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Virtual Fundamental Cycles in Symplectic Topology Mathematical Surveys and Monographs Volume 237 Virtual Fundamental Cycles in Symplectic Topology John W. Morgan, editor Dusa McDuff Mohammad Tehrani Kenji Fukaya Dominic Joyce SIMONSCENTER FOR GEOMETRY AND PHYSICS 10.1090/surv/237 Virtual Fundamental Cycles in Symplectic To p o l o g y Mathematical Surveys and Monographs Volume 237 Virtual Fundamental Cycles in Symplectic To p o l o g y John W. Morgan, editor Dusa McDuff Mohammad Tehrani Kenji Fukaya Dominic Joyce SIMONSCENTER FOR GEOMETRY AND PHYSICS EDITORIAL COMMITTEE Robert Guralnick, Chair Benjamin Sudakov Natasa Sesum Constantin Teleman 2010 Mathematics Subject Classification. Primary 53D45, 53D37, 58J10, 57R17, 57R18. For additional information and updates on this book, visit www.ams.org/bookpages/surv-237 Library of Congress Cataloging-in-Publication Data Names: Morgan, John, 1946– editor. | McDuff, Dusa, 1945– | Simons Center for Geometry and Physics (Stony Brook University) Title: Virtual fundamental cycles in symplectic topology / John Morgan, editor ; Dusa McDuff [and three others]. Description: Providence, Rhode Island : American Mathematical Society, [2019] | Series: Mathe- matical surveys and monographs ; volume 237 | “Simons Center for Geometry and Physics, Stony Brook, New York.” | Includes bibliographical references and index. Identifiers: LCCN 2018057233 | ISBN 9781470450144 (alk. paper) Subjects: LCSH: Symplectic geometry. | Geometry, Differential. | AMS: Differential geometry – Symplectic geometry, contact geometry – Gromov-Witten invariants, quantum cohomology, Frobenius manifolds. msc | Differential geometry – Symplectic geometry, contact geometry – Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category. msc | Global analysis, analysis on manifolds – Partial differential equations on manifolds; differential operators – Differential complexes. msc | Manifolds and cell complexes – Differential topology – Symplectic and contact topology. msc | Manifolds and cell complexes – Differential topology – Topology and geometry of orbifolds. msc Classification: LCC QA665 .V57 2019 | DDC 516.3/6–dc23 LC record available at https://lccn.loc.gov/2018057233 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 select pages 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. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2019 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains 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 https://www.ams.org/ 10987654321 242322212019 Contents Introduction by John W. Morgan ix Notes on Kuranishi Atlases by Dusa McDuff 1 1. Introduction 1 1.1. Outline of the main ideas 4 2. Kuranishi atlases with trivial isotropy 8 2.1. Smooth Kuranishi charts, coordinate changes and atlases 8 2.2. The Kuranishi category and virtual neighbourhood |K| 13 2.3. Tame topological atlases 20 2.4. Reductions and the construction of perturbation sections 33 3. Kuranishi atlases with nontrivial isotropy 44 3.1. Kuranishi atlases 45 3.2. Categories and tamings 54 3.3. Orientations 57 3.4. Perturbation sections and construction of the VFC 63 4. Constructing atlases 72 4.1. Sketch proof of Theorem A 72 4.2. Manipulating atlases 80 5. Atlases for orbifolds and orbibundles 82 5.1. Orbifolds 82 5.2. Nontrivial obstruction bundles 88 6. Order structures and products 93 6.1. Semi-additive atlases 93 6.2. From a good semi-additive atlas to the VFC 97 6.3. From semi-additive to tameable atlases 101 References 107 Gromov-Witten Theory via Kuranishi Structures by Mohammad F. Tehrani and Kenji Fukaya 111 Preface 111 1. Introduction 113 1.1. Moduli space of pseudoholomorphic maps 114 1.2. GW invariants 117 1.3. Semi-positive case 117 1.4. Virtual Fundamental Class 118 1.5. Outline 121 v vi CONTENTS 2. Preliminaries 122 2.1. Orbifolds 123 2.2. Orbibundles 127 2.3. Multisections 130 2.4. Perturbations 133 2.5. Resolution of multisections 136 2.6. Euler class 145 3. Abstract Kuranishi structures 151 3.1. Introductory remarks 151 3.2. Kuranishi structures 152 3.3. Dimensionally graded systems 156 3.4. Shrinking 162 3.5. Cobordism 166 3.6. Existence of DGS (proof of Theorem 3.5.3) 168 3.7. Deformations of Kuranishi maps 181 3.8. Construction of perfect EOB (proof of Theorem 3.7.11) 188 3.9. Kuranishi vector bundles 192 4. VFC for abstract Kuranishi spaces 193 4.1. The construction of a VFC in a thickening 193 4.2. VFC via evaluation maps 200 4.3. Cechˇ homology VFC 201 5. Moduli spaces of stable maps 202 5.1. Stable curves and stable maps 203 5.2. Orbifold structure of the Deligne-Mumford space 206 5.3. Gromov Topology 211 6. Kuranishi structure over moduli space of stable maps 214 6.1. Analytics preliminaries 215 6.2. Case of smooth stable domain 219 6.3. Case of stable nodal domain 225 6.4. Case of un-stable domain 233 6.5. Induced charts 236 6.6. Coordinate change maps 243 6.7. GW invariants 246 7. Examples 247 7.1. Degree zero maps 247 7.2. Elliptic surfaces 248 7.3. Genus zero maps in quintic 249 References 250 Kuranishi Spaces as a 2-category by Dominic Joyce 253 1. Introduction 253 2. Previous definitions of Kuranishi space 255 2.1. Fukaya–Oh–Ohta–Ono’s Kuranishi spaces 255 2.2. How FOOO Kuranishi spaces are used 258 2.3. McDuff–Wehrheim’s Kuranishi atlases 259 2.4. How MW Kuranishi atlases are used 261 CONTENTS vii 2.5. Dingyu Yang’s Kuranishi structures, and Hofer–Wysocki–Zehnder’s polyfolds 261 2.6. How polyfolds are used 263 3. Kuranishi neighbourhoods as a 2-category 264 3.1. Kuranishi neighbourhoods, 1-morphisms, and 2-morphisms 264 3.2. Making Kuranishi neighbourhoods into a 2-category 267 3.3. Properties of 1- and 2-morphisms 272 3.4. Relation to Fukaya–Oh–Ohta–Ono’s work 273 3.5. Relation to McDuff and Wehrheim’s work 274 3.6. Relation to d-orbifolds 275 4. The weak 2-category of Kuranishi spaces 276 4.1. Kuranishi spaces, 1-morphisms, and 2-morphisms 276 4.2. Making Kuranishi spaces into a 2-category 279 4.3. Manifolds, orbifolds, and m-Kuranishi spaces 281 4.4. Relation to FOOO, MW, DY, polyfolds, and d-orbifolds 284 5. Differential geometry of Kuranishi spaces 285 5.1. Isotropy groups, and tangent and obstruction spaces 285 5.2. W-transverse morphisms and fibre products 288 5.3. Submersions and w-submersions 289 Appendix A. Background from Category Theory and Algebraic Geometry 289 A.1. Basics of 2-categories 289 A.2. 2-functors between 2-categories 292 A.3. Fibre products in 2-categories 293 A.4. Sheaves and stacks on topological spaces 294 References 295 Introduction In 2013–2014, the Simons Center for Geometry and Physics hosted a one- year program to discuss various problems related to the foundations of that part of symplectic geometry which concerns the theory of moduli spaces of pseudo- holomorphic curves. An important part of this program were lecture courses on the ‘Virtual fundamental chain and cycle’ determined by these moduli spaces, a technique of central importance in symplectic geometry. This volume consists of written and expanded versions of these lecture courses. Our goal in bringing together experts in these techniques to give the lecture courses and in producing this follow-up volume is to facilitate a wider understanding of the various approaches to the basics of the virtual fundamental chain and cycle theory in symplectic geometry and the assumptions that each approach makes. We believe that future major advances in symplectic geometry will require use of these techniques without restrictive topological or geometric assumptions on the symplectic manifold. It is our hope that this volume will contribute to the wider application of these methods. The Foundational Work of Gromov and Floer. The method of using pseudo-holomorphic curves in the study of global symplectic geometry was intro- duced in a ground-breaking paper by M. Gromov, [3]. From the appearance of this paper until today, this approach has been the most important tool in global symplectic geometry. The method is based on the following facts: (i) Symplectic manifolds have compatible almost complex structures (though the manifold often does not have any complex structure). (ii) The equation for a map from an al- most complex structure on a smooth 2-manifold to an almost complex structure on a higher dimensional manifold is an elliptic equation so the usual elliptic theory applies. (iii) Since almost complex structures on smooth 2-manifolds are auto- matically integrable, pseudo-holomorphic maps from smooth 2-manifolds to almost complex manifolds are in fact holomorphic maps from complex curves1. As a result, the rich and highly developed holomorphic curve theory applies in the context of pseudo-holomorphic maps from a complex one-dimensional manifold to a symplectic manifold. Even though there are many almost complex structures compatible with a given symplectic structure, the space of all such is contractible. Together with Gromov’s compactness theorem for these pseudo-holomorphic maps, which relies essentially on the fact that we are mapping to a symplectic manifold not just an almost complex manifold, this implies that the cobordism class of the moduli space of pseudo- holomorphic maps from a complex one-dimensional manifolds to a given symplectic manifold is independent of the choice of the compatible almost complex structure.
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