Moment Maps, Cobordisms, and Hamiltonian Group Actions

http://dx.doi.org/10.1090/surv/098

Mathematical Surveys

and Monographs

Volume 98

Moment Maps, Cobordisms, and Hamiltonian Group Actions

Victor Guillemin Viktor Ginzburg Yael Karshon

American Mathematical Society

EDITORIAL COMMITTEE Peter S. Landweber Tudor Stefan Ratiu Michael P. Loss, Chair J. T. Stafford

2000 Mathematics Subject Classification. Primary 53Dxx, 57Rxx, 55N91, 57S15.

ABSTRACT. The book contains a systematic treatment of numerous symplectic geometry questions in the context of cobordisms of Hamiltonian group actions. The topics analyzed in the book in­clude abstract moment maps, symplectic reduction, the Duistermaat-Heckman formula, geometric quantization, and the quantization commutes with reduction theorem.

Ten appendices cover a broad range of related subjects: proper actions of compact Lie groups, equivariant cohomology, the Atiyah-Bott-Berline-Vergne localization theorem, the Kawasaki Riemann-Roch formula, and a variety of other results.

The book can be used by researchers and graduate students working in symplectic geometry and its applications.

Library of Congress Cataloging-in-Publication D a t a Guillemin, V., 1937-

Moment maps, cobordisms, and Hamiltonian group actions / Victor Guillemin, Viktor Ginz-burg, Yael Karshon.

p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 98) Includes bibliographical references and index. ISBN 0-8218-0502-9 (alk. paper) 1. Symplectic geometry. 2. Cobordism theory. 3. Geometric quantization. I. Karshon, Yael,

1964- II. Ginzburg, Viktor L., 1962- III. Title. IV. Mathematical surveys and monographs ; no. 98. QA665.G85 2002 516.3 /6-dc21 2002074590

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Contents

Chapter 1. Introduction 1. Topological aspects of Hamiltonian group actions 2. Hamiltonian cobordism 3. The linearization theorem and non-compact cobordisms 4. Abstract moment maps and non-degeneracy 5. The quantum linearization theorem and its applications 6. Acknowledgements

Part 1. Cobordism

Chapter 2. Hamiltonian cobordism 1. Hamiltonian group actions 2. Hamiltonian geometry 3. Compact Hamiltonian cobordisms 4. Proper Hamiltonian cobordisms 5. Hamiltonian complex cobordisms

Chapter 3. Abstract moment maps 1. Abstract moment maps: definitions and examples 2. Proper abstract moment maps 3. Cobordism 4. First examples of proper cobordisms 5. Cobordisms of surfaces 6. Cobordisms of linear actions

Chapter 4. The linearization theorem 1. The simplest case of the linearization theorem 2. The Hamiltonian linearization theorem 3. The linearization theorem for abstract moment maps 4. Linear torus actions 5. The right-hand side of the linearization theorems 6. The Duistermaat-Heckman and Guillemin-Lerman-Sternber

Chapter 5. Reduction and applications 1. (Pre-)symplectic reduction 2. Reduction for abstract moment maps 3. The Duistermaat-Heckman theorem 4. Kahler reduction 5. The complex Delzant construction 6. Cobordism of reduced spaces

vi C O N T E N T S

7. Jeffrey-Kirwan localization 82 8. Cutting 84

Part 2. Quantization 87

Chapter 6. Geometric quantization 89 1. Quantization and group actions 89 2. Pre-quantization 90 3. Pre-quantization of reduced spaces 96 4. Kirillov-Kostant pre-quantization 99 5. Polarizations, complex structures, and geometric quantization 102 6. Dolbeault Quantization and the Riemann-Roch formula 110 7. Stable complex quantization and Spinc quantization 113 8. Geometric quantization as a push-forward 117

Chapter 7. The quantum version of the linearization theorem 119 1. The quantization of C d 119 2. Partition functions 125 3. The character of Q(Cd) 130 4. A quantum version of the linearization theorem 134

Chapter 8. Quantization commutes with reduction 139 1. Quantization and reduction commute 139 2. Quantization of stable complex toric varieties 141 3. Linearization of [Q,R]=0 145 4. Straightening the symplectic and complex structures 149 5. Passing to holomorphic sheaf cohomology 150 6. Computing global sections; the lit set 152 7. The Cech complex 155 8. The higher cohomology 157 9. Singular [Q,R]=0 for non-symplectic Hamiltonian G-manifolds 159 10. Overview of the literature 162

Part 3. Appendices 165

Appendix A. Signs and normalization conventions 167 1. The representation of G on C°°(M) 167 2. The integral weight lattice 168 3. Connection and curvature for principal torus bundles 169 4. Curvature and Chern classes 171 5. Equivariant curvature; integral equivariant cohomology 172

Appendix B. Proper actions of Lie groups 173 1. Basic definitions 173 2. The slice theorem 178 3. Corollaries of the slice theorem 182 4. The Mostow-Palais embedding theorem 189 5. Rigidity of compact group actions 191

Appendix C. Equivariant cohomology 197 1. The definition and basic properties of equivariant cohomology 197

CONTENTS vii

2. Reduction and cohomology 201 3. Additivity and localization 203 4. Formality 205 5. The relation between H Q and H^ 208 6. Equivariant vector bundles and characteristic classes 211 7. The Atiyah-Bott-Berline-Vergne localization formula 217 8. Applications of the Atiyah-Bott-Berline-Vergne localization formula 222 9. Equivariant homology 226

Appendix D. Stable complex and Spinc-structures 229 1. Stable complex structures 229 2. Spinc-structures 238 3. Spinc-structures and stable complex structures 248

Appendix E. Assignments and abstract moment maps 257 1. Existence of abstract moment maps 257 2. Exact moment maps 263 3. Hamiltonian moment maps 265 4. Abstract moment maps on linear spaces are exact 269 5. Formal cobordism of Hamiltonian spaces 273

Appendix F. Assignment cohomology 279 1. Construction of assignment cohomology 279 2. Assignments with other coefficients 281 3. Assignment cohomology for pairs 283 4. Examples of calculations of assignment cohomology 285 5. Generalizations of assignment cohomology 287

Appendix G. Non-degenerate abstract moment maps 289 1. Definitions and basic examples 289 2. Global properties of non-degenerate abstract moment maps 290 3. Existence of non-degenerate two-forms 294

Appendix H. Characteristic numbers, non-degenerate cobordisms, and non-virtual quantization 301

1. The Hamiltonian cobordism ring and characteristic classes 301 2. Characteristic numbers 304 3. Characteristic numbers as a full system of invariants 305 4. Non-degenerate cobordisms 308 5. Geometric quantization 310

Appendix I. The Kawasaki Riemann-Roch formula 315 1. Todd classes 315 2. The Equivariant Riemann-Roch Theorem 316 3. The Kawasaki Riemann-Roch formula I: finite abelian quotients 320 4. The Kawasaki Riemann-Roch formula II: torus quotients 323

Appendix J. Cobordism invariance of the index of a transversally elliptic operator by Maxim Braverman 327

1. The Spin -Dirac operator and the Spinc-quantization 327 2. The summary of the results 329

viii C O N T E N T S

3. Transversally elliptic operators and their indexes 331 4. Index of the operator B a 333 5. The model operator 335 6. Proof of Theorem 1 336

Bibliography 339

Index 349

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Index

abstract moment map, 31 exact, 32 Hamiltonian, 32 non-degenerate, 289 polarized, 34 proper, 33

action, 173 effective, 174 faithful, 174 free, 174 locally free, 174 proper, 173

additivity techniques, 203 alcove, 21, 77 assignment, 259

ry-polarized, 261 associated with a moment map, 31 minimal stratum, 263 moment, 279

assignment cohomology, 279 relative, 283

Atiyah-Bott Lefschetz fixed point formula, 116

Bargmann space, 120 basic form, 64, 184 Borel construction, 197

Cartan model, 198 Chern class, 171, 315

equivariant, 212, 316 Chevalley's theorem, 200 cobordism, 34

complex oriented, 235 equivariant, 35 Hamiltonian

compact, 25 complex, 30 polarized, 28 proper, 27

non-degenerate, 308 proper, 35

compatible almost complex structure and symplectic

structures, 111 complex and symplectic structures, 103

convexity, 294

convexity theorem, 21 curvature, 169

equivariant, 172 curvature class, 171

equivariant, 172

Delzant construction

complex, 76 symplectic, 75

polytope, 75 space, 23, 70, 76

determinant line bundle, 113 Dirac axioms, 89 distinguishing cohomology class, 247 distribution, 103

complex, 103 Liouville, 27 real, 103

distributional character, 132 Dolbeault complex, 105 Duistermaat-Heckman

distribution, 27 formula, 60, 222 function, 70 integral, 24 measure, 17 polynomial, 72 theorem, 69

Ehresmann's lemma, 69 equivariant

characteristic classes, 212 characteristic numbers, 218, 304

mixed, 304 cohomology, 197 complex line bundles, 214 de Rham Theorem, 198 differential forms, 198 homology, 226 Mayer-Vietoris exact sequence, 203 Poincare duality, 223 vector bundles, 212

Fock space, 120 formality, 206, 290

349

350 INDEX

generating vector field, 177 group cohomology, 193 Guillemin-Lerman-Sternberg formula, 59

Hamiltonian G-manifold, 16 stable complex, 29

index theorem, 316 equivariant, 115, 318

Kawasaki invariants, 325 Kawasaki Riemann-Roch formula, 324 Kirwan's epimorphism, 292

C2 -cohomology, 106 lattice

group,168 weight, 168

linearization theorem for abstract moment maps, 51 Hamiltonian, 47 quantum, 134

Liouville measure, 16, 17 lit set, 153 local linearization theorem, 181 localization

Jeffrey-Kir wan, 82 localization theorem

Atiyah-Bott-Berline-Vergne, 218, 219 Borel's, 204

moment cone, 22 map, 15

polarized, 27 polytope, 21

Mostow-Palais embedding theorem, 189

orbit type, 187 principal, 188 stratification, 188

infinitesimal, 188

Pin c structure, 244 Poisson bracket, 19 polarization, 103 polarized function to g*, 34 polarizing vector, 57 polytope

simple, 75 pre-quantization, 91

data, 91 equivariant, 94 line bundle, 90

equivariant, 93 principal bundle, 178 push-forward, 219

equivariant, 221

[Q,R] = 0 , 139 quantization, 104

Dolbeault, 111 Spinc, 113

quantization commutes with reduction regular values, 140 singular values, 141

reduced space, 63 reduction

for abstract moment maps, 65 symplectic, 64

Riemann-Roch formula, 112 number, 83

Riemann-Roch theorem equivariant, 316 topological, 118

rigidity, 191 ring structure

on set of polarized cobordism classes, 37

shift formula, 114, 253 slice theorem, 180 Spinc Dirac operator, 113 Spinc structure, 113, 238

bundle equivalence, 240 destabilization, 252 homotopy, 241 homotopy classification, 246 metric, 242

splitting principle, 208, 315 stability theorem, 22 stabilizer, 174

infinitesimal, 178 stable complex structure, 229

bundle equivalence, 230 equivariant, 229 homotopy, 230

symplectic cutting, 84

tame G-manifold, 37 Todd class, 112, 315

equivariant, 317

uniqueness lemmas for abstract moment maps, 37, 51 for Hamiltonian actions on vector bundles,

49

weights, 22, 52 isotropy, 22 polarized, 53, 57, 126 real, 52

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M. Cole) , Rings, modules, and algebras in stable homotopy theory, 1997 46 S tephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and A d a m O. Hausknecht, Cogroups and co-rings in

categories of associative rings, 1996 44 J. Amoros , M. Burger, K. Corlette , D . Kotschick, and D . Toledo, Fundamental

groups of compact Kahler manifolds, 1996 43 James E. Humphreys , Conjugacy classes in semisimple algebraic groups, 1995 42 Ralph Freese, Jaroslav Jezek, and J. B. Nat ion , Free lattices, 1995 41 Hal L. Smith , Monotone dynamical systems: an introduction to the theory of

competitive and cooperative systems, 1995 40.5 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 5, 2002 40.4 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 4, 1999 40.3 Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the

finite simple groups, number 3, 1998

For a complete list of t i t les in this series, visit t he AMS Bookstore at w w w . a m s . o r g / b o o k s t o r e / .