedabove shows that such relations (2.2) are indeed possible with any givenm > 1, which is consistent with the known fact that A-harmonic functions do nothave the uniqueness property. However, an upper bound for m can be deducedfrom (2.2) using Harnack's inequality. This gives a pure potential-theoretic proofof Rickman's theorem [12, 19]. Notice that this proof does not depend on the deepresult that quasiregular maps are open and discrete. Lewis's paper [19] which usesnothing but Harnack's inequality opened a path for further generalizations of Rickman'stheorem. The strongest result in this direction was obtained by Holopainenand Rickman [17]. For simplicity, we state it only in the special case of quasiregularmaps whose domain is R n .Theorem 3 Let Y be an orientable Riemannian manifold of dimension n. If thereexists a K-quasiregular map R n —¥ Y, then the number of ends of Y has an upperbound that depends only on K and n.A more general result, with R n replaced by a Riemannian manifold subject tocertain conditions, is contained in [17].Notice that there are no restrictions on Y in this theorem. Conditions ofTheorem 3 will be satisfied if Y is a compact manifold with finitely many pointsremoved, so a if-quasiregular map from R n to a compact n-dimensional manifoldcan omit at most N(K,n) points.Now we turn to the second theorem of Picard mentioned in the Introduction:

684 A. EremenkoTheorem 4 If there exists a non-constant holomorphic map f : C —¥ S from thecomplex plane to a compact Riemann surface S, then the genus of S is at most 1.First extensions of this result to quasiregular maps in dimension n > 2 wereobtained by Gromov in 1981 [6, Ch. 6] who proved that the fundamental groupof a compact manifold of dimension n which receives a quasiregular map from R ncannot be too large. Gromov applied a geometric method, based on isoperimetricinequalities, which goes back to Ahlfors's approach in dimension 2. The strongestresult in this direction is the following theorem from [31]: If a compact manifoldY of dimension n > 2 receives a quasiregular map from R", then the fundamentalgroup of Y is virtually nilpotent and has polynomial growth of degree at most n.We notice that unlike this last result, Theorem 3 has nothing to do with thefundamental group of Y: removing a finite set from a compact manifold does notchange its fundamental group. Recently, Bonk and Heinonen [2] applied potentialtheoreticarguments, somewhat similar to those outlined above, to obtain new topologicalobstructions to the existence of quasiregular maps:Theorem 5 If Y is a compact manifold of dimension n which receives a K-quasiregularmap from R", then the dimension of the de Rham cohomology ring of Y isbounded by a constant that depends only on n and K.This result implies that for every K > 1 there exist simply connected compactmanifolds Y such that there are no if-quasiregular maps R n —¥ Y. The questionwhether there exists a compact simply connected manifold Y such that there areno quasiregular maps R n —¥ Y (with any K) remains open.For a compact manifold Y, the natural objects to pull back via / are differentialforms rather then functions. According to the "non-linear Hodge theory" [28], eachcohomology class of Y can be represented by a p-harmonic form, which satisfies anon-linear elliptic PDE. Such forms and their pullbacks to R n play a similar roleto the A-harmonic functions above.It is natural to conjecture that the theorem of Bonk-Heinonen remains validif the requirement that Y is compact is dropped. Such a generalization would alsoimply Theorem 3.3. Holomorphic curves in projective varietiesHere we return to the classical logarithmic potential theory, which allows moreprecise quantitative estimates.Points in the complex projective space P" are represented by their homogeneouscoordinates z = (ZQ : ... : z n ). Let Y C P" be an arbitrary projectivevariety. We consider divisors D on Y which are the zero sets of homogeneous formsP(ZQ,- • • ,z n ) restricted to Y. The degree of D is defined as the homogeneous degreeof P. Suppose that q of such divisors Dj of degrees dj are given, and they satisfythe condition that for some integer m < q — 1 every m + 1 of these divisors on Yhave empty intersection. We are going to study the distribution of preimages ofdivisors Dj under a holomorphic map / : C —¥ Y whose image is not contained

Value Distribution and Potential Theory 685in UDj. To such a map correspond n + 1 entire functions without common zeros:/ = (/o,.. •, fn)- Thus we are interested in the distribution of zeros of entirefunctions Pj o / = Pj(f 0 , ••-, /„).We introduce the subharmonic functionsu = 11/11 = \/|/o| 2 + •••+ \fn? and Uj = log \Pj o f\/dj.The assumption on intersections of Dj easily implies that\\l Uj — u\ < c for every J C {1,..., q}, such that card I = m + 1. (3.1)je/This relation is a generalization of (1.1). The rescaling procedure mentioned in Section1 permits to remove the constant c in (3.1) and obtain subharmonic functionswi,..., Vq and v in a disc which satisfy\J Vj = v, I C {1,.. .,q}, cardi = m + 1, (3.2)je/and such that v is not harmonic.If / omits q = 2m+1 divisors in Y, then all Vj in (3.2) will be harmonic (while vis not!) and it is easy to obtain a contradiction. Indeed, let Ej = {z : Vj(z) = v(z)}.Then (3.2) with q = 2m + 1 implies that for some J of cardinality m + 1, theintersection Cij e iEj has positive area. It follows by the uniqueness theorem thatall Vj for j £ I are equal. Applying (3.2) with this J we obtain that v = Vjfor j £ I, so v is harmonic, which gives a contradiction. Thus we obtain thefollowing generalization of Picard's theorem proved by V. Babets in 1983 for thecase Y = P", m = n, and under a stronger restriction on the intersection of divisors[7]-Theorem 6 Let Y be a projective variety. If a holomorphic map C —¥ Y omits2m + 1 divisors, such that the intersection of any m + 1 of them is empty, then fis constant.Notice that dimension of Y is not mentioned in this theorem. A more carefulanalysis of (3.2) and more sophisticated rescaling techniques yield a quantitativeresult of the type of the Nevanlinna's Second Alain Theorem. To state it, we recallthe definitions of Nevanlinna theory. If p is the Riesz measure of u, then theNevanlinna characteristic can be defined asrdtT(r,f)= / Mi>:M

686 A. EremenkoTheorem 7 Let Y be a projective variety, and q divisors Dj of degrees dj in Ysatisfy the intersection condition of Theorem 6. Let f : G —¥ Y be a holomorphicmap whose image is not contained in UjDj. ThenQj(q - 2m)T(r, /) < £ ~^N(r, Dj, f) + o(T(r, /)),j=iwhen r —t oo avoiding a set of finite logarithmic measure.3This theorem is stated in [13] only for the case Y = P", m = n but the sameproof applies to the more general statement. When m = n = 1 we obtain a roughform of the Second Alain Theorem of Nevanlinna; with worse error term, and moreimportantly, without the ramification term. A corollary from Theorem 7 is thedefect relation:2_,ô(Dj,f) < 2m, where ô(Dj,f) = 1 — lim sup 'J ' . (3.4)r-s-oo djl (r, f)The key result of potential theory used in the proof of Theorem 7 is of independentinterest [11]:Theorem 8 Suppose that a finite set of subharmonic functions {WJ} in a region inthe plane has the property that the pointwise minima w, A Wj are subharmonic forevery pair. Then the pointwise minimum of all these functions is subharmonicThis is derived in turn from the following:Theorem 9 Let Gi,G2,Gz be three pair-wise disjoint regions, and pi,p2,Pz theirharmonic measures. Then there exist Borei sets Ej c dGj such that Pj(Ej) =1, j = 1,2,3, and EiÊ 2Ê z = 0.For regions in R 2 (the only case needed for theorems 7 and 8) this is easyto prove: just take Ej to be the set of accessible points from Gj and notice thatat most two points can be accessible from all three regions [11]. It is interestingthat Theorem 9 holds for regions in R" for all n, but the proof of this (based onadvanced stochastic analysis rather then potential theory) is very hard [30].We notice that the number 2 in Picard's Theorem 1, as well as in Theorem7, thus admits an interpretation which seems to be completely different from thecommon one: with our approach it has nothing to do with the Euler characteristicof the sphere or its canonical class, but comes from Theorem 9. Recently, Siu[29] gave a proof of a result similar to Theorem 7 (with Y = P", m = n) usingdifferent arguments which are inspired by "Vojta's analogy" between Nevanlinnatheory and Diophantine approximation. However Siu's proof gives a weaker estimateem « 2.718m instead of 2m in (3.4), and his assumptions on the intersection ofdivisors are stronger than those in Theorem 7.The constant 2m in (3.4) is best possible. Aloreover, one can give a rathercomplete characterization of extremal holomorphic curves of finite lower order. Werecall that the lower order of a holomorphic curve isA = l iminfÂ>i/).r-s-oo logr

Value Distribution and Potential Theory 687Theorem 10 [8] Let Di,...,D q be divisors and f a curve satisfying all the hypothesesof Theorem 7. Suppose in addition that f has finite lower order and thatequality holds in the defect relation (3.4). Then(i) 2X is an integer, and X > 1,(ii) T(r,f) = r x £(r), where £(r) is a slowly varying function in the sense of Karamata:£(cr)/£(r) —¥ 1, r —t oo uniformly with respect to c £ [1,2],(iii) All defects are rational: ö(Dj,f) = Pj/X, where pj are integers whose sum is2mX.When m = n = 1, this result was conjectured by F. Nevanlinna [23]. Afterlong efforts, mainly by A. Pfluger, A. Edrei, W. Fuchs and A. Weitsman, D. Drasinfinally proved F. Nevanlinna's conjecture in [4]. The potential-theoretic methodpresented here permitted to give a simpler proof of Drasin's theorem, and then togeneralize the result to arbitrary dimension, as well as to obtain a stronger resultin dimension 1 which is discussed in the next section. The proof of Theorem 10, isbased on the following result about subharmonic functions:Theorem 11 Suppose that v,vi,...,v q , q> 2m + 1 are subharmonic functions inthe plane, which satisfy (3.2), and in addition v(z) < \z\ x , z £ G, and v(0) = 0.Then the functionih = 2_, v j — 2TOWj=iis subharmonic. If h is harmonic, then 2X is an integer andwhere c > 0 and a is a real constant.v(re %t ) = c|r| A |cosA(r. — a)\,4. Functions with small ramificationWe recall the definition of the ramification term in Nevanlinna theory. Supposethat the image /(C) of a holomorphic curve / : C —¥ P" is not contained inany hyperplane. This means that / 0 ,...,/„ in the homogeneous representationof / are linearly independent. Let rii (r, /) be the number of zeros in the disc{z : \z\ < r} of the Wronski determinant W(fo, • • •, f n ), and Ni(r, /) the averagedcounting function of these zeros as in (3.3). If n = 1, then m counts the number ofcritical points of /. The Second Alain Theorem of Cartan [18] says that for everyholomorphic curve / whose image does not belong to a hyperplane, and every finiteset of hyperplanes {ai,... ,a q } in general position, we have(q^n-l + o(lj)T(r,f) + Ni(r,f)

688 A. Eremenkoand ö(a,fi) was defined in (3.4). So, if n = 1, and the sum of deficiencies equals2, then 9(f) = 0. The work of F. Nevanlinna [23] mentioned in Section 3 actuallysuggests something stronger than he conjectured: that the weaker assumption9(f) = 0 for functions of finite lower order implies all conclusions (i)—(iii) of Theorem10. This stronger result was proved in [9]. It follows that for functions of finitelower order the conditions 9(f) = 0 and ^ 8(a, fi) = 2 are in fact equivalent. Thereis some evidence that this result might have the following extension to holomorphiccurves in P":Conjecture Let f be a holomorphic curve of finite lower order, whose image is notcontained in any hyperplane. If Ni(r) = o(T(r,f)),r —¥ oo, then X is a rationalnumber and assertion (ii) of Theorem 10 holds.This is not known even under a stronger assumption that the sum of deficienciesis n + 1.5. Cartan's conjectureAccording to a philosophical principle of Bloch and Valiron [1], to theoremsabout entire functions should correspond theorems about families of functions in theunit disc, in the same way as Landau's theorem corresponds to Picard's theorem.One can supplement Theorem 6 with an explicit estimate of derivative of a holomorphicmap from the unit disc to projective space that omits 2m +1 hypersurfacessatisfying the intersection condition of Theorem 6. To prove such generalization ofLandau's theorem, one replaces the use of the uniqueness theorem for harmonicfunctions by the corresponding quantitative result as in [22].In 1887 Borei proved an extension of Picard's theorem, from which Theorem6 and many other similar results (see, for example, [15]) can be derived:Theorem 12 (Borei) If fi, • • •, f p are entire functions without zeros, that satisfyfi + f 2 + ... + f P = 0, (5.1)then there is a partition of the set J = {fi,---,f p } into classes I, such that forevery I, all functions in I are proportional and their sum is zero.When p = 3 it is equivalent to the Picard's Little Theorem. The question iswhat kind of normality criterion corresponds to Theorem 12 in the same way asMontel's criterion corresponds to Picard's theorem. The following conjecture wasstated by H. Cartan in his thesis [3] (see also [18] for a comprehensive discussion ofthis conjecture).Conjecture A Let F be an infinite sequence of p-tuples / = (fi,- •• ,f p ) of holomorphicfunctions in the unit disc, such that each fj has no zeros, and (5.1) issatisfied.Then there exists an infinite subsequence F' of F and a partition of the setJ = {1,... ,p} into classes I, such that for / in F' and every class I we have:(*) there exists j £ I such that for every i £ I the ratios fi/fj are uniformlybounded on compact subsets of the unit disc, and î€/ fi/fj —t 0 uniformly oncompact subsets of the unit disc.

Value Distribution and Potential Theory 689One obtains this statement by replacing "proportional" by "have boundedratio" and "equals zero" by "tends to zero" in the conclusions of Borel's theorem.When p = 3, Conjecture A is equivalent to Montel's theorem.Let us call a subset J C J = {l,...,p} having the property (*) a C-class ofthe sequence F'. Cartan proved in [3] that under the hypotheses of Conjecture Athere exists an infinite subsequence F', such that either the whole set J constitutesa single C-class, or there are at least 2 disjoint C-classes in J. This result impliesthat Conjecture A is true for p = 4, which corresponds to holomorphic curves inP 2 omitting four lines. Indeed, it follows from (*) that each C-class contains atleast two elements, so if there are two disjoint C-classes they have to be a partitionof the set J of four elements. For p > 5, Cartan's result falls short of proving hisconjecture because the union of the two C-classes whose existence is asserted mightnot coincide with the whole set {1,... ,p}.It turns out that Conjecture A is wrong as originally stated, beginning fromp = 5 (that is in dimensions > 3). A simple counterexample was constructed in[10]). Nevertheless a small modification of the statement is valid in dimension 3:Conjecture B Under the assumptions of Conjecture A its conclusions hold is thedisc {z : \z\ < r p }, where r p < 1 is a constant that depends only on p.This was proved in [10] when p = 5, that is for holomorphic curves in P 3omitting 5 planes.References[i[2[3;[4;[6;[7;[9[io;A. Bloch, La conception actuelle de la théorie des fonctions entières etméromorphes, Ens. Math., 25 (1926) 83-103.Al. Bonk and J. Heinonen, Quasiregular mappings and cohomology, Acta math.,186 (2001) 219-238.H. Cartan, Sur les systèmes de fonctions holomorphes a variétés linéaires lacunaires,Ann. Sci. Ecole Norm. Super., 45 (1928) 255-346.D. Drasin, Proof of a conjecture of F. Nevanlinna concerning functions whichhave deficiency sum two, Acta math., 158 (1987) 1-94.A.A. Gol'dberg and I.V. Ostrovskii, Distribution of values of meromorphicfunctions, Aloscow, Nauka, 1970 (Russian).Al. Gromov, Metric structures for Riemannian and non-Riemannian spaces,Birkhäuser, Boston, 1999.A. Eremenko, A Picard type theorem for holomorphic curves, Period. Math.Hung., 38 (1999) 39-42.A. Eremenko, Extremal holomorphic curves for defect relations, J. d'Analysemath., vol. 74 (1998) 307-323.A. Eremenko, Aleromorphic functions with small ramification, Indiana Univ.Math. J., 42, 4 (1993), 1193-1218.A. Eremenko, Holomorphic curves omitting five planes in projective space,Amer. J. Math., 118 (1996), 1141-1151.A. Eremenko, B. Fuglede and Al. Sodin, On the Riesz charge of the lowerenvelope of delta-subharmonic functions. Potential Analysis, 1 (1992) 191-204.

690 A. Eremenko[12] A. Eremenko and J. Lewis, Uniform limits of certain A-harmonic functionswith applications to quasiregular mappings, Ann. Acad. Sei. Fenn., Ser. A. I,16 (1991) 361-375.[13] A. Eremenko and Al. Sodin, Value distribution of meromorphic functions andmeromorphic curves from the point of view of potential theory. St. PetersburgMath. J., 3 (1992), 109-136.[14] A. Eremenko and AI. Sodin, Proof of a conditional theorem of Littlewood on thedistribution of values of entire functions, Engl, transi.: Math USSR Izvestiya,30 (1988) 395-402.[15] Al. Green, Some Picard theorems for holomorphic maps to algebraic varieties.Amer. J. Math. 97 (1975), 43-75.[16] Ph. Griffiths and J. King, Nevanlinna theory and holomorphic mappings betweenalgebraic varieties, Acta Math., 130 (1973) 145-220.[17] I. Holopainen and S. Rickman, Ricci curvature, Harnack functions and Picardtype theorems for quasiregular mappings, Analysis and topology, 315-326.World Sci. Pubi., River Edge NJ, 1998.[18] S. Lang, Introduction to complex hyperbolic spaces, Springer, Berlin, 1987.[19] J. Lewis, Picard's theorem and Rickman's theorem by way of Harnack's inequality,Proc. AMS, 122 (1994) 199-206.[20] J. Lewis, On a conditional theorem of Littlewood for quasiregular entire functions,J. Anal. Math., 62 (1994) 169-198.[21] Alin Ru, Nevanlinna theory and its relation to Diophantine approximation,World Scientific, River Edge, NJ, 2001.[22] N. Nadirashvili, On a generalization of Hadamard's three-circle theorem, VestnikMoskovskogo Universiteta, Mat., 31 (1976) 39-42.[23] F. Nevanlinna, Über eine Klasse meromorpher Funktionen, 7-e congrès desmathématiciens Scandinaves, Oslo 1929, A. W. Broggers, Oslo, 1930.[24] Yu. Reshetnyak, Space mappings with bounded distortion, AA1S, Providence,RI, 1989.[25] S. Rickman, Quasiregular mappings, Springer, NY, 1993.[26] S. Rickman, An analogue of Picard's theorem for quasiregular mappings indimension three, Acta math. 154 (1985) 195-242.[27] A. Russakovskii and B. Shiffman, Value distribution for sequences of rationalmappings and complex dynamics, Indiana Univ. Math. J, 46 (1997) 897-932.[28] C. Scott, IP theory of differential forms on manifolds, Trans. AMS, 347 (1995),2075-2096.[29] Y.-T. Siu, Recent techniques in hyperbolicity problems, Several complex variables(Berkeley CA, 1995-1996), 429-508, Cambridge UP, 1999.[30] B. Tsirelson, Triple points: from non-Brownian filtrations to harmonic measures,G AFA, 7 (1997) 1096-1142.[31] N. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and geometry ongroups, Cambridge UP, Cambridge, 1992.

ICAl 2002 • Vol. II • 691-700The Branch Set of a QuasiregularMapping*Julia HeinonentAbstractWe discuss the issue of branching in quasiregular mapping, and in particularthe relation between branching and the problem of finding geometricparametrizations for topological manifolds. Other recent progress and openproblems of a more function theoretic nature are also presented.2000 Mathematics Subject Classification: 30C65, 57M12.Keywords and Phrases: Quasiregular map, Bi-Lipschitz map, Branch set.1. Branched coveringsA continuous mapping f : X —¥ Y between topological spaces is said to bea branched covering if / is an open mapping and if for each y £ Y the preimagef^1(y)is a discrete subset of X. The branch set Bf of / is the closed set of pointsin X where / does not define a local homeomorphism.Nonconstant holomorphic functions between connected Riemann surfaces areexamples of branched coverings. From the Weierstrassian (power series) point ofview this property of holomorphic functions is almost immediate. It is a deeper fact,due to Riemann, that the same conclusion can be drawn from the mere definitionof complex differentiability, or, equivalently, from the Cauchy-Riemann equations.Alost of this article discusses the repercussions of this fact.2. Quasiregular mappingsIn a 1966 paper [27], Reshetnyak penned a definition for mappings of boundeddistortion or, as they are more commonly called today, quasiregular mappings.* Supported by NSF grant DMS 9970427. 1 thank Mario Bonk and Alex Eremenko for theircriticism on earlier versions of this article. My warmest thanks go to Mario Bonk, Seppo Rickman,and Dennis Sullivan for collaboration, mentoring, and friendship.^Department of Mathematics, University of Michigan, MI 48109, USA. E-mail:juha@math.lsa.umich.edu

692 Juha HeinonenThese are nonconstant mappings / : 0 —t R n in the Sobolev space Wj o '"(0;R"),where O c R n is a domain and n > 2, satisfying the following requirement: thereexists a constant K > 1 such that\f'(x)\ n < KJ f (x) (2.1)for almost every x £ ii, where |/'(x)| denotes the operator norm of the (formal)differential matrix f'(x) with J/(x)=det/'(x) its Jacobian determinant. One alsospeaks about K-quasiregular mappings if the constant in (2.1) is to be emphasized. 1Requirement (2.1) had been used as the analytic definition for quasiconformalmappings since the 1930s, with varying degrees of smoothness conditions on/. Quasiconformal mappings are by definition quasiregular homeomorphisms, andReshetnyak was the first to ask what information inequality (2.1) harbours per se.In a series of papers in 1966-69, Reshetnyak laid the analytic foundations for thetheory of quasiregular mappings. The single deepest fact he discovered was thatquasiregular mappings are branched coverings (as defined above). It is instructiveto outline the main steps in the proof for this remarkable assertion, which akinto Riemann's result exerts significant topological information from purely analyticdata. For the details, see, e.g., [28], [29], [18].To wit, let / : 0 —t R n be K^quasiregular. Fix y £ R n and consider thepreimage Z = f^1(y).One first shows that the function u(x) = log \f(x) —y\ solvesa quasilinear elliptic partial differential equation-divA(x,Vu(xj)=0, A(x,Ç)-Ç~\Ç\ n , (2.2)in the open set 0 \ Z in the weak (distributional) sense. In general, A in (2.2)depends on /, but its ellipticity only on K and n. For holomorphic functions, i.e.,for n = 2 and K =1, equation (2.2) reduces to the Laplace equation —divVu = 0.Now u(x) tends to ôo continuously as x tends to Z. Reshetnyak developssufficient nonlinear potential theory to conclude that such polar sets, associatedwith equation (2.2), have Hausdorff dimension zero. It follows that Z is totallydisconnected,i.e., the mapping / is light. This is the purely analytic part of theproof. The next step is to show that nonconstant quasiregular mappings are sensepreserving.This part of the proof mixes analysis and topology. What remains is apurely topological fact that sense-preserving and light mappings between connectedoriented manifolds are branched coverings.Initially, Reshetnyak's theorem served as the basis for a higher dimensionalfunction theory. In the 1980's, it was discovered by researchers in nonlinear elasticity.In the following, we shall discuss more recent, different types of applications.3. The branch setBranched coverings between surfaces behave locally like analytic functions accordingto a classical theorem of Stoïlow. By a theorem of Chernavskiï, for every1 The definition readily extends for mappings between connected oriented Riemannian n-manifolds.

The Branch Set of a Quasiregular Alapping 693n > 2, the branch set of a discrete and open mapping between n-manifolds hastopological dimension at most n — 2. For branched coverings between 3-manifolds,the branch set is either empty or has topological dimension 1 [24], but in dimensionsn > 5 there are branched coverings between n-manifolds with branch set ofdimension n — 4, cf. Section 7. 2The branch set of a quasiregular mapping is a somewhat enigmatic object indimensions n > 3. It can be very complicated, containing for example many wildCantor sets of classical geometric topology [14], [15]. There is currently no theoryavailablethat would explain or describe the geometry of allowable branch sets, cf.Problems 2 and 4 in Section 7.In the next three sections, we shall discuss the problem of finding bi-Lipschitzparametrizations for metric spaces. It will become clear only later how this problemis related to the branch set.4. Bi-Lipschitz parametrization of spacesA homeomorphism f : X —¥ Y between metric spaces is bi-Lipschitz if thereexists a constant L > 1 such thatL- 1 d x (a,b) 2. If this is the case, let us say, for brevityand with a slight abuse of language, that X is locally bi-Lipschitz equivalent to R n .Now a separable metrizable space is a Lipschitz manifold (in the sense ofcharts) if and only if it admits a metric, compatible with the given topology, thatmakes the space locally bi-Lipschitz equivalent to R n [22]. The problem here isdifferent from characterizing Lipschitz manifolds among topological spaces, for themetric is given first, cf. [8], [39], [40], [41].To get a grasp of the difficulty of the problem, consider the following example:There exist finite 5-dimensional polyhedra that are homeomorphic to the standard 5-sphere S 5 , but not locally bi-Lipschitz equivalent to R 5 . This observation of Siebenmannand Sullivan [38] is based on a deep result of Edwards [9], which asserts thatthe double suspension £ 2 ff 3 of a 3-dimensional homology sphere H 3 , with nontrivialfundamental group, is homeomorphic to the standard sphere S 5 . (See also [6].)One can think of X = T, 2 H 3 as a join X = S 1 * H 3 , and it is easy to check that thecomplement of the suspension circle S 1 in X is not simply connected. Consequently,every homeomorphism / : X —t S 5 must transfer S 1 to a closed curve F = /(S 1 )whose complement in S 5 is not simply connected. A general position argument andFubini's theorem imply that, in this case, the Hausdorff dimension of F must be atleast 3. Hence / cannot be Lipschitz. In fact, / cannot be Holder continuous withany exponent greater than 1/3. It is not known what other obstructions there arefor a homeomorphism X —t S 5 , cf. [16, Questions 12-14].See [33] and [37] for surveys on parametrization and related topics.2 See [23] for a recent survey on dimension theory and branched coverings.

694 Juha Heinonen5. Necessary conditionsWhat are the obvious necessary conditions that a given metric space X mustsatisfy, if it were to be locally bi-Lipschitz equivalent to R n , n > 2? Clearly, Xmust be an n-manifold. Next, bi-Lipschitz mappings preserve Hausdorff measurein a quantitative manner, so in particular X must be n-rectifiable in the senseof geometric measure theory; moreover, locally the Hausdorff n-measure shouldassign to each ball of radius r > 0 in X a mass comparable to r n . Yet us saythatX is metrically n-dimensional if it satisfies these geometric measure theoreticrequirements.It is not difficult to find examples of metrically n-dimensional manifolds thatare not locally bi-Lipschitz equivalent to R n . The measure theory allows for cuspsand folds that are not tolerated by bi-Lipschitz parametrizations. Further geometricconstraints are necessary; but, unlike in the case of the measure theoretic conditions,it is not obvious what these constraints should be. A convenient choice is that oflocal linear contractibility: locally each metric ball in X can be contracted to a point3inside a ball with the same center but radius multiplied by a fixed factor.Still, a metrically n-dimensional and locally linearly contractible metric n-manifold need not be locally bi-Lipschitz equivalent to R n . The double suspensionof a homology 3-sphere with nontrivial fundamental group as described in the previoussection serves as a counterexample. In 1996, Semmes [34], [35] exhibited examplesto the same effect in all dimensions n > 3, and recently Laakso [21] crushedthe last hope that the above conditions might characterize at least 2-dimensionalmetric manifolds that are locally bi-Lipschitz equivalent to R 2 . However, unlike theexamples of Edwards and Semmes, Laakso's metric space cannot be embedded bi-Lipschitzly in any finite dimensional Euclidean space. Thus the following problemremains open:Problem 1 Let X be a topological surface inside some R* with the inheritedmetric. Assume that X is metrically 2-dimensional and locally linearly contractible.Is X then locally by-Lipschitz equivalent to R 2 ?In conclusion, perhaps excepting the dimension n = 2, more necessary conditionsare needed in order to characterize the spaces that are locally bi-Lipschitzequivalent to R". 4 The idea to use Reshetnyak's theorem in this connection originatesin two papers by Sullivan [40], [41], and is later developed in [17]. Recall thatin this theorem topological conclusions are drawn from purely analytic data. Nowimagine that such data would make sense in a space that is not a priori Euclidean.Then, if one could obtain a branched covering mapping into R", manifold pointswould appear, at least outside the branch set. We discuss the possibility to developthis idea in the next section.6. Cartan-Whitney presentations3 See [36] for analytic implications of this condition.4 There are interesting and nontrivial sufficient conditions known, but these are far from beingnecessary [42], [43], [2], [3], [5].

The Branch Set of a Quasiregular Alapping 695Let X be a metrically n-dimensional, linearly locally contractible n-manifoldthat is also a metric subspace of some R^. Suppose that there exists a bi-Lipschitzhomeomorphism / : X —t f(X) c R". Then / pulls back to X the standardcoframe of R", providing almost everywhere defined (essentially) bounded differential1-forms pi = f*dxi, i = 1,... ,n. To be more precise here, by Kirzsbraun'stheorem, / can be extended to a Lipschitz mapping / : H N —t R", and the 1-formsPi = f*dxi = dfi, i=l,...,n, (6.1)are well defined in H N as flat 1-forms of Whitney. Flat forms are forms with L°°coefficientssuch that the distributional exterior differential of the form also hasL°°-coefficients. The forms in (6.1) are closed, because the fundamental relationdf* = f*d holds true for Lipschitz maps.According to a theorem of Whitney [45, Chapter IX], flat forms (p,) have awell defined trace on X, and on the measurable tangent bundle of X, essentiallybecause of the rectifiability. 5 Because / = /|X has a Lipschitz inverse, there existsa constant c > 0 such that*(piA---Ap„)>c>0 (6.2)almost everywhere on X, where the Hodge star operator * is determined by thechosen orientation on X.Condition (6.2) was turned into a definition in [17]. We say that X admits localCartan- Whitney presentations if for each point p £ X one can find an n-tuple of flat1-forms p = (pi,..., p n ) defined in an R^-neighborhood of p such that condition(6.2) is satisfied on X near the point p.Theorem 1 [17] Let X c H N be a metrically n-dimensional, linearly locally contractiblen-manifold admitting local Cartan-Whitney presentations. Then X is locallybi-Lipschitz equivalent to R" outside a closed set of measure zero and of topologicaldimension at most n — 2.To prove Theorem 1, fix a point p £ X, and let p = (pi,... ,p n ) be a Cartan-Whitney presentation near p. The requirement that p be flat together with inequality(6.2) can be seen as a quasiregularity condition for forms. 6 We define amappingf(x) = / p (6.3)for x sufficiently near p, where [p, x] is the line segment in H N from p to x, andclaim that Reshetnyak's program can be run under the stipulated conditions on X.In particular, we show that for a sufficiently small neighborhood U of p in X, themap / : U —¥ R" given in (6.3) is a branched covering which is locally bi-Lipschitzoutside its branch set Bf, which furthermore is of measure zero and of topologicaldimension at most n — 2. It is important to note that p is not assumed to be closed,so that df ^ p in general.5 There is a technical point about orientation which we ignore here [17, 3.26].6 In fact, (6.2) resembles a stronger, Lipschitz version of ' (2.1) studied in [26], [40], [15]

696 Juha HeinonenIn executing Reshetnyak's proof, we use recent advances of differential analysison nonsmooth spaces [13], [20], [36], as well as the theory developed simultaneouslyin [15]. Incidentally, we avoid the use of the Harnack inequality for solutions, andtherefore a deeper use of equation (2.2); this small improvement to Reshetnyak'sargument was found earlier in a different context in [12].Theorem 1 provides bi-Lipschitz coordinates for X only on a dense open set.In general, one cannot have more than that. The double suspension of a homology3-sphere, as discussed in Section 4, can be mapped to the standard 5-sphere by afinite-to-one, piecewise linear sense-preserving map. By pulling back the standardcoframe by such map, we obtain a global Cartan-Whitney presentation on a spacethat is not locally bi-Lipschitz equivalent to R 5 . Similar examples in dimensionn = 3 were constructed in [14], [15], by using Semmes's spaces [34], [35]. On theother hand, we have the following result:Theorem 2 Let X c H N be a metrically 2-dimensional, linearly locally contractible2-manifold admitting local Cartan-Whitney presentations. Then X is locallybi-Lipschitz equivalent to R 2 .Theorem 2 is an observation of Al. Bonk and myself. We use Theorem 1together with the observation that, in dimension n = 2, the branch set consists ofisolated points, which can be resolved. The resolution follows from the measurableRiemann mapping theorem together with the recent work by Bonk and Kleiner [4].While Theorem 2 presents a characterization of surfaces in Euclidean space thatadmit local bi-Lipschitz coordinates, we do not know whether the stipulation aboutthe existence of local Cartan-Whitney presentations is really necessary (compareProblem 1 and the discussion preceding it).For dimensions n > 3, it would be interesting to know when there is no branchingin the map (6.3). In [17], we ask if this be the case when the flat forms (p,)of the Cartan-Whitney presentation belong to a Sobolev space H^ on X. Therelevant example here is the map (r, 9, z) >-¥ (r, 29, z), in the cylindrical coordinatesof R", which pulls back the standard coframe to a frame that lies in the Sobolevspace H têfor each e > 0. Indeed, it was shown in [11] that in R" every (Cartan-Whitney) pullback frame in Hfi 0 " c must come from a locally injective mapping.7. Other recent progress and open problemsIn his 1978 ICA1 address, Väisälä [44] asked whether the branch set of a C 1 -smooth quasiregular mapping is empty if n > 3. It was known that C n/, ( n_2 )-smoothquasiregular mappings have no branching when n > 3. The proof in [Ri, p. 12] ofthis fact uses quasiregularity in a rather minimal way. In this light, the followingrecent result may appear surprising:Theorem 3 [1] For every e > 0 there exists a degree two C 3ê -smooth quasiregularmapping / : S 3 —¥ S 3 with branch set homeomorphic to S 1 .We are also able to improve the previous results as follows:

The Branch Set of a Quasiregular Alapping 697Theorem 4 [1] Given n > 3 and K > 1, there exist e = e(n,K) > 0 and e' =e'(n, K) > 0 such that the branch set of every K-quasiregular mapping in a domainin R" has Hausdorff dimension at most n — e, and that every (7«/(«- 2 )- e smoothK-quasiregular mapping in a domain in R" is a local homeomorphism.The second assertion in Theorem 4 follows from the first, by way of Sard-typetechniques. The first assertion was known earlier in a local form where e > 0 wasdependent on the local degree [31]. Our improvement uses [31] together with thework [30] by Rickman and Srebro.The methods in [1] fall short in showing the sharpness of Theorem 4 in dimensionsn > 4 in two technical aspects. First, we would need to construct aquasiconformal homeomorphism of R" to itself that is uniformly expanding on acodimension two affine subspace; moreover, such a map needs to be smooth outsidethis subspace. In R 3 , it is easier to construct a mapping with expanding behavioron a line; moreover, every quasiconformal homeomorphism in dimension three canbe smoothened (with bounds) outside a given closed set [19].We finish with some open problems related to branching and quasiregularmappings. The problems are neither new nor due to the author.Problem 2 What are the possible values for the topological dimension of thebranch set of a quasiregular mapping?By suspending a covering map H 3 —t S 3 , where H 3 is as in Section 4, and usingEdwards's theorem, one finds that there exists a branched covering S 5 —¥ S 5 thatbranches exactly on S 1 C S 5 . It is not known whether there exists a quasiregularmapping S 5 —¥ S 5 with similar branch set. If no such map existed, we wouldhave an interesting implication to a seemingly unrelated parametrization problem;it would follow that no double suspension of a homology 3-sphere with nontrivialfundamental group admits a quasisymmetric homeomorphism onto the standard5-sphere, cf. [38], [16, Question 12].By work of Bonk and Kleiner [4], the bi-Lipschitz parametrization problemin dimension n = 2 is equivalent to an analytic problem of characterizing, up to abounded factor, the Jacobian determinants of quasiconformal mappings in R 2 . Anaffirmative answer to Problem 1 in Section 5 would give an affirmative answer tothe following problem.locally comparaProblem 3 (Compare [16, Question 2]) Is every Ai-weight in R 2ble to the Jacobian determinant of a quasiconformal mapping?An Ai-weight is a nonnegative locally integrable function whose mean-valueover each ball is comparable to its essential infimum over the ball. See [7], [32], [3],[16] for further discussion of this and related problems.Problem 4 [16, Question 28] Is there a branched covering / : S" —¥ S n , for somen > 3, such that for every pair of homeomorphisms , fi : S" —¥ S", the mapping4> o / o fi fails to be quasiregular?

698 Juha HeinonenBranched coverings constructed by using the double suspension are obviouscandidates for such mappings. In [15, 9.1], we give an example of a branchedcovering / : S 3 —¥ S 3 such that for every homeomorphism fi : S 3 —¥ S 3 , / o fifails to be quasiregular. The example is based on a geometric decomposition spacearising from Bing's double [34].We close this article by commenting on the lack of direct proofs for somefundamental properties of quasiregular mappings related to branching. For example,it is known that for each n > 3 there exists K(n) > 1 such that every K(n)-quasiregular mapping is a local homeomorphism [25], [28, p. 232]. All knownproofs for this fact are indirect, exploiting the Liouville theorem, and in particularthere is no numerical estimate for K(n). It has been conjectured that the windingmapping (r,9,z) >-¥ (r,29,z) is the extremal here (cf. Section 6). Thus, if one usesthe inner dilatation Ki(fi) of a quasiregular mapping, then conjecturally Ki(fi) < 2implies that Bf = 0 for a quasiregular mapping / in R" for n > 3 [29, p. 76].Ostensibly different, but obviously a related issue, arises in search of Bloch'sconstant for quasiregular mappings. Namely, by exploiting normal families, Eremenko[10] recently proved that for given n > 3 and K > 1, there exists bo =bo(n, K) > 0 such that every if-quasiregular mapping / : R" —t S n has an inversebranch in some ball in S n of radius bo- No numerical estimate for 6 0 is known. Aloregenerally, despite the deep results on value distribution of quasiregular mappings,uncovered by Rickman over the past quarter century, the affect of branching onvalue distribution is unknown, cf. [29, p. 96].References[1] Al. Bonk and J. Heinonen, in preparation.[2] Al. Bonk, J. Heinonen, and S. Rohde, Doubling conformai densities, J. reineangew. Alath., 541 (2001), 117-141.[3] Al. Bonk, J. Heinonen, and E. Saksman, The quasiconformal Jacobian problem,preprint (2002).[4] Al. Bonk and B. Kleiner, Quasisymmetric parametrizations of two-dimensionalmetric spheres, Inventiones Alath., (to appear).[5] Al. Bonk and U. Lang, Bi-Lipschitz parametrizations of surfaces, in preparation.[6] J. W. Cannon, The characterization of topological manifolds of dimension n >5, Proceedings ICA1 (Helsinki, 1978), Acad. Sei. Fenn. Helsinki, (1980), 449-454.[7] G. David and S. Semmes, Strong A^ weights, Sobolev inequalities and quasiconformalmappings, in Analysis and partial differential equations, LectureNotes in Pure and Appi. Alath., 122, Dekker, New York (1990), 101-111.[8] S. K. Donaldson and D. P. Sullivan, Quasiconformal 4-manifolds, Acta Alath.,163 (1989), 181-252.[9] R. D. Edwards, The topology of manifolds and cell-like maps, Proceedings ICA1(Helsinki, 1978) Acad. Sei. Fenn. Helsinki, (1980), 111-127.

The Branch Set of a Quasiregular Alapping 699[10] A. Eremenko, Bloch radius, normal families and quasiregular mappings, Proc.Amer. Alath. Soc, 128 (2000), 557-560.[11] J. Heinonen and T. Kilpeläinen, BLD-mappings in W 2 ' 2 are locally invertible,Alath. Ann., 318 (2000), 391-396.[12] J. Heinonen and P. Koskela, Sobolev mappings with integrable dilatation, Arch.Rational Alech. Anal., 125 (1993), 81-97.[13] J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces of controlledgeometry, Acta Alath., 181 (1998), 1-61.[14] J. Heinonen and S. Rickman, Quasiregular maps S 3 —¥ S 3 with wild branchsets, Topology, 37 (1998), 1-24.[15] J. Heinonen and S. Rickman, Geometric branched covers between generalizedmanifolds, Duke Alath. J., (to appear).[16] J. Heinonen and S. Semmes, Thirty-three yes or no questions about mappings,measures, and metrics, Conform. Geom. Dyn., 1 (1997), 1-12.[17] J. Heinonen and D. Sullivan, On the locally branched Euclidean metric gauge,Duke Alath. J., (to appear).[18] T. Iwaniec and G. Alartin, Geometric function theory and non-linear analysis,Oxford Alathematical Alonographs, Oxford University Press, Oxford (2001).[19] Al. Kiikka, Diffeomorphic approximation of quasiconformal and quasisymmetrichomeomorphisms, Ann. Acad. Sei. Fenn. Ser. A I Alath., 8 (1983), 251-256.[20] P. Koskela, Sobolev spaces and quasiconformal mappings on metric spaces, ProceedingsECA1 (Barcelona, 2000) Progress in Alath., 201, Birkhäuser (2001),457-467.[21] T. J. Laakso, Plane with A^-weighted metric not bilipschitz embeddable to R n ,Bull. London Alath. Soc, (to appear).[22] J. Luukkainen and J. Väisälä, Elements of Lipschitz topology, Ann. Acad. Sci.Fenn. Ser. A I Alath., 3 (1977), 85-122.[23] O. Alartio and V. I. Ryazanov, The Chernavskit theorem and quasiregular mappings,Siberian Adv. Alath., 10 (2000), 16-34.[24] O. Alartio and S. Rickman, Measure properties of the branch set and its imageof quasiregular mappings, Ann. Acad. Sei. Fenn. Ser. A I Alath., 541 (1973),1-15.[25] O. Alartio, S. Rickman, and J. Väisälä, Topological and metric properties ofquasiregular mappings, Ann. Acad. Sei. Fenn. Ser. A I Alath., 488 (1971), 1-31.[26] O. Alartio and J. Väisälä, Elliptic equations and maps of bounded length distortion,Alath. Ann., 282 (1988), 423-443.[27] Yu. G. Reshetnyak, Estimates of the modulus of continuity for certain mappings,Sibirsk. Alat. Z., 7 (1966), 1106-1114; English transi, in Siberian Alath.J., 7 (1966), 879-886.[28] Yu. G. Reshetnyak, Space mappings with bounded distortion, Translation ofAlathematical Alonographs, 73, American Alathematical Society, Providence(1989).[29] S. Rickman, Quasiregular mappings, Ergebnisse der Alathematik und ihrerGrenzgebiete, 26 Springer-Verlag, Berlin Heidelberg New York (1993).[30] S. Rickman and U. Srebro, Remarks on the local index of quasiregular mappings,

700 Juha HeinonenJ. Analyse Alath., 46 (1986), 246-250.[31] J. Sarvas, The Hausdorff dimension of the branch set of a quasiregular mapping,Ann. Acad. Sei. Fenn. Ser. A I Alath., 1 (1975), 297-307.[32] S. Semmes, Bi-Lipschitz mappings and strong A^ weights, Ann. Acad. Sci.Fenn. Ser. A I Alath., 18 (1993), 211-248.[33] S. Semmes, Finding structure in sets with little smoothness, Proceedings ICA1(Zürich, 1994) 1994) Birkhäuser, Basel (1995), 875-885.[34] S. Semmes, Good metric spaces without good parameterizations, Rev. Alat.Iberoamericana, 12 (1996), 187-275.[35] S. Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometricproblems about A^ weights, Rev. Alat. Iberoamericana, 12 (1996), 337-410.[36] S. Semmes, Finding curves on general spaces through quantitative topology,with applications to Sobolev and Poincaré inequalities, Selecta Alath. (N.S.), 2(1996), 155-295.[37] S. Semmes, Real analysis, quantitative topology, and geometric complexity,Pubi. Alat., 45 (2001), 265-333.[38] L. Siebenmann and D. Sullivan, On complexes that are Lipschitz manifolds, inGeometric topology (Proceedings Georgia Topology Conf., Athens, Ga. 1977).Edited by J. C. Cantrell, Academic Press, New York, N.Y. - London (1979),503-525.[39] D. Sullivan, Hyperbolic geometry and homeomorphisms, in Geometric topology(Proceedings Georgia Topology Conf., Athens, Ga. 1977). Edited by J. C.Cantrell, Academic Press, New York, N.Y. - London (1979), 543-555.[40] D. Sullivan, The exterior d, the local degree, and smooth-ability, in "Prospectsof Topology" (F. Quinn, ed.) Princeton Univ. Press, Princeton, New Jersey(1995).[41] D. Sullivan, On the foundation of geometry, analysis, and the differentiablestructure for manifolds, in "Topics in Low-Dimensional Topology" (A.Banyaga, et. al. eds.) World Scientific, Singapore-New Jersey-London-HongKong (1999), 89-92.[42] T. Toro, Surfaces with generalized second fundamental form in L 2 are Lipschitzmanifolds, J. Diff. Geom., 39 (1994), 65-101.[43] T. Toro, Geometric conditions and existence of bi-Lipschitz parameterizations,Duke Alath. J., 77 (1995), 193-227.[44] J. Väisälä, A survey of quasiregular maps in R", Proceedings ICA1 (Helsinki,1978) Acad. Sei. Fenn. Helsinki, (1980), 685-691.[45] H. Whitney Geometric Integration Theory Princeton University Press, Princeton,New Jersey (1957).

ICAl 2002 • Vol. II • 701-709Harmonic Measure and "Locally Flat"Domains*Carlos E. KenigAbstractWe will review work with Tatiana Toro yielding a characterization of thosedomains for which the harmonic measure has a density whose logarithm hasvanishing mean oscillation.2000 Mathematics Subject Classification: 31B25, 35R35, 42B35, 51M25.Keywords and Phrases: Harmonic measure, Locally flat domains, Vanishingmean oscillation.In this lecture, I will describe a series of joint works with Tatiana Toro onthe relationship between regularity properties of harmonic measure and Poissonkernels, and regularity properties of the underlying domains. Thus, consider adomain Q C R" +1 and the solution to the classical Dirichlet problem:{Au = 0 in Qu\ dQ = f£ c b (dfi),( }u £ C{,(0), where Cj is the class of bounded continuous functions. The maximumprinciple and the Riesz representation theorem yield the formulau(X,)= [ f(Q)dw x *(Q), X, GO,Janand the family of positive Borei probability measures {duj x *} is called harmonicmeasure. We sometimes fix X» £ Q and write du = doj x *. Note that, if 0 isa smooth domain, then du x *(Q) = -^-(Q,X„)da(Q), where G is the Green'sfunction for Q, da is surface measure, and -pM— denotes differentiation along theoutward unit normal. When Q is unbounded and v is a minimal harmonic function*Partially supported by the NSF.t University of Chicago, Department of Mathematics, Chicago, IL 60637, USA. E-mail:cek@math.uchicago.edu

702 C. E. Kenigin Q with u| = 0, we define doj°°, harmonic measure with pole at infinity, to bethe measure satisfyingpdw 00 = [ vAip, for p £ C 0 °°(O).an JnThe existence and uniqueness of v and OJ 00 (modulo multiplicative constants) canbe established, for instance, when Q is an unbounded NTA (non-tangentially accessible)domain abbreviation is clarified later, it might as well be here, (see [16] fordetails). For example, if 0 = R" +1 = {(x,t) : t > 0}, then v(x,t) = t and du>°° = dxon R". The work I will describe originated from trying to understand, as a --I 0,the classical theorem of Kellogg, which shows that, if 0 is of class C 1,a , 0 < a < 1,then dw = kda with log k £ C a ; and its "converse", the free boundary regularity ofAlt-Caffarelli [1], which states that, if 0 satisfies certain necessary weak conditions(to be more fully explained later) and du = kda with logfc £ C a , then Q must beof class C 1,a .To motivate our results, we recall real variable characterizations of C 1,a andC a :p £ C 1 ' a (W l )(0

Harmonie Aleasure and "Locally Flat" Domains 703where D denotes Hausdorff distance. Note that this is a significant condition onlyfor 5 < 1. We will always assume Ö < -fi^. We say that 0 is Reifenberg vanishingif, as r —t 0, we can take Ö —¥ 0. For instance, the domain above the graph of a A»function is Reifenberg vanishing. In general, Reifenberg vanishing domains are notlocal graphs; they do not have tangent planes or a "surface measure". This class ofdomains was introduced by Reifenberg [20] in his study of the Plateau problem forminimal surfaces in higher dimensions.In order to state our analogue of Kellogg's theorem in this setting, we needto introduce "multiplicative" analogues of (I)o- A measure p, supported on 90, isdoubling if, Vif ÇC R" +1 , there exists RK > 0 such that, if 0 < r < RK, thenp(B(2r, Q) n 90) < C p(B(r, Q) n 90).Such a p is called asymptotically optimal doubling (see [], []) for details) if it isdoubling andUm . nf p(B(rr,Q)ndn)_ i]m p(B(rr, Q) n 90) _ ^rôQGOfinif p(B(r,Q)ndü) rô Qe9nnK p(B(r,Q)ndü)for 0 < r < 1, K CC R". For example, if 0 is of class C 1,a and da denotessurface measure, then a(B(r, Q) n 90) = a n r n + 0(r n+a ), Q £ 90, and hence a isasymptotically optimal doubling. If logfc £ C a , then the same is true for du = kda.Our analog of Kellog's theorem is:Theorem 1. ([15]) If ii is a Reifenberg vanishing domain, then u (OJ°°) is asymptoticallyoptimal doubling.The proof uses the fact that oJ-Reifenberg flat domains are NTA domains ([9],[15]). One then uses the theory of the boundary behavior of harmonic functions onNTA domains ([9]) and comparisons to half-planes, using the Reifenberg vanishingcondition and the maximum principle.To understand a possible converse to Theorem 1, we recall a geometric measuretheory (GAIT) problem, first posed by Besicovitch: let p be a positive Radonmeasure on R" +1 such that, for each Q £ S (S the support of p) and each r > 0,we havep(B(r, Q)) = ar n , a>0 fixed. (B)Then, what can be said about pi Clearly, if dp = dx on R" C R" +1 , then (B)holds. Nevertheless, in 1987, D. Preiss found the following interesting example: letSc be the light cone x\ = x\ + x\ + x\, and dp = daj: c its surface measure. Then psatisfies (B). Aloreover, the general case of (B) is settled by the following remarkabletheorem of Kowalski-Preiss [17].Theorem. ([17]) Let p be a non-zero measure with property (B), and put S =suppM Ç K n+1 . If n = 1,2, then S = R". If n > 3, then either S = R" orS = Sc ® R" -3 , modulo rigid motions.The connection of the Preiss example to our problem comes from the fact([16]) that, if 0 = {x\ < x\ + x\ + x\}, OCR 4 , then dui 00 = daj: c (separation of

704 C. E. Kenigvariables) and, by Preiss's result, OJ°° is asymptotically optimal doubling, but, ofcourse, 0 is not Reifenberg vanishing, since it is -A^-Reifenberg flat, and no better.Our converse to Theorem 1 is now:Theorem 2. ([16]) Assume that O C R" +1 is an NTA, and thatoj (OJ°°) is asymptoticallyoptimal doubling. Ifn= 1,2, then 0 is Reifenberg vanishing. Ifn>3 and0 is ö-Reifenberg flat, ö < -fi^, then 0 is Reifenberg vanishing.This is in fact a GAIT result. It remains valid if OJ (OJ°°) is replaced by anyasymptotically optimal doubling measure p with support 90. The idea of the proofis to use a "blow-up" argument to reduce matters to the Kowalski-Preiss theorem.Further GAIT results along these lines, also in the higher codimension case, wereobtained by David-Kenig-Toro [4].We now turn to the results motivated by (II)o- Note that the unit normaln satisfies \n\ = 1, and so the BAIO condition on it is automatic, but the VAIOcondition is not. To put our work in perspective, we recall some of the history ofthe subject.A domain 0 C R 1+1 = R 2 is called a chord-arc domain is 90 is locally rectifiable,and, whenever Qi,Q2 £ 90, we have £(s(Qi,Q2)) < C\Qi — Q2I, where £denotes length and s(Qi,Qfi) is the shortest arc between Qi and Q2- 0 is calledvanishing chord-arc if, in addition, as Qi —^ Q2, the ratio,A AA tends to 1, uni-\W 1 W2 [formly on compact sets. The first person to study harmonic measure on chord-arcdomains in the plane was Lavrentiev ([18]), who proved:Theorem. ([18]) //OC R 1+1 is chord-arc, then doj = kda with logfc £ BAlO(da).(In fact, OJ £ A 00 (da), the Muckenhoupt class [7].)For vanishing chord-arc domains in the plane, Pommerenke [19] proved:Theorem. ([19]) Suppose that ii is a chord-arc domain in R 1+1 . Then 0 is vanishingchord-arc if and only if doj = kda with logfc £ VAlO(da).These results were obtained using function theory, so their proofs don't generalizeto higher dimensions. In higher dimensions, the first breakthrough came in thecelebrated theorem of B. Dahlberg [2], who showed that, if 0 C R" +1 is a Lipschitzdomain, then doj = kda with logfc £ BAIO (in fact, OJ £ A 00 (daj). One directionof Pommerenke's result was extended to higher dimensions by Jerison-Kenig [10],who showed that, if 0 is a C 1 domain, then log k £ VAIO. (In general, note that0 is of class C 1 need not imply that log k is continuous.) In order to explain ourresults and to clarify the connection with condition (II)o, we need to introduce someterminology. A domain 0 C R" +1 will be calld a chord-arc domain if it is an NTAdomain (see [9]) of locally finite perimeter (see [6]) and its boundary is Ahlfors regular,i.e., the surface measure a (which is Radon measure on 90, by the assumptionof locally finite perimeter) satisfies the inequalitiesC- X r n < a(B(r, Q) n 90) < Cr n(for Q ë An 90, K CC R" +1 and small r; or, if 0 is an unbounded NTA, forall Q £ 90 and r > 0). A fundamental result of David-Jerison [3] and Semmes

Harmonie Aleasure and "Locally Flat" Domains 705[21] is that Dahlberg's theorem extends to this case, i.e., that doj = kda withlogfc £ BAIO, and, in fact, OJ £ A 00 (da).We say that O C R" +1 is a "o"-chord-arc domain" if 0 if oJ-Reifenberg flat, 0 isof locally finite perimeter, the boundary of 0 is Ahlfors regular and the BAIO normof the unit normal n is bounded by Ö. We say that 0 is "vanishing chord-arc" if, inaddition, it is Reifenberg vanishing and n £ VMO(da).Remark. S. Semmes [22] has proved that, if 0 is a ^-chord-arc domain (underthe definition used above), then(1 + ^ r W " < a(B(r, Q) n 90) < (1 + y/S)a n r n ,where a n is the volume of the unit ball in R" and Ö < 8 n . Moreover, a combinationof the results in [22] and [16] shows that, if 0 is a oJ-Reifenberg flat domain whichis of locally finite perimeter, and for which a(B(r,Q) n 90) < (1 + 8)a n r n , thenthe BAIO norm of n is bounded by CVô for Ô < b~ n . Thus, the two notions introducedof "vanishing chord-arc" domains in the plane are the same, and a domainis vanishing chord-arc exactly when it is of locally finite perimeter, has an Ahlforsregular boundary, it is Reifenberg vanishing and satisfies n £ VAIO.isOur potential-theoretic result, which extends the work of Jerison-Kenig [10],Theorem 3. ([15]) If ii is a vanishing chord-arc domain, then OJ (OJ°°) has theproperty that doj = kda (doj°° = h da) with logfc £ VAIO (log h £ VAIOJ.This was proved by a combination of real variable arguments, potential-theoreticarguments, and the estimates in [10].In order to understand possible converses of this, extending the work of Pommerenketo higher dimensions, we will recall precisely the Alt-Caffarelli [1] resultwhich we alluded to earlier. In the language that we have introduced, their localregularity theorem can be stated as follows:Theorem. ([1]) Let ii be a set of locally finite perimeter whose boundary is Ahlforsregular. Assume that 0 is ô-Reifenberg flat, Ö < ó~ n . Suppose that doj = kda withlogfc G C a (90) (0 0 in O, u| = 0,Au = 0 in O and h = ||. Thus, knowledge of the regularity of the Cauchy data ofv (v\ dQ , §§| ö Q) yields regularity of 90 (or of n, the normal).The first connection between the above Theorem and the work of Pommerenkewas made by Jerison [8], who was also the first to formulate the higher-dimensionalanalogues of Pommerenke's theorem as end-point estimates as a —¥ 0 in the Alt-Caffarelli theorem. Jerison's theorem in [8] states that, if O is a Lipschitz domainand doj = kda with logfc continuous, then n £ VAIO. There is an error in Lemma4 of Jerison's paper. Nonetheless, in [16] we made strong use of the ideas in [8]. Inthe more recent version of our results [14], we bypass this approach.

706 C. E. KenigBefore stating our result, it is useful to classify the assumptions in the Alt-Caffarelli theorem. For this, we recall some examples:Examples. When n = 1, Keldysh-Lavrentiev [12] (see also [5]) constructed domainsin R 1+1 with locally rectifiable boundaries which ([5]) can be taken to beReifenberg vanishing and for which doj = da, i.e., k = 1, but which are not verysmooth.For instance, they fail to be chord-arc. These domains do not, of course,have Ahlfors regular boundaries. When n = 2, Alt-Caffarelli constructed a doublecone F in R 3 such that, for 0 the domain outside the cone, doj°° = da, i.e., k = 1.This is of course not smooth near the origin, the problem being that, while 0 isNTA and 90 is Ahlfors regular, 0 is not oJ-Reifenberg flat for small 8. When n = 3,the Preiss cone we saw before exhibits the same behavior.Our first result was:Theorem 4. ([16]) Assume that 0 Ç R" +1 is ô-chord-arc, 8 < 8 n , that OJ (OJ°°)is asymptotically optimal doubling and that logfc £ VAIO (log h £ VAIOJ. Thenn £ VAIO and 0 is vanishing chord-arcNotice, however, that, when comparing the hypothesis of Theorem 4 to theAlt-Caffarelli theorem two things are apparent : first, we are making the additionalassumption that OJ is asymptotically optimal doubling, and hence, in lightof Theorem 2, 0 is Reifenberg vanishing. Next, the "flatness" assumption in theAlt-Caffarelli theorem is $-Reifenberg flatness, while in Theorem 4 we make the apriori assumption that, in addition, the BAIO norm of n is smaller than 8. R...".This does not make much sense, ecently we have developed a new approach whichhas removed these objections. We have:Theorem 5. ([14]) Let ii be a set of locally finite perimeter whose boundary isAhlfors regular. Assume that 0 is 8-Reifenberg flat, 8 < 8 n . Suppose that doj = kda(doj°° = h da) with logfc G VMO(da) flog h £ VMO(da)). Then ft £ VMO(da)and ii is a vanishing chord-arc domain.Note that Theorems 3 and 5 together give a complete characterization of thevanishing chord-arc domains in terms of their harmonic measure, in analogy withPommerenke's 2-dimensional result, thus answering a question posed by Semmes[21].Our technique for the proof of Theorem 5 is to use a suitable "blow-up" toreduce matters to the following version of the "Liouville theorem" of Alt-Caffarelli(W, [13]):Theorem 6. ([1], [13]) Let ii be a set of locally finite perimeter whose boundaryis (unboundedly) Ahlfors regular. Assume that 0 is an unbounded 8-Reifenberg flatdomain, 8 < 8 n . Suppose that u and h satisfy:andI A« = 0 in 0u > 0 inii u\ ar ,=0. \auuAp = ph da, for p £ C 0 °° (R" +1 ).n Jan

Harmonie Aleasure and "Locally Flat" Domains 707Suppose that sup x€fi |Vu(x)| < 1 and h(Q) > 1 for (da-)a.ca half-space and u(x,x n+ i) = x n+ i.Q on dil. Then 0 isThis allows us to prove the crucial blow-up result, which we now describe.Let 0 be as in Theorem 5, and assume in addition that 0 is unbounded. Supposedoj°° = h da with log h £ VMO(da), and let u be the associated harmonic function.Let Qi £ 90 and assume that Qi —¥ Qoo £ 90 as i —¥ oo (without loss of generality,Qoo = 0). Let {r,}^1 be a sequence of positive numbers tending to 0, and putiii1 (0 - Qi düi1 (90 - QiUifiX)1u(r t X + Qi) and doj°° = hi(Q)dai(Q),where h t (Q) = aVB(r. ^ft^M^Q + Qi)- Then:Theorem 7. There exists a subsequence of {Oj} (which we will call again {Hi})satisfying:andOj —t OQO in the Hausdorff distance sense, uniformly on compact sets; (7.1)90j —t diioo in the Hausdorff distance sense, uniformly on compact sets; (7.2)«« —^ «oo uniformly on compact sets (7.3)AUQO = 0 in OQO«oo > 0 in ii r«no =0 in diirx(7.4)Furthermore(7.5)ando-i ->• CT«», (7.6)weakly as Radon measures. Here, CTQO = / ri n [dii^ and WQO denotes the harmonicmeasure of iloo with pole at oo (corresponding to «ooj- Moreover,SUp |V«oo(^)| < 12Gfioo(7.7)andhoo(Q)dojr,da r . '-(Q)>1 forHn -a.c Q £ 90 œ . (7.8)

708 C. E. KenigSince log/i G VA10(90), the average av B^.^hda is close to the value of log/iin a proportionally large subset of B(r, Q) n 90. This remark allows us to concludethat (7.6) holds, which is crucial to the application and which fails in general underjust (7.1) and (7.2).As an immediate application of Theorems 6 and 7, we obtain that Ooo is ahalf-plane. This already establishes that 0 is Reifenberg vanishing in Theorem 5.To establish that n is in VAIO, we assume otherwise, and obtain Q t —t Q œ , r t —t 0,such that av B (ri,Qi)\n — nB( ri ,Qi)\ 2 da > £ 2 , £ > 0. We consider the correspondingblow-up sequence, and let e n +i be the direction perpendicular to 90^. By thedivergence theorem and (7.1) and (7.2), we have for p £ Cg°(W n+1 ) thatand hencelim / p(Hi,e n+ i)dai = / pdx'Ofi;Jll"x{0}2—»OOso that (7.6) yieldslim -j / pdai - - j Lp\rti — e n+ i\ 2 dai > = / pdx,l—»oo Ofi; 2 J dQ . J jR»x{0}lim / p\n,i — e n+ iYdai = 0.2—»OOTaking p > XB(I,Q) yields the corresponding bound for the integral on 90j C\B(1,0).Butand henceav B(l,0)nOfiil n * — e »+l|aa i — ay B(r i ,Qi)\ n — e n +i| OCT,, xl/2 , xl/2£< lim (av B(r Qi) |n^n B ( r Q;)! 2^) < 2 lim (av B(r ^cfin - e„+i| 2 do-) ,i—»oo \ / i—»oo \ /a contradiction. This concludes the proof.References[1] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problemwith free boundary, J. Reine Angew. Math., 325 (1981), 105^144.[2] B. Dahlberg, On estimates for harmonic measure, Arch. Rat. Mech. Anal., 65(1977), 272^288.[3] G. David and D. Jerison, Lipschitz approximation to hypersurfaces, harmonicmeasure and singular integrals, Indiana Univ. Math. J., 39 (1990), 831A345.[4] G. David, C. Kenig, and T. Toro, Asymptotically optimally doubling measuresand Reifenberg flat sets with vanishing constant, CPAM, 54 (2001), 385^449.[5] P. Düren, The theory of H p spaces, Academic Press, New York, 1970.[6] L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions,Studies in Advanced Alathematics, CRC Press, 1992.[7] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities andrelated topics, Alath. Studies, no. 116, North Holland, 1985.

Harmonie Aleasure and "Locally Flat" Domains 709[8] D. Jerison, Regularity of the Poisson kernel and free boundary problems, ColloquiumMathematicum, 60-61 (1990), 547-567.[9] D. Jerison and C. Kenig, Boundary behavior of harmonic functions in nontangentiallyaccessible domains, Adv. in Math., 46 (1982), 80-147.[10] , The logarithm of the Poisson kernel of a C 1 domain has vanishingmean oscillation, Trans. Amer. Math. Soc, 273 (1982), 781-794.[11] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm.Pure Appi. Math., 14 (1961), 415-126.[12] Al. V. Keldysh and Al. A. Lavrentiev, Sur la représentation conforme des domaineslimités par des courbes rectifiables, Ann. Sci. Ecole Norm. Sup., 54(1937), 1-38.[13] C. Kenig and T. Toro, On the free boundary regularity theorem of Alt andCaffarelli, preprint.[14] , Poisson kernel characterization of Reifenberg flat chord-arc domains,to appear, Ann. Sci. Ec Norm. Sup.[15] , Harmonic measure on locally flat domains, Duke Math. J., 87 (1997),509-551.[16] , Free boundary regularity for harmonic measures and Poisson kernels,Ann. of Math., 150 (1999), 369-154.[17] O. Kowalski and D. Preiss, Besicovitch-type properties of measures and submanifolds,J. Reine Angew. Math., 379 (1987), 115-151.[18] Al. Lavrentiev, Boundary problems in the theory of univalent functions, MathSb. (N S.) I, 43 (1936), 815-844.[19] Ch. Pommerenke, On univalent functions, Bloch functions and VAIOA, Math.Ann., 236 (1978), 199-208.[20] E. Reifenberg, Solution of the Plateau problem for m-dimensional surfaces ofvarying topological type, Acta Math., 104 (1960), 1-92.[21] S. Semmes, Analysis vs. geometry on a class of rectifiable hypersurfaces, IndianaUniv. J., 39 (1990), 1005-1035.[22] , Chord-arc surfaces with small constant I, Adv. in Math., 85 (1991),198-293.

ICAl 2002 • Vol. II • 711-720Solving Pseudo-Differential EquationsNicolas Lerner*AbstractIn 1957, Hans Lewy constructed a counterexample showing that verysimple and natural differential equations can fail to have local solutions. A geometricinterpretation and a generalization of this counterexample were givenin 1960 by L.Hörmander. In the early seventies, L.Nirenberg and F.Treves proposeda geometric condition on the principal symbol, the so-called condition(ip), and provided strong arguments suggesting that it should be equivalentto local solvability. The necessity of condition (ip) for solvability of pseudodifferentialequations was proved by L.Hörmander in 1981. The sufficiency ofcondition (ip) for solvability of differential equations was proved by R.Bealsand C.Fefferman in 1973. For differential equations in any dimension and forpseudo-differential equations in two dimensions, it was shown more preciselythat (ip) implies solvability with a loss of one derivative with respect to theelliptic case: for instance, for a complex vector field X satisfying (ip), f e Lf oc ,the equation Xu = f has a solution u e L'f oc .In 1994, it was proved by N.L. that condition (ip) does not imply solvabilitywith loss of one derivative for pseudo-differential equations, contradictingrepeated claims by several authors. However in 1996, N.Dencker proved thatthese counterexamples were indeed solvable, but with a loss of two derivatives.We shall explore the structure of this phenomenon from both sides: on theone hand, there are first-order pseudo-differential equations satisfying condition(ip) such that no Lf oc solution can be found with some source in Lf oc . Onthe other hand, we shall see that, for these examples, there exists a solutionin the Sobolev space #,"*.The sufficiency of condition (ip) for solvability of pseudo-differential equationsin three or more dimensions is still an open problem. In 2001, N.Denckerannounced that he has proved that condition (ip) implies solvability (with aloss of two derivatives), settling the Nirenberg-Treves conjecture. Althoughhis paper contains several bright and new ideas, it is the opinion of the authorof these lines that a number of points in his article need clarification.2000 Mathematics Subject Classification: 35S05, 35A05, 47G30.Keywords and Phrases: Solvability, Pseudo-Differential equation, Condition*University of Rennes, Université de Rennes 1, Irmar, Campus de Beaulieu, 35042 Rennescedex, France. E-mail: lerner@univ-rennesl.fr

712 N. Lerner1. From Hans Lewy to Nirenberg-Treves'condition (^)Year 1957.The Hans Lewy operator L 0 , introduced in [20], is the following complex vectorfield in R 3 d d dL Q = - Vi- \-i(xi+ix 2 )^—• (1.1)OXi OX2 OX3There exists / £ C°° such that the equationL 0 u = f (1.2)has no distribution solution, even locally. This discovery came as a great shock forseveral reasons. First of all, L 0 has a very simple expression and is natural as theCauchy-Riemann operator on the boundary of the pseudo-convex domain{(zi,z 2 ) £ C 2 ,|zi| 2 + 2Im0 2 < 0}.Moreover L 0 is a non-vanishing vector field so that no pathological behaviour relatedto multiple characteristics is to be expected. In the fifties, it was certainly theconventional wisdom that any "reasonable" operator should be locally solvable, andobviously (1.1) was indeed very reasonable so the conclusion was that, once more,the CW should be revisited. One of the questions posed by such a counterexamplewas to find some geometric explanation for this phenomenon.1960.This was done in 1960 by L.Hörmander in [7] who proved that if p is thesymbol of a differential operator such that, at some point (x, £) in the cotangentbundle,p(x.fi) = 0 and {Rep,lmp}(x.fi) > 0, (1.3)then the operator P with principal symbol p is not locally solvable at x; in fact,there exists / £ C°° such that, for any neighborhood V of x the equation Pu = fhas no solution u £ V(V). Of course, in the case of differential operators, thesign > 0 in (1.3) can be replaced by ^ 0 since the Poisson bracket {Rep, Imp} isthen an homogeneous polynomial with odd degree in the variable £. Nevertheless,it appeared later (in [8]) that the same statement is true for pseudo-differentialoperators, so we keep it that way. Since the symbol of -iL 0 is £1 —X2^3+i(^2+xi^),and the Poisson bracket {£1 —X2Ç3, £2 + Ï1C3} = 2£3, the assumption (1.3) is fulfilledfor L 0 at any point x in the base and the nonsolvability property follows. This givesa necessary condition for local solvability of pseudo-differential equations: a locallysolvableoperator P with principal symbol p should satisfy{Rep, lmp}(x, £) < 0 at p(x.fi) = 0. (1.4)Naturally, condition (1.4) is far from being sufficient for solvability (see e.g. thenonsolvable Af 3 below in (1.5)). After the papers [20], [7], the curiosity of the

Solvability 713mathematical community was aroused in search of a geometric condition on theprincipal symbol, characterizing local solvability of principal type operators. It isimportant to note that for principal type operators with a real principal symbol,such as a non-vanishing real vector field, or the wave equation, local solvability wasknown after the 1955 paper of L.Hörmander in [6]. In fact these results extend quiteeasily to the pseudo-differential real principal type case. As shown by the Hans Lewycounterexample and the necessary condition (1.4), the matters are quite differentfor complex-valued symbols.1963.It is certainly helpful to look now at some simple models. For t, x £ R, withthe usual notationsD t = -W t , (\Dfi\u)(0 = |£|«(£),where « is the ar-Fourier transform of u, I £ N, let us consider the operators definedbyM, = D t + it l D x , Ni = D t + it l \D x \. (1.5)It is indeed rather easy to prove that, for k £ N, M 2 k, N 2 k, A r 2* fc+1 are solvablewhereas M 2 k+i, N 2 k+i are nonsolvable. In particular, the operators Afi, A r i satisfy(1.3). On the other hand, the operator A r 1* = D t — it\D x \ is indeed solvable sinceits adjoint operator A r i verifies the a priori estimater||A r i«|| L 2( R 2 ) > ||«|| L 2( R 2 ) ,for a smooth compactly supported u vanishing for \t\ > T/2. No such estimate issatisfied by N£u since its a:-Fourier transform is-idf.v — it\Ç\v = (—i)(dt.v + t\Ç\v),where v is the ar-Fourier transform of u. A solution of Nfu = 0 is thus given bythe inverse Fourier transform of e - * ^/ 2 , ruining solvability for the operator A r i.A complete study of solvability properties of the models M/ was done in [23] byL.Nirenberg and F.Treves, who also provided a sufficient condition of solvabilityfor vector fields; the analytic-hypoellipticity properties of these operators were alsostudied in a paper by S.Alizohata [21].1971.The ODE-like examples (1.5) led L.Nirenberg and F.Treves in [24-25-26] toformulate a conjecture and to prove it in a number of cases, providing strong groundsin its favour. To explain this, let us look simply at the operatorL = D t + iq(t,x,D x ), (1.6)where q is a real-valued first-order symbol. The symbol of L is thus r + iq(t,x.fi).The bicharacteristic curves of the real part are oriented straight lines with directiond/dt; now we examine the variations of the imaginary part q(t, x, £) along theselines. It amounts only to check the functions t >-¥ q(t,x,Ç) for fixed (#,£). The

714 N. Lernergood cases in (1.5) (when solvability holds) are t 2k Ç, —t 2k+1 \Ç\: when t increasesthese functions do not change sign from — to +. The bad cases are t 2k+1 \Ç\: when tincreases these functions do change sign from — to +; in particular, the nonsolvablecase (1.3), tackled in [8], corresponds to a change of sign of Imp from — to + at asimple zero. The general formulation of condition (ip) for a principal type operatorwith principal symbol p is as follows: for all z £ C, lm(zp) does not change signfrom — to + along the oriented bicharacteristic curves of Re(zp). It is a remarkableand non-trivial fact that this condition is invariant by multiplication by an ellipticfactor as well as by composition with an homogeneous canonical transformation.The Nirenberg-Treves conjecture, proved in several cases in [24-25-26], such as fordifferential operators with analytic coefficients, states that, for a principal typepseudo-differential equation, condition (ip) is equivalent to local solvability.The paper [25] introduced a radically new method of proof of energy estimatesfor the adjoint operator L* based on a factorization of q in (1.6): whenever 0}, the forward Cauchy problem for (1.8) iswell posed, whereas in {&(#,£) < 0}, well-posedness holds for the backward Cauchyproblem. This remark led L.Nirenberg and F.Treves to use as a multiplier in theenergy method the sign of the operator with symbol 6. They were also able toprovide the proper commutator estimates to handle the remainder terms generatedby this operator-theoretic method. Although a factorization (1.7) can be obtainedfor differential operators with analytic regularity satisfying condition (rp), such afactorization is not true in the C°° case. Incidentally, one should note that fordifferential operators, condition (ip) is equivalent to ruling out any change of sign ofImp along the bicharacteristics of Rep (the latter condition is called condition (Pj);this fact is due to the identity p(x, —£) = ( — l) m p(x, £), valid for an homogeneouspolynomial of degree m in the variable £.Using the Malgrange-Weierstrass theorem on normal forms of complex-valuednon-degenerate C°° functions and the Egorov theorem on quantization of homogeneouscanonical transformations, there is no loss of generality considering only firstorder operators of type (1.6). The expression of condition (ip) for L is then verysimplesince it readsq(t, x, £) < 0 and s > t =^> q(s,x.fi) < 0. (1.9)Note that the expression of condition (P) for L is simply q(t,x.fi)q(s,x.fi) > 0.Aluch later in 1988, N.Lerner [14] proved the sufficiency of condition (ip) for localsolvability of pseudo-differential equations in two dimensions and as well forthe classical oblique-derivative problem [15]. The method of proof of these results

Solvability 715is based upon a factorization analogous to (1.7) but where b(x.fi) is replaced byß(t,x)|£| and ß is a smooth function such that t >-¥ ß(t,x) does not change signfrom + to — when t increases. Then a properly defined sign of ß(t, x) appears as anon-decreasing operator and the Nirenberg-Treves energy method can be adaptedto this situation.1973.At this date, R.Beals and C.Fefferman [1] took as a starting point the previousresults of L.Nirenberg and F.Treves and, removing the analyticity assumption, theywere able to prove the sufficiency of condition (P) for local solvability, obtainingthus the sufficiency of condition (ip) for local solvability of differential equations.The key ingredient was a drastically new vision of the pseudo-differential calculus,defined to obtain the factorization (1.7) in regions of the phase space much smallerthan cones or semi-classical "boxes" {(#,£), |x| < 1,|£| < h^1}.Considering thefamily {q(t,x.fi)} f ,_ 11, of classical homogeneous symbols of order 1, they define,via a Calderón-Zygmund decomposition, a pseudo-differential calculus dependingon the family {q(t, •)}, in which all these symbols are first order but also such that,at some level to, some ellipticity property of q(to, •) or V x^q(to, •) is satisfied. Condition(P) then implies easily a factorization of type (1.7) and the Nirenberg-Trevesenergy method can be used. It is interesting to notice that some versions of thesenew pseudo-differential calculi were used later on for the proof of the Fefferman-Phong inequality [5]. In fact, the proof of R.Beals and C.Fefferman marked the daywhen microlocal analysis stopped being only homogeneous or semi-classical, thanksto methods of harmonic analysis such as Calderón-Zygmund decomposition madecompatible with the Heisenberg uncertainty principle.1978.Going back to solvability problems, the existence of C°° solutions for C°°sources was proved by L.Hörmander in [9] for pseudo-differential equations satisfyingcondition (P). For such an operator P of order m, satisfying also a non-trappingcondition, a semi-global existence theorem was proved, with a loss of 1+e derivatives,with e > 0. Following an idea given by R.D.Aloyer [22] for a result in two dimensions,L.Hörmander proved in [10] that condition (ip) is necessary for local solvability:assuming that condition (ip) is not satisfied for a principal type operator P, he wasable to construct approximate non-trivial solutions u for the adjoint equation P*u =0, which implies that P is not solvable. Although the construction is elementary forthe model operators N 2 k+i in (1.5) (as sketched above for A r i in our 1963 section),the multidimensional proof is rather involved and based upon a geometrical opticsmethod adapted to the complex case. The details can be found in the proof oftheorem 26.4.7' of [11].We refer the reader to the paper [13] for a more detailed historical overview ofthis problem. On the other hand, it is clear that our interest is focused on solvabilityin the C°° category. Let us nevertheless recall that the sufficiency of condition (ip)in the analytic category (for microdifferential operators acting on microfunctions)was proved by J.-M.Trépreau [27] (see also [12], chapter vu).

716 N. Lerner2. Counting the loss of derivativesCondition (tp) does not imply solvability with loss of one derivative.Let us consider a principal-type pseudo-differential operator L of order m.We shall say that L is locally solvable with a loss of p derivatives whenever theequation Lu = f has a local solution u in the Sobolev space H s+m^' i for a source/ in H s . Note that the loss is zero if and only if L is elliptic. Since for the simplestprincipal type equation d/dxi, the loss of derivatives is 1, we shall consider that 1is the "ordinary" loss of derivatives. When L satisfies condition (P) (e.g. if L is adifferential operator satisfying condition (ip)), or when L satisfies condition (ip) intwo dimensions, the estimatesC\\L*u\\ H ->\\u\\ H , +m -i, (2.1)valid for smooth compactly supported u with a small enough support, imply localsolvability with loss of 1 derivative, the ordinary loss referred to above. For manyyears,repeated claims were made that condition (ip) for L implies (2.1), that issolvability with loss of 1 derivative. It turned out that these claims were wrong, asshown in [16] by the following result (see also section 6 in the survey [13]).Theorem 2.1. There exists a principal type first-order pseudo-differential operatorL in three dimensions, satisfying condition (fi), a sequence Uk of Cfi" functions withsuppig C {x £ R 3 , |x| < 1/k} such thatIKIIL2(R3) = 1, um \\L*Uk\\ L 2(m) = 0- (2-2)As a consequence, for this L, there exists / £ L 2 such that the equationLu = f has no local solution u in L 2 . We shall now briefly examine some of themain features of this counterexample, leaving aside the technicalities which can befound in the papers quoted above. Let us try, with (t, x, y) £ R 3 ,L = D t - ia(t)(D x + H(t)V(x)\D y \), (2.3)with H = 1 R+ , C°°(R) 9 V > 0, C°°(R) 9 a > 0 flat at 0. Since the functionq(t,x,y.fi,n) = —a(£)(£ + H (t)V (x)\n\) satisfies (1.9) as the product of the nonpositivefunction —a(t) by the non-decreasing function t >-¥ £ + H(t)Y r (x)\r]\, theoperator L satisfies condition (ip). To simplify the exposition, let us assume thata = 1, which introduces a rather unimportant singularity in the t-variable, let usreplace \D y \ by a positive (large) parameter A, which allows us to work now onlywith the two real variables t,x and let us set W = AV. We are looking for anon-trivial solution u(t,x) of L*u = 0, which means thenBxU for'* < °'*" ~ 1 (D x + W(x))u, for t > 0.du-{The operator D, x + W is unitarily equivalent to D, x : with A'(x) = W(x), we haveD x -V W(x) = e^tA^D x e %A^x\ so that the negative eigenspace of the operator

Solvability 717D, x + W(x) is {v £ L 2 (R),supp e iA v C R_}. Since we want u to decay whent —¥ ±00, we need to choose vi,v 2 £ L 2 (M), such that{ etD"Vi, suppwTcR+ for t < 0,et-(D*+w) V2 _ suppe M v 2 CK_ fort>0.(2.4)We shall not be able to choose vi = 1)2 in (2.4), so we could only hope for L*u tobe small if \\v 2 — WI||L 2 (R) is small. Thus this counterexample is likely to work if theunit spheres of the vector spacesE+ = {v £ L 2 (R), supp v C R+ } and Efi = {v £ L 2 (R), supp tßv CL}are close. Note that since W > 0, we get Eff\Efiproducts, we have= {0}: in fact, with L 2 (R) scalarv£E+ 0 0} or in {Imp < 0}. This is no longer the case when condition(ip) holds, although the bicharacteristics are not allowed to pass from {Imp < 0} to{Imp > 0}. The situation of having a bicharacteristic of Rep staying in {Imp = 0}will generically trigger the drift phenomenon mentioned above when condition (P)does not hold. So the counterexamples to solvability with loss of one derivative arein fact very close to operators satisfying condition (P).A related remark is that the ODE-like solvable models in (1.5) do not catch thegenerality allowed by condition (ip). Even for subelliptic operators, whose tranposedare of course locally solvable, it is known that other model operators than M 2 k, A r jcan occur. In particular the three-dimensional models Dt + it 2k (D x +t 2l+1 x 2m \D y \),where k, l, m are non-negative integers are indeed subelliptic and are not reducibleto (1.5) (see chapter 27 in [11] and the remark before corollary 27.2.4 there).

718 N. LernerSolvability with loss of two derivatives.Although theorem 2.1 demonstrates that condition (ip) does not imply solvabilitywith loss of one derivative, the counterexamples constructed in this theoremare indeed solvable, but with a loss of two derivatives, as proven by N.Dencker in1996 [2]. The same author gave a generalization of his results in [3] and later on,analogous results were given in [17].A measurable function p(t, x, £) defined on 1 x I" x I" will be called in thenext theorem a symbol of order m whenever, for all (a, ß) £ N" x N"sup \(d^d f lp)(t,x,0\(l + ^\r m+m 0, and r(t,x,£) be a (complex-valued)symbol of order 0. Then the operatorL=D t + ia(t, x, D x )b(t, x, D x ) + r(t, x, D x ) (2.7)is locally solvable with a loss of two derivatives. Since the counterexamples constructedin theorem 2.1 are in fact of type (2.7), they are locally solvable with a lossof two derivatives.In fact, for all points in R" +1 , there exists a neighborhood V, a positive constantC such that, for all u £ C£°(V)C\\L*u\\ H o>\\u\\ H -i. (2.8)This estimate actually represents a loss of two derivatives for the first-order L; theestimate with loss of 0 derivative would be ||L*U||ìJO > ||«||ffi, the estimate withloss of one derivative would be ||L*U||ìJO > ||w||if°> and both are false, the firstbecause L* is not elliptic, the second from theorem 2.1. The proof of theorem 2.2is essentially based upon the energy method which boils down to compute for allT £ RRe(L*u, iBu + iH(t — T)u) L 2( R „+i)where B = b(t,x,D x ). Some complications occur in the proof from the rather weakassumption df.b > 0 and also from the lower order terms. Anyhow, the correctmultiplier is essentially given by b(t,x,D x ). Theorem 2.2 can be proved for muchmore general classes of pseudo-differential operators than those given by (2.6). Asa consequence, it can be extended naturally to contain the solvability result undercondition (P) (but with a loss of two derivatives, see e.g. theorem 3.4 in [17]).Miscellaneous results.Let us mention that the operator (1.6) is solvable with a loss of one derivative(the ordinary loss) if condition (ip) is satisfied (i.e. (1.9)) as well as the extraconditionIdx^ÔI 2 !^-1 + l%(M,£)| 2 |£| < C\d t q(t,x,Ç)\ when q(t,x,Ç) = 0.

Solvability 719This result is proved in [18] and shows that "transversal" changes of sign do notgenerate difficulties. Solvability with loss of one derivative is also true for operatorssatisfying condition (ip) such that the changes of sign take place on a Lagrangeanmanifold, e.g. operators (1.6) such that the sign of q(t,x,Ç) does not depend on£, i.e. q(t,x,Ç)q(t,x,n) > 0 for all (t,x,Ç,n). This result is proved in section 8 of[13] which provides a generalization of [15] where the standard oblique-derivativeproblem was tackled. On the other hand, it was proved in [19] that for a first-orderpseudo-differential operator L satisfying condition (ip), there exists a L 2 boundedperturbation R such that L + R is locally solvable with loss of two derivatives.3. Conclusion and perspectivesThe following facts are known for principal type pseudo-differential operators.Fl. Local solvability implies (ip).F2. For differential operators and in two dimensions, (ip) implies local solvability.F3. (ip) does not imply local solvability with loss of one derivative.F4. The known counterexamples in (F3) are solvable with loss of two derivatives.The following questions are open.QI. Is (ip) sufficient for local solvability in three or more dimensions?Q2. If the answer to QI is yes, what is the loss of derivatives?Q3. In addition to (ip), which condition should be required to get local solvabilitywith loss of one derivative?Q4. Is analyticity of the principal symbol and condition (ip) sufficient for localsolvability?The most important question is with no doubt QI, since, with Fl, it would settlethe Nirenberg-Treves conjecture. From F3, it appears that the possible loss in Q2should be > 1. In 2001, N.Dencker announced in [4] a positive answer to QI, withanswer 2 in Q2. His paper contains several new and interesting ideas; however, theauthor of this report was not able to understand thoroughly his article.The Nirenberg-Treves conjecture is an important question of analysis, connectinga geometric (classical) property of symbols (Hamiltonians) to a priori inequalitiesfor the quantized operators. The conventional wisdom on this problem turned outto be painfully wrong in the past, requiring the most careful examination of futureclaims.References[1] R.Beals, C.Fefferman, On local solvability of linear partial differential equations,Ann. of Alath. 97 (1973), 482-498.[2] N.Dencker, The solvability of non-L 2 -solvable operators, Saint Jean de Alontsmeeting (1996).[3] , Estimates and solvability, Arkiv.Mat. 37 (1999), 2, 221-243.[4] , On the sufficiency of condition (ip), preprint (october 2001).[5] C.Fefferman, D.H.Phong, On positivity of pseudo-differential equations, Proc.Nat. Acad. Sci. 75 (1978), 4673-4674.

720 N. Lerner6] L.Hörmander, On the theory of general partial differential operators, Acta Alath.94 (1955), 161-248.7] , Differential equations without solutions, Alath.Ann. 140 (1960), 169-173.8] , Pseudo-differential operators and non-elliptic boundary value problems,Ann. of Alath. 83 (1966), 129-209.9] , Propagation of singularities and semiglobal existence theorems for (pseudo-) differential operators of principal type, Ann.of Alath. 108 (1978), 569-609.10] , Pseudo-differential operators of principal type, Singularities in boundaryvalueproblems, D.Reidel Pubi.Co., Dortrecht, Boston, London, 1981.11] , The analysis of linear partial differential operators I IV, Springer Verlag,1983-85.12] , Notions of convexity, Birkhäuser, 1994.13] , On the solvability of pseudodifferential equations, Structure of solutionsof differential equations (Al.Alorimoto, T.Kawai, eds.), World Sci. Publishing,River Edge, NJ, 1996, 183-213.14]N.Lerner, Sufficiency of condition (ip) for local solvability in two dimensions,Ann.of Alath. 128 (1988), 243-258.15] , An iff solvability condition for the oblique derivative problem, SéminaireEDP, Ecole Polytechnique (1990-91), exposé 18.16] , Nonsolvability in L 2 for a first order operator satisfying condition (ip),Ann.of Alath. 139 (1994), 363-393.17] , When is a pseudo-differential equation solvable?, Ann. Fourier 50(2000), 2(spéc.cinq.), 443-460.18] , Energy methods via coherent states and advanced pseudo-differentialcalculus, Alultidimensional complex analysis and partial differential equations( , 177-201, eds.), AA1S.19] , Perturbation and energy estimates, Ann.Sci.ENS 31 (1998), 843-886.20]H.Lewy, An example of a smooth linear partial differential equation withoutsolution, Ann.of Alath. 66, 1 (1957), 155-158.21]S.Alizohata, Solutions nulles et solutions non analytiques, J.Alath.Kyoto Univ.1 (1962), 271-302.22]R.D.Aloyer, Local solvability in two dimensions: necessary conditions for theprincipal type case, Alimeographed manuscript, University of Kansas (1978).23]L.Nirenberg, F.Treves, Solvability of a first order linear partial differential equation,Comm.Pure Appi.Alath. 16 (1963), 331-351.24] , On local solvability of linear partial differential equations. I.Necessaryconditions, Comm.Pure Appi.Alath. 23 (1970), 1-38.25] , On local solvability of linear partial differential equations. II. Sufficientconditions, Comm.Pure Appi.Alath. 23 (1970), 459-509.26] , On local solvability of linear partial differential equations. Correction,Comm.Pure Appl.Math. 24 (1971), 279-288.27]J.-M.Trépreau, Sur la résolubilité analytique microlocale des opérateurs pseudodifférentielsde type principal, Thèse, Université de Reims (1984).

ICAl 2002 • Vol. II • 721-732Singular Integrals Meet ModulationInvarianceC. Thiele*AbstractMany concepts of Fourier analysis on Euclidean spaces rely on the specificationof a frequency point. For example classical Littlewood Paley theorydecomposes the spectrum of functions into annuii centered at the origin. Inthe presence of structures which are invariant under translation of the spectrum(modulation) these concepts need to be refined. This was first done byL. Carleson in his proof of almost everywhere convergence of Fourier seriesin 1966. The work of M. Lacey and the author in the 1990's on the bilinearHilbert transform, a prototype of a modulation invariant singular integral,has revitalized the theme. It is now subject of active research which will besurveyed in the lecture. Most of the recent related work by the author is jointwith C. Muscalu and T. Tao.2000 Mathematics Subject Classification: 42B20, 47H60.Keywords and Phrases: Fourier analysis, Singular integrals, Multilinear.1. Multilinear singular integralsA basic example for the notion of singular integral is a convolution operatorTf(x) = K*f(x) = J K(x-y)f(y)dy (1.1)whose convolution kernel K is not absolutely integrable. If K was absolutely integrablethen we had trivially an a priori estimatel|tf*/llp

722 C. ThieleA basic point of singular integral theory is that an estimate of the form (1.2)may prevail for 1 of ||if||i on the righthand side, if K is not absolutely integrable and the integral (1.1) is only defined ina distributional (principal value) sense. The most prominent example on the realline (indeed, all operators in this article will act on functions on the real line) is theHilbert transform with K(x) = 1/x.Taking formally Fourier transforms, one can write (1.1) as multiplier operator:37(0 = K(Of(0 =: m(£)/(£). (1.3)For the purpose of this survey a sufficiently interesting class of singular integrals isdescribed in terms of the multiplier m by imposing the symbol estimatesfor a = 0,1,2. We define the dual bilinear form(d/dO a m(0 < C|£|-° (1.4)A(fi,f 2 )= (Tfi(xj)f2(x)dx= fitti) M&mfo) da (1.5)J-'Ci+C2=owhere da is the properly normalized Lebesgue measure on the hyperplane £i+£2 = 0.The natural generalization of estimate (1.2) using duality of L p spaces then takesthe form|A(/i,/2)|j),- G Z ê a f amu y °f functions suchthat rrij := ipj is supported in the ball B(0,2 J ) of radius 2 J around 0, vanisheson B(0, 2 J_2 ), and satisfies the symbol estimates (1.4) uniformly in j. By squarefunction estimate we mean the inequalityii(£i/*î 2 ) 1/2 iii>i,j(£i)i/>2,j(^£i) for two families ipij and ip 2 ,j as in thesquare function estimate. Then we haveA(/i,/ 2 )I YI / (Ä * '

with multipliers m satisfyingSingular Integrals Aleet Modulation Invariance 723Ö a m(£') 1/n. Then it is easy to arrange (see Figure "Cone") thesupport of rrij to be in(B(0,2 j ) \ B(0,2 j - 2 j) x (B(0,2 j+n ) \ B(0,2 j - n j) x B(0,2 i+n ) n - 3 .Using smoothness of the multiplier rrij we may use Fourier expansion to writeit as rapidly converging sum of multipliers of elementary tensor formnÄ,i(£i)Ä,i(£ 2 )nî(eo1=3with £„ = — Y^j=i C»-i- The symbol estimates prevail for these elementary tensors,and thus we observe(d/d0 a (fiu)(0

724 C. ThieleFigure 1: "Cone"< en IKE i/» * i kA 2 ) i/2 \\ Pl n II SU P m/« * ^.;iiip. ^ cu n/'iiif •J=l i J=3J = lHere we have used for I = 1,2 the square function estimate (1.7) and for I > 2 theequally fundamental Hardy Littlewood maximal inequalitywhich is valid due to (1.11).2. Modulation invarianceI|SUP|/*

Singular Integrals Aleet Modulation Invariance725Ci V.Figure 2: "Circles"derivative of m can by translated to a point far away from the origin, until the valueof the derivative, which remains constant at the translated points, contradicts (1.9).A natural replacement for (1.9) in the presence of modulation symmetry alongvectors in F has been introduced by Gilbert/Nahmod [6]:ll Pj-We remark that it is unknown whether the condition dim(F) < n/2 can berelaxed in this theorem.The forms A have dual multilinear operators. Theorem 2.1 implies a prioriestimates for these multilinear operators. Aloreover, these multilinear operatorssatisfy estimates which cannot be formulated in terms of IP estimates for A. Let(pi,..., p n ) bea tuple of real numbers or oo such that at most one of these numbers

726 C. Thieleis negative. If all of them are nonnegative, we say A is of type (pi,... ,p n ) If (1-10)holds. If one of them, say pj, is negative, then we define the dual operator T byA(/i,...,/„) = / T(fi,..., fjî, f j+ i,..., f n )(x)fj(x) dx.We then say that A is of type (pi,... ,p n ) if\\T(fl,-- -,fj-l,fj+l,.. -,fn)\\ p >. < C|| WfiWpiwhere p'j = Pj/(pj — 1)- Observe 0 < p'j < 1. The following theorem is again dueto [6] (n = 3) and [16]:Theorem 2.2 Let Y and A be as in Theorem 2.1. Then A is of type (pi,... ,p n )if J2j Ì/Pj= 1; °'t most one of the pj is negative, none of the pj is in [0,1], andn — 2dim(F) + r1/p^ + • • • + l/p ir < ^for all 1 < ii < • • • of a modulation invariant form A is when n = 3 and m(£i, £2)is constant on both sides of a line F but not globally constant. With proper choiceof constants this form can be written asA a (/i,/2,/ 3 ) = /B a (fi,f 2 )(x)fi(x)dxwith the bilinear Hilbert transformB a = p.v. / fi(x - t)f 2 (x - at)- dtand a (projective) parameter a determining the direction of the line F. Theorems2.1 and 2.2 in this special case are due to [10] and [11].For the bilinear Hilbert transform nondegeneracy specializes to the conditiona $ {0,1,00}, and the conclusion of both theorems can be summarized to\\B a (fi,f2)\\ P 0M\{-e,e]also satisfy (2.4) provided a is not degenerate.f(x — t)g(x — at)- dtt

Singular Integrals Aleet Modulation Invariance 727(0,0,1)\eVFigure 3: "Hexagon"/This is stronger than the bounds for the bilinear Hilbert transform itself.The main difference in proving the theorems in this section compared to thediscussion in Section is that it is not sufficient to split the functions fu into frequencypartssupported in B(0,2 J ) \ B(0,2 J_2 ). The special role that is attributed to thezero frequency by this splitting is obsolete in the modulation invariant setting.Instead one has to consider frequency bands of fu away from the origin and verynarrow,such as intervals [N — e, N + e] for large N and small e. Geometricallythese bands can be viewed as the projections of the circles in Figure "Circles" ontothe projected coordinate axes. Handling thin frequency bands requires a new set oftechniques. Prior to the work [10] and [11] these techniques have been pioneered in[2] and [5] where the Carleson operatorCf(x) = sup\p.v. I e %y( -f(x — y) — dy\z J yhas been estimated. Note that this operator is modulation invariant, C(fi) =C(M„fi). See also [12]. Alost theorems discussed in this survey have a simplerbut significant model theorem in the dyadic setting, see for example [17], [22],3. Uniform estimatesTheorem 2.1 excludes certain degenerate subspaces F. For some degenerate Fthe multilinear forms split into simpler objects and one can provide L p estimates

728 C. Thielealso in these degenerate cases; we will give examples below. This raises the questionwhether one can prove bounds on A uniformly in the choice of F, as F approachesone of these degenerate cases.Substantial progress on this question has only been made in the case dim(F) =1.Theorem 3.1 Let n > 3 and (ni, • • • ,n n ) be a unit vector spanning the space Y,and assume r\j ^ 0 for all j. Define the metricd(x,y) := supi

Singular Integrals Aleet Modulation Invariance 729CiFigure 4: "Ellipses"We mention that closely related to the topic of uniform estimates for thebilinear Hilbert transform is that of bilinear multiplier estimates for multiplierswhich are singular along a curve rather than a line, provided the curve is tangentto a degenerate direction. Results for such multipliers have been found by Muscalu[15] and Grafakos/Li [8].We conclude this section with a remark on the history of the bilinear Hilberttransform. Calderon is said to have considered the bilinear Hilbert transform in the1960's while studying what has been named Calderon's first commutator. This isthe bilinear operatorC(A, fi)(x) = p.v. J A{^Z^y) f(y)ay-It can be viewed as a bilinear operator in the derivative A' of A and the function/, and as such has a multplier form as in (1.8). To see this, we can write C(A,fi)in terms of A' as a superposition of bilinear Hilbert transforms:C(A,f)(x) =p.v. / A'(x-V a(y - xj) f(y)dadylox-yB a (f,A')(x)da.The estimate Calderon was looking for wasl|C(A,/)[|2

730 C. ThieleThus he needed good control over the constant C a as a approaches 0 or 1. However,even finiteness of C a was not known to Calderon. Sufficiently good control over C awas first established in [23].The multiplier of C(A',fi) is more regular than that of the bilinear Hilberttransform, and Calderon, quitting his attempts to estimate the bilinear Hilberttransform, proved estimate (3.2) by refinements of the methods in Section (see [1]).4. More multilinear operatorsTheorem 2.1 discusses multipliers singular at a single subspace F'. Cut andpaste arguments easily allow to generalize the theorem to the case of multiplierssingular at finitely many subspaces IV,..., IV, provided each subspace satisfiesthe dimension and non-degeneracy conditions of Theorem 2.1.Interesting phenomena occur for multipliers singular at several subspaces IY,..., Y'k' which do not satisfy the conditions of Theorem 2.1. Some operators correspondingto multipliers singular at degenerate subspaces can be written in terms ofpointwise products and lower degree operators and thus can be trivially shown tosatisfy IP estimates. If m is singular at several such subspaces, the trivial splittingmay no longer be possible, and one has to do a much more subtle analysis.We consider the special case when the spaces IY,... ,IY are hyperplanes andthe multiplier is the characteristic function of one of the infinite simplices beencut out of R n by these hyperplanes, see Figure "Wedge". A basic example is thetrilinear operatorT(h,h, hm = fand its associated fourlinear formâiÇi

Singular Integrals Aleet Modulation Invariance 731C2 = C3Ci = C2Figure 5: "Wedge"An example for a degenerate choice of (0:1,0:2,0:3) is (1,-1,1). In this casethere is a negative result [19]:Theorem 4.2 For cti = l,a 2 = —1,0:3 = 1 the a priori estimatedoes not hold.3|r(/l,/2,/3)l|2/3

732 C. Thiele[6] Gilbert J., Nahmod A., Boundedness of bilinear operators with non-smoothsymbols Alath. Res. Lett. 7, 767^778. [2000][7] Grafakos L., Li X, Uniform bounds for the bilinear Hilbert transform I, preprint.[2000][8] Grafakos L., Li X, The disc as multiplier preprint. [2000][9] Al. Lacey, The bilinear maximal function maps into IP for 2/3 of Polynomials andEigenfunctionsS. Zelditch*AbstractWe review some recent results on asymptotic properties of polynomials oflarge degree, of general holomorphic sections of high powers of positive linebundles over Kahler manifolds, and of Laplace eigenfunctions of large eigenvalueon compact Riemannian manifolds. We describe statistical patterns inthe zeros, critical points and L p norms of random polynomials and holomorphicsections, and the influence of the Newton polytope on these patterns.For eigenfunctions, we discuss L p norms and mass concentration of individualeigenfunctions and their relation to dynamics of the geodesic flow.2000 Mathematics Subject Classification: 35P20, 30C15, 32A25, 58J40,60D05, 81S10, 14M25.Keywords and Phrases: Random polynomial, Holomorphic section of positiveline bundle, Distribution of zeros, Correlation between zeros, Bergman-Szego kernels, Newton polytope, Laplace eigenfunction, Spectral projections,I/ p -norms, Quantum ergodicity.1. IntroductionIn many measures of 'complexity', eigenfunctions ^/Ä(p\ = \(p\ of first orderelliptic operators behave like polynomials p(x) = X^| Q |

I p : — \ J ( Z i , . . . , Z m ) — / J C Q - Z ^ • • • z m , c a £ ( L - j - .734 S. ZelditchIn this article, we review some recent results on the asymptotics of polynomialsand eigenfunctions, concentrating on our work in collaboration with P. Bleher, A.Hassell, B. Shiffman, C. Sogge, J. Toth and Al. Zworski. A unifying feature is theasymptotic properties of reproducing kernels, namely Szegö kernels HN(Z, tv) in thecase of polynomials, and spectral projections E\(x,y) for intervals [A, À+ 1] in thecase of eigenfunctions of sJ~K. For other recent expository articles, see [9, 26].2. PolynomialsThere are several sources of interest in random polynomials. One is the desireto understand typical properties of real and complex algebraic varieties, and howthey depend on the coefficients of the defining equations. Another is their use as amodel for the local behavior of more general eigenfunctions. A third is that theymay be viewed as the eigenvectors of random matrices. Just as random matricesmodel the spectra of 'quantum chaotic' systems, so random polynomials model theireigenfunctions.2.1. SU(m + 1) polynomials on CP m and holomorphic sectionsComplex polynomials of degree of Polynomials and Eigenfunctions 735The problems we discuss in this section involve the geometry of zeros of sectionss £ H°(M, L N ) of general positive line bundles. There is a similar story for criticalpoints.• Problem 1 How are the simultaneous zeros Z s = {z : Si(z) = • • • = Sk(z) =0} of a k-tuple s = (si,..., Sk) of typical holomorphic sections distributed?• Problem 2 How are the zeros correlated? When k = m, the simultaneouszeros form a discrete set. Do zeros repel each other like charged particles? Orbehave independently like particles of an ideal gas? Or attract like gravitatingparticles?By the distribution of zeros we mean either the current of integration over Z s ormore simply the Riemannian (2m—2fc)-volume measure (\Z s \,ip) = J z ipdVol2 m -2k •By the n-point zero correlation functions, we mean the generalized functionsK^k(z 1 ,...,z n )dz =E\Z s \ n ,where \Z s \ n denotes the product of the measures \Z S \ on the punctured productM n = {(z 1 ,..., z n ) £ M x • • • x M : z p ^ z q for p ^ q} and where dz denotes theproduct volume form on M n .The answer to Problem 1 is that zeros almost surely become uniformly distributedrelative to the curvature OJ of the line bundle [18]. Curvature causes sectionsto oscillate more rapidly and hence to vanish more often. Alore precisely, weconsider the space S = Y\'^=1 SH 0 (M,L N ) of random sequences, equipped withthe product measure measure p = Iljv-i ßN- An element in S will be denoteds = {SM}- Then, j^Z s —t OJ, as N -t oo for almost every s.The answer to Problem 2 is more subtle: it depends on the dimension. Weassume fc = m so that almost surely the simultaneous zeros of the fc-tuple of sectionsform a discrete set. We find that these zeros behave almost independently if theyare of distance > -4= apart for D >• 1. So they only interact on distance scalesof size -j=. Since also the density of zeros in a unit ball Bi(zo) around zo growslike N m , we rescale the zeros in the l/-\/ÏV-ball B 1 i^(zo) by a factor of v'ÏV toget configurations of zeros with a constant density as N -t oo. We thus rescale thecorrelation functions and take the scaling limitsK^m(z\...,z n )=Nlim K 1 k (zo)- n K*k(zo + ^=,...,zo+ ^=). (2.1)JV-S-OO y/N VNIn [1], we proved that the scaling limits of these correlation functions were universal,i.e. independent of M, L, OJ, h. They depend only on the dimension m of themanifold and the codimension fc of the zero set.In [2], we found explicit formulae for these universal scaling limits. In the casen = 2, K^^z 1 ,z 2 ), depends only on the distance between the points z 1 ,z 2 , sinceit is universal and hence invariant under rigid motions. Hence it may be written as:We refer to [1] for details.K-2km\ z ,Z") = K-kn%(\Z — Z"\) . (2-2)

736 S. ZelditchTheorem 1 [2] The pair correlation functions of zeros when k = m are given byf ±i r 4-2m + 0( r 8-2m) ; 8S r -) 0K m m(r) = { „ 2 (2.3)[ 1 + 0(e- Cr ), (C > 0) as r -• oo.When m = l,K mm (r) —¥ 0 as r —^ 0 and one has "zero repulsion." Whenm = 2, K mm (r) —¥ 3/4 as r —^ 0 and one has a kind of neutrality. With m > 3,K-mrn (r) / oo as r -I 0 and there is some kind of attraction between zeros. Aloreprecisely, in dimensions greater than 2, one is more likely to find a zero at a smalldistance r from another zero than at a small distance r from a given point; i.e.,zeros tend to clump together in high dimensions.One can understand this dimensional dependence heuristically in terms ofthe geometry of the discriminant varieties V^ C H°(M,L N ) m of systems S =(si,... ,s m ) of m sections with a 'double zero'. The 'separation number' sep(F)of a system is the minimal distance between a pair of its zeros. Since the nearestelement of V 1^ to F is likely to have a simple double zero, one expects: sep(F) ~^Jdist(F,T> 1 ff). Now,the degree ofD^ is approximately N m . Hence, the tube (T>^) tof radius e contains a volume ~ e 2 N m . When e ~ N~ m l 2 , the tube should coverPH°(M,L N ). Hence, any section should have a pair of zeros whose separation is~ A r_TO / 4 apart. It is clear that this separation is larger than, equal to or less thanN^1/ 2 accordingly as m = l,m = 2,m > 3.2.3. Bergman-Szegö kernelsA key object in the proof of these results is the Bergman-Szegö kernel Y1M(X, y),i.e. the kernel of the orthogonal projection onto H°(M,L N ) with respect to theKahler form OJ. For instance, the expected distribution of zeros is given by EJV(^/) =^fi^-ddlogYl]y(z,z) + OJ. Of even greater use is the joint probability distribution(JPD) DM(X 1 , ... ,a;";^1,....fi""-^1,...,z n ) of the random variables x 3 '(s) =s(zJ), ÇJ(s) = Vs(z 3 ), which may be expressed in terms of njv and its derivatives.In turn, the correlation functions may be expressed in terms of the JPD byK N (z\...,z n ) = f D N (0,t,z)Y% =1 (Mm 2 d^)dC[l].The scaling asymptotics of the correlation functions then reduce to scalingasymptotics of the Bergman-Szegö kernel: In normal coordinates {ZJ} at Po £ Mand in a 'preferred' local frame for L, we have [1]:7 r UV——11TT/njv ( rn -\ ;=,®— :r>rn H ;=,V—\)uNm "^ y/N N \/N N Jei(»-

A P • • = M E1[ - P ° ) C CmAsymptotics of Polynomials and Eigenfunctions 7372.4. Polynomials with fixed Newton polytopeThe well-known Bernstein-Kouchnirenko theorem states that the number ofsimultaneous zeros of (a generic family of) m polynomials with Newton polytopeP equals m\Vol(P). Recall that the Newton polytope Pf of a polynomial is theconvex hull of its support Sf = {a £ Z TO : c a ^ 0}. Using the homogenization mapf —¥ F, the space of polynomials / whose Newton polytope Pf contained in P maybeidentified with a subspaceH 0 (CV m ,O(p),P) = {F£ H 0 (CV m ,O(pj) :P f CP} (2.4)ofH 0 (CV m ,O(pj).The problem we address in this section is:• Problem 3 How does the Newton polytope influence on the distribution ofzeros of polynomials?Again, one could ask the same question about L 2 mass, critical points and soon and obtain a similar story. In [19] we explore this influence in a statistical andasymptotic sense. The main theme is that for each property of polynomials understudy, P gives rise to classically allowed regions where the behavior is the same asif no condition were placed on the polynomials, and classically forbidden regionswhere the behavior is exotic.Let us define these terms. If P C R is a convex integral polytope, then theclassically allowed region for polynomials in H°(CW m ,ö(p),P) is the set.-i'' 1(where P° denotes the interior of P), and the classically forbidden region is its(II 2 i i 2 \-,finl n 2, • • •, -IYTJ[|2 I is the moment map ofCP.The result alluded to above is statistical. Since we view the polytope P ofdegree p as placing a condition on the Gaussian ensemble of SU(P) polynomialsof degree p, we endow H°(CW m ,ö(p),P) with the conditional probability measure1S\Pdj6\p(s)= -ê-^dX, «=Â Q^, (2.5)a£P " "where the coefficients A Q are again independent complex Gaussian random variableswith mean zero and variance one.Our simplest result concerns the the expected density E|p(Z/ li ... i / m ) of thesimultaneous zeros of (fi, • • •, f m ) chosen independently from H°(CW m ,ö(p),P).It is the measure on C* TO given byV\p(Z fu ..., f J(U)d%\p(fi)--- I'd % \ P (f m ) [#{z £ U : fi(z) = • • • = f m (z) = 0}] , (2.6)

738 S. Zelditchfor U C

Asymptotics of Polynomials and Eigenfunctions 739Our first concern is with L p norms of L 2 -normalized eigenfunctions. We measurethe growth rate of L p norms by L p (X,g) = sup^^nîi 2=1 ||. By the localWeyl law, E x (x,x) = EA„

740 S. Zelditch[Pi,Pj] = 0 and whose symbols define a moment map V := (pi,... ,p n ) satisfyingdpi A dp2 A • • • A dp n Ôona dense open set QcT*M-0. Since {Pi,Pj} = 0, thefunctions pi,---,p n generate a homogeneous Hamiltonian R"-action whose orbitsfoliate T*M — 0. We refer to this foliation as the Liouville foliation.We consider the IP norms of the L 2 -normalized joint eigenfunctions Pj*fix =Xjipx- The spectrum of A often has bounded multiplicity, so the behaviour of jointeigenfunctions has implications for all eigenfunctions.Theorem 5 [22, 23] Suppose that the Laplacian A g of(M,g) is quantum completelyintegrable and that the joint eigenfunctions have uniformly bounded L°° norms.Then (AT, g) is a flat torus.This is a kind of quantum analogue of the 'Hopf conjecture' (proved by Burago-Ivanov) that metrics on tori without conjugate points are flat. In [23], a quantitativeimprovement is given under a further non-degeneracy assumption. Unless (M,g) isa flat torus, the Liouville foliation must possess a singular leaf of dimension < n.Yet £ denote the minimum dimension of the leaves. We then construct a sequenceof eigenfunctions satisfying:n-l , (n-»(p-2) ,\\Vk\\L~>C(e)X k * ', \\v>k\\L P > C(e)X k 4 * , (2 < p)for any e > 0. It is easy to construct examples were £ = n — l, but it seems plausiblethat in 'many' cases £ = 1. To investigate this, one would study the boundary facesof the image V(T*M — 0) of T * M — 0 under a homogeneous moment map. For arelated study in the case of torus actions, see Lerman-Shirokova [12].3.3. Quantum ergodicityQuantum ergodicity is concerned with the sums (A £ \P 0 (Af)):S P (X)= J2 \(A

Asymptotics of Polynomials and Eigenfunctions 7410(N(X)/(logX) p j) [28]. The asymptotics S 2 (X) ~ B(A)X have recently been obtainedby Luo-Sarnak [13] for Hecke eigenfunctions of the modular group, exploitingthe connections with L-functions. These asymptotics (though not the coefficient)are predicted by the random polynomial model. Other strong bounds in the arithmeticcase were obtained by Kurlberg-Rudnick for eigenfunctions of certain quantizedtorus automorphisms [10]. Bourgain-Lindenstrauss [3] and Wolpert [25] havedeveloped the 'non-scarring' result of [16] to give entropy estimates of possiblequantum limit measures in arithmetic cases.A natural problem is the converse:• Problem 6 What can be said of the dynamics if S P (X) = o(N(X)jiquantum ergodicity imply classical ergodicity?DoesIt is known that classical ergodicity is equivalent to this bound plus estimateson off-diagonal terms [21]. The existence of KAA1 quasimodes (due to Lazutkin[11], Colin de Verdiere [5], and Popov [15]) makes it very likely that KAA1 systemsare not quantum ergodic, nor are (M,g) which have stable elliptic orbits.A further problem which may be accessible is:• Problem 7 How are the nodal sets {(p v = 0} distributed in the limit v —t oo?In [14] (for elliptic curves) and [19] (general Kahler manifolds) it is proved thatthe complex zeros of quantum ergodic eigenfunctions become uniformly distributedrelative to the volume form. Can one prove an analogue for the real zeros?References[1] P. Bleher, B. Shiffman and S. Zelditch, Universality and scaling of correlationsbetween zeros on complex manifolds, Invent. Math. 142 (2000), 351^395.[2] P. Bleher, B. Shiffman and S. Zelditch, Correlations between zeros and supersymmetry,Commun. Alath. Phys. 224 (2001) 1, 255-269.[3] J. Bourgain and E. Lindenstrauss, Entropy of quantum limits (preprint, 2002).[4] Boutet de Alonvel, L.; Sjstrand, J. Sur la singularit des noyaux de Bergmanet de Szeg.Journes: quations aux Drives Partielles de Rennes (1975), 123^164.Astérisque, No. 34-35, Soc. Alath. France, Paris, 1976.[5] Colin de Verdire, Yves Quasi-modes sur les varits Riemanniennes. Invent.Alath. 43 (1977), no. 1, 15^52.[6] H. Donnelly, C. Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds.Invent. Alath. 93 (1988), no. 1, 161-183.[7] P. Gerard and E. Leichtnam, Ergodic properties of eigenfunctions for theDirichlet problem, Duke Alath. J. 71 (1993), 559^607.[8] A. Hassell and S. Zelditch, Quantum ergodicity and boundary values of eigenfunctions(preprint, 2002).[9] D. Jakobson, N. Nadirashvili, and J. Toth, Geometry of eigenfunctions (toappear in Russian Alath Surveys).[10] P. Kurlberg and Z. Rudnick, Value distribution for eigenfunctions of desymmetrizedquantum maps. Internat. Alath. Res. Notices 2001, no. 18, 985^1002.

742 S. Zelditch[11] V. F. Lazutkin, KAM theory and semiclassical approximations to eigenfunctions.With an addendum by A. I. Shnirelman. Ergebnisse der Alathematikund ihrer Grenzgebiete 24. Springer-Verlag, Berlin, 1993.[12] E. Lerman and N. Shirokova, Completely integrable torus actions on symplecticcones, Alath. Res. Letters 9 (2002), 105-116.[13] W. Luo and P. Sarnak, Ergodicity of eigenfunctions on SL(2, Z) \ H, II (inpreparation).[14] Nonnenmacher, S.; Voros, A. Chaotic eigenfunctions in phase space. J. Statist.Phys. 92 (1998), no. 3-4, 431-518.[15] G. Popov, Invariant tori, effective stability, and quasimodes with exponentiallysmallerror terms. I. Birkhoff normal forms, Ann. Henri Poincaré 1 (2000),223-248.[16] Z. Rudnick and P. Sarnak, The behaviour of eigenstates of arithmetic hyperbolicmanifolds. Comm. Alath. Phys. 161 (1994), no. 1, 195-213.[17] Yu. G. Safarov, Asymptotics of a spectral function of a positive elliptic operatorwithout a nontrapping condition, Funct. Anal. Appi. 22 (1988), no. 3, 213-223.[18] B. Shiffman and S. Zelditch, Distribution of zeros of random and quantumchaotic sections of positive line bundles. Comm. Alath. Phys. 200 (1999), no.3, 661-683.[19] B. Shiffman and S. Zelditch, Random polynomials with prescribed Newtonpolytope I (preprint, 2002).[20] C. Sogge and S. Zelditch, Riemannian manifolds with maximal eigenfunctiongrowth (Duke Alath J.).[21] T. Sunada, Quantum ergodicity. Progress in inverse spectral geometry, 175—196, Trends Alath., Birkhuser, Basel, 1997.[22] J. A. Toth and S. Zelditch, Riemannian manifolds with uniformly boundedeigenfunctions, Duke Alath. J. Ill (2002), 97-132.[23] J. A. Toth and S. Zelditch, L p estimates of eigenfunctions in the completelyintegrable case (preprint, 2002).[24] J. Al. VanderKam, L°° norms and quantum ergodicity on the sphere. IMRN 7(1997), 329-347.[25] S. A. Wolpert, The modulus of continuity for F 0 (m)\ll semi-classical limits.Comm. Alath. Phys. 216 (2001), no. 2, 313-323.[26] S. Zelditch, From random polynomials to symplectic geometry, in XHIth InternationalCongress of Mathematical Physics, International Press (2001), 367-376.[27] S. Zelditch, Quantum ergodicity of C* dynamical systems. Comm. Alath. Phys.177 (1996), no. 2, 507-528.[28] S. Zelditch, On the rate of quantum ergodicity. I. Upper bounds. Comm. Alath.Phys. 160 (1994), 81-92.[29] S. Zelditch, Szegö kernels and a theorem of Tian. IMRN 6 (1998), 317-331.[30] S. Zelditch and Al. Zworski, Ergodicity of eigenfunctions for ergodic billiards.Comm. Alath. Phys. 175 (1996), no. 3, 673-682.

ICAl 2002 • Vol. II • 743-753Some Results Related to Group Actionsin Several Complex VariablesXiangyu Zhou*AbstractIn this talk, we'll present some recent results related to group actions inseveral complex variables. We'll not aim at giving a complete survey aboutthe topic but giving some our own results and related ones.We'll divide the results into two cases: compact and noncompact transformationgroups. We emphasize some essential differences between the twocases. In the compact case, we'll mention some results about schlichtness ofenvelopes of holomorphy and compactness of automorphism groups of someinvariant domains. In the noncompact case, we'll present our solution of thelongstanding problem --- the so-called extended future tube conjecture whichasserts that the extended future tube is a domain of holomorphy. Invariantversion of Cartan's lemma about extension of holomorphic functions from thesubvarities in the sense of group actions will be also mentioned.2000 Mathematics Subject Classification: 32.Key words and phrases: Domain of holomorphy, Plurisubharmonic function,Group actions.1. Fundamentals of several complex variablesAbout one century ago, Hartogs discovered that there exist some domainsin several complex variables on which any holomorphic functions can be extendedto larger domains, being different with one complex variable. This causes a basicconcept - domain of holomorphy.Definition. A domain of holomorphy in C n is a domain on which thereexists a holomorphic function which can't be extended holomorphically across anyboundary points.A domain in C" is called holomorphically convex, if given any infinite discretepoint sequence Zk there exists a holomorphic function / s.t. f(zk) is unbounded (or\f(x v )\ —¥ +oo). Consequently, there exists a holomorphic function which tends to•"Institute of Mathematics, AMSS, Chinese Academy of Sciences, Beijing; Department of Mathematics,Zhejiang University, Hangzhou, China. E-mail: xyzhou@math08.math.ac.cn

744 Xiangyu Zhou+00 at the boundary. By Cartan-Thullen's theorem, a domain in C" is a domainof holomorphy if and only if the domain is Stein, i.e., holomorphically convex.Definition. A function ip with value in [—00, +00) on the domain D in C nis called plurisubharmonic (p.s.h.): if (i) ip is upper semicontinuous (i.e., {ip < c}is open for each eel, or equivalently lim z^Zo ip(z) < ip(zo) for z 0 £ D); (ii) foreach complex line L := {zo + tr : z 0 £ D},ip\i Jn £ l is subharmonic w.r.t. one complexvariable t.An equivalent definition in the sense of distributions is that iddip is a positivecurrent; in particular, when tp> is C 2 , this means Levi form f A A- ) > 0 everywhere.In other words, dJdip > 0, where J is the complex structure. (If iddip > 0,then (p is called strictly p.s.h.)Example. For a bounded domain or a domain biholomorphic to a boundeddomain, the Bergman kernel K(z,z) is strictly p.s.h..A pseudoconvex domain in C" is a domain on which there exists a p.s.h.function which tends to +00 at the boundary. It's easy to see that a holomorphicalconvex domain is pseudoconvex, since |/| 2 is plurisubharmonic function where / isgiven in the consequence of the definition of a Stein domain.A deep characterization of a domain of holomorphy is given by a solution toLevi problem which is the converse of the above statement.Fact. A domain D in C" is a domain of holomorphy if and only if the domainis pseudoconvex.A natural corresponding concept of the domain of holomorphy in the settingof complex manifolds (complex spaces) is the so-called Stein manifold (Stein space),which is defined as a holomorphically convex and holomorphically separable complexmanifold (space) . A complex manifold (or space with finite embedding dimension)is Stein if and only if it is a closed complex submanifold (or subvariety) in some C",and if and only if there exists a strictly p.s.h. exhaustion function which is R-valued(i.e., the value —00 is not allowed). A complex reductive Lie group, in particular acomplex semisimple Lie group, is a Stein manifold.We know that a domain of holomorphy or a Stein manifold are defined byspecialholomorphic functions which are usually hard to construct in several complexvariables. However, a pseudoconvex domain is defined by a special p.s.h. functionwhich is a real function and then relatively easy to construct. Construction ofvarious holomorphic objects in several complex variables and complex geometry isa fundamental and difficult problem. An important philosophy here is reducingthe construction of holomorphic functions to the construction of plurisubharmonicfunctions, because of the solution of Levi problem and Hörmander's L 2 estimatesfor 3 and other results.2. Group actions in several complex variablesDefinition. A group action of the group G on a set X is given by a mappingip : G x X —t X satisfying the following: 1) e • x = x, 2) (ab) • x) = a • (b • x), wheree is the identity of the group, a,b, £ G,x £ X, a • x := ip(a,x).

Some Results Related to Group Actions in Several Complex Variables 745A group action on a set can be restricted on various cases. When the set isa topological space and the group is a topological group, the action is continuous,then one gets a topological transformation group; when the space is a metric space,the transformation preserves the metric, then one gets a motion group; when theset is a differentiable manifold and the group is a Lie group, the action is differentiable,then one gets a Lie transformation group; when the set is a vector space, thetransformation preserves the vector space structure, then one gets a linear transformationgroup; when the set is an algebraic variety (or a scheme), the group is analgebraic group, and the action is algebraic, one gets an algebraic transformationgroup; when the set is a complex space, the transformation is holomorphic, andthe action is real analytic, then one gets a (real) holomorphic transformation group(note that in this case, if the action is continuous then it is also real analytic); ifthe set is a complex space, the group is a complex Lie group, and the the action isholomorphic, then one gets a complex (holomorphic) transformation group.In this talk, we're mainly concerned with the last case. We consider a complexLie group Gc with a real form GR acting holomorphically on a complex manifold(also called holomorphic Gc- manifold) and a G^-invariant domain. It's knownthat a complex reductive Lie group has a unique maximal compact subgroup up toconjugate as its real form, but it also has many noncompact real forms.A group action on a set can be regarded as a representation of the group on thewhole group of transformations. An effective group action means the representationis faithful, so it corresponds to a (closed) subgroup of the whole transformationgroup.Actually, many domains in several complex variables such as Hartogs, circular,Reinhardt and tube domains can be formulated in the setting of group actions.Examples, a) Hartogs and circular domains: consider the Hartogs actionof C* with the real form S 1 on C: C x C" -) C n given by (t, (z t ,-- -,z n j) -•(tzi, z 2 , • • •, z n ), then Hartogs domain is ^-invariant domain; consider the circularaction of C* with the real form S 1 on C": C'xC -I C" given by (t, (zi, • • •, z n j) -t(tzi,tz 2 , • • • ,tz n ), then circular domain is SMnvariant domain.b) Reinhardt domains: consider the Reinhardt action of (C*) n on C" given byvvî, ' ' ', in), \Zi, - - -, z n )) y \%iZi, - - -, t n z n ),then Reinhardt domain is (S' 1 )"-invariant domain. One can similarly defines matrixReinhardt domainsc) tube domains: consider the action of R" on C" given by (r, z) —¥ r + z, thenR"-invariant domain is tube domain.d) future tube: let AT 4 be the Alinkowski space with the Lorentz metric:x-y = xnyn-xiyi-x 2 y 2 -X3y3, where a; = (xo,xi,x 2 ,xz),y = (yo,yi,y 2 ,yz) £ -R 4 ;let V + and V~ = —Y r+ be the future and past light cones in R 4 respectively, i.e.Y r± = {y £ M : y 2 > 0, ±yn > 0}, the corresponding tube domains r 1 * 1 = T v =R 4 + iY r± in C 4 are called future and past tubes; let L be the Lorentz group, i.e.L = 0(1,3), L has four connected components, denote the identity component ofL by Lfi, which is called the restricted Lorentz group, i.e. Lfi = 30+(1,3); letL(C) be the complex Lorentz group, i.e.L = 0(1,3,C) = 0(4, C),L(C) has two

746 Xiangyu Zhouconnected components, denote the identity component of L(C) by L + (C), calledthe proper complex Lorentz group which has the restricted Lorentz group as itsreal form. Considering the linear action of L + (C) on C 4 , the future (or past) tubeis ^-invariant.Denote the Appoint future tube by T N = T^1 X • • • x T^1 A r -times, let L + (£)act diagonally on C 4JV , i.e. for z = (z {1 \- • • ,zW) £C 4N ,Az= (Az {1 \ • • •, AzW)where A £ L + (£), then T N is L' + ,-invariant.e) matrix Reinhardt domains: let C n [m x rn] = {(Zi, • • • ,Z n ): Zj £ C[m xrn]} be the space of n-tuples of ro x ro matrices. A domain Da C n [m x ro] iscalled matrix Reinhardt if it is invariant under the diagonal U(m) x U(m) action(U, V)(Zi, •• •, Z n ) H> (UZiV, • • •, UZ„y). These domains are the usual Reinhardtdomains in the case TO = 1. Diag(£>) is defined as the intersection of D with thediagonal matrices (Zi, • • •, Z n ) £ C n [TO x rn]Slice theoryWhen G is a Lie transformation group properly acting on a smooth manifoldX (e.g. when G is compact), one has a satisfactory slice theory about the structureof a neighborhood of an orbit. This theory was extended to the case of an affinereductive group action regularly on an affine variety by D. Luna ([20]) and the caseof a complex reductive Lie group G action holomorphically on a Stein space X by-Snow ([27]). In these cases, the structure of a neighborhood of a closed orbit is finelydetermined.We state the result for reduced Stein spaces. Let G-a; be a closed orbit,then there exists a locally closed G x -invariant Stein subspace B containing x s.t.the natural map from the homogeneous fiber bundle G Xa* B over G/G x = G • x isbiholomorphic onto a 7r-saturated open Stein subset of X, where TT : X —t X//G isthe categorical quotient (or GIT quotient) which exists as a Stein space. Here B iscalled a slice at x. The slice B is transversal to the closed orbit G • x. When X isregular at x, then B can be chosen to be regular.As a consequence of the slice theorem, one has a stratification of the categoricalquotient X//G at least when X is a Stein manifold. The stratum with maximaldimension is Zariski open in X//G and is contained in the regular part of X//G.This is called principal stratum. The inverse of the principal stratum under n : X —tX//G consists of all G-closed orbits satisfying that they are of maximal dimensionk among the dimensions of all G-closed orbits and their corresponding isotropygroups are of minimum number of components. Such orbits are called principalclosed orbits, and the corresponding isotropy groups are called principal. Whenk = dim G, then X is called having FPIG.3. Some results on compact holomorphic transformationgroupsThe relationship between orbit connectedness, orbit convexity, and holomorphicalconvexity goes back to the beginning of this century, when several complexvariables was born. Due to Hartogs, Reinhardt, H.Cartan and others, one alreadyknew some classical relations between completeness, logarithmic convexity and holo-

Some Results Related to Group Actions in Several Complex Variables 747morphical convexity for circular domains, Hartogs domains, and Reinhardt domains.The orbit connectedness and orbit convexity are defined in a general setting (forarbitrary compact connected Lie group), which correspond to completeness andlogarithmic convexity when one restricts to the above domains.There are some fundamental relationships between orbit connectedness andorbit convexity with holomorphically convexity and envelope of holomorphy forinvariant domains.Definition. Let Gc be a connected complex Lie group, GR be a connectedclosed real form ofGc- Let X be a holomorphic Gc-space, D c X be a Gu-invariantset, we call D orbit connected, if for b z : Gc -^ X,g H> g- z,b~ 1 (D) is connected foreach z £ D. When (GC,GR) is a geodesic convex pair/i.e. the map LìO(GR) X GR 9(v, g) —t exp(iv)g £ Gc is a homeomorphism, cf. [3]), D is called orbit convex iffor each z £ D, and v £ ìLì6(GR) s.t. exp(u) £ b 1 z (D) it follows exp(to) £ b 1 z (D)for all t£ [0,1].Roughly speaking, orbit connectedness means that Gcx n I? is connected forevery x £ D.One has known for a long time that the envelope of holomorphy of a domainin C" (or more general a Riemann domain over C") exists uniquely as a Riemanndomain over C". There is a difficult problem of univalence: When is the envelope ofholomorphy of a domain in C" itself a domain in C" ? We have the following criteriafor the univalence of the envelope of holomorphy for certain invariant domains:Theorem 1 ([36]). Let X be a Stein manifold, K c be a complex reductive Liegroup holomorphically acting on X, where K is a connected compact Lie group andK c be its universal compiexification. Let D c X be a K-invariant orbit connecteddomain. Then the envelope of holomorphy E(D) of D is schlicht and orbit convex ifand only if the envelope of holomorphy E(K C -D) of K c -D is schlicht. Furthermore,in this case, E(K C • D) = K c • E(D).When K = S 1 and the action is circular (or a-circular) and Hartogs, thecorresponding concepts of orbit connectedness for such domains were introducedseparately and the above results were obtained and stated separately by CasadioTarabushi and Trapani in [1,2].When K = (S 1 )"" and the action is Reinhardt, the result is well known as aclassical result about Reinhardt domain which asserts that any Reinhardt domainin (C*) n has schlicht envelope of holomorphy.Some other results were also included in the above theorem. So our result canalso be regarded as an extension of their results and the classical result on Reinhardtdomains in a unified way.In the proof, a theorem due to Harish-Chandra on the infinite dimensionalrepresentation of Lie groups plays an important role.We also give some examples of orbit connected domains. Let X = K c /L c ,the action of K c on X be given by the left translations. When L is connected or(K, L) is a symmetric pair, then any if-invariant domain is orbit connected. Thesimplest example is Reinhardt domains in (C*) n .The origin of orbit connectedness could at least go back to [28].Example. A theorem of V.Bargmann, D. Hall and A.S. Wightman (cf.

748 Xiangyu ZhouWightman [32], Jost [12], Streater-Wightman [28]) asserts that r^ is orbit connected.We also consider the homogeneous embeddings of K c /L c . Yet X be a smoothhomogeneous space embedding of K c /L c , D c X be a if-domain. Assume that Lis connected or (K, L) is a symmetric pair. Then E(D) is schlicht and orbit convex.In particular, every matrix Reinhardt domain of holomorphy D is orbit convex.Since an orbit convex matrix Reinhardt domain has a path connected Diag(D), soa matrix Reinhardt domain of holomorphy has a connected Diag(D).Theorem 2 ([37]). Let K be a connected compact Lie group, L be a closed(not necessarily connected) subgroup of K. Let K c and L c be respectively universalcomplexification of K and L. Suppose that D is K-invariant relatively compactdomain in K c /L c (Here the action of K c is given by left translations). Then (i)Aut(D) is a compact Lie group; (ii) Any proper holomorphic self-mapping of D isbiholomorphic if K is semisimple or a direct product of a semisimple compact Liegroup and a compact torus.By a result of Alatsushima, K c /L c is a Stein manifold which is a holomorphicK c - manifold w.r.t. left translation action.The motivations of the present work are two-folds: the result (i) is to extend amain result of [4], where the same result was obtained by requiring a restrictive conditionthat (K, L) is a symmetric pair,i.e., K/L is a compact Riemannian symmetricspace; the result (ii) is to extend a classical result which asserts that proper selfmapping of the relatively compact Reinhardt domains in (C*) n is biholomorphic.The proof is involved with many famous results such as Alostow decompositiontheorem, H. Cartan's theorem about compactness of automorphism groups,Andreotti-Frankel's theorem on homology group of a Stein manifold, the holomorphicversion of de Rham's theorem on a Stein manifold, a result of Milnor's aboutCW complex, a result from iteration theory, Poincaré duality theorem, degree theoryfor proper mappings, covering lifting existence theorem, and a result aboutcompact semisimple Lie groups et al.4. Extended future tube conjectureLet's keep the notation in Example d of the section 2. The set T' N := {Az :z £ T^,A £ L + (C)} is called the extended future tube.The extended future tube conjecture, which arose naturally from axiomaticquantum field theory at the end of 1950's, asserts that the extended future tube T' Nis a domain of holomorphy for N > 3. This conjecture turns out to be very beautifuland natural. In their papers, Vladimirov and Sergeev said that the importance ofthe conjecture is also due to the fact that there are some assertions in QFT, suchas the finite covariance theorem of Bogoliubov-Vladimirov, proved only assumingthat this conjecture is true.According to the axiomatic quantum field theory (cf. [12,13,28]), one maydescribephysical properties of a quantum system using the Wightman functionswhich correspond to holomorphic functions in r^ invariant w.r.t. the diagonalaction of EX. This sort of functions have the following extension property.

Some Results Related to Group Actions in Several Complex Variables 749BHW Theorem (due to Bargman, Hall, and Wightman 1957). An Zq_-invariantholomorphic function on r^ can be extended to an L + (C)-invariant holomorphicfunction on T' N (cf. [12,13,28]).In the proof, the orbit connectedness of r^ play a key role. With this andIdentity Theorem, one can easily define the invariant holomorphic extension.So, a natural question arises, i.e., can these holomorphic functions be extendedfurther? Or, is T' N holomorphic convex w.r.t. L + (C)-invariant holomorphic function?After some argument, this is equivalent to ask if T' N is a domain of holomorphy.Streater's theorem. A holomorphic function on the Dyson domain TJU%UJ(where J := T' N n M 4N is the set of Jost points which was proved to exist andcharacterized by R. Jost) can be extended to a holomorphic function on T' N (cf.[12,28]).So, a natural question is to construct the envelope of holomorphy of the Dysondomain r| U % U J (This question is mentioned in the article "Quantum fieldtheory" of the Russian's great dictionary "Encyclopedia of Alathematics"). Thatthe extended future tube conjecture holds is equivalent to that this envelope ofholomorphy is exactly the extended future tube T' N .The conjecture have been mentioned as an open problem in many classical([12,28]) and recent references ([11,21-24,28-31]) and references therein. In [38,39],we found a route to solve the conjecture via Kiselman-Loeb's minimum principleand Luna's slice theory. Let's recall the minimum principle.Minimum principleLet X be a complex manifold, Gc a connected complex Lie group, GR aconnected closed real form of Gc- Denote ip : Gc —t GC/GR, and p : X x Gc —t Xthe natural projections.Gc acts on X x Gc on the right by:(X" x Gc) x Gc —y X x Gc((x,g),h) i—• (x,gh)Yet 0 c X x Gc be a right GR-invariant domain and have connected fibres ofp; and u £ C°°(ii) be a right GR-invariant function, u naturally induces a smoothfunction ù(x,ip(gj) on Q := (idx,ip)(ty-Suppose that (1) u is p.s.h on Q, (2) Va; £ p(ii),u(x,-) is strictly p.s.h. onii x = 0 C\p^1(x),and (3) u(x, •) is exhaustive on Ù x = ip(iì x ), then the minimumprinciple asserts that v(x) = inf u(x,g) is C°° and p.s.h. on p(ii).Remark. CO. Kiselman in [14] first obtained the minimum principle whenX = C",Gc = C TO ,GR = JTOC , J.J. Loeb in [18] generalized Kiselman's result tothe present general case.It's easy to construct invariant p.s.h. functions w.r.t. compact Lie groupvia "averaging technique". However, such a technique doesn't hold again for noncompact Lie group.Observation. Let G be a real Lie group which acts on C" linearly. Let Dbe a Bergman hyperbolic domain which is G-invariant. Then the Bergman kernelKD(Z,W) satisfies KD(Z,Z) = Kp,(g • z,~g~ r z) for g £ G, when G is compact; whenG is semisimple, we have KD(Z,W) = Kp,(g • z,g • tv).

750 Xiangyu ZhouBrief proof is as follows. Since G linearly act on C", one has a representationG —¥ GL(n,C); if G is semisimple, then the image of G must be in SL(n,C); ifG is compact, the image of G is in U(n). Using the transformation formula forthe Bergman kernels and noting that the determinant of the Jacobian of the mapz —¥ g • z is 1 for semisimple case, and is in S 1 for compact case, then we can getthe result.We consider the following question: Let X be a Stein manifold, Gc be aconnected complex reductive Lie group acting on X s.t. the action is holomorphic,GR a connected real form of Gc- Let D c X be a GR-invariant orbit connectedStein domain, is Gc • D also Stein?When GR is compact, the answer is positive (cf. [22]). This is a special caseof Theorem 1 in the section 3.The extended future tube conjecture is a special case of the question, whereX = C 4N ,Gc = L+(C),GR = L\,D = T+,GC-D = T' NConsider X x Gc —> X, p(x, g) = g^1• x. Suppose that there is a suitable GRinvariants.p.s.h. function ip on D. We have a p.s.h. function u(x,g) = ip(g^1 • x)on Q = p^1(D).Define ip(x) = inf u(x,g) for x £ p(ii), where p : X x Gc —t X isg&l,,given by p(x,g) = x, and fl x := {g £ G c : (X,g) £ fl}.In order to prove ip(x) is p.s.h. on p(Q) = Gc • D, we can use the minimumprinciple due to Kiselman-Loeb.Observation. ii x is connected if and only if D is orbit connected.In order to use the minimum principle, we still need to check two assumptions:(i) u(x, •) is s. p.s.h. on ii x ; (ii) «(a;,-) is exhaustion on Ù x , where u(x, ip(gj) is definedon Q = (id,ip)(ii) c X x GC/GR and is induced by u,ip : Gc —ï GC/GR, Ù X =ip(iì x ). Usually speaking, assumption (i) fails on the whole 0. However, when Xhas FPIG, then the assumption (i) is fulfilled on a Zariski open subset of 0. LetX' := {x £ X : Gear is closed, (Gc) x is principal and finite }, then, by the slicetheory, A = X\A' is a Gc-invariant analytic subset of X. Let D' = D n X', Q' :=p~ 1 (D'), then the assumption (i) is satisfied on Q'. If the assumption (ii) is alsosatisfied on Q', then we can use the minimum principle on Q' and get that ip(x) isp.s.h. on p(iY) = Gc • D\A since ip(x) is upper semicontinuous on Gc • D, by theextension theorem for p.s.h. functions, ip(x) can be extended to a p.s.h. functionon Gc • D.If we can prove that the extended p.s.h. function is weak exhaustion, thenGc • D is Stein.As a consequence of our observations, we deduce that the general question istrue for pseudoconvex pair (GC,GR) (i.e., there exists a GR-invariant p.s.h. functionon Gc which is exhaustion on Gc/GR(cf.[17]), which include the case whenGR is compact and nilpotent (cf. [17]). However it's pity that (L + (C),Zq_) is not apseudoconvex pair.In the case of the extended future tube conjecture, we proved that the assumption(ii) in the minimum principle is satisfied and the constructed functionis weak exhaustion. These are the main technical difficulties. We overcome themand finished our proof via a consideration of the matrix form of the conjecture andexplicit calculations based on Hua's work and matrix techniques ([9,19]).

Some Results Related to Group Actions in Several Complex Variables 751Theorem [38,39]. The extended future tube conjecture is true.A.G. Sergeev posed an interesting idea to attack the mentioned question. Heassumed an invariant version of Cartan's lemma: if A C D is a GR-invariant analyticsubset, / £ 0(A) G - : -, then there exists an F G 0(D) G: s.t. F\A = f. If this is thecase, we can prove that n(D) is Stein in X//Gc- In order to prove it, it's sufficientto prove n(D) is holomorphically convex. Let {y n } C n(D) be an arbitrary discreteset. Then {ir^1(yn )}riD is a GR-invariant analytic subset in D. By the assumption,then there exists a GR-invariant holomorhic function F on D s.t. F\ n -i (y n ) = n.Since 0(n(Dj) = Ö(D) G '•-, then we get a holomorphic function on n(D) which isunbounded on {y n }. This means that n(D) is holomorphically convex, and thenir^1(ir(Dj)is also Stein. When 7r _1 (7r(I?)) = Gc • D, i.e., Gc • D is 7r-saturated,then Gc • D is Stein.It seems to be hard to prove directly the invariant version of Cartan's lemmafor a noncompact Lie group GR, although it's trivially the case for a compact Liegroup. Actually, we have the following:Proposition ([41]). Suppose, furthermore, Gc • D is n-saturated. Then theinvariant version of Cartan's lemma holds if and only if Gc • D is Stein.However, we recently observed that it should be possible to directly give ananswer to the above question based on L 2 -methods and group actions.References[1] E. Casadio Tarabushi, S. Trapani: Envelopes of holomorphy of Hartogs andcircular domains. Pacific J. Math. 149 (1991), no. 2, 231-249.[2] E. Casadio Tarabushi, S. Trapani: Construction of envelopes of holomorphyfor some classes of special domains. J. Geom. Anal. 4 (1994), no. 1, 1-21.[3] G. Coeuré, J.J. Loeb: Univalence de certaines enveloppes d'holomorphie.(French) C R. Acad. Sci. Paris Sér. Alath. 302 (1986), no. 2, 59-61.[4] G. Fels, L. Geati: Invariant domains in complex symmetric spaces, J.rein undangew. Math. 454 (1994), 97-118.[5] H. Grauert: Selected papers, With commentary by Y. T. Siu et al. Springer-Verlag, 1994.[6] H. Grauert, R. Remmert: Coherent analytic sheaves, Springer-Verlag, BerlinHeidelberg, 1984.[7] H. Grauert, R. Remmert: Theory of Stein spaces, Grundl. 236, Springer-Verlag, 1979.[8] L. Hörmander: Introduction to complex analytic in several variables, thirdrevised ed., North-Holland Alathematical Library, Vol.7, North-Halland, Amsterdom,1991.[9] L.-K. Hua: Harmonic analysis of functions of several complex variables in theclassical domains, (in Chinese) Science Press, Beijing, 1958; English translation,Amer.Alath.Soc, Providence, RI, 1963.[10] Al. Jarnicki, P. Pflug: Extension of holomorphic functions, de Gruyter Expositionsin Alathematics, 34. Walter de Gruyter Co., Berlin, 2000.

752 Xiangyu Zhou[11] Al. Jarnicki, P.Pflug: On the extended tube conjecture. Manuscripta Math.,89 (1996), no. 4, 461-470.[12] R. Jost: The general theory of quantized fields. Amer. Alath. Soc, Providence,R. I., 1965.[13] David Kazhdan: Introduction to QFT . in Quantum fields and strings: acourse for mathematicians. Vol. 1, 2. pp.377-418 American AlathematicalSociety, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ,1999.[14] CO. Kiselman: The partial Legendre transformation for plurisubharmonicfunctions. Invent. Math. 49, 137-148 (1978).[15] CO. Kiselman: Plurisubharmonic functions and potential theory in severalcomplex variables, in "Developments of Mathematics 1950-2000", ed. by J.-P.Pier, pp.655-714, Birkhauser-verlag, 2000.[16] AI. Lassalle: Séries de Laurent des fonctions holomorphes dans la complexificationd'un espace symétrique compact. (French) Ann. Sci. école Norm. Sup.(4) 11 (1978), no. 2, 167-210.[17] J.J. Loeb: Pseudo-convexite des ouverts invariants et convexité geodesiquedans certains espaces symétriques.Sern. Lelong-Skoda, Lect. Notes inMath.1198, 172-190.[18] J.J. Loeb: Action d'une forme réele sur un groupe de Lie complexe. Ann. Inst.Fourier, fasse 4, t.35(1985).[19] Qikeng Lu: Classical manifolds and classical domains (in Chinese), ShanghaiScientific and Technical Press, 1963.[20] D. Luna: Slices étales. Bull. Soc. Math. France. Alem 33, 81-105(1973).[21] A.G. Sergeev: Around the extended future tube conjecture, Lect. Notes inMath., v.1574, Springer-Verlag, 1994.[22] A. G. Sergeev, P. Heinzner: The extended matrix disk is a domain of holomorphy.Math. USSR Izvestija, Vol.38(1992), no.3.[23] A. G. Sergeev, V.S. Vladimirov: Complex analysis in the future tube, inEncyclopedia of Math. Sci., Vol.8 (Several Complex Variables, II), Springer-Verlag, 1994.[24] A.G. Sergeev, X.Y. Zhou: On invariant domains of holomorphy (in Russian).Proc. of Steklov Math. Institute, Tom 203, 159-172, 1994.[25] A.G. Sergeev, X.Y. Zhou: Extended future tube conjecture. Proc. of SteklovMath. Institute, Tom 228, 32-51, 2000.[26] Y.-T. Siu: Pseudoconvexity and the problem of Levi. Bull. Amer. Math. Soc.84 (1978), no. 4, 481-512.[27] D. Al. Snow: Reductive Group Actions on Stein Spaces. Math. Ann. 259,79-97 (1982).[28] R. F. Streater, A.S. WightmamPCT, Spin and statistics, and all that. Benjamin,Reading, Alass, 1964.[29] V. S. Vladimirov: Analytic functions of several complex variables and quantumfield theory. Proc. of the Steklov Inst, of Math. 1978, Issue 1, 69-81.[30] V. S. Vladimirov: Several complex variables in mathematical physics. Sem.Lelong-Skoda, Lecture Notes in Math., Vol.919, 1982, 358-386.

[31[32[33[34'[35[36[37[38;[39[4o;[41Some Results Related to Group Actions in Several Complex Variables 753V. S. Vladimirov, V. V. Zharinov: Analytic methods in mathematicial physics.Proc. of the Steklov Inst, of Math. 1988, Issue 2, 117-137.A. S. Wightman: Quantum field theory and analytic function of several complexvariables, J. Indian Math. Soc. (N.S.) 24(1960/1961), 625-677.B. I. Zav'yalov, V.B.Trushin: On the extended n-point tube, Teoret. Mat. Fiz.27, 1(1976), 3-15.X.Y. Zhou: On matrix Reinhardt domains. Math.Ann. 287, 35-46(1990).X.Y. Zhou: On orbit convexity of certain torus invariant domain of holomorphy.Dokl. AN SSSR, T.322, N.2, 1992, 262-267.X.Y. Zhou: On orbit connectedness, orbit convexity, and envelopes of holomorphy.Izvestiya Ross.Akad.Nauk, Series Alath. T.58, N.2, 1994, 196-205.X.Y. Zhou: On invariant domains in certain homogeneous spaces. Ann. LTnst.Fourier, T.47, N.4, 1997, 1101-1115.X.Y. Zhou: A proof of the extended future tube conjecture(in Russian).Izvestiya Ross.Akad.Nauk, Series Alath. T.62, N.l, 1998, 211-224.X.Y. Zhou: The extended future tube is a domain of holomorphy. Math.Research Letters 5, 185-190(1998).X.Y. Zhou: Quotients, invariant version of Cartan's lemma, and the minimumprinciple. Proc. of first ICCM., 335-343, Amer. Alath. Soc. and InternationalPress, 2001.X.Y. Zhou: Invariant version of Cartan's lemma and complexicification ofinvariant domains (in Russian). Dokl. Ross. Akad. Nauk, vol.366, no.5, 1999,608-612.

Section 9. Operator Algebras andFunctional AnalysisSemyon Alesker: Algebraic Structures on Valuations, Their Properties andApplications 757P. Biane: Free Probability and Combinatorics 765D. Bisch: Subfactors and Planar Algebras 775Liming Ge: Free Probability, Free Entropy and Applications to vonNeumann Algebras 787V. Lafforgue: Banach KK-theory and the Baum-Connes Conjecture 795R. Latala: On Some Inequalities for Gaussian Measures 813

ICAl 2002 • Vol. II • 757-764Algebraic Structures on Valuations,Their Properties and ApplicationsSemyon Alesker*AbstractWe describe various structures of algebraic nature on the space of continuousvaluations on convex sets, their properties (like versions of Poincaréduality and hard Lefschetz theorem), and their relations and applications tointegral geometry.2000 Mathematics Subject Classification: 46, 47.Keywords and Phrases: Valuations, Convex sets, Kinematic formulas, ReductiveLie group.0. IntroductionThe theory of continuous valuations on convex sets generalizes, in a sense, boththe measure theory and the theory of the Euler characteristic. Roughly speaking oneshould think of a continuous valuation on a real linear space V as a finite additivemeasure on a class of compact nice subsets of V (say piecewise smooth submanifoldswith corners) which satisfy the following additional property (instead of the usualsigma-additivity): the restriction of to the subclass of convex compact domainswith smooth boundary extends by continuity to the class /C(V) of all convex compactsubsets of V. Here the continuity is understood in the sense of the Hausdorff metricon /C(V). Remind that the Hausdorff metric dn on /C(V) depends on the choiceof the Euclidean metric on V and it is defined as follows: dn(A,B) := infje >0|A c (B) e and B C (A) e }, where (U) e denotes the e-neighborhood of a set U.This condition of continuity turns out to be very strong restriction and has a lot ofconsequences on purely algebraic level. These properties will be discussed in thispaper. The simplest examples of such valuations are any smooth measure on V andthe Euler characteristic. Also it turns out that one of the main tools used recentlyin investigations of valuations is the representation theory of real reductive groupsand the Beilinson-Bernstein theory of D-modules.Now let us give the formal definition of valuation.* Department of Mathematics, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel. E-mail:semyon@post.tau.ac.il

758 Semyon Alesker0.1.1 Definition, a) A function

Algebraic Structures on Valuations, Their Properties and Applications 7591.1.2 Proposition ([5]). The space PVal(V) of polynomial continuous valuationsis dense in the space of all continuous valuations CVal(V).The proof of this proposition is rather simple; it is a tricky use of a formof the Peter-Weyl theorem (for the orthogonal group 0(nj), and in particular theconvexity is not used in any essential way.Let us remind the basic definition of a smooth vector for a representation ofa Lie group. Let p be a continuous representation of a Lie group G in a Fréchetspace F. A vector £ £ F is called G-smooth if the map g H> p(g)Ç is infinitelydifferentiable map from G to F. It is well known the the subset F sm of smoothvectors is a G-invariant linear subspace dense in F. Moreover it has a naturaltopology of a Fréchet space (which is stronger than that induced from F), and therepresentation of G is F sm is continuous.We will especially be interested in polynomial valuations which are GL(V)-smooth. This space will be denoted by (PVal(V)) sm .Example. Let p be a measure on V with a polynomial density with respectto the Lebesgue measure. Let A £ IC(V) be a strictly convex compact subset withsmooth boundary. ThenMip £ Q(fi~ x W) of twovaluations £ Q(V), fi £ Q(W). Let S (fi S n).Now let us define a product on Q(V). Yet A : V < L -¥ V x V denote the diagonalimbedding. For , ip £ Q(V) let

760 Semyon Alesker2. Translation invariant continuous valuationsFor a linear finite dimensional real vector space V let us denote by Val(V) thespace of translation invariant continuous valuations on V. This is a Fréchet spacewith respect to the topology of uniform convergence on compact subsets of /C(V).In this section we will discuss properties of this space.2.1. Irreducibility theorem and Poincaré dualityIt was shown by P. McMullen [17] that the space Val(V) of translation invariantcontinuous valuations on V has a natural grading given by the degree ofhomogeneity of valuations. Let us formulate this more precisely.2.1.1 Definition. A valuation 0

Algebraic Structures on Valuations, Their Properties and Applications 761By the results of Section 1 (Val(V)) sm is a subalgebra of (PVal(V)) sm . It iseasy to see that the algebra structure is compatible with the grading, namelyIn particular we have(Vali(V)) sm CS) (Valj(V)) sm —• (Val ì+ j(Vj) sm .(Vali(Vj) sm ® (Val n -i(Vj) sm —• Dens(V).A version of the Poincaré duality theorem says that this is a perfect pairing. Aloreprecisely2.1.4 Theorem ([5]). The induced mapis an isomorphism.(Vali(Vj) sm —• (Val n -i(V)*) sm ® Dens(V)2.2. Even translation invariant continuous valuationsLet us denote by Val ev (V) the subspace of even translation invariant continuousvaluations. Then clearly (Val ev (V)) sm is a subalgebra of (Val(V)) sm . It turnsout that it satisfies a version of the hard Lefschetz theorem which we are going todescribe.Let us fix on V a scalar product. Let D denote the unit ball with respect tothis product. Let us define an operator A : Val(V) —y Val(V). For a valuation £ Val(V) set(AcP)(K):=^\ e=Q cP(K+ eD).(Note that by a result of P. McMullen [17] (K + sD) is a polynomial in e > 0 ofdegree at most n.) It is easy to see that A preserves the parity of valuations anddecreases the degree of homogeneity by 1. In particularA:Vair(V)^Val^Li(V).The following result is a version of the hard Lefschetz theorem.2.2.1 Theorem ([4]). Let k > n/2. ThenA2k-n . (Vall v (V)y m —• (Val e fi_ k (Vj) smis an isomorphism. In particular for 1 of of this result is based on the solution of the cosine transformproblem due to J. Bernstein and the author [6], which is the problem from(Gelfand style) integral geometry motivated by stochastic geometry and going backto G. Alatheron [16].

762 Semyon Alesker2.3. Valuations invariant under a groupLet G be a subgroup of GL(V). Yet us denote by Val G (V) the space of G-invariant translations invariant continuous valuations. From the results of [2] and[4] follows the following result.2.3.1 Theorem. Let G be a compact subgroup ofGL(V) acting transitively on theunit sphere. Then Val G (V) is a finite dimensional graded subalgebra of (Val(V)) sm .It satisfies the Poincaré duality, and if —Id £ G it satisfies the hard Lefschetztheorem.It turns out that Val G (V) can be described explicitly (as a vector space) forG = SO(n), 0(n), and U(n). In the first two cases it is the classical theorem ofHadwiger [10], the last case is new (see [4]). In order to state these results we haveto introduce first sufficiently many examples.Let 0 be a compact domain in a Euclidean space V with a smooth boundary90. Yet n = dim V. For any point s £ 90 let ki(s),..., fc n _i (s) denote the principalcurvatures at s. For 0 < i < n — 1 defineVi(Q) := - (n T 1 ) f {k h ,..., *,•„_!_, }da,n\n-l-ijwhere {kj x ,..., kj n _ 1 _ i } denotes the (n — l — z)-th elementary symmetric polynomialin the principal curvatures, da is the measure induced on 90 by the Euclideanstructure. It is well known that V» (uniquely) extends by continuity in the Hausdorffmetric to /C(V). Define also F„(0) := vol (Si). Note that Vo is proportional to theEuler characteristic x- It ' 1S weu known that Vo, Vi,..., V n belong to Val°^(V).It is easy to see that \fi is homogeneous of degree k. The famous result of Hadwigersays2.3.2 Theorem (Hadwiger, [10]). LetV be n-dimensional Euclidean space. Thevaluations Vn,Vi,.. .,V n form a basis of Val s °( n fiV)(= Val°( n fiVj).Now let us describe unitarily invariant valuations on a Hermitian space. LetW be a Hermitian space, i.e. a complex vector space equipped with a Hermitianscalar product. Let m := dim^W (thus dim^W = 2m). For every non-negativeintegers p and k such that 2p < k < 2m let us introduce the following valuations:ThenUk, P £J dQUk,p(K)= / V k - 2p (KC\E)-dE.JEÂGr m - pVal" {m) (W).2.3.3 Theorem ([4]). Let W be a Hermitian vector space of complex dimensionm. The valuations Uk, p with 0 of the spaceVal? m \W).It turns out that the proof of this theorem is highly indirect, and it useseverything known about even translation invariant continuous valuations including

Algebraic Structures on Valuations, Their Properties and Applications 763the solution of McMullen's conjecture, cosine transform, hard Lefschetz theorem forvaluations, and also results of Howe and Lee [11] on the structure of certain GL n (H)-modules. Namely in order to describe explicitly the (finite dimensional) space ofunitarily invariant valuations it is necessary to study the (infinite dimensional)GL R (IA")-module Val ev (W).Note that as algebra Val s °( n fiV) is isomorphic to

764 Semyon AleskerReferences[1] Alesker, Semyon; Continuous rotation invariant valuations on convex sets. Ann.of Alath. (2) 149 (1999), no. 3, 977^1005.[2] Alesker, Semyon; On P. McMullen's conjecture on translation invariant valuations.Adv. Alath. 155 (2000), no. 2, 239^263.[3] Alesker, Semyon; Description of translation invariant valuations on convex setswith solution of P. McMullen's conjecture. Geom. Funct. Anal. 11 (2001), no.2, 244^272.[4] Alesker, Semyon; Hard Lefschetz theorem for valuations, unitarily invariantvaluations, and complex integral geometry, in preparation.[5] Alesker, Semyon; The multiplicative structure on valuations on convex sets, inpreparation.[6] Alesker, Semyon; Bernstein, Joseph; Range characterization of the cosine transformon higher Grassmannians. math.MG/0111031[7] Chern, Shiing-shen; On the kinematic formula in integral geometry. J. Alath.Alech. 16 (1966), 101-118.[8] Chern, Shiing-shen; On the kinematic formula in the Euclidean space of ndimensions. Amer. J. Alath. 74, (1952). 227^236.[9] Griffiths, Phillip A.; Complex differential and integral geometry and curvatureintegrals associated to singularities of complex analytic varieties. Duke Alath.J. 45 (1978), no. 3, 427^512.[10] Hadwiger, Hugo; Vorlesungen über Inhalt, Oberfläche und Isoperimetrie.Springer-Verlag, Berlin-Göttingen-Heidelberg 1957.[11] Howe, Roger; Lee, Soo Teck; Degenerate principal series representations ofGL n (C) and GL„(R). J. Funct. Anal. 166 (1999), no. 2, 244^309.[12] Klain, Daniel A.; A short proof of Hadwiger's characterization theorem. Mathematika42 (1995), no. 2, 329^339.[13] Klain, Daniel A.; Rota, Gian-Carlo; Introduction to geometric probability.Lezioni Lincee. [Lincei Lectures] Cambridge University Press, Cambridge, 1997.[14] Pukhlikov, A. V.; Khovanskii, A. G.; Finitely additive measures of virtualpolyhedra. (Russian) Algebra i Analiz 4 (1992), no. 2, 161-185; translation inSt. Petersburg Alath. J. 4 (1993), no. 2, 337^356.[15] Pukhlikov, A. V.; Khovanskii, A. G.; The Riemann-Roch theorem for integralsand sums of quasipolynomials on virtual polytopes. (Russian) Algebra i Analiz4 (1992), no. 4, 188^216; translation in St. Petersburg Alath. J. 4 (1993), no.4, 789^812.[16] Alatheron, G.; Un théorème d'unicité pour les hyperplans poissoniens. J. Appi.Probability 11 (1974), 184^189.[17] McMullen, Peter; Valuations and Euler-type relations on certain classes ofconvex polytopes. Proc. London Alath. Soc. (3) 35 (1977), no. 1, 113-135.[18] McMullen, Peter; Continuous translation-invariant valuations on the space ofcompact convex sets. Arch. Alath. (Basel) 34 (1980), no. 4, 377^384.[19] Santaló, Luis A.; Integral geometry and geometric probability. With a forewordby Alark Kac. Encyclopedia of Alathematics and its Applications, Vol. 1.Addison-Wesley Publishing Co., Reading, Alass.-London-Amsterdam, 1976.[20] Schneider, Rolf; Simple valuations on convex bodies. Alathematika 43 (1996),no. 1, 32^39.

ICAl 2002 • Vol. II • 765^774Free Probability and CombinatoriesP. Biane*AbstractA combinatorial approach to free probability theory has been developpedby Roland Speicher, based on the notion of noncrossing cumulants, a freeanalogue of the classical theory of cumulants in probability theory. We reviewthis theory, and explain the connections between free probability theory andrandom matrices. We relate noncrossing cumulants to classical cumulants andalso to characters of large symmetric groups. Finally we give applications tothe asymptotics of representations of symmetric groups, specifically to theLittlewood-Richardson rule.2000 Mathematics Subject Classification: 46L54, 05E10, 60B15.Keywords and Phrases: Free probability, Symmetric group, Noncrossingpartitions.1. IntroductionFree probability has been introduced by D. Voiculescu [21] as a means of studyingthe group von Neumann algebras of free groups, using probabilistic techniques.His theory has become very successful when he discovered a deep relation withthe theory of random matrices, and solved some old questions in operator algebra,see [4], [7], [24] for an overview. A purely combinatorial approach to Voiculescu'sdefinition of freeness has been given by R. Speicher [19], [20], building on G. C.Rota's [16] approach to classical probability. It is based on the notion of noncrossingpartitions, also known as "planar diagrams" in quantum field theory, andprovides unifying concepts for many computations in free probability. Noncrossingpartitions turn out to be connected with the geometry of the symmetric group, andthis leads to some new understanding of the asymptotic behaviour of the charactersand representations of large symmetric groups. Our aim is to survey these results,we shall start with the basic definition of freeness, then explain its connection torandom matrix theory. In the third section we review Speicher's theory. In thefourth section we show how noncrossing cumulants arise naturally in connection* Département de Mathématiques et Applications, École Normale Supérieure, 45 rue d'Ulm75005 Paris, France. E-mail: Philippe.Biane@ens.fr

766 P. Bianewith classical cumulants associated with random matrices, and with characters ofsymmetric groups. Finally in section 5 we explain the asymptotic behaviour ofrepresentations of symmetric groups in terms of free probability concepts.2. Freeness and random matricesThe usual framework for free probability is a von Neumann algebra A, equippedwith a faithful, tracial, normal state r. To any self-adjoint element X £ A one canassociate its distribution, the probability measure on the real line, uniquely determinedby the identity r(X n ) = f- R x n p(dx) for all n > 1. This makes it natural tothink of the elements of A as noneommutative random variables, and of r as an expectationmap, and one usually calls noneommutative probability space such a pair(A,T). Although a great deal of the theory, especially the combinatorial side, canbe developped in a purely algebraic way, assuming only that A is a complex algebrawith unit, and r a complex linear functional, we shall stick to the von Neumannframework in the present exposition.Given (A, T), one considers a family {AJ; i £ 1} of von Neumann subalgebras.This family is called a free family if the following holds: for any k > 1 and fc-tuplecti,..., ak £ A such that• each aj belongs to some algebra A tj ,• T(OJ) = 0 for all j,with ii ^ i 2 ,i 2 ^ h, • • •, iu-i ^ ik,one has r(ai... ak) = 0.Moreover, a family of elements of A is called free if the von Neumann algebraseach of them generates form a free family. Freeness is a noneommutativenotion analogous to the independence of a-fields in probability theory, but whichincorporates also the notion of algebraic independence.Observe that if cti and a 2 are free elements in (A,T), and one defines thecentered elements ô, = a, — r(a,)l then one can conputer(aia 2 ) = r(âiâ 2 ) -V r(ai)r(a 2 ) = r(ai)r(a 2 )where the freeness condition has been used to get r(âiÔ2) = 0. Actually, if {Affi £1} is a free family, it is not dificult to see that one can compute the value of r onany product of the form cti... ak, where each aj belongs to some of the A t 's, interms of the quantities r(aj t ... a-j l ) where all the elements aj 1 ,..., a-j l belong tothe same subalgebra. This implies that the value of r on the algebra generatedby the family {Affi £ 1} is completely determined by the restrictions of r toeach of these subalgebras. However the problem of finding an explicit formula isnontrivial, and this is where combinatorics comes in. We shall describe Speicher'stheory of noncrossing cumulants, which solves this problem, in the next section, butbefore that we explain how free probability is relevant to understand large randommatrices.Consider n random N x N matrices X} ,..., X n , of the formXf ] = UjDf ] U* (2.1)

Free Probability and Combinatorics 767where D- ';j = l,...,n are diagonal, hermitian, nonrandom matrices and Uj areindependent unitary random matrices, each distributed with the Haar measure onthe unitary group YJ(N). In other words we have fixed the spectra of the X>but their eigenvectors are chosen at random. The n-tuple Xf ,..., X n canbe recovered, up to a global unitary conjugation X> H> UX t 'U*, (where Udoes not depend on i), from its mixed moments, i.e. the set of complex numbersjjTr(X h .. .X ik ) where ii,...ik are arbitrary sequences of indices in {l,...,n}.In particular the spectrum of any noneommutative polynomial of the X> can berecovered from these data. A most remarkable fact is that if we assume that theindividual moments j^Tr((Xl ) k ) converge as N tends to infinity, then the mixedmoments -^Tr(X> i .. .X ik ) converge in probability, and their limit is obtainedby the prescriptions of free probability.Theorem 1. Let (A,T) be a noneommutative probability space with free selfadjointelements Xi,...,X n , satisfying r(Xf) = limjv-s.00 ^Tr((X t ) k ), for alli and k, then, in probability, j^Tr(X ii ' ...X ik ') ^JV-S-OO T(X ì1 ...X ik ), for allii,--.,ik-This striking result was first proved by D. Voiculescu [23], and has lead to theresolution of many open problems about von Neumann algebras, upon which weshall not touch here.3. Noncrossing partitions and cumulantsA partition of the set {1,... ,n} is said to have a crossing if there exists aquadruple (i,j, k,l), with 1 of the partition and j and I belong to another class. If a partition has nocrossing, it is called noncrossing. The set of all noncrossing partitions of {1,... ,n}is denoted by NC(n). It is a lattice for the refinement order, which seems to havebeen first systematically investigated in [10].Let (A,T) be a non-commutative probability space, then we shall define afamily RS") of n-multilinear forms on A, for n > 1, by the following formulaHere, for n £ NC(n), one has definedr(ai...a n )= ^ R[n](ai,.. .,a n ). (3.1)TreNC(n)R[n](ai,...,a n )= JJ R,U v V(a v )where ay = (a,j 1 ,. • • ,aj k ) if V = {ji,-- -,jk} is a elass of the partition n, withji < J2 < • • • < jk and |F| = fc is the number of elements of V. In particularR[l n ] = R^ if l n is the partition with only one class. Thus one has, for n = 3,T(aia 2 a 3 ) = R^ (oi ,a 2 ,a 3 ) + R^ (oi, a 2 )R^ (o 3 ) + R {2) (ai,a 3 )R^ (a 2 )+RW (a 2 , a 3 )R^ (oi) + R^ (ai)R^ (oa)^1)(a 3 ).V67T

768 P. BianeObserve thatr(ai ... a n ) = R^(cti,..., a n ) + terms involving R^ for fc < nso that the R^ are well defined by (3.1) and can be computed by induction on n.They are called the noncrossing (or sometimes free) cumulant functionals on A.The formula (3.1) can be inverted to yieldR (n) (ai,...,a n )= Y Moeb([n,l n ])T[n](ai,...,a n ).TreNC(n)Here T[TT] (cti,..., a n ) = Yl V€n T~(aj x ... aj k ) where V = {ji,..., jk} are the classesof IT, and Moeb is the Möbius function of the lattice NC(n), see [20].For example, one hasfiW(ai) = r(ai); R^(ai,a 2 ) = r(aia 2 ) - r(ai)r(a 2 );R^(ai,a 2 ,a 3 ) = r(aia 2 a 3 ) - r(ai)r(a 2 az) - r(a 2 )r(aiaz)-r(az)r(aia 2 ) + 2r(ai)T(a 2 )r(az).Note that when the lattice of all partitions is used instead of noncrossing partitions,then one gets the usual family of cumulants (see Rota [16]), with another Alöbiusfunction.The connection between noncrossing cumulants and freeness is the followingresult from section 4 of [19].Theorem 2. Let {Afi £ 1} be a free family of subalgebras of (A,T), andai,...,a n £ A be such that aj belongs to some A tj for each j £ {1,2,..., n}. Thenone has R^ (ai,..., a n ) = 0 if there exists some j and k with ij ^ ik •This result leads to an explicit expression for r(ai.. .a n ), where ai,...,a nis an arbitrary sequence in A, such that each aj belongs to one of the algebrasAfi £ I. By Theorem 2, in the right hand side of (3.1), the terms correspondingto partitions n having a class containing two elements j, k such that aj and a^belong to distinct algebras give a zero contribution. Thus we have to sum overpartitions in which all j's belonging to a certain block of the partition are suchthat aj belongs to the same algebra. Since we can express noncrossing cumulantsin terms of moments we get the formula for r(ai ... a n ) in terms of the restrictionsof r to each of the subalgebras A t . Noncrossing cumulants are a powerful toolfor making computations in free probability, see [11], [12], [13], [14], [18], for someapplications. We give a simple illustration below.Let Xi and X 2 be two self-adjoint elements which are free, then the distributionof Xi + X 2 , depends only on the distributions of Xi and X 2 and can becomputed as follows. Let R^(Xi,... ,Xi) and R^(X 2 ,... ,X 2 ), for n > 1, be thenoncrossing cumulants of Xi and X 2 , then one can expand R^ (Xi + X 2 ,...,Xi +X 2 ) by multilinearity as ^. • R^ (X tl ,..., X in ) where the sum is over all sequencesof 1 and 2. By Theorem 2, all terms vanish except R^nfiXi,...,Xi) andRW (X 2 ,..., X 2 ). It follows thatR (n) (Xi + X 2 ,... ,Xi + X 2 ) = R (n) (Xi,.. .,Xi) + R (n) (X2,... ,X 2 )

Free Probability and Combinatorics 769allowing the computation of the moments of Xi + X 2 , hence its distribution, interms of the distributions of Xi and X 2 . It remains to give a compact form to therelation between moments and noncrossing cumulants. For any self-adjoint elementX with distribution p, letbe its Cauchy transform, and let1 °° r 1G x (z)=- + Yz- k - 1 T(X k )= / p(dx)zitiK(z) = - +be the inverse series for composition.Theorem 3. [19]Y R kZ kk=0Jnz-xOne has R k = R (k) (X,..., X) for all fc.The operation which associates to the two distributions of Xi and X 2 thedistribution of their sum is called the free convolution of measures on the real line,and was introduced by D. Voiculescu, who first considered the coefficients Rk andproved the formula for the free convolution of two measures, using very differentmethods [22].Combining theorems 1 and 2, given two large random matrices of known spectraone can predict the spectral distribution of their sum, with a good accuracyand probability close to 1. It is illuminating to look at the following example. Thehistogram below is made of the 800 eigenvalues of a random matrix of the formHi + n 2 where Hi and n 2 are two orthogonal projections onto some random subpacesof dimension 400 in C 800 , chosen independently. The curve y = —, 40T\JX(2-x)which corresponds to the large N limit predicted by free probability has been drawn.0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

770 P. Biane4. Noncrossing cumulants, random matrices andcharacters of symmetric groupsBesides free probability theory, noncrossing partitions appear in several areasof mathematics. We indicate some relevant connections. The first is with thetheory of map enumeration initiated by investigations of theoretical physicists intwo-dimensional quantum field theory. The noncrossing partitions appear thereunder the guise of planar diagrams, the Feynman diagrams which dominate thematrix integrals in the large N limit. This is of course related to the fact that largematrices model free probability. We shall not discuss this further here, but refer to[26] for an accessible introduction. Another place where noncrossing partitions playa role, which is closely related to the preceding, is the geometry of the symmetricgroup, more precisely of its Cayley graph. Consider the (unoriented) graph whosevertex set is the symmetric group £„, and such that {01,02} is an edge if and onlyif a~ 1 0 2 is a transposition, i.e. this is the Cayley graph of £„ with respect to thegenerating set of all transpositions. The distance on the graph is given byd(ai,a 2 ) = n — number of orbits of aT x a 2 := |07" 1 0 2 |.The lattice of noncrossing partitions can be imbedded in £„ in the following way[10], given a noncrossing partition of {1,..., n}, its image is the permutation a suchthat a(i) is the element in the same class as i, which follows i in the cyclic order12...n. One can check [1] that the image of NC(n) is the set of all permutationssatisfying |cr| + |0 _1 c| = \c\ where c is the cyclic permutation c(i) = i+ 1 mod(n), inother words, this set consists of all permutations which lie on a geodesic from theidentity to c in the Cayley graph. These facts are at the heart of the connectionsbetween free probability, random matrices and symmetric groups. As an illustrationwe shall see how free cumulants arise from asymptotics of both random matrixtheory and symmetric group representation theory.Recall that cumulants (also called semi-invariants, see e.g. [17]) of a randomvariable X with moments of all orders, are the coefficients in the Taylor expansionof the logarithm of its characteristic function, i.e.logE[e itx ] = J2(ity ,Cn(X)n=0We shall consider random variables of the following form y W = NX[ -fi whereX( N ) = UDÛ* is a random matrix chosen as in (2.1) and X} 1' is its upper leftcoefficient. Assume now that the moments of X^converge±-Tr((X^) k ) - ^ ^ [ x k p(dx)for some probability measure p on R, with noncrossing cumulants R n (p), then onehashm -^-C n (YW) = -R n (p).iv-s-oo N zn

Free Probability and Combinatorics 771This was first observed by P. Zinn-Justin [25], a proof using representation theoryhasbeen found by B. Collins [6].We have related noncrossing cumulants to usual cumulants via random matrixtheory, we shall see that that noncrossing cumulants are also useful in evaluatingcharacters of symmetric groups. The precise relation however is not obvious at firstsight.Let us recall a few facts about irreducible representations of symmetric groups.It is well known that they can be parametrized by Young diagrams. In the followingit will be convenient to represent a Young diagram by a function u : R —¥ R suchthat (jj(x) = \x\ for |x| large enough, and a; is a piecewise affine function, with slopes±1, see the following picture which shows the Young diagram corresponding to thepartition 8 = 3 + 2 + 2+1.Xl J/1 X2 y 2 x 3 t/3 XiAlternatively we can encode the Young diagram using the local minima andlocal maxima of the function OJ, denoted by xi,..., Xk and 2/1, • • •, 2/fc-i respectively,which form two interlacing sequences of integers. These are (-3,-1,2,4) and (-2,1,3)respectively in the above picture. Associated with the Young diagram there is aunique probability measure m w on the real line, such thatR1,(dx)for all z £ C \ R.This probability measure is supported by the set {xi,...,Xk} and is called thetransition measure of the diagram, see [8]. Let a denote the conjugacy class in £„of a permutation with fc 2 cycles of length 2, fc 3 of length 3, etc.. Here fc 2 ,fc3,...are fixed while we let n —¥ 00. Denote by Xw the normalized character of £„associated with the Young diagram OJ, then the following asymptotic evaluationholds uniformly on the set of A-balanced Young diagrams, i.e. those whose longestrow and longest column are less that Ay/n (where A is some constant > 0),xAo-) = f[n- jk >Rf +1 (ou) + 0(r -i-\*\ß\ (4.1)Note that Rk is scaled by A* if we scale the diagram u by a factor A, therefore thefirst term in the right hand side is of order 0(n^- , i^+1 ' kj ' 2 ~^- , ' : ' kj ) = 0(n~^^2),

772 P. Bianethis gives the order of magnitude of the character of a fixed conjugacy group for anA-balanced diagram.In [2] a proof of (4.1) has been given, using in an essential way the Jucys-Murphy operators. Another proof, leading to an exact formula for characters ofcycles due to S. Kerov [9], was shown to me later by A. Okounkov [15], see [5].5. Representations of large symmetric groupsThe asymptotic formula (4.1) shows in particular that irreducible characters ofsymmetric groups become asymptotically multiplicative i.e. for permutations withdisjoint supports 01 and 0 2 , one hasxAwz) = XîJX^2) + Ofa- 1 -!!/ 2 ) (5.1)uniformly on A-balanced diagrams. Conversely, given a central, normalized, positivedefinite function on £„, a factorization property such as (5.1) implies that thepositive function is essentially an irreducible character [3]. Alore precisely, recallthat a central normalized positive definite function ip on £„ is a convex combinationof normalized characters, and as such it defines a probability measure on the setof Young diagrams. For any e, 5 > 0, for all n large enough, if an approximatefactorization such as (5.1) holds for ip, then there exists a curve OJ, such that themeasure on Young diagrams associated with ip puts a mass larger than 1 — 5 onYoung diagrams which lie in a neighbourhood of this curve, of width e\/n. Thereforeone can say that condition (5.1) on a positive definite function implies that therepresentation associated with this function is approximately isotypical, i.e. almostall Young diagrams occuring in the decomposition have a shape close to a certaindefinite curve.Using this fact it is possible to understand the asymptotic behaviour of severaloperations in representation theory. Consider for example the operation of induction.One starts with two irreducible representations of symmetric groups S ni , £„ 2 ,corresponding to two Young diagrams OJI and OJ 2 . One can then induce the productrepresentation u;i®u; 2 of S ni x £„ 2 to £„ 1+ „ 2 . This new representation is reducibleand the multiplicities of irreducible representations can be computed using a combinatorialdevice, the Littlewood-Richardson rule. This rule however gives littlelight on the asymptotic behaviour of the multiplicities. Using the factorizationconcentrationresult, one can prove that when rii and n 2 are very large, but ofthe same order of magnitude, then there exists a curve, which depends on ui andOJ 2 , and such that the typical Young diagram occuring in the decomposition of theinduced representation, is close to this curve. As we saw in section 4, one can associatea probability measure on the real line to any Young diagram. The descriptionof the typical shape of Young diagram which occurs in the decomposition of theinduced representation is easier if we use this correspondance between probabilitymeasuresand Young diagrams, indeed the probability measure associated with theshape of the typical Young diagram corresponds to the free convolution of the twoprobability measures [2].

Free Probability and Combinatorics 773There are analogous results for the restriction of representations from largesymmetric groups to smaller ones. There the corresponding operation on probabilitymeasureis called the free compression, it corresponds at the level of the large matrixapproximation, to taking a random matrix with prescribed eigenvalue distribution,as in section 2, and extracting a square submatrix. Finally there are also results forKronecker tensor products of representations. Here a central role is played by thewell known Kerov-Vershik limit shape, whose associated probability measure is thesemi-circle distribution with density 7y-\j4 — x 2 on the interval [—2,2], see [2].References[I] P. Biane, Some properties of crossings and partitions. Discrete Math. 175(1997), no. 1-3, 41-53.[2] P. Biane, Representations of symmetric groups and free probability. Adv. Math.138 (1998), no. 1, 126-181.[3] P. Biane, Approximate factorization and concentration for characters of symmetricgroups. Internat. Math. Res. Notices no. 4 (2001), 179-192.[4] P. Biane, Entropie libre et algèbres d'opérateurs. Séminaire Bourbaki Exposé889, Juin 2001.[5] P. Biane, Free cumulants and characters of symmetric groups. Preprint, 2001.[6] B. Collins, Aloments and cumulants of polynomial random variables on unitarygroupsPreprint, 2002.[7] L. Al. Ge, these Proceedings.[8] S. V. Kerov, Transition probabilities of continual Young diagrams and theMarkov moment problem Funct. Anal. Appi. 27 (1993), 104-117.[9] S.Kerov, talk at IHP, January 2000.[10] G. Kreweras, Sur les partitions non croises d'un cycle. Discrete Math. 1 (1972),no. 4, 333-350.[II] A. Nica, R. Speicher, Commutators of free random variables, Duke Math. J.92 (1998), no. 3, 553-592.[12] A. Nica, R. Speicher, On the multiplication of free A r -tuples of noneommutativerandom variables, Amer. J. Math. 118 (1996), no. 4, 799-837.[13] A. Nica, R Speicher, iï-diagonal pairs—a common approach to Haar unitariesand circular elements. Free probability theory (Waterloo, ON, 1995), 149-188,Fields Inst. Commun., 12, Amer. Alath. Soc, Providence, RI, 1997.[14] A. Nica, D. Shlyakhtenko, R. Speicher, iï-diagonal elements and freeness withamalgamation. Canad. J. Math. 53 (2001), no. 2, 355-381.[15] A. Okounkov, private communication.[16] G.-C. Rota, On the foundations of combinatorial theory. I. Theory of Albiusfunctions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964) 340-368.[17] A. N. Shiryaev, Probability Graduate texts in Alathematics 95, Springer, 1991.[18] P. Sniady, R. Speicher, Continuous family of invariant subspaces for i?-diagonaloperators. Invent. Math., 146 (2001), no. 2, 329-363.[19] R. Speicher, Alultiplicative functions on the lattice of noncrossing partitionsand free convolution. Math. Ann., 298 (1994), no. 4, 611-628.

774 P. Biane[20] R. Speicher, Combinatorial theory of the free product with amalgamation andoperator-valued free probability theory, Alem. Amer. Alath. Soc. 132 (1998), no.627.[21] D. Voiculescu, Symmetries of some reduced free product G*-algebras. Operatoralgebras and their connections with topology and ergodic theory (Bu§teni, 1983),Lecture Notes in Alath., 1132: 556-588, Springer, Berlin-New York, 1985.[22] D. Voiculescu, Addition of non-commuting random variables, J. OperatorTheory 18 (1987) 223-235[23] D. Voiculescu, Limit laws for random matrices and free products. Inv. Math.104,(1983) 201-220.[24] D. Voiculescu, Free probability theory: random matrices and von Neumannalgebras. Proceedings of the International Congress of Mathematicians. Vol. 1,2 (Zrich, 1994), 227-241, Birkhuser, Basel, 1995.[25] P. Zinn-Justin, Universality of correlation functions of Hermitian random matricesin an external field. Comm. Alath. Phys. 194 (1998), no. 3, 631-650.[26] A. Zvonkin, Alatrix integrals and map enumeration: an accessible introduction.Math. Comput. Modelling 26 (1997), no. 8-10, 281-304.

ICAl 2002 • Vol. II • 775-785Subfactors and Planar AlgebrasD. Bisch*AbstractAn inclusion of Hi factors N c M with finite Jones index gives rise to a powerfulset of invariants that can be approached successfully in a number of different ways.We describe Jones' pictorial description of the standard invariant of a subfactor asa so-called planar algebra and show how this point of view leads to new structureresults for subfactors.2000 Mathematics Subject Classification: 46L37, 46L60, 82B20, 81T05.Keywords and Phrases: Von Neumann algebras, Subfactors, Planar algebras.1. IntroductionAbelian von Neumann algebras are simply algebras of bounded, measurablefunctions on a measure space. A general (non-abelian) von Neumann algebra canbe viewed as an algebra of "functions" (operators) on a non-commutative measurespace. The building blocks of what one might call non-commutative probabilityspaces are the so-called Hi factors M, that is those von Neumann algebras withtrivial center that are infinite dimensional and possess a distinguished tracial state(the analogue of a non-commutative integral). The "smallest" Hi factor is thehyperfinite Hi factor which is obtained as the closure in the weak operator topologyof the canonical anti-commutation relations (CAR) algebra of quantum field theory.A Hi factor comes always with a natural left representation on L 2 (M), the noncommutativeL 2 -space associated to M. See for instance [13].Vaughan Jones initiated in the early 80's the theory of subfactors as a "Galoistheory" for inclusions of Hi factors. A subfactor is an inclusion of Hi factors N C Msuch that the dimension of M as left A r -Hilbert module is finite. This dimensionis called the Jones index [M : N] ([19]) and one would expect by classical resultsof Murray and von Neumann that it takes on any real number > 1. One of theearly results in the theory of subfactors was Jones' spectacular rigidity theoremwhich says that this index is in fact quantized [19]: if [M : N] < 4, then it hasto be of the form 4 cos 2 5-, for some n > 3. Since Jones' early work the theory of*Department of Mathematics, Vanderbilt University, Nashville, TN 37240 and TJCSB, Departmentof Mathematics, Santa Barbara, CA 93106, USA. E-mail: bisch@math.vanderbilt.edu

776 D. Bischsubfactors has developed into one of the most exciting and rapidly evolving areasof operator algebras with numerous applications to different areas of mathematics(e.g. knot theory with the discovery of the Jones polynomial [20]), quantum physicsand statistical mechanics. Subfactors with finite Jones index have an amazinglyrichmathematical structure and an interplay of analytical, algebraic-combinatorialand topological techniques is intrinsic to the theory.2. SubfactorsA subfactor can be viewed as a group-like object that encodes what one mightcall generalized symmetries of the data that went into its construction. To decodethis information one needs to compute the higher relative commutants, a systemof inclusions of certain finite dimensional C*-algebras naturally associated to thesubfactor. This system is an invariant of the subfactor, the so-called standardinvariant, which contains in many natural situations precisely the same informationas the subfactor itself ([30], [32], [33]). Here is one way to construct the standardinvariant: If N c M denotes an inclusion of Hi factors with finite Jones index,and ei is the orthogonal projection L 2 (M) —t L 2 (N), then we define Mi to be thevon Neumann algebra generated by M and ei on L 2 (M). Mi is again a Hi factorand M c Mi has finite Jones index as well so that the previous construction canbe repeated and iterated [19]. One obtains a tower of Hi factors Ac M c Mi CAf 2 c ... associated to N c M, together with a remarkable sequence of projections(ej)j>i, the so-called Jones projections, which satisfy the Temperley-Lieb relationsand give rise to Jones' braid group representation [19], [20]. The (trace preserving)isomorphism class of the system of inclusions of (automatically finite dimensional)centralizer algebras or higher relative commutantsC = N' n N c N' n M c AT n Mi c N' n M 2 cu u uC = M ' n M c M ' n Mi c M ' n M 2 cis then the standard invariant GN,M of the subfactor N c M. Each row of inclusionsis given by a sequence of Bratteli diagrams, which can in fact be reconstructed froma single, possibly infinite, bipartite graph. Hence one obtains two graphs (one foreach row), the so-called principal graphs of N c M, which capture the inclusionstructure of the above double-tower of higher relative commutants. It turns out thatif M is hyperfinite and N c M has finite depth (i.e. the principal graphs are finitegraphs) [30], [32] or more generally if N c M is amenable [33], then the standardinvariant determines the subfactor. In this case the subfactor can be reconstructedfrom the finite dimensional data given by GN,M- In particular, subfactors of thehyperfinite Hi factor R with index < 4 are completely classified by their standardinvariant and an explicit list can be given (see for instance [14], [16] or [33]). If theJones index becomes > 6 such an explicit list is out of reach as the work in [6], [11]and [12] shows: there are uncountably many non-isomorphic, irreducible infinite

Subfactors and Planar Algebras 777depth subfactors of R with Jones index 6 and the same standard invariant! Partiallists of irreducbile subfactors with index between 4 and 6 have been obtained bydifferentmethods (see for instance [1], [5], [6], [17], [35], [36], [37], [38]), but muchwork remains to be done.There are several distinct ways to analyze the standard invariant of a subfactor(see [2], [4], [14], [22], [30], [33]). For instance, in the bimodule approach ([13], [30],see also [4], [14], [18]) GN,M is described as a graded tensor category of naturalbimodules associated to the subfactor. GN,M can thus be viewed as an abstractsystem of (quantum) symmetries of the mathematical or physical situation fromwhich the subfactor was constructed. It is in fact a mathematical object whichgeneralizes for instance discrete groups and representation categories of quantumgroups ([37], [38]). A variety of powerful and novel techniques have been developedover the last years that make it possible to compute and understand the standardinvariant of a subfactor. A key result is Popa's abstract characterization of thestandard invariant [34]. Popa gives a set of axioms that an abstract system ofinclusions of finite dimensional C*-algebras needs to satisfy in order to arise as thestandard invariant of some (not necessarily hyperfinite) subfactor. This result makesit possible to analyze the structure of subfactors, which are infinite dimensional,highly non-commutative objects, by investigating the finite dimensional structuresencoded in their standard invariants.3. Planar algebrasJones found in [22] a powerful formalism to handle complex computations withGN,M- He showed that the standard invariant of a subfactor has an intrinsic planarstructure (this will be made precise below) and that certain topological argumentscan be used to manipulate the operators living in the higher relative commutants ofthe subfactor. The standard invariant is a so-called planar algebra. To explain thisnotion let us first define the planar "operad" following [22]. Elements of the planaroperad are certain classes of planar k-tangles which determine multilinear operationson the vector spaces underlying the higher relative commutants associated to a finiteindex subfactor.A planar fc-tangle consists of the unit disk D in the complex plane togetherwith several interior disks Di, D 2 ,..., D n . The boundary of D is marked with 2kpoints and each Dj has 2kj marked points on its boundary. These marked pointsare connected by strings in D, which meet the boundary of each disk transversally.We also allow (finitely many) strings which are closed curves in the interior of D.The main point is that all strings are required to be disjoint (hence planarity) andto lie in the complement of the interiors of the Dfis. Additional data of a planarofc-tangle is a checkerboard shading of the connected components of £>\Uj=i Dj,and a choice of a white region at every Dj (which corresponds to a choice of thefirst marked point on the boundary of each Dj). The planar operad V is definedto consist of all orientation-preserving diffeomorphism classes of planar fc-tangles

778 D. Bisch(for all k > 0), where the diffeomorphisms leave the boundary of D fixed but areallowed to move the interior disks. V becomes a colored operad [22] (see [28]). Anexample of a 4-tangle is depicted in the next figure:Note that there are two classes of planar 0-tangles according to the shading ofthe tangle near the boundary of D.Two planar tangles T and S can be composed in a natural way if the number ofboundary points of S matches the number of boundary points of one of the interiordisks Dj of T: To obtain the composed tangle T °j S shrink S and paste it insideDj so that the shadings and marked white regions match up. Join the strings at theboundary of Dj, smooth them and erase the boundary of Dj. It is clear that thisoperation is well-defined (the checkerboard shading and choice of a white regionat each disk avoid rotational ambiguity) and that it depends only on the isotopyclass of each tangle. Note that there may be several different ways of composingtwo given tangles, each composition yielding potentially distinct planar tangles. Anexample of such a composition is given in the next figure (insert S in the disk D 2ofT):T = T o 2 S =An abstract planar algebra is then defined to be an algebra over this planaroperad ([28]). Alore concretely, an abstract planar algebra V is the disjoint unionof vector spaces V = P htte JJ pbiack Y[ n>Q P n plus a morphism from the planaroperad to the (colored) operad of multilinear maps between these vector spaces. Inother words a planar algebra structure on V is a procedure that assigns to eachplanar fc-tangle T (with interior disks Dj having 2kj boundary points, 1 < j < n)a multilinear map Z(T) : i\ x • • • x Pk n —¥ Pk in such a way that compositionof tangles is compatible with the usual composition of maps (naturality of composition).Note that the F^'s are automatically associative algebras since the tangle

Subfactors and Planar Algebras 779in the next figure (drawn in the case k = 5) defines an associative multiplicationPk x Pk —t Pk (associativity follows from naturality of the composition).Observe that this is a purely algebraic structure - the definition can be madefor (possibly infinite dimensional) vector spaces over an arbitrary field. The keypointis of course that this structure appears naturally in the theory of subfactors.In order to connect with subfactors several additional conditions will be requiredin the definition of a planar algebra. A planar algebra (or subfactor planar algebrato emphasize the operator algebra context) will be an abstract planar algebra suchthat dim Pk < oo for all k, dim p htte = dim p^lack = l and such that the partitionfunction Z associated to the planar algebra is positive and non-degenerate. Thepartition function is roughly obtained as follows: If T is a 0-tangle, then Z(T)is a scalar since it is an element in the 1-dimensional space P htte resp. P^lack .Note that every planar algebra comes with two parameters Oi = Z((Q)) and ô 2 =Z(®), which we require to be fi^ 0 (the inner circles are strings, not boundariesof disks!). In the case of a subfactor planar algebra we have ô = ôi = ô 2 (whichis equivalent to extremality of the subfactor [31]). In fact ó = [M : A r ] 1 / 2 in thiscase. There is an intrinsic way to define an involution on the planar algebra arisingfrom a subfactor which makes the partition function into a sesquilinear form onthe standard invariant. Positivity of the partition function Z means then positivityof this form. Note that Z gives in particular the natural trace on the standardinvariant of the subfactor. The main result of [22] is then the following theorem.Theorem 3.1. The standard invariant GN,M of an extremal subfactor N c Mis a subfactor planar algebra V = (P n ) n >o with P n = N' n M n _i.This theorem says in particular that planar tangles always induce multilinearmaps ("planar operations") on the standard invariant of a subfactor. As a consequenceone obtains a diagrammatic formalism that can be employed to manipulatethe operators in N' n M n _i and intricate calculations with these operators can becarried out using simple topological arguments. This point of view has been turnedin [9], [10] into a powerful tool to prove general structure theorems for subfactors,and to analyze the rather complex combinatorial structure of the standard invariantof a subfactor. It has led to a generators and relations approach to subfactors. Seealso [23], [24] for more on this.The two most fundamental examples of subfactor planar algebras are theTemperley-Lieb systems of [19] (see also [22]) and the Fuss-Catalan systems of [7](see section 4).

780 D. BischObserve that by construction planar algebras are closely related to invariantsfor graphs, knots and links and to the pictorial formalism commonly used in thetheory of integrable lattice models in statistical mechanics.4. Fuss-Catalan algebrasJones and I discovered in [7] a new hierarchy of finite dimensional algebras,which arise as the higher relative commutants of subfactors when intermediate subfactorsare present. These algebras have a number of interesting combinatorialproperties and they have recently been used to construct new integrable latticemodels and new solutions of the Yang-Baxter equation ([15], [29]).We show in [7] that a chain of fc — 1 intermediate subfactors N c Pi C-P 2 C ...Pk-i C M leads to a tower of algebras (FC n (ai,... ,ak)) n>Q , whichdepend on fc complex parameters ai,...,ak- The dimensions of these algebras aregiven by the generalized Catalan numbers or Fuss-Catalan numbers j^+ìi « ")and we therefore call these algebras the Fuss-Catalan algebras. If no intermediatesubfactor is present, i.e. F = N or Fj = M for all i, then one finds the wellknownTemperley-Lieb algebras (case fc = 1) [19]. The additional symmetry comingfrom the intermediate subfactor is captured completely by these new algebras andit is proved in [7] (see also [8]) that they constitute the minimal symmetry presentwhenever an intermediate subfactor occurs. See also [26].Let us explain in more detail what happens in the case of just one intermediatesubfactor. We consider N c F c M, an inclusion of Hi factors with finite Jonesindex, and construct the associated tower of of Hi factors as in section 2. One ob-Pi e\ P'2 62tains an inclusion of Hi factors N c P C M c Pi C Mi C F 2 c M 2 c ..., wherethe Pi's are the orthogonal projections from L 2 (Afj_i) onto L 2 (Fj_i) (P 0 = P,M 0 = M) and the intermediate subfactors F are the von Neumann algebras generatedby Mj_i and p,. The algebra IA n (a, ß) = Alg(l,ei,... ,e n _i,pi,... ,p n _i),generated by the e,'s and the p,'s, is a subalgebra of N' n M n _i. It can be shownto depend only on the two indices a = [P : N] and ß = [M : P], and not on theparticular position of F in N c M. The projections e, and pj satisfy again somerather nice commutation relations (see [7] for details). In order to describe thestructure of these algebras let us for the moment consider the complex vector spaceFC n (a, b), spanned by labelled, planar diagrams of the form2n marked pointsa b b a a b b ab a a b b awhere a, b £ C\{0} are fixed. There is a natural multiplication of these diagrams,which makes FC n (a, 6) into an associative algebra (see [25]). To obtain Di • D 2 put

Subfactors and Planar Algebras 781the basis diagram Di on top of D 2 so that the labelling matches, remove the middlebar and all closed loops. Multiply the resulting diagram with factors of a resp. 6according to the number of removed a-loops resp. 6-loops. An example is depictedin the next figure.a b b a a b b a= abbb a a b ba-Ü--Ûbba a b baCounting diagrams shows that dim FC„ (a, 6)(3n\ the n-th Fuss-2n+l \ n >••Catalan number [7]. Clearly FC n (a,i>) embeds as a subalgebra of FC n+ i(a,6) byaddingtwo vertical through strings to the right of each basis diagram of FC n (a, 6).A diagrammatic technique, called the middle pattern analysis in [7], can be used tocompute the structure of these algebras completely in the semi-simple case. Oneobtains that the structure of the tower FCi (a, 6) C FC 2 (a, 6) C ... of Fuss-Catalanalgebras is given by the Fibonacci graph [7].KThe algebras lA n (a,ß) that we are interested in can then be shown to beisomorphic to FC n (a,b), where a ß = b", if the indices a and ß are generic,i.e. > 4. In the non-generic case lA n (a,ß) is a certain quotient of FC n (a,b) (see[7] for the details).There is a natural 2-parameter Markov trace on the Fuss-Catalan algebrasand the trace weights are calculated explicitly in [7]. In the special case of theTemperley-Lieb algebras this Alarkov trace is the one discovered by Jones in [19].The Fuss-Catalan tower together with this Alarkov trace satisfies Popa's axioms in[34] and hence, one can conclude from [34] that for every pair (a, ß) of possibleJones indices, there is a subfactor whose standard invariant is given precisely bythe corresponding Fuss-Catalan system (FC n (\/ä, Vß)) n>0 - One obtains in thisway uncountably many new subfactors. A complete set of generators and relationsfor the Fuss-Catalan algebras is also determined in [7].

782 D. BischIt should be evident that the Fuss-Catalan algebras can be viewed as planaralgebras generated by a single element in F 2 = N' n Mi, namely by the Jonesprojection pi onto the intermediate subfactor. This projection can be characterizedabstractly [3] and it satisfies a remarkable exchange relation ([9], [27]), which playsan important role in the work described in the next section.5. Singly generated planar algebrasAny subset S of a planar algebra V generates a planar subalgebra as thesmallest graded vector space containing S and closed under planar operations. Fromthis point of view the simplest subfactors will be those whose planar algebra isgenerated by the fewest elements satisfying the simplest relations, while the indexmay be arbitrarily large. If S is empty we obtain the Temperley-Lieb algebra.The next most complicated planar algebras after Temperley-Lieb should be thosegenerated by a single element R which is in the fc-graded subspace Pk for somefc > 0. We call such an element a k-box. In [22] the planar algebra generated by asingle 1-box was completely analyzed so the next case is that of a planar algebragenerated by a single 2-box. This means that the dimension of F 2 is at least 3 sothe first case to try to understand is when dim F 3 = 3. This dimension conditionby itself imposes many relations on V but probably not enough to make a completeenumeration a realistic goal. However, if one imposes dimF 3 < 15, then apart froma degenerate case, this forces enough relations to reduce the number of variablesgoverning the planar algebra structure to be finite in number ([22], see also [9]). Itseems therefore reasonable to try to find all subfactor planar algebras V generatedby a single element in F 2 subject to the two restrictions dim F 2 = 3 and dim F 3 = dwith d < 15.In [9] we solved this problem when d < 12. In fact, using planar algebratechniques we prove a much more general structure theorem for subfactors.Theorem 5.1. Let N c M be an inclusion of Hi factors with 3 < [M : N]

Subfactors and Planar Algebras 783The dimension conditions imply that a subfactor whose standard invariant isa planar algebra of the form b) or c) satisfies the hypothesis of Theorem 5.1 andhence must have an intermediate subfactor. Since the Fuss-Catalan planar algebrais the minimal symmetry associated to an intermediate subfactor it then followseasily that the planar algebra has to be one of these.It is quite natural to expect that increasing the dimension of F 3 should resultin a larger number of examples of planar algebras since there are more a prioriundetermined structure constants in the action of planar tangles on V. Thus theresult in [10] that there is a single subfactor planar algebra satisfying the aboverestrictions with d = 13 is a complete surprise. The planar algebra which arisesis that of a subfactor obtained as follows. Take an outer action of the dihedralgroup D 5 on a type Hi factor R and let M be the crossed product R x D 5 andN be the subfactor R x Z 2 . This particular subfactor has played a significant rolein the development of subfactors and relations with knot theory and statisticalmechanics. In [21] it was noted that there is a solvable statisitical mechanicalmodel associated with it and that it corresponds to an evaluation of the Kauffmanpolynomial invariant of a link. We prove in [10] the followingTheorem 5.3. Let V = (Pk)k>o be a subfactor planar algebra generated bya non-trivial element in F 2 (i.e. an element not contained in the Temperley-Liebsubalgebra of F 2 J subject to the conditions dimF 2 = 3 and dimF 3 = 13. Then Vis the standard invariant of the crossed product subfactor R x Z 2 c R x D 5 . Thusthere is precisely one subfactor planar algebra V subject to the above conditions.Note that this subfactor can be viewed as a Birman-Alurakami-Wenzl subfactor(associated to the quantum group of Sp(4,M.) at a 5-th root of unity, see [36]). Wenote here that the standard invariants V = (Pk)k>o of all BA1W subfactors aregenerated by a single non-trivial operator in F 2 and that they satisfy the conditiondimF 3 < 15.The proof of this theorem uses in a crucial way theorem 5.1 and the tightrestrictions imposed by compatibility of the rotation of period 3 on F 3 and thealgebra structure.The next phase of this enumeration project will be to tackle the case d = 14.Here we know that the quantum Sp(4, R) specialization of the BA1W algebra willgive examples with a free parameter. We do expect however, that the general ideasof [9] and [10] will enable us to enumerate all such subfactor planar algebras.References[1] Al. Asaeda & U. Haagerup, Exotic subfactors of finite depth with Jones indices(5 + >/Ï3)/2 and (5 + vTf)/2, Comm. Alath. Phys. 202 (1999), 1-63.[2] T. Banica, Representations of compact quantum groups and subfactors, J.Reine Angew. Math. 509 (1999), 167^198.[3] D. Bisch, A note on intermediate subfactors, Pacific Journal of Math. 163(1994), 201-216.

784 D. Bisch[4] D. Bisch, Bimodules, higher relative commutants and the fusion algebra associatedto a subfactor, The Fields Institute for Research in Math. SciencesCommun. Series, vol. 13, AAIS, Providence, Rhode Island, 1997, 13-63.[5] D. Bisch, An example of an irreducible subfactor of the hyperfinite Hi factorwith rational, noninteger index, J. Reine Angew. Math. 455 (1994), 21-34.[6] D. Bisch & U. Haagerup, Composition of subfactors: new examples of infinitedepth subfactors, Ann. scient. Ec Norm. Sup. 29 (1996), 329-383.[7] D. Bisch & V.F.R. Jones, Algebras associated to intermediate subfactors,Invent. Math. 128 (1997), 89-157.[8] D. Bisch & V.F.R. Jones, A note on free composition of subfactors, "Geometryand Physics", vol. 184, Alarcel Dekker, Lecture Notes in Pure and AppliedAlathematics, 1997, 339-361.[9] D. Bisch & V.F.R. Jones, Singly generated planar algebras of small dimension,Duke Math. Journal 101 (2000), 41-75.[10] D. Bisch & V.F.R. Jones, Singly generated planar algebras of small dimension,Part II, Advances in Math, (to appear).[11] D. Bisch & S. Popa, Examples of subfactors with property T standard invariant,Geom. Funct. Anal. 9 (1999), 215-225.[12] D. Bisch & S. Popa, A continuous family of non-isomorphic irreducible hyperfinitesubfactors with the same standard invariant, in preparation..[13] A. Connes, Noneommutative geometry, Academic Press, 1994.[14] D. Evans & Y. Kawahigashi, Quantum symmetries on operator algebras, OxfordUniversity Press, 1998.[15] P. Di Francesco, New integrable lattice models from Fuss-Catalan algebras,Nuclear Phys. B 532 (1998), 609-634.[16] F. Goodman & P. de la Harpe & V.F.R. Jones, Coxeter graphs and towersof algebras, Springer Verlag, A1SRI publications, 1989.[17] U. Haagerup, Principal graphs of subfactors in the index range 4 < [M : N]

Subfactors and Planar Algebras 785[25] L. Kauffman, State models and the Jones polynomial, Topology 26 (1987),395-407.[26] Z. Landau, Fuss-Catalan algebras and chains of intermediate subfactors, PacificJ. Math. 197 (2001), 325-36.[27] Z. Landau, Exchange relation planar algebras, preprint (2000).[28] J.P. May, Definitions: operads, algebras and modules, Contemporary Mathematics202 (1997), 1-7.[29] Al. J. Alartins & B. Nienhuis, Applications of Temperley-Lieb algebras toLorentz lattice gases, J. Phys. A 31 (1998), L723^L729.[30] A. Ocneanu, Quantized group string algebras and Galois theory for operatoralgebras, in Operator Algebras and Applications 2, London Math. Soc. Lect.Notes Series 136 (1988), 119-172.[31] Al. Pimsner & S. Popa, Entropy and index for subfactors, Ann. scient. EcNorm. Sup. 19 (1986), 57-106.[32] S. Popa, Classification of subfactors: reduction to commuting squares, Invent.Math. 101 (1990), 19-43.[33] S. Popa, Classification of amenable subfactors of type II, Acta Math. 172(1994), 352-445.[34] S. Popa, An axiomatizaton of the lattice of higher relative commutants, Invent.Math. 120 (1995), 427-445.[35] A. Wassermann, Operator algebras and conformai field theory III, Invent.Math. 92 (1998), 467-538.[36] H. Wenzl, Quantum groups and subfactors of type B, C and D, Comm.Math. Phys 133, 383-432.[37] H. Wenzl, C* tensor categories from quantum groups, J. Amer. Math. Soc.11 (1998), 261-282.[38] F. Xu, Standard A-lattices from quantum groups, Invent. Math. 134 (1998),455-187.

ICAl 2002 • Vol. II • 787^794Free Probability, Free Entropy andApplications to von Neumann AlgebrasLiming Ge*This talk is organized as follows: First we explain some basic concepts in noncommutativeprobability theory in the frame of operator algebras. In Section 2, wediscuss related topics in von Neumann algebras. Sections 3 and 4 contain some ofthe key ideas and results in free probability theory. Last section states some of theimportant applications of free probability theory.1. Non-commutative probability spacesIn general, a non-commutative probability space is a pair (A,T), where A isa unital algebra (over the field of complex numbers C) and r a linear functionalwith T(I) = 1, where J is the identity of A. Elements of A are called randomvariables. Since positivity is a key concept in (classical) probability theory, thiscan be captured by assuming that A is a * algebra and r is positive (i.e., a state).Elements of the form A* A are called positive (random variables).A state r is a trace if T(AB) = T(BA). We often require that r be a faithfultrace (r corresponds to the classical probability measure, or the integral given bythe measure). In this talk, we always assume that A is a unital * algebra over C andr a faithful state on A. Subalgebras of A are always assumed unital * subalgebras.Examples of noneommutative probability spaces often come from operatoralgebras on a Hilbert space and the states used here are usually vector states.A C*-probability space is a pair (A,T), where A is a unital C*-algebra (normclosed subalgebra of B(H)) and r is a state on A.A W*-probability space is a pair (M,T) consisting of a von Neumann algebraM (strong-operator closed C*-subalgebra of B(lfij) and a normal (i.e., countablyadditive) state r on ;M.The following are some more basic concepts:Independence: In a noneommutative probability space (A,T), a family {Aj} ofsubalgebras Aj of A is independent if the subalgebras commute with each other and,for n £ N, T(AI • • • A n ) = T(A{) • • • r(A n ) for all Ak in Aj k and jk fi 1 ji wheneverfc#l.* Academy of Mathematics and System Science, CAS, Beijing 100080, China. Department ofMathematics UNH, Durham, NH 03824, USA. E-mail: liming@math.unh.edu

788 Liming GèThis independence gives a "tensor-product" relation among subalgebras Aj : ifA is generated by Aj, then A — ®jAj (in the case of C*- or W*-probability spaces,the tensor-product shall reflect the corresponding topological structures on A andAfi. _Distributions and moments: Given (A, T), for A in A, we define a map PA '•C[x] —¥ C by PA(P(X)) = r(p(Aj). Then PA is the distribution of A. For Ai,..., A nin A, the joint distribution pA 1: ...,A n '• C(#i,..., x n ) —¥ C is given byPA u ...,A n (p(xi,- • -,x n j) = r(p(Ai,.. .,A„fi).Ifp is a monomial, r(p(Ai,... ,A n j) is called a (p-)moment. When random variablesare non self-adjoint, one also considers (joint) * distributions of random variables,that can be defined in a similar way. In this case, there is a natural identificationof C(#i,... ,x n ,x*,... ,x* n ) with the semigroup algebra CS 2n , where S 2n is the freesemigroup on 2n generators. Alonomials are given by words in S 2n .Conditional Expectations: Suppose B is a subalgebra of A. A conditional expectationfrom A onto B is a B-bimodule map (a projection of norm one in the case of C*-algebras) of A onto B so that the restriction on B is the identity map.Alany other concepts in probability theory and measure theory can be generalizedto operator algebras, especially von Neumann algebras which can be regardedas non-commutative measure spaces. For basic operator algebra theory, we refer to[KR] and [T].2. GNS representation and von Neumann algebrasGiven a C*-probability space (A,T), one defines an inner product (A,B) =T(B*A) on A. Yet L 2 (A,T) be the Hilbert space obtained by the completion ofA under the L 2 -norm given by this inner product. Then A acts on L 2 (A,T) byleftmultiplication. This representation of A on the Hilbert space L 2 (A,T) is calledthe GNS representation. In a similar way, one can define L P (A,T), where ||A|| P =r(|A| p ) 1 / p = T((A*A) p / 2 ) 1 / p . The von Neumann algebra generated by A (or thestrong-operator closure of Â) is sometimes denoted by L°°(A, T) (C L P (A,T), p >1). All von Neumann algebras admit such a form. Any von Neumann algebra is a(possibly, continuous) direct sum of "simple" algebras, or factors (algebras with atrivial center). Von Neumann algebras that admit a faithful (finite) trace are saidto be finite. The classification of (infinite-dimensional) finite factors has becomethe central problem in von Neumann algebras.Alurray and von Neumann [A1N] also separate factors into three types:Type I: Factors contain a minimal projection. They are isomorphic to fullmatrix algebras M n (C) or B(7fi).Type II: Factors contain a "finite" projection but without minimal projections:it is said to be of type Hi when the identity J is a finite projection; of type J/QOwhen J is infinite. Every type IIQO is the tensor product of B(fii) with a factor oftype Hi.

Free Probability, Free Entropy and Applications to von Neumann Algebras 789Type III: Every (non zero) projection is infinite.Examples of von Neumann algebras: 1) Let A = CG for some discrete groupG, % be l 2 (G) and r be the vector state given by the vector that takes value 1 atg and 0 elsewhere. Then r is a trace, l 2 (G) = L 2 (A,T) and the weak (or strong)operator closure of A is called the group von Neumann algebra, denoted by CG-We have that CQ is a factor if and only if each conjugacy class of G other than theidentity is infinite (i.e.a). For example, the free group F n (n > 2, on n generators)is such an i.c.c. group.2) Suppose (ii,p) is a measure space with a a-finite measure p, G is a groupand a is a measurability preserving action of G on Q. Formally, we have an algebraL°°(Q,p)G (= Â) similar to the group algebra definition: (ip(x)g)(ip(x)fi) =(fi(x)ip(g^1 (xj)gh, for >p,ip £ L°°(Q,p) and g,h £ G. Assume that G acts freely(i.e., for any g in G with j / e , the set {x £ 0 : g(x) = x} has measure zero).Define an action of A on the Hilbert space ©„ G G L 2 (Q,p)g by left multiplication( which is induced by the multiplication in A), where L 2 (Q,p)g is an isomorphiccopy of L 2 (Q, p). Then the von Neumann algebra generated by A is called the crossproduct von Neumann algebra, denoted by L°°(Q,p) x Q G. This cross product isa factor if and only if a is ergodic. If a is ergodic and not a measure preservingaction, then L°° (ii, p) x a G is a factor of type III. Type II factors are obtained frommeasure preserving actions (with Q a non atomic measure space) and the finitenessof p gives rise to type Hi factors.The above 1) and 2) are the two basic constructions of von Neumann algebras(given by Alurray and von Neumann [MN]). A. Connes [C] shows that thereare finite factors that cannot be constructed by 1). It was a longstanding openproblem whether every (finite) von Neumann algebra can be obtained by using theconstruction in 2). Using free probability theory, especially the notion of free entropy,Voiculescu [V2] gives a negative answer to this question. We shall discusssome details later in the talk.In recent years, the focus of studies of von Neumann algebras is centered onfactors of type Hi • Many of the unsolved problems in operator algebras are alsoreduced to this class. We end this section with two of the (still) open problems fromthe list of 20 questions asked by Kadison in 1967.1. The weak-operator closure of the left regular representation of the free (nonabelian)group on two or more generators is a factor of type Hi • Are these factorsisomorphic for different numbers of generators?2. Is each factor generated by two self-adjoint operators? —each von Neumannalgebra? —the factor arising from the free group on three generators? —is eachvon Neumann algebra finitely generated?Three of those 20 problems were answered in the last ten years by using freeprobability and free entropy. We explain some of the theory involved in the followingtwo sections.3. Free independence

790 Liming GèSuppose (A,T) is a C*-probability space. We assume that r is a trace. Afamily A,,, i £ I, of unital * subalgebras of A are called free with respect to r ifT(AIA 2 • • • A n ) = 0 whenever Aj £ A h , ti fi 1 • • • fi 1 t n (*i and t 3 may be the same)and T(AJ) = 0 for 1 < j < n and every n in N. A family of subsets (or elements)of A are said to be free if the unital * subalgebras they generate are free.Note that freeness is a highly noneommutative notion, the non-commutativity(or algebraic freeness) of free random variables is encoded in the definition. Recallsome basic concepts in free probability theory.Semicircular elements: The Gaussian laws in classical theory is replaced bythe semicircular laws. The semicircular law centered at a and of radius r is thedistribution p a>r : C[x] —¥ C such thatoi-a+rßa,r(v>(xj) = —2 /

Free Probability, Free Entropy and Applications to von Neumann Algebras 791called the vacuum vector and UJ-H the vacuum state. If Hi and H 2 are orthogonalsubspaces of Ho, then C*(l(Hij) and C*(l(H 2 j) are free (with respect to OJ-H)- If his a unit vector in Ho, then (1(h) + 1(h)*)/2 is semicircular with distribution p 0: i-Gaussian Random Matrices: Let X(s,n) = (fij(s,n)) in M n (L°°[0,1]) be realrandom matrices, where n £ N and « G S for some index set S. Assume thatfij(s,n) = fji(s,n) and {/y(«,n) : i, j,s} (given each n) is a family of independentGaussian (0,1/n) random variables. Let D n be a constant diagonal matrix inM n (R) having a limit distribution (as n —¥ oo). Then {X(s,n)} U {D n } is asymptoticallyfree as n —¥ oo and {X(s,n) : s £ S} converges in distribution to a freesemicircular family. As a corollary, Voiculescu shows that Cp 5 ® Af 2 (C) = Cp 2 .Moreover Cp r ® M n (C) = Cp r_ 1 , for any real number r, and Cp r * Cp, — Cp r+B •Now we know that either Cp r , r > 1, are all isomorphic to each other or they areall non isomorphic factors (see [D] and [R]).Further studies of free probability theory have been pursued by many peoplein several directions, such as infinitely divisible laws, free brownian motion, etc. (werefer to [B], [VDN] and [HP] for details).4. Free entropyFree entropy is a non commutative analogue of classical Shannon entropy. Firstwe recall the definition of entropy and its basic properties.Classical Entropy: Yet (0,S,p) be a probability space with probability measure pand fi, ...,/„: Q —t R be random variables. Suppose (p is the density function onR" corresponding to the joint distribution of fi,..., /„. Then the entropy:H(fi,---,fn) = - (p(h,... ,t n )log(p(ti,... ,t n )dti •••dt n .Here are two important properties of entropy: H (fi,..., /„) = H(fi)-\ V H(f n )if and only if fi,...,f n are independent; when assuming that E(f 2 ) = 1, H ismaximal if and only if fi,..., f n are Gaussian independent (0,1) random variables.Free entropy: Yet Xi,...,X n be self-adjoint random variables in (A,T). For anye > 0, when fc large, there may be self-adjoint matrices Ai,..., A n in Mk(C) suchthat "the algebra generated by Xj 's looks like the algebra generated by Aj 's withine. " Alore precisely, for any e > 0, large m £ N and any monomial p in C(#i,..., x n )with degree less than or equal to ro, choose fc large enough so that(*) |r(_p(Xi,...,X„))-r fc (_p(Ai,...,A„))|

792 Liming Gèm,k,e) to denote YR(XI,. .. ,X n ;m,k,e). Yet vol be the euclidean volume in realeuclidean space (Mfc(C) s-a- ) n (here "s.a." denote the self-adjoint part and theeuclidean norm ||A|| 2 = Tr(A 2 )). Now we define, successively,x(Xi,.. .,X n ;m,k,e) = log vol (Y(Xi,... ,X n ; TO, k,ej),X(Xi ,...,X n ; TO, e) = lim sup(fc _2 x(Xi,..., X n ; TO, k,e)-V - log fc),X(Xi,...,X„) =inf{x(Xi,...,X n ;m,e) : m £ N,e > 0}.We call x(Xi,- • •, X n ) the free entropy of (Xi,..., X n ).The following are some basic properties of free entropy (proved in [VI]):(i)x(AA,...,X„)

Free Probability, Free Entropy and Applications to von Neumann Algebras 793Furthermore, we prove the following result which shows the existence of aseparable prime factor (the one that is not the tensor product of two factors of thesame type).Theorem 4. ([G2]) If M = Mi®M 2 for some infinite dimensional finite vonNeumann algebras Mi and M 2 , then, for any self-adjoint generators Xi,...,X nofM,ö(Xi,...,X n )

794 Liming Ge[V3] D. Voiculescu, Free entropy dimension < 1 for some generators of property T factorsof type Ih, J- Reine Angew. Alath. 514 (1999), 113-118.[VDN] D. Voiculescu, K. Dykema and A. Nica, "Free Random Variables," CRAI MonographSeries, vol. 1, AA1S, Providence, R.I., 1992.

ICAl 2002 • Vol. II • 795^812Banach KK-theory andthe Baum-Connes ConjectureV. Lafforgue*AbstractThe report below describes the applications of Banach KK-theory to a conjectureof P. Baum and A. Connes about the K-theory of group C*-algebras,and a new proof of the classification by Harish-Chandra, the construction byParthasarathy and the exhaustion by Atiyah and Schmid of the discrete seriesrepresentations of connected semi-simple Lie groups.2000 Mathematics Subject Classification: 19K35, 22E45, 46L80.Keywords and Phrases: Kasparov's KK-theory, Baum-Connes conjecture,Discrete series.This report is intended to be very elementary. In the first part we outline themain results in Banach KK-theory and the applications to the Baum-Connes conjecture.In the second part we show how the Baum-Connes conjecture for connectedsemi-simple Lie groups can be applied to recover the classification of the discreteseries representations.1. Banach KK-theory and the Baum-Connes conjectureThere are many surveys on Kasparov's KK-theory and the Baum-Connesconjecture (see [4, 48, 49, 21, 27, 13, 54]) and on Banach KK-theory ([49, 38]).1.1. Generalized Fredholm modulesWe wish to define A-linear Fredholm operators (where A is a Banach algebra),with an index in K 0 (A). If A = C, this index should be the usual index of C-linearFredholm operators in K 0 (C) = Z.•"Institut de Mathématiques de Jussieu, 175 rue du Chevaleret, 75013 Paris, France. E-mail:vlafForg@math.jussieu.fr

796 V. LafforgueWe define a Banach algebra as a (non necessarily unital) C-algebra A thatis complete for a norm ||.|| satisfying ||aò|| < ||a||||b|| for any a,b £ A. If A andB are Banach algebras a morphism 6 : A —t B is an algebra morphism such thatP( a )[| < IMI f° r an y a € A.K 0 and Ki are two covariant functors from the category of Banach algebrasto the category of abelian groups. If X is a locally compact space and Co(X) thealgebra of continuous functions vanishing at infinity, Kn(Cn(X)) and Ki(Cn(X))are the Atiyah-Hirzebruch K-theory groups. For technical reasons we shall restrictourselves to unital Banach algebras in this subsection.Let A be a unital Banach algebra.A right A-module E is finitely generated projective if and only if it is a directsummand in A n for some integer n. The set of isomorphism classes of right finitelygeneratedprojective A-modules is a semigroup because the direct sum of two rightfinitely generated projective A-modules is a right finitely generated projective A-module. Then Kn(A) is the universal group associated to this semigroup (i.e. thegroup of formal differences of elements of the semigroup). If 9 : A —t B is amorphism of unital Banach algebras, and E is a right finitely generated projectiveA-module then E (E)A B is a right finitely generated projective B-module and thisdefines 0» : K 0 (A) -• K 0 (B).There is another definition of Kn(A) for which the functoriality is even moreobvious : Kn(A) is the quotient of the free abelian group generated by all idem-(p 0^potents p in Mk(A) for some integer k, by the relations[p] + [q] forVO fiany idempotents p £ Mk(A) and q £ MfiA) and [p] = [q] if p, q are idempotents ofMk(A) and are connected by a path of idempotents in Mk(A) and [0] = 0 where 0is the idempotent 0 in Mk(A). The link with the former definition is that any idempotentp £ Mk(A) acts on the left on A k as a projector P and ImP is a right finitelygeneratedprojective A-module (it is a direct summand in the right A-module A k ).The following construction was performed for C*-algebras by Alischenko andKasparov, in connection with the Novikov conjecture ([43, 28]). We adapt it toBanach algebras.A right Banach A-module is a Banach space (with a given norm \\.\\E) equippedwith a right action of A such that 1 G A acts by identity and ||a;a||^ < Hx^HaH^for any x £ E and a £ A. Let E and F be right Banach A-modules. A morphismu : E —t F of right Banach A-modules is a continuous C-linear map such thatu(xa) = u(x)a for any x £ E and a £ A. The space CA(E, F) of such morphisms is aBanach space with norm ||«|| = sup x€£; ,||X||E=I IIWMIIF- A morphism u £ CA(E, F)is said to be "A-rank one" if u = w o v with v £ CA(E, A) and w £ CA(A, F). Thespace KA(E,F) of A-compact morphisms is the closed vector span of A-rank onemorphisms in CA(E,F). If E = F, CA(E) = CA(E,E) is a Banach algebra andKA(E) = KA(E,E) is a closed ideal in it.Definition 1.1.1 A Fredholm module over A is the data of a Z/2 graded rightBanach A-module E and an odd morphismT £ CA(E) such thatT 2 —Id# £ KA(E).In other words E = E 0 ® Ei, T = ( J and u £ CA(EQ,EI) and v £

Banach KK-theory and the Baum-Connes Conjecture 797CA(EI,E 0 ) satisfy vu — Id# 0 £ ICA(EQ) and uv — Id^ £ KA(EI).If (E, T) is a Fredholm module over A and 9 : A —t B a unital morphism then(E (E)A B,T ® 1) is a Fredholm module over B (here E (E)A B is the completion ofE ®^s B for the maximal Banach norm such that ||x ® b\\ < ||X||E:||&||B for x £ Eand 6 £ B).Yet A[0,1] be the Banach algebra of continuous functions from [0,1] to A withthe norm ||/|| = sup tG r 01 n ||/(i)|U and 9n,9i : A[0,1] —t A the evaluations at 0 and1. Two Fredholm modules on A are said to be homotopic if they are the images by9o and 9i of a Fredholm module over A[0,1].Theorem 1.1.2 There is a functorial bijection between Kn(A) and the set of homotopyclasses of Fredholm modules over A, for any unital Banach algebra A.Let (En,Ei,u,v) be a Fredholm module over A. Its index, i.e. the correspondingelement in K 0 (A), is constructed as follows. It is possible to find n £ N andw £ K,A(A U , EI) such that (u, w) £ CA(EQ ® A n , E{) is surjective. Its kernel is thenfinitely generated projective and the index is the formal difference of Ker((u,wj)and A n .An ungraded Fredholm module over A is the data of a (ungraded) right Banachmodule E over A, and T £ CA(E) such that T 2 — Id# £ KA(E). There is afunctorialbijection between KfiA) and the set of homotopy classes of ungraded Fredholmmodules.For a non-unital algebra A, K 0 (A) = Ker(K 0 (Ä) -+ K 0 (C) = Z) and KfiA) =Ki(A) where A = A © CI. In particular every idempotent in Mk(A) gives a classin Kn(A) but in general not all classes in Kn(A) are obtained in this way. Thedefinition of a Fredholm module should be slightly modified for non-unital Banachalgebras, but the theorem 1.1.2 remains true.1.2. Statement of the Baum-Connes conjectureLet G be a second countable, locally compact group. We fix a left-invariantHaar measure dg on G. Denote by C c (G) the convolution algebra of complex-valuedcontinuous compactly supported functions on G. The convolution of /,/' £ C C (G)is given by / * f'(g) = J G f(h)f'(hr 1 g)dh for any g £ G.When G is discrete and dg is the counting measure, C c (G) is also denoted byCC? and if e g denotes the delta function at g £ G (equal to 1 at g and 0 elsewhere),(e g ) ge G is a basis of CG and the convolution product is given by e g e g i = e gg i.The completion of C C (G) for the norm ||/||LI = J G \f(g)\dg is a Banach algebraand is denoted by L 1 (G).For any / £ C C (G) let A(/) be the operator /' H> / * /' on L 2 (G). Thecompletion of C C (G) by the operator norm ||/|| re d = \\Hf)\\c(L 2 (G)) ' 1S ealled thereduced G*-algebra of G and denoted by C* ed (G). If G is discrete (e g ') g i e G is anorthonormal basis of L 2 (G) and X(e g ) : e g i >-¥ e gg i.For any / £ C C (G), ||/||j,i > ||/|| re d and L 1 (G) is a dense subalgebra ofC* ed (G). We denote by i : L 1 (G) —¥ C* ed (G) the inclusion.

798 V. LafforgueAssume now that M is a smooth compact manifold, and M a Galois coveringof M with group G (if M is simply connected, G = 7Ti(M)). Let En and Ei be twosmooth hermitian finite-dimensional vector bundles over M and u an order 0 ellipticpseudo-differential operator from L 2 (M, En) to L 2 (M, E{). Since u is elliptic thereis an order 0 pseudo-differential operator v : L 2 (M,E{) —t L 2 (M,E 0 ) such thatWi 2 (M,E(]) — vu anc l Wj,2( M]El ) — uv have order < —1 and therefore are compact.Let £ be the quotient of M x C* ed (G) by the diagonal action of G (G acting onC* ed (G) by left translations) : £ is a flat bundle of right C* ed (G)-modules over AT,whose fibers are isomorphic to C* ed (G). Then L 2 (M,E 0 ® £) and L 2 (M,Ei ® £)are right Banach (in fact Hilbert) modules over C* ed (G) and it is possible to lift uand v to ü and w so that (L 2 (M, E Q ®£),L 2 (M, Ei ®£),ü,v) is a Fredholm moduleover C* ed (G), whose index lies in K 0 (C* ed (Gj) and the index does not depend onthe choice of the liftings.The operator u represents a "if-homology class" in K 0 (M), and using theclassifying map M —t BG, it defines an element of K 0:C (BG), the K-homology withcompact support of the classifying space BG. For any discrete group G we candefine a morphism of abelian groups K*^C(BG) —¥ K„,(C* ed (Gj) (* = 0,1). Thismorphism is the Baum-Connes assembly map when G is discrete and torsion free.When G is not discrete or has torsion, the index construction can be performedstarting from a proper action of G (instead of the free and proper action of Gon M in the last paragraph), and therefore we have to introduce the space EGthat classifies the proper actions of G. Using Kasparov equivariant KK-theory, theG-equivariant K-homology K c fi(E_G) with G-compact support (* = 0,1) may bedefined, and there is an assembly mapMred : K^(EG) -+ KfiC; ed (Gj).In the same way we can define p,p\ : K c fi(E_G) —t K^(L 1 (Gj) and p re d = i* ° pp 1 -Baum-Connes conjecture [3, 4] : If G is a second countable, locally compactgroup then the assembly map p re d : K c fi(E_G) —t K*(C* ed (Gj) is an isomorphism.Bost conjectured : If G is a second countable, locally compact group (andhas reasonable geometric properties) then the assembly map p,p\ : K c fi(E_G) —tK^(L 1 (Gj) is an isomorphism.In many cases K c fi(E_G) can be computed. For instance if G is a discretetorsion free subgroup of a reductive Lie group 17 and if is a maximal compactsubgroup of 17, then a possible EG is H/K and K c fi(E_G) is the K-homology withcompact support of G\H/K. This group may be computed thanks to Mayer Vietorissequences. See part 2 for the case where G is a Lie group.1.3. ÜTÜT-theoryFor any G*-algebras A and B, Kasparov [28, 31] defined an abelian groupKK(A, B), covariant in B and contravariant in A. There is a product KK(A, B) ®

Banach KK-theory and the Baum-Connes Conjecture 799KK(B,C) -• KK(A,C). Moreover KK(C,A) = K Q (A) and therefore the productgives a morphism KK(A, B) -+ Hom(i\o(A), K 0 (Bj). The definition of KK(A, B)is like definition 1.3.1 below, but with Hilbert modules instead of Banach modules.For any Banach algebras A and B, we define [37, 49] an abelian groupKK bAn (A,B), covariant in B and contravariant in A. There is no product, buta morphism KK bAn (A,B) —t Hom(Kn(A),Kn(B)). Assume that B is unital (otherwisethe definition has to be slightly modified).Definition 1.3.1 E bAn (A,B) is the set of isomorphism classes of data (E,n,T),where E is a 7L/27L-graded right Banach module, n : A —¥ Cp(E) is a morphismof Banach algebras and takes values in even operators, and T £ Cp(E) is odd andsatisfies a(T 2 - ld E ) £ K B (E) and aT -Ta£ K B (E) for any a £ A.Then KK bAn (A,B) is the set of homotopy classes in E bAn (A,B), where thehomotopy relation is defined using E bAn (A, B[0,1]).Remark : E bAn (C, B) is the set of isomorphism classes of Fredholm modules overB and KK bAn (C,B) = K 0 (B).If p is an idempotent in A, and (E,n,T) £ E bAn (A,B), the image of [p] £Ko(A) by the image of [(E, TT, T)] £ KK bAn (A, B) in Hom(K 0 (A), K 0 (B)) is definedto be the index of the Fredholm module over B equal to (lmn(p), irfifiTirfi))). Whenp is an idempotent in Mk(A), we use the image of p by Mk(A) —¥ Cp(E k ). This isenough to define the morphism KK bAn (A,B) —t Hom(Kn(A),Kn(B)), when A isunital.The same definition with ungraded modules gives KK bAn (A, B), and, with thenotation KK = KKn, we have a morphism KK bAn (A, B) —t Hom(Kj (A), K i+ j (Bj),where all the indices are modulo 2.1.4. Status of injectivity and the element 7The injectivity of the Baum-Connes map p re d (and therefore of ppi) is knownfor the following very large classes of groups :a) groups acting continuously properly isometrically on a complete simply connectedriemannian manifold with controlled non-positive sectional curvature, andin particular closed subgroups of reductive Lie groups ([29, 31]),b) groups acting continuously properly isometrically on an affine building and inparticular closed subgroups of reductive p-adic groups ([32]),c) groups acting continuously properly isometrically on a discrete metric space withgood properies at infinity (weakly geodesic, uniformly locally finite, and "bolic"[33, 34]), and in particular hyperbolic groups (i.e. word-hyperbolic in the sense ofGromov),d) groups acting continuously amenably on a compact space ([22]).In the cases a),b),c) above, the proof of injectivity provides an explicit idempotentendomorphism on K*(C* ed (Gj) whose image is the image of p re d (and the

800 V. Lafforguesame for ppi). In case d), J.-L. Tu has also constructed such an endomorphism,but in a less explicit way.To state this we need to understand a baby case of Kasparov's equivariantKK-groups. Let G be a second countable, locally compact group. We denote byEG (C, C) the set of isomorphism classes of triples (H,n,T) where 17 is a Z /2-gradedHilbert space, n a unitary representation of G on 17 (such that for any x £ H,g H> gx is continuous from G to 17) and T an odd operator on 17 such T 2 — Id His compact and •K(g)T / ïï(g^1)— T is compact and depends norm continuously ong £ G. Then KKQ (C, C) is the quotient of EG(C,C) by homotopy. Kasparovprovedthat KKG(C, C) has a ring structure (using direct sum for the addition andtensor products together with a quite difficult construction for the multiplication).If7T is a unitary representation of G on a Hilbert space Hn and Hi = 0 then(17, n, 0) £ EG(C, C) if and only if H 0 has finite dimension. If moreover Hn = C and7T is the trivial representation of G, the class of (H,n,Ö) is the unit of KKG(C,C)and is denoted by 1. If G is compact the classes of (H,n,Ö) with Hi = 0 (anddimifo < +oo) generate KKG(C,C) and KKG(C,C) is equal to the representationring of G.The important fact is that there is a "descent morphism"j r e d : KK G (C,C) -+ End(KfiC; ed (Gjj).In fact it is a ring homomorphism and j re d(l) = 1&K,(C* (G))- It is defined as thecomposite of two maps KK G (C, C) -¥ KK(C* ed (G),C* ed (Gj) -t End(K r .(C* ed (G)j).The construction of j re d is due to Kasparov. The construction of jpi to be explainedbelow is an adaptation of it.The following extremely important theorem also contains earlier works ofAlishchenko and Solovjev.Theorem 1.4.1 (Kasparov, Kasparov-Skandalis [31, 32, 33, 34]) If G belongs toone of the classes a),b),c) above, the geometric conditions in a),b) or c) allow toconstruct an idempotent element 7 £ KKG(C,C) such that p re d is injective and itsimage is equal to the image of the idempotent j re d(7) € End(K„,(C* ed (Gjj).1.5. Homotopies between 7 and 1We assume that G belongs to one of the classes a),b),c). Then the injectivityof p ie d is known and the surjectivity is equivalent to the equality j re d(7) = Id GEnd(KfiC; ed (G)j).Theorem 1.5.1 We have 7 = 1 in KK G (C,C)1. G is a free group (Cuntz, [14]) or a closed subgroup of SO(n, 1) (Kasparov,[30]) or of SU(n, 1) (Julg-Kasparov, [25]) or of SL 2 (V) with F a local nonarchimedianfield (Julg- Valette, [24]),2. G acts isometrically and properly on a Hilbert space (Higson-Kasparov [20,27]).if

Banach KK-theory and the Baum-Connes Conjecture 801In fact the second case contains the first one.If G has property (T) and is not compact, 7 7^ 1 in KKG(C, C) : it is impossibleto deform 1 to 7 in EG (C, C) because the trivial representation is isolated amongunitary representations of G if G has property (T) and 7 can be represented by(17, n, T) such that 17 has no invariant vector (and even 17 is tempered). All simplereal or p-adic groups of rank > 2, and Sp(n, 1) and F^_ 20^, and all their lattices,have property (T) (see [19]).It is then natural to broaden the class of representations in order to breakthe isolation of the trivial one. In [26] Julg proposed to use uniformly boundedrepresentations on Hilbert spaces (to solve the case of Sp(n, 1)).For any non compact group G the trivial representation is not isolated amongisometric representations in Banach spaces (think of the left regular representationon L P (G), p going to infinity).Definition 1.5.2 Let E An G (C,C) be the set of isomorphism classes of triples(E,n,T) with E a Z/2-graded Banach space endowed with an isometric representationof G (such that g H> gx is continuous from G to E for any x £ E), T £ Cc(E)an odd operator such that T 2 — Id E belongs to Kc(E) and ir(g)TiT(g^1)— T belongsto K-c(E) and depends norm continuously on g £ G.Then KK G An (C,C) is defined as the quotient of E G An (C,C) by homotopy.Since any unitary representation of G on a Hilbert space 17 is an isometricrepresentation on the Banach space 17, there is a natural morphism of abeliangroups KK G (C,C) -• KK bAn (C,C).To state our main theorem, we need to look at slightly smaller classes than a)and c) above. We call these new classes a') and c'). They are morally the same,and in particular they respectively contain all closed subgroups of reductive Liegroups, and all hyperbolic groups (for general hyperbolic groups see [42], and [37]for a slightly different approach).Theorem 1.5.3 [37, 49] For any group G in the classes a'), b), or c'), we have7= 1 in KK bAn (CC).In fact the statement is slightly incorrect, we should allow representationswith a slow growth, but this adds no real difficulty. The proof of this theorem isquite technical. Let me just indicate some ingredients involved. If G is in class a')then G acts continuously isometrically properly on a complete simply connectedriemmannian manifold X with controlled non-positive sectional curvature, and Xis contractible (through geodesies) and the de Rham cohomology of X (withoutsupport) is C in degree 0 and 0 in other degrees. It is possible to put norms on thespaces of differential forms (on which G acts) and to build a parametrix for the deRham operator (in the spirit of the Poincaré lemma) in order to obtain a resolutionof the trivial representation, and in our language an element of E bAn (C, C) equal to1 in KK bAn (C, C). The norms we use are essentially Sobolev L°° norms. Then itis possible to conjugate the operators by an exponential of the distance to a fixedpoint in X and then to deform these norms to Hilbert norms (through L p norms,p £ [2, +00]) and to reach 7.

802 V. LafforgueIf G belongs to class b) the de Rham complex is replaced by the simplicialhomology complex (with L 1 norms) on the building. If G belongs to class c') a Ripscomplex plays the same role as the building in b).It is not possible to apply directly this theorem to the Baum-Connes conjecturebecause there is no obvious descent map KK bAn (C,C) —¥ End(K„,(C* ed (Gjj), andin the next subsection we shall see the difficulties encountered and the way onebypasses them in a few cases.On the other hand, we may apply this theorem to Bost conjecture, becausethere is descent map j L i : KK bAn (C,C) -• KK bAn (L 1 (G),L 1 (G)).We explain it when G is discrete. Let (E,TT,T) £ E An G (C,C). We denoteby L 1 (G,E) the completion of E ® CG for the norm || ^2 ge G x Ì9) ® e =sll12 g eG\\ x (s)\\ E - Then L 1 (G,E) is a right Banach L 1 (G)-module by the formula(x ® e g )e g i = x ® e gg i and there is a Banach algebra morphism n : L 1 (G) —¥Cpi(G)(L 1 (G,Ej) by the formulan(e g i)(x®e g ) = ir(g')(x)®e g i g . Then (L 1 (G,£'),7r,T® 1) € E bAn (L 1 (G),L 1 (Gj) gives the desired class in KK bAn (L 1 (G),L 1 (G)).This and section 1.3 imply the Bost conjecture in many cases.Theorem 1.5.4 For any group G in the classes a'), b) or c'), p L i : K c fi(E_G) —tK^(L 1 (Gj) is an isomorphism.1.6. Unconditional completionsLet G be a second countable, locally compact group. Let A(G) be a Banachalgebra containing C c (G) as a dense subalgebra. We write -4(G) instead of A fornotational convenience. We ask for a necessary and sufficient condition such thatthere is a "natural" descent map j A : KK bAn (C,C) -• KK bAn (A(G),A(Gj).In order to simplify the argument below, we will assume G to be discrete.Let E be a Banach space with an isometric representation of G. Then E ® CGhas a right CG-module structure given by (x ® e g )e g i = x ® e gg i and there isa morphism n : CG —¥ Endos (E ® CG) given by the formula n(e g ')(x ® e g ) =ir(g')(x) ® e g 'g. We look for a completion A(G,E) of E ® CG by a Banach normsuch that A(G, E) is a right Banach „4(G)-module and n extends to a morphism ofBanach algebras n : A(G) —¥ C^G)(A(G,Ej).In order to have enough „4(G)-rank one operators, it is quite natural to assumethat the norm on A(G, E) satisfies : for any x £ E and £ £ Cc(E, C), if we denoteby R x : CG -• E ® CG the map e g H> x ® e g and by S(_ : E ® CG -^ CG themap y ® e g >-+ Ç(y)e g , we have \\R x (f)\\A(G,E) < IMbll/IU(G) for any f £ CGand IIScMIU^) < ||C||£ :: (iî,c)||a;m( Giiï) for any u £ E®CG. Now fix x £ Eand £ £ Cc(E,C) and denote by 1 the unit in G. For any / = X^GG fÌ9) e eaCG, S^(n(fi)(R x (eijj) is XôGG^( 7r (5)(- c ))/(5) e s m g ^ E sG G c(g)f(g)e g .In this way we obtain the following necessary condition : for any x £ E and£ £ Cc(E,C) the Schur multiplication by the matrix coefficient g H- Ç(n(g)(x)) isbounded from A(G) to itself and its norm (in Cc(A(Gjj) is less than ||x||s||C||£ .-{E,Q •But for any L°°-function c on G we can find an isometric representation n of G on

Banach KK-theory and the Baum-Connes Conjecture 803a Banach space E and x £ E and £ £ Cc(E,C) such that ||X||E:||£||£.-{E,Q = ll c IU°°and c(g) = Ç(n(g)x) for any g £ G (take E = L 1 (G), x = ó~i, Ç = c). Thereforea necessary condition is that A(G) is an unconditional completion in the followingsense.Definition 1.6.1 A Banach algebra A(G) (with a given norm \\-\\A(G)) containingC C (G) as a dense subalgebra is called an unconditional completion if the normll/IU(G) of f £ C C (G) only depends on g ^ \f(g)\, G -• K+.Remark that L 1 (G) is an unconditional completion of C C (G) but C* ed (G) isnot.In fact this condition is also sufficient to construct the descent map. For thesake of simplicity, we still assume that G is discrete. If A(G) is an unconditionalcompletion of CG, and (E,n,T) is in E bAn (C,C), we define A(G,E) as the completionof E ® CG for the norm || E S GG X (#) ® e sll = II E S GG IkG?)!!^ eslU(G) an dA(G, E) is a right Banach module over A(G) and there is a morphism n : A(G) —¥£A(G)(A(G,E)), and (A(G,E),n,T®1) £ E bAn (A(G),A(G)).In this way, for any unconditional completion A(G) of CG, we have a descentmap jf n : KK An G (C,C) -• KK bAn (A(G),A(G)) -• End(KfiA(G))). We canalso define an assembly map PA '• K^(EG) —¥ K*(A(G)). If A(G) is an involutivesubalgebra of C* ed (G), and i : A(G) —¥ C* ed (G) denotes the inclusion, p ie d = ì*°PA-Theorem 1.6.2 ([37]) For any group G in the classes a'), b) or c'), and for anyunconditional completion A(G) of C C (G), PA '• K c fi(E_G) —t K*(A(G)) is an isomorphism.Yet A, B be Banach algebras and i : A —¥ B an injective morphism of Banachalgebras. We say that A is stable under holomorphic functional calculus in B if anyelement of A has the same spectrum in A and in B. If A is dense and stable underholomorphic functional calculus in B then z» : KfiA) —t K*(B) is an isomorphism(see the appendix of [6]).Corollary 1.6.3 For any group G in the classes a'), b) or c'), if C C (G) admits anunconditional completion A(G) which is an involutive subalgebra of C* ed (G) and isstable under holomorphic functional calculus in C* ed (G), then p re d '• K^(EG) —¥Kx(C* ed (Gj) is an isomorphism.This condition is fulfilled fora) hyperbolic groups,b) cocompact lattices in a product of a finite number of groups among Lie or p-adicgroups of rank one, SL 3 (¥) with F a local field (even M) and E 6^_ 26^,c) reductive Lie groups and reductive groups over non-archimedian local fields.In case c), A(G) is a variant of the Schwartz algebra of the group ([37]). In thiscase the Baum-Connes conjecture was already known for linear connected reductivegroups (Wassermann [55]) and for the p-adic GL n (Baum, Higson, Plymen [5]). Incase a),b) this result is based on a property first introduced by Haagerup for the

804 V. Lafforguefree group and called (RD) (for rapid decay) by Jolissaint ([23]). In case a),b) Ghas property (RD) : this is due to Haagerup for free groups ([16]), Jolissant for"geometric hyperbolic groups", de la Harpe for general hyperbolic groups ([18]),Ramagge, Robertson and Steger for SL% of a non-archimedian local field ([47]), theauthor for SL 3 (R) and SL 3 (C) ([39]), Chatterji for SL 3 (M) and E 6{ _ 26) ([10]), andthe remark that it holds for products is due to Ramagge, Robertson and Steger([47]) in a particular case, and independantly to Chatterji ([10]) and Talbi ([50])in general. A discrete group G has property (RD) if there is a lenght function£ : G —¥ R+ (i.e. a function satisfying i(g^v)= 1(g) and £(gh) < 1(g) + 1(h) forany g,h £ G) such that for s £ R+ big enough, the completion H S (G) of CG forthe norm || E f(g)e g \\ H *(G) = II EC 1 + %)) s /(^KIIL^G) is contained in G r * ed (G).Then, for s big enough, H S (G) is a Banach algebra and an involutive subalgebra ofC* ed (G) and is dense and stable under holomorphic functional calculus ([23, 39]); itis obvious that H S (G) is an unconditional completion of CG.As a consequence of this result the Baum-Connes conjecture has been provenfor all almost connected groups by Chabert, Echterhoff and Nest ([9]).1.7. Trying to push the method furtherIn order to prove new cases of the surjectivity of the Baum-Connes map (whenthe injectivity is proven and the 7 element exists) we should look for a densesubalgebra -4(G) of C* ed (G) that is stable under holomorphic functional calculusand a homotopy between 7 and 1 through (perhaps special kind of) elements ofE bAn (C, C) which all give a map K„,(A(Gj) —¥ K*(C* ed (Gj) by the descent construction.Thanks to the discussion in subsection 1. a necessary condition for this is thatfor any (E, n, T) in the homotopy between 7 and 1, for any x £ E and £ £ Cc(E, C),the Schur multiplication by the matrix coefficient g H> Ç(ir(g)(xj) is bounded fromA(G) to C* ed (G) and has norm < ||X||E:||£||£ r {E,Q- So we should first look for ahomotopy between 7 and 1 such that the fewest possible matrix coefficients appear.For groups acting properly on buildings, this homotopy can be shown to exist. Theproblem for general discrete groups properly acting on buildings is to find a subalgebrav4(G) of C* ed (G) that is stable under holomorphic functional calculus andsatisfies the condition with respect to these matrix coefficients. The first step (thecrucial one I think) should be to find a subalgebra -4(G) of C* ed (G) that is stableunder holomorphic functional calculus and satisfies the following condition : thereis a integer n, a distance d on the building and a point xo on the building such thatthe Schur product by the characteristic function of {g £ G,d(xo,gxo) < r} fromA(G) to C* ed (G) has norm less than (1 + r) n , for any r £ R+.1.8. The Baum-Connes conjecture with coefficientsLet G be a second countable, locally compact group and A a G-Banach algebra(i.e. a Banach algebra on which G acts continuously by isometric automorphismsg : a >-¥ g (a)). The space C C (G,A) of A-valued continuous compactly supportedfunctions on G is endowed with the following convolution product : f * f'(g) =JG fWh(f'(h^19J)dhand the completion L 1 (G, A) of C C (G, A) for the norm ||/|| =

Banach KK-theory and the Baum-Connes Conjecture 805J G 11/(5) I ÌAdg is a Banach algebra. Alore generally for any unconditional completion„4(G), we define A(G,A) tobe the completion of C C (G, A) for the norm ||/|U(G,/1) =\\9^\\f(g)\\A\\ A{GrFor any G-Banach algebras A and B, we define in [37] an abelian groupKK bAn (A, B). This is a contravariant functor in A and a covariant functor in B.When G = 1 this is equal to KK bAn (A, B). For any unconditional completion „4(G)of C C (G), there is descent morphism KK bAn (A,B) -• KK bAn (A(G,A),A(G,Bj).These constructions are adaptations of the classical constructions for C*-algebras : for any G-G*-algebra A (i.e. G acts continuously by G*-algebras automorphismson A) we have a natural G*-algebra C* ed (G,A) containing L 1 (G,A)as a dense subalgebra. If B is another G-G*-algebra, Kasparov defined an abeliangroup KKG(A,B). This is a contravariant functor in A and a covariant functor inB. When G = 1 this is equal to KK(A,B). There is an associative and distributiveproduct KKQ(A,B) ® KKQ(B,C) —¥ KKQ(A,C) and a descent morphismKK G (A,B) -+ KK(C; ed (G,A),C; ed (G,Bj).Let K c fi(E_G,A), * = 0,1, be the inductive limit over G-invariant G-compactsubsets Z of EG of KKa,*(Cn(Z),A). Then the assembly mapPred,A '• K* (EG, A) -t K*(C* ed (G, A))is defined in [4] and similar maps ppi } A, and more generally PA,A for any unconditionalcompletion A(G), can be defined.The Baum-Connes conjecture "with coefficients" claims that p re A,A is an isomorphismand the Bost conjecture "with coefficients" claims that ppiî is an isomorphism.Theorems 1.4.1, 1.5.4, 1.6.2 are still true with arbitrary coefficients.The surjectivity of the Baum-Connes conjecture with coefficients has beencounter-exampled recently (Higson, Lafforgue, Ozawa, Skandalis, Yu) using a randomgroup constructed by Gromov ([15]) but Bost conjecture with coefficients stillstands. If the Baum-Connes conjecture with coefficients is true for a group, it istrue also for all its closed subgroups; the Baum-Connes conjecture with coefficientsis also stable under various kinds of extensions (Chabert [7], Chabert-Echterhoff [8],Oyono [44], and Tu [51]).Kasparov's equivariant KK-theory was generalized to groupoids by Le Gall[31, 40, 41] and this generalized KK-theory was applied by Tu in [52, 53] to thebijectivity of the Baum-Connes map for amenable groupoids and the injectivity for(the holonomy groupoids of) hyperbolic foliations. It is possible to generalize alsoBanach KK-theory and unconditional completions. In this way we obtain the Baum-Connes conjecture for any hyperbolic group, with coefficients in any commutativeG*-algebra, and also for foliations with compact basis, admitting a (strictly) negativelycurved longitudinal riemannian metric, and such that the holonomy groupoidis Hausdorff and has simply connected fibers (not yet published).2. Discrete series representations of connectedsemi-simple Lie groups

806 V. LafforgueIn this part we examine how the Baum-Connes conjecture for a connectedsemi-simple Lie group with finite center can be used to establish the construction ofthe discrete series by Dirac induction ([17, 45, 1]). That this is morally true is knownfrom the beginning of the conjecture (see for instance [12]). In the proof we shallintroduce 3 ingredients : these are classical facts stated here without proof. Partsof the argument apply to more general groups (not connected, not semi-simple).This work owes its existence to Paul Baum. He asked me to study the problemand we discussed a lot.2.1. Dirac operatorsLet G be a Lie group, with a finite number of connected components, and K amaximal compact subgroup. We assume that there exists a G-invariant orientationon G/K. For the sake of simplicity, we assume that G/K admits a G-invariant spinstructure (it is true anyway for a two fold covering of G). Alore precisely let p be acomplementary subspace for the Lie algebra 6 of If in the Lie algebra g of G. Wechoose p such that it is invariant for the adjoint action of K and we endow it with alf-invariant euclidian metric. The above assumption means that the homomorphismK —t SO(p) lifts to Spin(p). We denote by S the associated spin representation ofK. If dim(G/K) is even, S is Z/2Z-graded. We write i = dim(G/K) [2].We denote by R(K) the (complex) representation ring of K and for any finitedimensional representation V of K we denote by [V] its class in R(K).Yet V be a finite dimensional representation of K. Yet Ey be the right Banach(in fact Hilbert) module over C* ed (G) (Z/2Z-graded if i = 0 [2]) whose elements arethe ÜT-invariant elements in V* ® S* ® C* ed (G), where K acts by left translationson C* ed (G). Yet Dy be the unbounded C* ed (G)-linear operator on Ey equal toEl® c(pi) ® pi, where the sum is over i, (p t ) is an orthonormal basis of p, pidenotes also the associated right invariant vector field on G, and c(p») is the Cliffordmultiplication by p». Let Ty = P y 2 . Then we define [dy] £ Ki(C* ed (Gj) to bethe class of the Fredholm module (Ey,Ty) over C* ed (G).In other words, Ey is the completion of the space of smooth compactly supportedsections of the bundle on K\G associated to the representation V* ® S* ofK, for the norm ||w|| = sup /GL 2 ( G),||/|| i2(G)= i \\w* f\\L*{{v*®s*)x K G), and Dy is theDirac operator, twisted by V*.Connes-Kasparov conjecture. The group morphism p re d '• R(K) —¥ Ki(C* ed (Gj)defined by [V] H> [dy] is an isomomorphism, and K i+ i(C* ed (Gj) = 0.This is a special case of the Baum-Connes conjecture because we may takeEG = G/K and thus Kf(EG) = R(K) and Kf +l (EG) = 0. It was checked for Gconnected reductive linear in [55] and the Baum-Connes conjecture was proved forany reductive group in [37] (see c) of the corollary 1.6.3 above).The following lemma has been suggested to me by Francois Pierrot. Assumethat i is even. Let moreover 17 be a unitary tempered admissible representationof G. This implies that we have a G*-homomorphism C* ed (G) —¥ K.(H). For anyelement x £ K 0 (C* ed (Gj) we denote by (H,x) £ Z the image of x by K 0 (C* ed (Gj) —¥Kn(K,(H)) = Z. If x is the class of an idempotent p £ C* ed (G), the image of p in

Banach KK-theory and the Baum-Connes Conjecture 807K,(H) is a finite rank projector, whose rank is(H,x).Lemma 2.1.1 We have (H, [dy]) = dim(F* ® S* ® H) K .2.2. Dual-Dirac operatorsFrom now on we assume that G is a connected semi-simple Lie group with finitecenter and we still assume that G/K has a G-invariant spin structure. Kasparovhasconstructed an element n £ Hom(Ki(C* ed (Gj),R(Kj) (coming from an elementof KKi(C* ed (G), C* ed (Kj), itself coming from an element of KKQ,ì(C, CO (G/K))).Kasparov has shown that n o p ied = Id#(jq [29, 31].Here is the detail of the construction. The G-invariant riemannian structureon G/K given by the chosen If-invariant euclidian metric on p has non-positive curvature.Let p be the distance to the origin and £ = d(\/l + p 2 )- Yet V be a finitedimensional complex representation of If, endowed with an invariant hermitian metric.Let Hy be the space of L 2 sections of the hermitian G-equivariant fibre bundleon G/K associated to the representation of K on S ® V and let c^y be the Cliffordmultiplication by £. In other words Hy is the subspace of If-invariant vectors inL 2 (G) ® S® V, where K acts by right translations L 2 (G), and c^y is the restrictionto this subspace of the tensor product of the Clifford multiplication by £ on L 2 (G)®Swith Id y. Left translation by G on G/K or on L 2 (G) gives rise to a (G*-)morphismn v : C* ed (G) -• Cc(Hy) and (H v ,ny,c^v) defines n v £ KK bAn (C* ed (G),C) (infact in KKi(C* ed (G),Cj). We denote by [ny] £ Hom(lfj(G* ed (G)),Z) the associatedmap, and n = EvbAAp 7 ] € Hom(Ki(C* ed (Gj),R(Kj), where the sum is overthe irreducible representations of K.Since the Connes-Kasparov conjecture is true, p re d '• R(K) —¥ Ki(C* ed (Gj)and n : Ki(C* ed (Gj) —¥ R(K) are inverse of each other and K i+ i(C* ed (Gj) = 0.Let 17 be a discrete series representation of G, i.e. an irreducible unitaryrepresentationwith a positive mass in the Plancherel measure. We recall that thisis equivalent to the fact that some (whence all) matrix coefficient c x (g) = (x, n(g)x),x £ H, ||x|| = 1, is square-integrable. Then ||c œ ||^2(G) is indépendant of x, and itsinverse is the formal degree dn of 17, which is also the mass of 17 in the Plancherelmeasure. We introduce a first ingredient.Ingredient 1. All discrete series representations of G are isolated in thetempered dual.In other words, all matrix coefficients belong to C* ed (G). In fact a standardasymptotic expansion argument shows that for any lf-finite vector x £ H, c x belongsto the Schwartz algebra ([17], II, corollary 1 page 77).Therefore there exists an idempotent p £ C* ed (G) such that the image inL 2 (G) of the image of p by the left regular representation is H* as a representationof G on the right. In fact we can take p = dnci for any x £ H, \\x\\ = 1, wherec^(g) = c x (g). The class of p in K 0 (C* ed (Gj) only depends on 17 and we denote itby [H]. It is easy to see that i : (DH^ —t K 0 (C* ed (Gj), (UH)H ^ E if ïI H[H], wherethe sums are over the discrete series representations of G, is an injection. Indeed,if 17 and H' are discrete series representations of G, (H', [H]) = 1 if 17 = 17' and 0otherwise.

808 V. LafforgueAs a corollary we see that if i = 1 [2], G has no discrete series representations.From now on we assume i = 0 [2].The first part of the following lemma was suggested to me by Georges Skandalis.Let 17 be a discrete series representation of G. We write r]([H]) = Ev ^v[V]in R(K) where the sum is finite and over the irreducible representations of K (inthe notation above, ny = [ny]([H]j).Lemma 2.2.1 IfV is an irreducible representation of K, ny =and therefore ny = (H, [dy]).dim(H*®S®Y r ) KWe have 1 = (17, [17]) = (H, p red o n([H])) = Z v n v (H,[d v ]) = Ev«v-Therefore one of the ny is ±1 and the others are 0.Alternatively we can consider the morphisms®yZ[F] = R(K) ^ K 0 (C: ed (G)) 4 Jjz where TT(X) = ({H,x)) HandH® ff Z 4 Ko(C* ed (Gj) 4 R(K) = @ V Z[V\where the sums are over the irreducible representations V of K and the discreteseries representations 17 of G. Their product TT O p ied or]oi = noiis equal to theinclusion of ®#Z in f\ H Z and their matrices in the base ([V])y and the canonicalbase of ®#Z are transpose of each other. Therefore each column of the matrix ofn o i contains exactly one non-zero coefficient, which is equal to ±1. A posteriori, ntakes its values in ®#Z.Corollary 2.2.2 The discrete series representations of G are in bijection with asubset of the set of isomorphism classes of irreducible representations of K. Theirreducible representation V of K associated to a discrete series representation His such that V = ±(H ® S*) as a formal combination of irreducible representationsof K, and H occurs in the kernel of the twisted Dirac operator Dy.Corollary 2.2.3 7/rankG ^ ranklf, G has no discrete series.In this case S* is 0 in R(K) (Barbasch and Moscovici [2] (1.2.5) page 156) :this was indicated to me by Henri Moscovici.2.3. A trace formulaFrom now on we assume that rank G = rank K. Yet T a maximal torus in K(therefore also in G). Choose a Weyl chamber for the root system of g and choosethe Weyl chamber of the root system of 6 containing it. Let V be an irreduciblerepresentation of K, p its highest wheight, and À = p + PK where PK is the halfsum of the positive roots of 6.We recall that the unbounded trace Tr : C* ed (G) —¥ R, / H> /(l) gives rise toa group morphism Ko(C* ed (Gj) —¥ R. When 17 is a discrete series representationof G, Tr([17]) is the value at 1 of p = dnc^ for some x £ H, \\x\\ = 1, and thereforeit is the formal degree dn of 17 and is > 0.

Banach KK-theory and the Baum-Connes Conjecture 809Ingredient 2. Tr([dy]) = ELe* (pod' wriere * is the set of simple roots ofthe chosen positive root system in g, and p is the half sum of the positive roots ofthis system.In this formula is used a right normalization of the Haar measure (if G is linearit is the one for which the maximal compact subgroup of the complexification ofG has measure 1). This formula is proven in [11] by a heat equation method, andin [1] by Atiyah's L 2 -index theorem.Corollary 2.3.1 If X is singular for g, [V] does not correspond to a discrete seriesrepresentation ofG.Ingredient 3. For any x £ K 0 (C* ed (Gj) such that Tr(x) ^ 0, there is adiscrete series representation 17 such that (H, x) fi^ 0.By the Plancherel formula, if G is the tempered spectrum of G, Tr(x) =J G (H,x) dH. We have to prove that, for almost all 17 outside the discrete series,(17, x) = 0. There are several possible arguments :• almost all 17 outside the discrete series are induced from a parabolic subgroupand belong to a family of representations indexed by some W, but (17', x) isconstant when H' varies in this family and goes to 0 when H' goes to infinity,• write x = [dy] for some V, then the 17 outside the discrete series with (H, x) fi^0 have measure 0 by [1] pl5 (3.19), p50 (9.8) and p51 (9.12) or by [11] p318-320.Corollary 2.3.2 If X is not singular for g, [V] does correspond to a discrete seriesrepresentation, whose formal degree isihjilla€* (p,a)We have recovered some results proved in [17], [45] and [1].References[1] Al. Atiyah and W. Schmid, A geometric construction of the discrete series forsemisimple Lie groups, Invent. Alath. 42 (1977), 1-62.[2] D. Barbasch and H. Moscovici, L 2 -index and the Selberg trace formula, J. Funct.Anal. 53 (1983), no. 2, 151-201.[3] P. Baum and A. Connes, Geometric K-theory for Lie groups and foliations,Preprint (1982), Enseign. Alath. (2) 46 (2000), no. 1-2, 3-42.[4] P. Baum and A. Connes and N. Higson, Classifying space for proper actionsand K-theory of group C* -algebras, G*-algebras: 1943-1993 (San Antonio, TX,1993), Contemp. Alath., 167, Amer. Alath. Soc. (1994), 240-291.[5] P. Baum, N. Higson and R. Plymen, A proof of the Baum-Connes conjecturefor p-adic GL(n), C. R. Acad. Sci. Paris Sér. I, 325, (1997), 171-176.[6] J.-B. Bost, Principe d'Oka, K-théorie et systèmes dynamiques non commutatifs,Invent. Alath., 101 (1990), 261-333.[7] J. Chabert, Baum-Connes conjecture for some semi-direct products, J. ReineAngew. Alath., 521, (2000) 161-184.

810 V. Lafforgue[8] J. Chabert and S. Echterhoff, Permanence properties of the Baum-Connes conjecture,Doc. Alath. 6 (2001), 127-183.[9] J. Chabert, S. Echterhoff and R. Nest, The Connes-Kasparov conjecture foralmost connected groups, Preprint, University of Münster (2001).[10] I. Chatterji, Property (RD) for cocompact lattices in a finite product of rank oneLie groups with some rank two Lie groups, to appear in Geometriae Dedicata.[11] A. Connes and H. Moscovici, The L 2 -index theorem for homogeneous spaces ofLie groups, Ann. of Alath. (2) 115 (1982), no. 2, 291-330.[12] A. Connes and H. Moscovici, L 2 -index theory on homogeneous spaces anddiscrete series representations, Operator algebras and applications, Part I(Kingston, Ont., 1980), pp. 419-433, Proc. Sympos. Pure Alath., 38, Amer.Alath. Soc, Providence, R.I., 1982.[13] A. Connes, Non commutative geometry, Academic Press, (1994).[14] J. Cuntz, K-theoretic amenability for discrete groups, J. Reine Angew. Alath.344 (1983), 180-195.[15] Al. Gromov, Spaces and questions, Geom. Funct. Anal. 2000, Special Volume,Part I, 118-161.[16] U. Haagerup, An example of a nonnuclear C* -algebra which has the metricapproximation property, Inv. Alath., 50, (1979), 279-293.[17] Harish-Chandra, Discrete series for semisimple Lie groups, I and II, ActaAlath. 113 (1965) 241-318 and 116 (1966), 1-111.[18] P. de la Harpe, Groupes hyperboliques, algèbres d'opérateurs et un théorème deJolissaint, C. R. Acad. Sci. Paris Sér. I, 307 (1988), 771-774.[19] P. de la Harpe and A. Valette, La propriété (T) de Kazdhan pour les groupeslocalement compacts, Astérisque 175, (1989).[20] N. Higson and G. Kasparov, E-theory and KK-theory for groups which actproperly and isometrically on Hilbert space., Invent. Alath. 144, 1, (2001), 23-74.[21] N. Higson, The Baum-Connes conjecture, Proc. of the Int. Cong, of Alath., Vol.II (Berlin, 1998), Doc. Alath., (1998), 637-646.[22] N. Higson, Bivariant K-theory and the Novikov conjecture, Geom. Funct. Anal.10 (2000), no. 3, 563-581.[23] P. Jolissaint, Rapidly decreasing functions in reduced C* -algebra of groups,Trans. Amer. Alath. Soc, 317, (1990), 167-196.[24] P. Julg and A. Valette, K-theoretic amenability for SL 2 (Q P ), and the actionon the associated tree, J. Funct. Anal., 58, (1984), 194-215.[25] P. Julg and G. Kasparov, Operator K-theory for the group SU(n, 1), J. ReineAngew. Alath., 463, (1995), 99-152.[26] P. Julg, Remarks on the Baum-Connes conjecture and Kazhdan's property T,Operator algebras and their applications, Waterloo (1994/1995), Fields Inst.Commun., Amer. Alath. Soc. 13, (1997), 145-153.[27] P. Julg, Travaux de N. Higson et G. Kasparov sur la conjecture de Baum-Connes, Séminaire Bourbaki. Vol. 1997/98, Astérisque, 252, (1998), No. 841,4, 151-183.[28] G. G. Kasparov, The operator K-functor and extensions of C*-algebras, Alath.

Banach KK-theory and the Baum-Connes Conjecture 811USSR Izv. 16 (1980), 513-572.[29] G. G. Kasparov, K-theory, group C* -algebras, and higher signatures (conspectus),Novikov conjectures, index theorems and rigidity, Vol. 1 (Oberwolfach,1993), 101-146, London Alath. Soc. LNS 226, (1981).[30] G. G. Kasparov Lorentz groups: K-theory of unitary representations andcrossed products, Dokl. Akad. Nauk SSSR, 275, (1984), 541-545.[31] G. G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent.Alath. 91 (1988), 147-201.[32] G. G. Kasparov and G. Skandalis, Groups acting on buildings, operator K-theory, and Novikov's conjecture, ÜT-Theory, 4, (1991), 303-337.[33] G. Kasparov and G. Skandalis, Groupes boliques et conjecture de Novikov,Comptes Rendus Acad. Sc, 319 (1994), 815-820.[34] G. Kasparov and G. Skandalis, Groups acting properly on bolic spaces and theNovikov conjecture, to appear in Annals of Alath.[35] V. Lafforgue, Une démonstration de la conjecture de Baum-Connes pourles groupes réductifs sur un corps p-adique et pour certains groupes discretspossédant la propriété (T), C. R. Acad. Sci. Paris Sér. I, 327, (1998), 439-444.[36] V. Lafforgue, Compléments à la démonstration de la conjecture de Baum-Connes pour certains groupes possédant la propriété (T), C. R. Acad. Sci. ParisSér. I, 328, (1999), 203-208.[37] V. Lafforgue, KK-théorie bivariante pour les algèbres de Banach et conjecturede Baum-Connes, Invent. Alath. 149 (2002) 1, 1-95.[38] V. Lafforgue, Banach KK-theory and the Baum-Connes conjecture, EuropeanCongress of Alathematics (Barcelona 2000), Birkäuser, Volume 2.[39] V. Lafforgue, A proof of property (RD) for cocompact lattices of SL(3,R) andSL(3,C), J. Lie Theory 10 (2000), no. 2, 255-267.[40] P.-Y. Le Gall, Théorie de Kasparov équivariante et groupoïdes, C. R. Acad.Sci. Paris Sér. I, 324, (1997), 695-698.[41] P.-Y. Le Gall, Théorie de Kasparov équivariante et groupoïdes. I, ÜT-Theory,16, (1999), 361-390.[42] I. Mineyev and G. Yu, The Baum-Connes conjecture for hyperbolic groups,Invent. Alath. 149 (2002) 1, 97-122[43] A. S. Alishchenko, Infinite-dimensional representations of discrete groups, andhigher signatures, Alath. USSR Izv. 8, no 1, (1974), 85-111.[44] H. Oyono-Oyono, Baum-Connes conjecture and group actions on trees, K-Theory 24 (2001), no. 2, 115-134.[45] R. Parthasarathy, Dirac operator and the discrete series, Ann. of Alath. (2) 96(1972), 1-30.[46] Al. V. Pimsner, KK-groups of crossed products by groups acting on trees, Invent.Alath., 86, (1986), 603-634.[47] J. Ramagge, G. Robertson, T. Steger, A Haagerup inequality for Ai x Ài andA 2 buildings, Geom. Funct. Anal., 8 (1998), 702-731.[48] G. Skandalis Kasparov's bivariant K-theory and applications, ExpositionesAlath., 9 (1991), 193-250.[49] G. Skandalis Progrès récents sur la conjecture de Baum-Connes, contribution

812 V. Lafforguede Vincent Lafforgue, Séminaire Bourbaki, No. 869 (novembre 1999).[50] Al. Talbi, Inégalité de Haagerup et géométrie des groupes, Thèse, Université deLyon I (2001).[51] J.-L. Tu, The Baum-Connes conjecture and discrete group actions on trees,«"-Theory, 17, (1999), 303-318.[52] J.-L. Tu, La conjecture de Baum-Connes pour les feuilletages moyennables,«"-Theory, 17, (1999), 215-264.[53] J.-L. Tu, La conjecture de Novikov pour les feuilletages hyperboliques, K-Theory, 16, (1999), 129-184.[54] A. Valette An introduction to the Baum-Connes conjecture, ETHZ Lectures inAlathematics, Birkhuser (2002).[55] A. Wassermann, Une démonstration de la conjecture de Connes-Kasparov pourles groupes de Lie linéaires connexes réductifs, C. R. Acad. Sci. Paris Sér. I,304 (1987), 559-562.

ICAl 2002 • Vol. II • 813-822On Some Inequalities for GaussianMeasuresfv. IjcttBilctAbstractWe review several inequalities concerning Gaussian measures - isoperimetricinequality, Ehrhard's inequality, Bobkov's inequality, S-inequality andcorrelation conjecture.2000 Mathematics Subject Classification: 60E15, 60G15, 28C20, 26D15.Keywords and Phrases: Gaussian measure, Isoperimetry, Ehrhard's inequality,Convex bodies, Correlation.1. IntroductionGaussian random variables and processes always played a central role in theprobability theory and statistics. The modern theory of Gaussian measures combinesmethods from probability theory, analysis, geometry and topology and isclosely connected with diverse applications in functional analysis, statistical physics,quantum field theory, financial mathematics and other areas. Some examples of applicationsof Gaussian measures can be found in monographs [4, 18, 20] and [23].In this note we present several inequalities of geometric nature for Gaussianmeasures. All of them have elementary formulations, but nevertheless yield manyimportantand nontrivial consequences. We begin in section 2 with the alreadyclassicalGaussian isoperimetric inequality that inspired in the 70's and 80's thevigorous development of concentration inequalities and their applications in thegeometry and local theory of Banach spaces (cf. [19, 24, 32]). In the sequel wereview several more recent results and finish in section 6 with the discussion of theGaussian correlation conjecture that remains unsolved more than 30 years.A probability measure p on a real separable Banach space F is called Gaussianif for every functional x* £ F* the induced measure po (x*)^1is a one-dimensionalGaussian measureM(a, a 2 ) for some a = a(x*) £ R and a = a(x*) > 0. Throughoutthis note we only consider centered Gaussian measures that is the measures such•"Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland. E-mail:rlatala@mimuw.edu.pl

814 R. Latalathat a(x*) = 0 for ail x* £ F*. A random vector with values in F is said to beGaussian if its distribution is Gaussian. Every centered Gaussian measure on R"is a linear image of the canonical Gaussian measure -y n , that is the measure onR" with the density dj n (x) = (27r) - "/ 2 exp(^|x| 2 /2)dx, where |x| = \/Yî=ix ì-Infinite dimensional Gaussian measures can be effectively approximated by finitedimensional ones using the following series representation (cf. [18, Proposition 4.2]):If p is a centered Gaussian measure on F and gi,g 2 , • • • are independent A'(0,1)random variables then there exist vectors xi, x 2 ,... in F such that the series X =Si-i x ì9ì ' 1S convergent almost surely and in every IP, 0 of the standard normal A'(0,1)r.v., that is1 f' x$(x) = 7i(—oo,x) = / er y ' 2 dy, ôo < x < oo.V 2-ÏÏ i-ooFor two sets A,B in a Banach space F and t £ M. we will write tA = {tx : x £ A}and A + B = {x + y : x £ A,y £ B}. A set A in F is said to be symmetric if-A = A.Alany results presented in this note can be generalized to the more generalcase of Radon Gaussian measures on locally convex spaces. For precise definitionssee [4] or [7].2. Gaussian isoperimetryFor a Borei set A in R" and t > 0 let A t = A + tB% = {x £ R" : |x - a\ 0. (2.1)Theorem 2.1 has an equivalent differential analog. To state it let us definefor a measure p on R" and any Borei set A the boundary p-measure of A by theformulaM+(A)=liminf M( - 4t)^(- 4) .Aloreover let ip(x) = $'(x) = (27T) -1 / 2 exp(^x 2 /2) and letI(t) = V>°$- 1 (t),t£[0,l]be the Gaussian isoperimetricfunction.

On Some Inequalities for Gaussian Aleasures 815The equivalent form of Theorem 2.1 is that for all Borei sets A in R"7+(-4)>l( 7 »(-4)). (2.2)The equality in (2.2) holds for any affine halfspace.For a probability measure p on R" we may define the isoperimetricof p byls(p)(p) = inf{p + (A) : p(A) =p}, 0 of application of (2.1) (see [20, Lemma3.1]).Corollary 2.2 Let X be a centered Gaussian random vector in a separableBanach space (F,\\ • ||). Then for any t > 0whereP(|[|X[| - Aled([|X[|)| >t) (Yol n (Aj) x (vol n (Bj) 1 - x for A G [0,1].Gaussian measures satisfy the similar log-concavity property, that is the inequalityln(p(XA + (1 - X)Bj) > Xln(p(Aj) + (1 - A) ln(p(Bj), X £ [0,1] (3.1)holds for any Gaussian measure p on a separable Banach space F and any Boreisets A and B in F (cf. [5]). However the log-concavity of the measure does notimply the Gaussian isoperimetry.In the early 80's A. Ehrhard [9] gave a different proof of the isoperimetricinequality (2.1) using a Gaussian symmetrization procedure similar to the Steinersymmetrization. With the same symmetrization tool Ehrhard established a newBrunn-Alinkowski type inequality, stronger than (3.1), however only for convex sets.Theorem 3.1(Ehrhard's inequality) If p is a centered Gaussian measure ona separable Banach space F and A, B are Borei sets in F, with at least one of themconvex, then^QfiXA + (1- X)Bj) > A#- 1 (M(A)) + (1 - A)#- 1 (M(B)) for X £ [0,1]. (3.2)For both sets A and B convex Ehrhard's inequality was proved in [9]. Thegeneralization to the case when only one of the sets is convex was established in[16].

816 R. FatalaIt is not hard to see that Theorem 3.1 implies the isoperimetric inequality(2.1). Indeed we have for any Borei set A in R"Q-H-YniAt)) = #- 1 ( 7 „(A(A- 1 A) + (1 - A)((l - A)" 1 *^")))> A#- 1 ( 7 „(A- 1 A)) + (1 - A)#- 1 ( 7 „((l - A)- 1^")) A^T ^(^(A)) + t.Conjecture 3.1 Inequality (3.2) holds for any Borei sets in F.Ehrhard's symmetrization procedure enables us to reduce Conjecture 3.1 tothe case F = R and p = 71. We may also assume that A and B are finite unions ofintervals. At the moment the conjecture is known to hold when A is a union of atmost 3 intervals.Ehrhard's inequality has the following Prekopa-Leindler type functional version.Suppose that A G (0,1) and f,g,h:R n —¥ [0,1] are such thatthenV œ ,, GRn ^(MAx + (1 - X)yj) > A#- 1 (/(x)) + (1 - A)*- 1^))* _1 (/ hchJiXQ-HÏ /d 7 „) + (lÂ)#- 1 (/' gd ln ). (3.3)We use here the convention $ _1 (0) = —oo,^1(Y)= 00 and ^00 + 00 = ^00. Atthe moment the above functional inequality is known to hold under the additionalassumption that at least one of the functions $ _1 (/),$ _1 ( I hd-/ n + t})0valid for all Lipschitz functions h : R" —¥ R with the Lipschitz seminorm ||/I||Lì P =sup{\h(x) — h(y)\ : x,y £ W 1 } < 1. However the logarithmic Sobolev inequalitydoesnot imply the isoperimetric inequality.

On Some Inequalities for Gaussian Aleasures 817The formulation of the functional form of Gaussian isoperimetry was given byS.G. Bobkov [2].Theorem 4.1 For any locally Lipschitz function f : R" —¥ [0,1] and p = -y nwe haveH I fdp) < f VHf) 2 + |V/| 2 d M . (4.2)ii» JR»Theorem 4.1 easily implies the isoperimetric inequality (2.2) by approximatingthe indicator function I A by Lipschitz functions. On the other hand if we apply(2.2) to the set A = {(x,y) £ R" x R : $(y) < f(x)} in R" +1 we get (4.2). It isalso not hard to derive the logarithmic Sobolev inequality (4.1) as a limit case ofBobkov's inequality (cf. [1]): one should use (4.2) for / = eg 2 (with g bounded)and let e tend to 0 (I(t) ~ ty?2log(l/t) as t -t 0+).The crucial point of the inequality (4.2) is its tensorization property. To stateit precisely let us say that a measure p on R" satisfies Bobkov 's inequality if theinequality (4.2) holds for all locally Lipschitz functions / : R" —¥ [0,1]. Easyargumentshows that if pi are measures on R m , i = 1,2, that satisfy Bobkov'sinequality then the measure pi ® p 2 also satisfies Bobkov's inequality.The inequality (4.2) was proved by Bobkov in an elementary way, based onthe following "two-point" inequality:i( a -^) < y nap+( a -^) 2 +y m 2 +( V)2 (43)valid for all a,b £ [0,1]. In fact the inequality (4.3) is equivalent to Bobkov'sinequality for p = |óAi + ^öi and the discrete gradient instead of Vf. Using thetensorization property and the central limit theorem Bobkov deduces (in the similarway as Gross in his proof of (4.1)) (4.2) from (4.3).Using the co-area formula and Theorem 4.1 F. Barthe and Al. Alaurey [1]gave interesting characterization of all absolutely continuous measures that satisfyBobkov's inequality.Theorem 4.2 Let c > 0 and p be a Borei probability measure on the Riemannianmanifold M, absolutely continuous with respect to the Riemannian volume.Then the following properties are equivalent(i) For every measurable A c M, pfi(A) > cI(p(Aj);(ii) For every locally Lipschitz function f : M —¥ [0,1]H I fdp) < f J 1(f) 2 + \\Vf\ 2 dp.JM JM V c-Theorem 4.2 together with the tensorization property shows that if ls(pi) > ci,i = 1,2..., then also Is(î ® ... ® p n ) > ci. In general it is not known how toestimate Is(î ®...® p n ) in terms of ls(pi) even in the case when all pi's are equal(another important special case of this problem was solved in [3]) .5. S-inequality

818 R. FatalaIn many problems arising in probability in Banach spaces one needs to estimatethe measure of balls in some Banach space F. In particular one may ask what isthe slowest possible grow of the Gaussian measure of balls in F or more general ofsome fixed convex symmetric closed set under dilations. The next theorem, provedby R. Latala and K. Oleszkiewicz [17], gives the positive answer to the conjectureposed in an unpublished manuscript of L. A. Shepp (1969).Theorem 5.1(S-inequality) Let p be a centered Gaussian measure on a separableBanach space F. If A is a symmetric, convex, closed subset of F and P C Fis a symmetric strip, that is P = {x £ F : |x*| < 1} for some x* £ F*, such thatp(A) = p(P) thenp(tA) > p(tP) fort>landp(tA) < p(tP)forO 1,where \P _1denotes the inverse of*(x) = >yi(-x,x) = —= j e^y2/2 dy.V 2-K J-xThe crucial tool in the proof of S-inequality is the new modified isoperimetricinequality. Let us first define for a convex symmetric set A in R"w(A) = 2sup{r : B(0,r) C A}.It is easy to see that for a symmetric strip P, w(P) is equal to the width of P andfor a symmetric convex set Aw(A) = inf{w(F) : A C P, P is a symmetric strip in R"}. (5.1)Thus w(A) can be considered as the width of the set A. The following isoperimetrictypetheorem holds true.Theorem 5.2 If -y n (A) = 7„(F), where P is a symmetric strip and A is aconvex symmetric set in R", thenw(A) 1 +(A)>w(P) 1 +(P). (5.2)The main advantage of the inequality (5.2) is that one may apply here thesymmetrization procedure and reduce Theorem 5.2 to the similar statement for2-dimensional convex sets symmetric with respect to some axis.

On Some Inequalities for Gaussian Aleasures 819It is not hard to see that Theorem 5.2 implies Theorem 5.1. Indeed, let usdefine for any measurable set B in R", 7#(t) = -f n (tB) for t > 0. Taking thederivatives of both sides of the inequalities in Theorem 5.1 one can see that it isenough to show7„(A) = ln (P)^ i A (ì) > Ì P (Ì) (5.3)for any symmetric convex closed set A and a symmetric strip P = {\xi\ < p}. Yetw = w(A), so B(0,w) C A. Then for t > 1 and x G A we have B(t^x, (t—Yjw/i) =* _1 x + (1 - t^1)B(0,w) C A, so B(x, (t - l)w) C tA. Hence A (t _ 1)ttJ c tA andHowever for the strip Pi A (i)>unt(A) = w(Ay f t(A).fi P (l) = \ßpe- p2/2 = w(Pht(P)V TTand the inequality (5.3) follows by Theorem 5.2.It is not clear if the convexity assumption for the set A in Theorem 5.2 isnecessary (obviously w(A) for nonconvex symmetric sets A should be defined by(5.1)). One may also ask if the symmetry assumption can be released (with thesuitable modification of the definition of the width for nonsymmetric sets). Alsofunctional versions of Theorems 5.1 and 5.2 are not known.As was noticed by S. Szarek S-inequality implies the best constants in comparisonof moments of Gaussian vectors (cf. [17]).Corollary 5.3 If X is a centered Gaussian vector in a separable Banach space(F, || • ||) then(EHXIH 1^ < ^(EIIXII«) 1 /« forc qanyp>q>0,wherec p = (E\gif) 1/P= 72(^F(^)) 1 /P.-fin 2Another interesting problem connected with the S-inequality was recentlyposedby W. Banaszczyk (private communication): Is it true that under the assumptionsof Theorem 5.1p(s x t 1 - x A) > p(sA) x p(tAfi- x , X £ [0,1] (5.4)for any closed convex symmetric set A in F and s, t > 0? Combining the facts thatthe function $ _1 (p(tAj) is concave (Theorem 3.1) and the function ^^1(p(tAj)is nondecreasing (Theorem 5.1) one can show that (5.4) holds if p(sA),p(tA) > c,where c < 0.85 is some absolute constant.It is of interest if Theorem 5.1 can be extended to the more general class ofmeasures. The following conjecture seems reasonable.Conjecture 5.1 Let v be a rotationally invariant measure on W 1 , absolutelycontinuous with respect to the Lebesgue measure with the density of the form f(\x\)

820 R. Fatalafor some nondecreasing function f : R+ —¥ [0, oo). Then for any convex symmetricset A in R" and any symmetric strip P in R" such that v(A) = v(P) the inequalityv(\A) > v(XP) is satisfied for A > 1.To show Conjecture 5.1 it is enough to establish the following conjecture concerningthe volumes of the convex hulls of symmetric sets on the n — 1-dimensionalunit sphere S""^1.Conjecture 5.2 Let a n -i be a Haar measure on S^1,A be a symmetricsubset of S""^1 and P = {x £ S""^1 : \xi\ < t} be a symmetric strip on S""^1 suchthat a n -i(A) = a n -i(P), then vol„(conv(A)) > vol n (conv(P)).It is known that both conjectures hold for n < 3 (cf. [30]).6. Correlation conjectureThe following conjecture is an object of intensive efforts of many probabilistssince more then 30 years.Conjecture 6.1 If p is a centered Gaussian measure on a separable Banachspace F thenp(A(lB)>p(A)p(B) (6.1)for all convex symmetric sets A, B in F.Various equivalent formulations of Conjecture 6.1 and history of the problemcan be found in [27]. Standard approximation argument shows that it is enoughto show (6.1) for F = R" and p = -y n . For n = 2 the solution was given byL. Pitt [26], for n > 3 the conjecture remains unsettled, but a variety of specialresults are known. Borell [8] established (6.1) for sets A,B in a certain class of(not necessary convex) sets in R", which for n = 2 includes all symmetric sets. Aspecial case of (6.1), when one of the sets A,B is a symmetric strip of the form{x £ F : |x*(x)| < 1} for some x* G F*, was proved independently by C. G. Khatri[14] and Z. Sidâk [28] (see [11] for an extension to elliptically contoured distributionsand [31] for the case when one of the sets is a nonsymmetric strip). Recently, theKhatri-Sidâk result has been generalized by G. Hargé [12] to the case when one ofthe sets is a symmetric ellipsoid.Theorem 6.1 If p is a centered Gaussian measure on W 1 , A is a symmetricconvex set in R" and B is a symmetric ellipsoid, that is the set of the form B ={x G R" : (Cx, x) < 1} for some symmetric nonnegative matrix C, thenThe following weaker form of (6.1)p(AnB) > p(A)p(B).p(A HB)> p(XA)p(y / l-X 2 B), 0 < A < 1was established for A = -fi in [27] and for general A in [21]. The Khatri-Sidâk resultand the above inequality turn out to be very useful in the study of the so-calledsmall ball probabilities for Gaussian processes (see [22] for a survey of results in thisdirection).

On Some Inequalities for Gaussian Aleasures 821The correlation conjecture has the following functional form:fgdp > / fdp / gdp (6.2)for all nonnegative even functions f,g such that the sets {/ > t} and {g > t} areconvex for all t > 0. Y. Hu [13] showed that the inequality (6.2) (that we would liketo have for log-concave functions) is valid for even convex functions /, g £ L 2 (F, p).References[i[2[3;[4;[6;[9[io;[n[12:[is;[w;[15[16F. Barthe, B. Alaurey, Some remarks on isoperimetry of Gaussian type, Ann.Inst. H Poincaré Probab. Statist. 36 (2000), 419-434.S.G. Bobkov, An isoperimetric inequality on the discrete cube, and an elementaryproof of the isoperimetric inequality in Gauss space, Ann. Probab. 25(1997), 206-214.S.G. Bobkov, C. Houdré, Isoperimetric constants for product probability measures,Ann. Probab. 25 (1997), 184-205.V.l. Bogachev, Gaussian Measures, American Alathematical Society, Providence,RI, 1998.C. Borell, Convex measures on locally convex spaces, Ark. Mat. 12 (1974),239-252.C. Borell, The Brunn-Alinkowski inequality in Gauss space, Invent. Math., 30(1975), 207-216.C. Borell, Gaussian Radon measures on locally convex spaces, Math. Scand. 38(1976), 265-284.C. Borell, A Gaussian correlation inequality for certain bodies in R n , Math.Ann. 256 (1981), 569-573.A. Ehrhard, Symétrisation dans l'espace de Gauss, Math. Scand., 53 (1983),281-301.L. Gross, Logaritmic Sobolev inequalities, Amer. J. Math. 97 (1975), 1061-1083.S. Das Gupta, ALL. Eaton, I. Olkin, Al. Perlman, L.J. Savage, Al. Sobel, Inequalitieson the probability content of convex regions for elliptically contoureddistributions, Proc. Sixth Berkeley Symp. Math. Statist. Prob. vol. II, 241-264,Univ. California Press, Berkeley, 1972.G. Hargé, A particular case of correlation inequality for the Gaussian measure,Ann. Probab. 27 (1999), 1939-1951.Y. Hu, Ito-Wiener chaos expansion with exact residual and correlation, varianceinequalities, J. Theoret. Probab. 10 (1997), 835-848.CG. Khatri, On certain inequalities for normal distributions and their applicationsto simultaneous confidence bounds, Ann. Math. Stat. 38 (1967), 1853-1867.S. Kwapien, J. Sawa, On some conjecture concerning Gaussian measures ofdilatations of convex symmetric sets, Studia Math. 105 (1993), 173-187.R. Latala, A note on the Ehrhard inequality, Studia Math. 118 (1996), 169-174.

822 R. Latala[17] R. Latala, K. Oleszkiewicz, Gaussian measures of dilatations of convex symmetricsets, Ann. Probab. 27 (1999), 1922-1938.[18] Al. Ledoux, Isoperimetry and Gaussian Analysis, Lectures on probability theoryand statistics (Saint-Flour, 1994), 165-294, Lecture Notes in Alath. 1648,Springer, Berlin, 1996.[19] Al. Ledoux, The concentration of measure phenomenon, American AlathematicalSociety, Providence, RI, 2001.[20] Al. Ledoux, Al. Talagrand, Probability on Banach Spaces. Isoperimetry andprocesses, Springer-Verlag, Berlin, 1991.[21] WV. Li, A Gaussian correlation inequality and its applications to small ballprobabilities, Electron. Comm. Probab. 4 (1999), 111-118.[22] WV. Li, Q.A1. Shao, Gaussian processes: inequalities, small ball probabilitiesand applications, Stochastic Processes: Theory and Methods, Handbook ofStatistics vol. 19, 533-597, Elsevier, Amsterdam 2001.[23] ALA. Lifshits, Gaussian random functions, Kluwer Academic Publications,Dordrecht, 1995.[24] V.D. Alilman, G. Schechtman, Asymptotic theory of finite-dimensional normedspaces, Lecture Notes in Alath. 1200, Springer-Verlag, Berlin, 1986.[25] R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978),1182-1238.[26] L. Pitt, A Gaussian correlation inequality for symmetric convex sets, Ann.Probability, 5 (1977), 470-474.[27] G. Schechtman, T. Schlumprecht, J. Zinn, On the Gaussian measure of intersection,Ann. Probab. 26 (1998), 346-357.[28] Z. Sidâk, Rectangular confidence regions for the means of multivariate normaldistributions, J. Amer. Statist. Assoc. 62 (1967), 626-633.[29] V.N. Sudakov, B.S. Tsirel'son, Extremal propertiesof half-spaces for sphericallyinvariantmeasures (in Russian), Zap. Nauchn. Sem. L.O.M.I. 41 (1974), 14-24.[30] V.N. Sudakov, V.A. Zalgaller, Some problems on centrally symmetric convexbodies (in Russian) Zap. Nauchn. Sem. L.O.M.I. 45 (1974), 75-82.[31] S. Szarek, E. Werner, A nonsymmetric correlation inequality for Gaussian measure,J. Multivariate Anal. 68 (1999), 193-211.[32] Al. Talagrand, Concentration of measure and isoperimetric inequalities in productspaces, IHES Pubi. Math. 81 (1995), 73-205.

Author IndexAlesker, Semyon 757Andrews, B 221Bartnik, Robert 231Bestvina, Alladen 373Biane, P 765Bigelow, S 37Biran, P 241Bisch, D 775Bondal, A 47Bouscaren, E 3Bray, Hubert L 257Chekanov, Yu. V 385Chen, Xiuxiong 273Cogdell, J.W 119Cohen, H 129Delorme, Patrick 545Denef, J 13Ding, Weiyue 283Eremenko, A 681Esnault, Hélène 471Etingof, Pavel 555Fontaine, Jean-Alare 139Furuta, Al 395Gaitsgory, D 571Ge, Liming 787Giroux, Emmanuel 405Göttsche, L 483Harris, Alichael 583Heinonen, Juha 691Hesselholt, Lars 415Huber, A 149Ionel, Eleny-Nicoleta 427Kato, Kazuya 163Kenig, Carlos E 701Kings, G 149Klyachko, Alexander 599Kobayashi, Toshiyuki 615Kudla, Stephen S 173Lafforgue, V 795Lascar, D 25Latala, R 813Lerner, Nicolas 711Levine, Al 57Li, P 293Loeser, F 13Long, Yiming 303Alazur, Barry 185Alehta, Vikram Bhagvandas .. 629Aleinrenken, E 637Alukai, Shigeru 495Nazarov, Alaxim 643Orlov, D 47Pandharipande, R 503Petrunin, Anton 315Piatetski-Shapiro, LI 119Praeger, Cheryl E 67Reid, Ailles 513Rong, Xiaochun 323Rost, Alarkus 77Rubin, Karl 185Schechtman, Vadim 525Schwartz, Richard Evan 339Seidel, Paul 351Sela, Z 87Shahidi, Freydoon 655Stafford, J. T 93Tamarkin, Dimitri 105Teichner, Peter 437Thiele, C 721Tillmann, Ulrike 447Totaro, B 533Ullmo, Emmanuel 197Vignéras, Alarie-France 667Wang, Shicheng 457Wooley, Trevor D 207Zelditch, S 733Zhang, Weiping 361Zhou, Xiangyu 743

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