Virtual Fundamental Cycles in Symplectic Topology

Mathematical Surveys and Monographs Volume 237

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

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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 introduced 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 almost 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. Such a cobordism class is actually an invariant of the symplectic structure, not just of the almost complex structure or its homotopy class. Combined with physicist’s view of the topological sigma model, see [7], this dis- covery by Gromov produces celebrated invariants of symplectic manifolds, which are now called Gromov-Witten invariants. There is also the quantum cup product on cohomology deﬁned using these invariants. Some of the basic properties,

1Typically there is no holomorphic map from a complex manifold of dimension > 1toan almost complex manifold. ix x INTRODUCTION especially the associativity of the quantum cup product, were first observed by physicists working on String Theory. Another important application of Gromov’s theory of pseudo-holomorphic curves is Floer homology. A. Floer observed that the gradient line of the classical action functional of periodic Hamiltonian system can be identified with a pseudo- holomorphic curve (with an order zero perturbation term). This makes it possible to apply the theory of pseudo-holomorphic curves to the problems of studying periodic solutions of periodic Hamiltonian systems (which is one of the most important problems in symplectic geometry), see [1]. Using similar techniques he also studied the problem of giving a lower bound for the number of intersection points of two Lagrangian submanifolds. In symplectic geometry nowadays there are various ‘Floer-type’ homology theories defined using moduli spaces of pseudo-holomorphic curves and their variants.

Early applications of the theory. In early days of the development of these theories, the moduli spaces of pseudo-holomorphic curves were ﬁrst studied under various additional simplifying assumptions. A typical assumption is monotonicity of the symplectic manifold (X, ω), meaning the existence of a positive number c such that the equality

(1) [ω],α = cc1(TX),α holds for any α ∈ π2(X), or semi-positivity, meaning that

c1(TX),α≥0 for any α ∈ π2(X) with [ω],α > 0. In the case of Floer homology associated with Lagrangian submanifolds Li ⊂ X, i =1, 2 typical assumptions are monotonicity (a similar equality to (1) above) or exactness, meaning the existence of one form θ on X such that dθ = ω and [θ] = 0 in the de-Rham cohomology of the Li.) These assumptions are used to establish the transversality of the moduli space of pseudo-holomorphic curves. Here, transversality means that the linearization of the Cauchy-Riemann equation is a surjective linear operator. Under this condition the relevant moduli space of pseudo-holomorphic curves is a smooth manifold and its dimension is calculated by the topological data using the Atiyah-Singer index theorem or the Riemann-Roch theorem. In the case of monotone or semi-positive symplectic manifolds (or monotone or exact Lagrangian submanifolds), a standard perturbation argument shows that transversality holds for generic compatible almost complex structure. Beyond these cases, an obstruction to being able to achieve the transversality arises from ramiﬁed multiple covers of transversal solutions. The problem is that the formal dimension of these covers can be negative even though the formal dimension of the underlying simple curve is non-negative. The main motivation for introducing the virtual fundamental chain and cycle technique (the main theme of this book) is to resolve this problem.

An Example. Let us give an explicit example of this phenomenon. We consider the moduli space of pseudo-holomorphic maps u: S2 → X from a two-sphere with three marked points, taken to be 0, 1, ∞, to a symplectic manifold X of real dimension 2 2n, and denote by α the homology class u∗[S ]. We assume that c1(X),α > 0. INTRODUCTION xi

The dimension of this moduli space is

2c1(TX),α +2n.

We denote this moduli space by M0,3(α). (Here the ﬁrst subscript, 0, is the genus of the curve and the second subscript, 3, is the number of marked points on the curve.) The map u → (u(0),u(1),u(∞)) ∈ X × X × X is called the evaluation map.WetakecyclesQ1,Q2,Q3 ⊂ X such that 3 (2) (2n − dim Qi)=2c1(TX),α +2n. i=1 Then, the formal dimension of the moduli space

M0,3(α; Q1,Q2,Q3)={u ∈M0,3(α) | (u(0),u(1),u(∞)) ∈ Q1 × Q2 × Q3} is 0. If this moduli space and its compactiﬁcation are cut out transversally, then the sum over the (ﬁnite set of) points of the moduli space of signs coming from orientations is a typical example of the Gromov-Witten invariant. Indeed, because of the assumption c1(TX),α > 0, for a generic compatible almost complex structure the moduli space M0,3(α; Q1,Q2,Q3)iscutouttransversally,sothatitisa smooth manifold of dimension 0. However, it is not in general true that this moduli space is compact for such a generic choice of almost complex structure. Gromov’s compactness theorem holds and implies that if we consider a sequence ui of elements of M0,3(α; Q1,Q2,Q3) we may choose a subsequence so that its limit becomes a stable map of a holomorphic curve to X. An example of stable curve which appears as a limit is drawn in Figure 1. It

Q2 Q3 z∞ z1 Q1

z0 um ub

zn zn

Figure 1. Limiting Conﬁguration

2 2 2 is a union of curves um : S → X, ub : S → X. The domain S of um comes with four special points (z0,z1,z∞,zn)wherez0,z1,z∞ are limits of 0, 1, ∞ in the domain of ui, respectively. The domain of ub comes with one special point zn,and we require um(zn)=ub(zn). So we may regard the source of our map u∞ as a union 2 2 of two spheres jointed at zn = zn. We put αm =(um)∗([S ]) and αb =(ub)∗([S ]), giving αm + αb = α. Then (um; z0,z1,z∞,zn)and(ub; zn) represent elements of the moduli spaces M0,4(αm; Q1,Q2,Q3)andM0,1(αb), respectively. The ‘virtual’ dimensions of the moduli spaces are given by 3 dim M0,4(αm; Q1,Q2,Q3)=2c1(TX),αm +2n +2− (2n − dim Qi), (3) i=1 dim M0,1(αb)=2c1(TX),αb +2n − 4. xii INTRODUCTION

The numbers +2 and −4 in the above dimension formulae are explained as 2 follows. The source curve (S ; z0,z1,z∞,zn)ofum has a 2-dimensional moduli 2 space:Thesourcecurve(S ; zn)ofub has an automorphism group of dimension 4. The condition um(zn)=ub(zn) adds 2n constraints. On the other hand, by Equations (2) and (3)

dim M0,4(αm; Q1,Q2,Q3)+dimM0,1(αb) − 2n = −2. This implies that if all the moduli spaces involved are transversal then this conﬁg- uration does not appear. In order to prove the compactness of M0,3(α; Q1,Q2,Q3) we need these limiting moduli spaces to be cut out transversally (so that they are empty). HoweveritisnotingeneraltruethatM0,1(αb)istransversalforgenericalmost complex structure. We consider the case when αb = kαb with 2 c1(TX),αb < 0. Then, it can happen that there is a pseudo-holomorphic curve with − ≥ 2 c1(TX),αb +2n 4 0and − − 2 c1(TX),αb +2n 4=2c1(TX),kαb +2n 4 < 0. − One explicit numerical example is n =3, c1(TX),αb = 1, and k =2. M ∅ Because of the ﬁrst inequality, we can not show 0,1(αb)= by dimension counting, and indeed it may be non-empty for every almost complex structure. 2 → M But if (u , 0), u : S X represents an element of 0,1(αb)then(u , 0) with k u (z)=u (z ) represents an element of M0,1(αb). In particular M0,1(αb)isnon- empty. Thus, the space M0,1(αb) with negative ‘virtual’ dimension is non-empty for every almost complex structure and so is not cut out transversally for any such structure. This is a typical example of the problem of multiple covers of negative formal dimension. Avoiding this problem is the reason one often makes certain (some- times restrictive) assumptions in the applications of pseudo-holomorphic curves in symplectic geometry.

The Virtual Fundamental Chain and Cycle Approach. Because it is not possible to use perturbation of the almost complex structure to produce the fundamental class of the moduli space when there are multiple covers of formally negative dimension, one is led to introduce the virtual fundamental chain or cycle technique. This approach resolves this problem in practically all the situations that appear in the study of pseudo-holomorphic curves in symplectic geometry. How- ever, the method comes with the drawback that one is forced to use the rational numbers as a coefficient ring. Namely, Gromov-Witten invariants obtained in this way are necessarily rational numbers rather than integers and coefficients for the Floer homology is the field of rational numbers. (The reason why this approach does not work over integers is similar to the reason that Euler number of an orbifold is, in general, a rational number rather than an integer.) Letuselaboratethispointabitmorebelow.Anorbifoldisaspacethatlooks like U/Γ locally, where U is an open ball centered at the origin of Rn and Γ is a finite group acting orthogonally on Rn. We consider an orbifold X which is a 2 3 2 union of C/Z2 and C/Z3.Weidentify[z ] ∈ C/Z2 with [w ]=[1/z ] ∈ C/Z3. This orbifold X has two singular points which correspond to z =0∈ C/Z2 and INTRODUCTION xiii w =0∈ C/Z3, respectively. There exists a vector field on X which vanishes only at these two singular points. Because of the presence of non-trivial isotropy group we need to count these zeros with multiplicity 1/2and1/3, respectively. Therefore the orbifold Euler number of X is 1 1 5 + = . 2 3 6 There are various versions of virtual fundamental chain or cycle technique. Some versions work with infinite dimensional spaces directly, other versions are based on local, finite dimensional reductions. All three articles in this book work in the context of finite dimensional reductions. In these versions the ‘virtual fundamental chain or cycle technique’ consists of the following three steps. Step 1: Represent a moduli space locally as a zero set of certain section s of an orbibundle E over an orbifold U. Step 2: Regard such a local description of the moduli space as a ‘coordinate chart’ of certain geometric object and introduce an appropriate notion of ‘coordinate change’. Step 3: Either (a) ‘perturb’ the section s in a way compatible with the coordinate changes so that it becomes transversal to 0 and then glue the zero sets of perturbed sections in ‘various coordinate charts’ by the coordinate change (this is the method used in the first two articles) or (b) develop and use general techniques from ‘derived topology’ to produce a fundamental chain or cycle from the geometric object in Step 2. An approach along these ‘derived’ lines is described in the third article. The application of the material in that article to produce the fundamental cycle and chain will appear in an article in preparation by Joyce. It is worth pointing out that the arguments for Step 3 use only the formal properties of the geometric objects constructed in Step 2.

Brief descriptions of the articles in this volume. As we indicated, the goal of all the articles in this volume is to produce a fundamental cycle and chain associated to the moduli space of pseudo-holomorphic maps of curves to a symplectic manifold X equipped with a compatible almost complex structure. The starting point is a local description of the moduli space as the zeros of a smooth section of a smooth finite dimensional vector bundle over a smooth finite dimensional base modulo a finite group action – the Kuranishi chart. If the sections were transverse, the moduli space would inherit the structure of a finite dimensional smooth orbifold, which is oriented and thus has a (rational) fundamental class. But one must deal with non- transverse sections, especially in the case when the finite groups are non-trivial.

The article by Dusa McDuff describes her joint work with Katrin Wehrheim, [6]. Their goal is to produce the virtual fundamental cycle for the moduli space of pseudo-holomorphic maps. As they say in the introduction, “However, in practice, [the moduli space] usually has a more complicated structure since the operator is not transverse to zero. Intuitively the fundamental class is therefore the zero set of a suitable perturbation of the Fredholm operator: all the difficulty in constructing it lies in finding a suitable framework in which to build this perturbation.” The framework described in this article is for the transitions between Kurannishi charts, formulated in what they call a weak SS Kuranishi atlas. In this article they show xiv INTRODUCTION that any moduli space of genus zero pseudo-holomorphic curves in a compact symplectic manifold with a compatible almost complex structure has such an atlas and any two are compatible. From such an atlas, they produce a weighted branched topological manifold and a virtual fundamental class in the Cech homology of this branched manifold, unique up to cobordism. This then gives the virtual fundamental class for such moduli spaces. The case of moduli spaces of higher genus curves is covered by papers referenced in this article.

The second article is written by Mohammad Farajzadeh Tehrani partially using materials taken from a series of talks by Kenji Fukaya at the Simons Center for Geometry and Physics and also using material from papers written by Fukaya-Y.- G. Oh, H. Ohta and K. Ono [2]. The first half of the article is on the abstract theory of Kuranishi structures. After the definition of a Kuranishi structure is explained, the author provides a detailed account of the construction of the virtual fundamental class on a space with a Kuranishi structure. The first part of this process is a construction of a system with a finite number of charts (called a dimensionally graded system, DGS). (The initial Kuranishi structure can, and usually does, consist of an infinite number of charts.) DGS is a special case of a so called ‘good coordinate system’ appearing in [2]. The construction of a compatible system of multi-sections (multi-valued perturbations) on the those finitely many charts (transversal to zero) then are described. The triangulation of the zero set of such multi-section is given and as a result one is able to define a singular homology class, that is the virtual fundamental class. In the second half of this article, the construction of a Kuranishi structure on the moduli space of pseudo holomorphic curves is discussed. The author starts by explaining such moduli spaces and their topologies. Examples are given to show how the construction works. This article provides a comprehensive introduction to the construction of the Gromov-Witten invariant for general symplectic manifold.

The third article is a survey by Joyce of his paper [5] and his in-progress mul- tivolume book “Kuranishi spaces and Symplectic Geometry.” Preliminary versions of volumes I, II of this book are available at: http://people.maths.ox.ac.uk/ joyce/Kuranishi.html. This article proposes a new definition of Kuranishi space that has the nice property that the resulting objects form a 2-category, denoted Kur. Any Fukaya–Oh–Ohta–Ono Kuranishi space X, as in the second article, can be made into a Kuranishi space X in this sense, uniquely up to equivalence in Kur. The same holds for McDuff and Wehrheim’s ‘Kuranishi atlases’ described in the first article in this volume and also holds for the Hofer, Wysocki and Zehnder’s ‘Polyfold Fredholm structures’, [4]. The Kuranishi spaces as defined in this article, i.e., the objects of Kur,areob- jects in derived differential geometry, a concept introduced by the author in earlier work. Derived differential geometry is the study of classes of derived manifolds and derived orbifolds that the author calls ‘d-manifolds’ and ‘d-orbifolds’, respectively. There is an equivalence of 2-categories Kur ∼ dOrb,wheredOrb is the 2-category of d-orbifolds. So Kuranishi spaces are really a form of derived orbifolds. As such they have their own differential geometry, with notions of orientation, immersions, INTRODUCTION xv submersions, transverse fibre products, and the other standard properties of good geometric categories.

References [1] A. Floer, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), no. 4, 575–611. MR987770 [2] K. Fukaya, Y-G. Oh, H. Ohta, K. Ono, Technical details on Kuranishi structure and virtual fundamental chain, preprint arXiv:1209.4410 [3] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347, DOI 10.1007/BF01388806. MR809718 [4]H.H.W.Hofer,Polyfolds and Fredholm theory, Lectures on geometry, Clay Lect. Notes, Oxford Univ. Press, Oxford, 2017, pp. 87–158. MR3676594 [5] D. Joyce, A new definition of Kuranishi space, preprint arXiv:1409.6908. [6] D. McDuff and K. Wehrheim, The fundamental class of smooth Kuranishi atlases with trivial isotropy, J. Topol. Anal. 10 (2018), no. 1, 71–243, DOI 10.1142/S1793525318500048. MR3737511 [7] E. Witten, Topological sigma models, Comm. Math. Phys. 118 (1988), no. 3, 411–449. MR958805

John W. Morgan October 7, 2017

Selected Published Titles in This Series

237 Dusa McDuff, Mohammad Tehrani, Kenji Fukaya, and Dominic Joyce, Virtual Fundamental Cycles in Symplectic Topology, 2019 236 Bernard Host and Bryna Kra, Nilpotent Structures in Ergodic Theory, 2018 235 Habib Ammari, Brian Fitzpatrick, Hyeonbae Kang, Matias Ruiz, Sanghyeon Yu, and Hai Zhang, Mathematical and Computational Methods in Photonics and Phononics, 2018 234 Vladimir I. Bogachev, Weak Convergence of Measures, 2018 233 N. V. Krylov, Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations, 2018 232 Dmitry Khavinson and Erik Lundberg, Linear Holomorphic Partial Differential Equations and Classical Potential Theory, 2018 231 Eberhard Kaniuth and Anthony To-Ming Lau, Fourier and Fourier-Stieltjes Algebras on Locally Compact Groups, 2018 230 Stephen D. Smith, Applying the Classification of Finite Simple Groups, 2018 229 Alexander Molev, Sugawara Operators for Classical Lie Algebras, 2018 228 Zhenbo Qin, Hilbert Schemes of Points and Infinite Dimensional Lie Algebras, 2018 227 Roberto Frigerio, Bounded Cohomology of Discrete Groups, 2017 226 Marcelo Aguiar and Swapneel Mahajan, Topics in Hyperplane Arrangements, 2017 225 Mario Bonk and Daniel Meyer, Expanding Thurston Maps, 2017 224 Ruy Exel, Partial Dynamical Systems, Fell Bundles and Applications, 2017 223 Guillaume Aubrun and Stanislaw J. Szarek, Alice and Bob Meet Banach, 2017 222 Alexandru Buium, Foundations of Arithmetic Differential Geometry, 2017 221 Dennis Gaitsgory and Nick Rozenblyum, A Study in Derived Algebraic Geometry, 2017 220 A. Shen, V. A. Uspensky, and N. Vereshchagin, Kolmogorov Complexity and Algorithmic Randomness, 2017 219 Richard Evan Schwartz, The Projective Heat Map, 2017 218 Tushar Das, David Simmons, and Mariusz Urbański, Geometry and Dynamics in Gromov Hyperbolic Metric Spaces, 2017 217 Benoit Fresse, Homotopy of Operads and Grothendieck–Teichmüller Groups, 2017 216 Frederick W. Gehring, Gaven J. Martin, and Bruce P. Palka, An Introduction to the Theory of Higher-Dimensional Quasiconformal Mappings, 2017 215 Robert Bieri and Ralph Strebel, On Groups of PL-homeomorphisms of the Real Line, 2016 214 Jared Speck, Shock Formation in Small-Data Solutions to 3D Quasilinear Wave Equations, 2016 213 Harold G. Diamond and Wen-Bin Zhang (Cheung Man Ping), Beurling Generalized Numbers, 2016 212 Pandelis Dodos and Vassilis Kanellopoulos, Ramsey Theory for Product Spaces, 2016 211 Charlotte Hardouin, Jacques Sauloy, and Michael F. Singer, Galois Theories of Linear Difference Equations: An Introduction, 2016 210 Jason P. Bell, Dragos Ghioca, and Thomas J. Tucker, The Dynamical Mordell–Lang Conjecture, 2016 209 Steve Y. Oudot, Persistence Theory: From Quiver Representations to Data Analysis, 2015

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/survseries/. The method of using the moduli space of pseudo-holomorphic curves on a symplectic manifold was introduced by Mikhail Gromov in 1985. From the appearance of Gromov’s original paper until today this approach has been the most important tool in global symplectic geometry. To produce numerical invariants of these manifolds using this method requires constructing a fundamental cycle associated with moduli spaces. This volume brings together three approaches to constructing the “virtual” fundamental cycle for the moduli space of pseudo-holomorphic curves. All approaches are based on the idea of local Kuranishi charts for the moduli space. Workers in the field will get a comprehensive understanding of the details of these constructions and the assumptions under which they can be made. These techniques and results will be essential in further applications of this approach to producing invariants of symplectic manifolds.

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