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El. knyga: Virtual Fundamental Cycles in Symplectic Topology

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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.

This book is published in cooperation with Simons Center for Geometry and Physics.
Introduction ix
John W. Morgan
Notes on Kuranishi Atlases
1(110)
Dusa McDuff
1 Introduction
1(7)
1.1 Outline of the main ideas
4(4)
2 Kuranishi atlases with trivial isotropy
8(36)
2.1 Smooth Kuranishi charts, coordinate changes and atlases
8(5)
2.2 The Kuranishi category and virtual neighbourhood |K|
13(7)
2.3 Tame topological atlases
20(13)
2.4 Reductions and the construction of perturbation sections
33(11)
3 Kuranishi atlases with nontrivial isotropy
44(28)
3.1 Kuranishi atlases
45(9)
3.2 Categories and tamings
54(3)
3.3 Orientations
57(6)
3.4 Perturbation sections and construction of the VFC
63(9)
4 Constructing atlases
72(10)
4.1 Sketch proof of Theorem A
72(8)
4.2 Manipulating atlases
80(2)
5 Atlases for orbifolds and orbibundles
82(11)
5.1 Orbifolds
82(6)
5.2 Nontrivial obstruction bundles
88(5)
6 Order structures and products
93(18)
6.1 Semi-additive atlases
93(4)
6.2 From a good semi-additive atlas to the VFC
97(4)
6.3 From semi-additive to tameable atlases
101(6)
References
107(4)
Gromov-Witten Theory via Kuranishi Structures
111(142)
Mohammad F. Tehrani
Kenji Fukaya
Preface
111(2)
1 Introduction
113(9)
1.1 Moduli space of pseudoholomorphic maps
114(3)
1.2 GW invariants
117(1)
1.3 Semi-positive case
117(1)
1.4 Virtual Fundamental Class
118(3)
1.5 Outline
121(1)
2 Preliminaries
122(29)
2.1 Orbifolds
123(4)
2.2 Orbibundles
127(3)
2.3 Multisections
130(3)
2.4 Perturbations
133(3)
2.5 Resolution of multisections
136(9)
2.6 Euler class
145(6)
3 Abstract Kuranishi structures
151(42)
3.1 Introductory remarks
151(1)
3.2 Kuranishi structures
152(4)
3.3 Dimensionally graded systems
156(6)
3.4 Shrinking
162(4)
3.5 Cobordism
166(2)
3.6 Existence of DGS (proof of Theorem 3.5.3)
168(13)
3.7 Deformations of Kuranishi maps
181(7)
3.8 Construction of perfect EOB (proof of Theorem 3.7.11)
188(4)
3.9 Kuranishi vector bundles
192(1)
4 VFC for abstract Kuranishi spaces
193(9)
4.1 The construction of a VFC in a thickening
193(7)
4.2 VFC via evaluation maps
200(1)
4.3 Cech homology VFC
201(1)
5 Moduli spaces of stable maps
202(12)
5.1 Stable curves and stable maps
203(3)
5.2 Orbifold structure of the Deligne-Mumford space
206(5)
5.3 Gromov Topology
211(3)
6 Kuranishi structure over moduli space of stable maps
214(33)
6.1 Analytics preliminaries
215(4)
6.2 Case of smooth stable domain
219(6)
6.3 Case of stable nodal domain
225(8)
6.4 Case of un-stable domain
233(3)
6.5 Induced charts
236(7)
6.6 Coordinate change maps
243(3)
6.7 GW invariants
246(1)
7 Examples
247(3)
7.1 Degree zero maps
247(1)
7.2 Elliptic surfaces
248(1)
7.3 Genus zero maps in quintic
249(1)
References
250(3)
Kuranishi Spaces as a 2-category
253(36)
Dominic Joyce
1 Introduction
253(2)
2 Previous definitions of Kuranishi space
255(9)
2.1 Fukaya--Oh--Ohta--Ono's Kuranishi spaces
255(3)
2.2 How FOOO Kuranishi spaces are used
258(1)
2.3 McDuff--Wehrheim's Kuranishi atlases
259(2)
2.4 How MW Kuranishi atlases are used
261(1)
2.5 Dingyu Yang's Kuranishi structures, and Hofer--Wysocki--Zehnder's polyfolds
261(2)
2.6 How polyfolds are used
263(1)
3 Kuranishi neighbourhoods as a 2-category
264(12)
3.1 Kuranishi neighbourhoods, 1-morphisms, and 2-morphisms
264(3)
3.2 Making Kuranishi neighbourhoods into a 2-category
267(5)
3.3 Properties of 1- and 2-morphisms
272(1)
3.4 Relation to Fukaya--Oh--Ohta--Ono's work
273(1)
3.5 Relation to McDuff and Wehrheim's work
274(1)
3.6 Relation to d-orbifolds
275(1)
4 The weak 2-category of Kuranishi spaces
276(9)
4.1 Kuranishi spaces, 1-morphisms, and 2-morphisms
276(3)
4.2 Making Kuranishi spaces into a 2-category
279(2)
4.3 Manifolds, orbifolds, and m-Kuranishi spaces
281(3)
4.4 Relation to FOOO, MW, DY, polyfolds, and d-orbifolds
284(1)
5 Differential geometry of Kuranishi spaces
285(4)
5.1 Isotropy groups, and tangent and obstruction spaces
285(3)
5.2 W-transverse morphisms and fibre products
288(1)
5.3 Submersions and w-submersions
289(1)
Appendix A Background from Category Theory and Algebraic Geometry
289(6)
A.1 Basics of 2-categories
289(3)
A.2 2-functors between 2-categories
292(1)
A.3 Fibre products in 2-categories
293(1)
A.4 Sheaves and stacks on topological spaces
294(1)
References 295
John W. Morgan, Simons Center for Geometry and Physics, Stony Brook, NY.
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