Table Of ContentAMS/IP
Studies in
Advanced
Mathematics
S.-T. Yau, Series Editor
American Mathematical Society • International Press
Lagrangian Intersection
Floer Theory
AMS/IP
Studies in
Advanced
Mathematics
Volume 46.1
Lagrangian Intersection
Floer Theory
Anomaly and Obstruction, Part I
Kenji Fukaya
Yong-Geun Oh
Hiroshi Ohta
Kaoru Ono
American Mathematical Society • International Press
Shing-tung Yau, General Editor
2000 Mathematics Subject Classification. Primary 53D12, 53D40;
Secondary 14J32, 81 T30, 37 JlO, 18D50, 55P62.
For additional information and updates on this book, visit
www. ams.org/bookpages/a msip-46
Library of Congress Cataloging-in-Publication Data
Lagrangian intersection floer theory : anomaly and obstruction / Kenji Fukaya ... (et al.).
p. cm. - (AMS/IP studies in advanced mathematics; v. 46}
Includes bibliographical references and index.
ISBN 978-0-8218-4831-9 (set : alk. paper) - ISBN 978-0-8218-4836-4 (pt. 1 : alk. paper) -
ISBN 978-0-8218-4837-1 (pt. 2 : alk. paper)
1. Floer homology. 2. Lagrangian points. 3. Symplectic geometry. I. Fukaya, Kenji, 1959-
QA665.L34 2009
516.3'6-dc22
2009025925
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting
for them, are permitted to make fair use of the material, such as to copy a chapter for use
in teaching or research. Permission is granted to quote brief passages from this publication in
reviews, provided the customary acknowledgment of the source is given.
Republication, systematic copying, or multiple reproduction of any material in this publication
is permitted only under license from the American Mathematical Society and International Press.
Requests for such permission should be addressed to the Acquisitions Department, American
Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests
can also be made by e-mail to reprint-permission©ams. org.
@2009 by the American Mathematical Society and International Press. All rights reserved.
The American Mathematical Society and International Press retain all rights
except those granted to the United States Government.
Printed in the United States of America.
§ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at http: I /www.ams.org/
Visit the International Press home page at URL: http: I /www.intlpress.com/
10987654321 14 13 12 11 10 09
Contents
Volume I
Preface xi
Chapter 1. Introduction 1
1.1 What is Floer (co)homology 1
1.2 General theory of Lagrangian Floer cohomology 5
1.3 Applications to symplectic geometry 13
1.4 Relation to mirror symmetry 16
1.5 Chapter-wise outline of the main results 25
1.6 Acknowledgments 35
1. 7 Conventions 36
Chapter 2. Review: Floer cohomology 39
2.1 Bordered stable maps and the Maslov index 39
2.1.1 The Maslov index: the relative first Chern number 39
2.1.2 The moduli space of bordered stable maps 43
2.2 The Novikov covering and the action functional 49
2.2.1 The r-equivalence 50
2.2.2 The action functional and the Maslov-Morse index 51
2.3 Review of Floer cohomology I: without anomaly 53
2.3.1 The £2-gradient equation of A 53
2.3.2 Floer's definition: Z -coefficients. 57
2
2.3.3 Bott-Morse Floer cohomology 59
2.4 Review of Floer cohomology II: anomaly appearance 60
2.4.1 The Floer cochain module 61
2.4.2 The Floer moduli space 62
2.4.3 The Novikov ring AR(L) 66
2.4.4 Monotone Lagrangian submanifolds 69
2.4.5 Appearance of the primary obstruction 71
Chapter 3. The A algebra associated to a Lagrangian submanifold 77
00
3.1 Outline of Chapter 3 77
3.2 Algebraic framework on filtered A algebras 86
00
3.2.1 A algebras and homomorphisms 86
00
3.2.2 Filtered A algebras and homomorphisms 89
00
3.3 Algebraic framework on the homotopy unit 94
3.3.1 Definition of the homotopy unit 94
3.3.2 Unital (resp. homotopy unital) A homomorphisms 97
00
3.4 A deformation of the cup product 97
00
v
vi CONTENTS
3.5 The filtered A algebra associated to a Lagrangian submanifold 102
00
3.6 Bounding cochains and the A Maurer-Cartan equation. 107
00
3.6.1 Bounding cochains and deformations 108
3.6.2 Obstruction for the existence of bounding cochain 111
3.6.3 Weak unobstructedness and existence of Floer cohomology 114
3.6.4 The superpotential and M(C) 117
3. 7 A bimodules and Floer cohomology 120
00
3.7.1 Algebraic framework 120
3. 7.2 A bimodule homomorphisms 123
00
3.7.3 Weak unobstructedness and deformations 125
3.7.4 The filtered A bimodule C(£(1),£(D);Ao,nov) 126
00
3. 7.5 The Bott-Morse case 137
3. 7.6 Examples 151
3.7.7 The multiplicative structure on Floer cohomology 155
3.8 Inserting marked points in the interior 156
3.8.1 The operator p 156
3.8.2 Applications to vanishing of the obstruction classes ok(L) 159
3.8.3 Outline of the construction of the operator p 161
3.8.4 The operator q 165
3.8.5 Bulk deformation of filtered A structures 168
00
3.8.6 Outline of the construction of the operator q 175
3.8.7 The operator t and the A00 bimodule 178
3.8.8 Construction of the operator t 181
3.8.9 Generalization of the operator p 182
3.8.10 Proof of parts of Theorems B, C and G 188
Chapter 4. Homotopy equivalence of A algebras 191
00
4.1 Outline of Chapters 4 and 5 191
4.2 Homotopy equivalence of A algebras: the algebraic framework 197
00
4.2.1 Models of [O, 1) x C 197
4.2.2 Homotopies between A homomorphisms 205
00
4.2.3 The unital or homotopy-unital cases 208
4.3 Gauge equivalence of bounding cochains 211
4.3.l Basic properties and the category .fj2( 211
00
4.3.2 Mweak(C) and its homotopy invariance 215
4.3.3 Mweak,der(L) and its homotopy invariance 216
4.4 Uniqueness of the model of [O, 1] x C 217
4.4.1 Induction on the number filtration I 218
4.4.2 AK structures and homomorphisms 219
4.4.3 Induction on the number filtration II 220
4.4.4 Unital case I: the unfiltered version 223
4.4.5 Coderivation and Hochschild cohomology 226
4.4.6 Induction on the energy filtration 230
4.4. 7 Unital case II: the filtered version 232
4.5 Whitehead theorem in A algebras 233
00
4.5.1 Extending AK homomorphisms to AK+l homomorphisms 234
4.5.2 Proof of Theorem 4.2.45 I: the number filtration 236
4.5.3 Unital case: the unfiltered version 237
CONTENTS vii
4.5.4 Extending filtered A homomorphism modulo T>.; to modulo
00
T>.;+1 239
4.5.5 Proof of Theorem 4.2.45 II: the energy filtration 241
4.6 Homotopy equivalence of A algebras: the geometric realization 242
00
4.6.1 Construction of A homomorphisms 242
00
4.6.2 Homotopies between A homomorphisms 249
00
4.6.3 Compositions 257
4.6.4 Homotopy equivalence and the operator q I: changing the cycle
in the interior 259
4.6.5 Homotopy equivalence and the operator q II: invariance of sym-
plectic diffeomorphisms 1 261
4.6.6 Homotopy equivalence and the operator q III: invariance of
symplectic diffeomorphisms 2 264
Chapter 5. Homotopy equivalence of A bimodules 267
00
5.1 Novikov rings 267
5.1.1 Reduction to universal Novikov ring 267
5.1.2 Hamiltonian independence of the Novikov ring 270
5.1.3 Floer cohomologies over A(L(0),£C1);£o) and Anov 272
5.2 Homotopy equivalences of A bimodules: the algebraic framework 275
00
5.2.1 Weakly filtered A bimodule homomorphisms 275
00
5.2.2 Deformations of A bimodule homomorphisms 276
00
5.2.3 Homotopies between A bimodule homomorphisms 282
00
5.2.4 Gauge invariance and the category f)Ql (C1, Co) 288
00
5.2.5 Obstructions to defining A bimodule homomorphisms I 291
00
5.2.6 Whitehead theorem for A bimodule homomorphisms 292
00
5.2.7 Obstructions to defining A bimodule homomorphisms II 294
00
5.3 Homotopy equivalences of A bimodules: the geometric realiza-
00
tion 296
5.3.1 Construction of filtered A bimodule homomorphisms 296
00
5.3.2 Moving Lagrangian submanifolds by Hamiltonian isotopies 306
5.3.3 Homotopies between bimodule homomorphisms 313
5.3.4 Compositions of Hamiltonian isotopies and of bimodule homo-
morphisms 319
5.3.5 An energy estimate. 321
5.3.6 The operators q, t and homotopy equivalence 326
5.3. 7 Wrap-up of the proof of invariance of Floer cohomologies 327
5.4 Canonical models, formal super schemes and Kuranishi maps 330
5.4.1 Canonical models, Kuranishi maps and bounding cochains 330
5.4.2 The canonical models of filtered A bimodules 336
00
5.4.3 Filtered A bimodules and complex of coherent sheaves 337
00
5.4.4 Construction of the canonical model 339
5.4.5 Including the operator q 347
5.4.6 Wrap-up of the proofs of Theorems F, G, M, N and Corollaries
0, p 349
Chapter 6. Spectral sequences 355
6.1 Statement of the results in Chapter 6 355
viii CONTENTS
6.1.1 The spectral sequence 355
6.1.2 Non-vanishing theorem and a Maslov class conjecture 357
6.1.3 Applications to Lagrangian intersections 360
6.2 A toy model: rational Lagrangian submanifolds 362
6.3 The algebraic construction of the spectral sequence 366
6.3.1 c.f.z. 367
6.3.2 d.g.c.f.z. (differential graded c.f.z.) 369
6.3.3 Construction and convergence 371
6.4 The spectral sequence associated to a Lagrangian submanifold 375
6.4.1 Construction 375
6.4.2 A condition for degeneration: proof of (D.3) 375
6.4.3 Non-vanishing theorem: proof of Theorem 6.1.9. 377
6.4.4 Application to the Maslov class conjecture: proofs of Theorems
6.1.15 and 6.1.17 381
6.4.5 Compatibility with the product structure 382
6.5 Applications to Lagrangian intersections 385
6.5.1 Proof of Theorem H 385
6.5.2 Proof of Theorem I 385
6.5.3 Torsion of the Floer cohomology and Hofer distance: Proof of
Theorem J 388
6.5.4 Floer cohomologies of Lagrangian submanifolds that do not in-
tersect cleanly 393
6.5.5 Unobstructedness modulo TE 395
Volume II
Chapter 7. Tuansversality 397
7.1 Construction of the Kuranishi structure 398
7.1.1 Statement of the results in Section 7.1 398
7.1.2 Kuranishi charts on M;;'_;~n,reg(,B): Fredholm theory 401
7.1.3 Kuranishi charts in the complement of M;;'_;~n,reg(,B): gluing 404
7.1.4 Wrap-up of the proof of Propositions 7.1.1 and 7.1.2 418
7.1.5 The Kuranishi structure of M}!1-t'in(M',L',{Jp}p : ,B;top(p)):
A map analog of Stasheff cells 425
00
7.2 Multisections and choice of a countable set of chains 435
7.2.1 Tuansversality at the diagonal 436
7.2.2 Inductive construction of compatible system of multisections in
the Bott-Morse case 437
7.2.3 Perturbed moduli space running out of the Kuranishi neighbor-
hood I 444
7.2.4 Statement of results 445
7.2.5 Proof of Proposition 7.2.35 449
7.2.6 Filtered An,k structures 458
7.2.7 Construction of filtered An,K structures 461
7.2.8 Perturbed moduli space running out of the Kuranishi neigbor-
hood II 466
7.2.9 Construction of filtered An,K homomorphisms 468
