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Kuranishi Structures and Virtual Fundamental ChainsSuurem pilt
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The package of Gromov’s pseudo-holomorphic curves is a major tool in global symplectic geometry and its applications, including mirror symmetry and Hamiltonian dynamics. The Kuranishi structure was introduced by two of the authors of the present volume in the mid-1990s to apply this machinery on general symplectic manifolds without assuming any specific restrictions. It was further amplified by this book’s authors in their monograph Lagrangian Intersection Floer Theory and in many other publications of theirs and others. Answering popular demand, the authors now present the current book, in which they provide a detailed, self-contained explanation of the theory of Kuranishi structures.

Part I discusses the theory on a single space equipped with Kuranishi structure, called a K-space, and its relevant basic package. First, the definition of a K-space and maps to the standard manifold are provided. Definitions are given for fiber products, differential forms, partitions of unity, and the notion of CF-perturbations on the K-space. Then, using CF-perturbations, the authors define the integration on K-space and the push-forward of differential forms, and generalize Stokes' formula and Fubini's theorem in this framework. Also, “virtual fundamental class” is defined, and its cobordism invariance is proved.

Part II discusses the (compatible) system of K-spaces and the process of going from “geometry” to “homological algebra”. Thorough explanations of the extension of given perturbations on the boundary to the interior are presented. Also explained is the process of taking the “homotopy limit” needed to handle a system of infinitely many moduli spaces. Having in mind the future application of these chain level constructions beyond those already known, an axiomatic approach is taken by listing the properties of the system of the relevant moduli spaces and then a self-contained account of the construction of the associated algebraic structures is given. This axiomatic approach makes the exposition contained here independent of previously published construction of relevant structures. 

Arvustused

Although the authors have dealt with this subject in some previous monographs and in numerous articles, it seemed necessary to have a book that could collect all the necessary and well-organized material. This is the reason why the two above-mentioned authors, together with Yong-Geun Oh and Hiroshi Ono, have decided to write this book that we review now. this book (an authentic tour de force) it is an indispensable reference for every researcher in symplectic geometry and topology. (Manuel de León, zbMATH 1482.53002, 2022)

1 Introduction
1(40)
1.1 Background: Why Virtual Fundamental Chains?
3(4)
1.2 The Story of Kuranishi Structures and Virtual Fundamental Chains
7(6)
1.3 Main Results of Part I
13(13)
1.3.1 Elementary Material
13(2)
1.3.2 Multisections
15(4)
1.3.3 CF-Perturbation and Integration
19(3)
1.3.4 Stokes' Formula
22(1)
1.3.5 Smooth Correspondence and Composition Formula
23(2)
1.3.6 Proof of Existence Theorems
25(1)
1.3.7 Virtual Fundamental Chain Over Q
25(1)
1.4 Related Works
26(15)
1.4.1 Papers Appearing in the Year 1996
27(1)
1.4.2 Issues in Developing the Theory of Virtual Fundamental Chains and Cycles
27(5)
1.4.3 The Works of the Authors of This Book
32(1)
1.4.4 The Development from the Work by Li-Tian [ LiTi2] and Liu-Tian [ LiuTi]
33(1)
1.4.5 The Work by Joyce
34(2)
1.4.6 Polyfolds
36(1)
1.4.7 The Work by Pardon
37(1)
1.4.8 Other Works
38(3)
2 Notations and Conventions
41(8)
Part I Abstract Theory of Kuranishi Structures, Fiber Products and Perturbations
3 Kuranishi Structures and Good Coordinate Systems
49(18)
3.1 Kuranishi Structures
49(4)
3.2 Good Coordinate Systems
53(1)
3.3 Embedding of Kuranishi Structures I
54(8)
3.4 Notes on Various Versions of the Definitions
62(5)
3.4.1 Tangent Bundle Condition
63(1)
3.4.2 Global Quotient
63(1)
3.4.3 Germ of Kuranishi Chart
63(1)
3.4.4 Definition 3.15 (7), (8)
64(1)
3.4.5 `Hausdorffness' Issue
64(3)
4 Fiber Product of Kuranishi Structures
67(12)
4.1 Fiber Product
67(5)
4.2 Boundaries and Corners I
72(2)
4.3 A Basic Property of Fiber Products
74(5)
5 Thickening of a Kuranishi Structure
79(16)
5.1 Background to Introducing the Notion of Thickening
79(2)
5.2 Definition of Thickening
81(1)
5.3 Embedding of Kuranishi Structures II
82(9)
5.4 Support System and Existence of Thickening
91(4)
6 Multivalued Perturbations
95(18)
6.1 Multisections and Multivalued Perturbations
95(4)
6.1.1 Multivalued Perturbations on an Orbifold
95(3)
6.1.2 Multivalued Perturbations on a Good Coordinate System
98(1)
6.2 Properties of the Zero Set of Multivalued Perturbations
99(6)
6.3 Transversality of the Multisection
105(2)
6.4 Embedding of Kuranishi Structures and Multivalued Perturbations
107(3)
6.5 General Strategy of Construction of Virtual Fundamental Chains
110(3)
7 CF-Perturbations and Integration Along the Fiber (Pushout)
113(34)
7.1 Introduction to Chaps. 7, 8, 9, 10, and 12
113(3)
7.2 CF-Perturbation on a Single Kuranishi Chart
116(7)
7.2.1 CF-Perturbation on a Kuranishi Chart Restricted to One Orbifold Chart
116(4)
7.2.2 CF-Perturbation on a Single Kuranishi Chart
120(3)
7.3 Integration Along the Fiber (Pushout) on a Single Kuranishi Chart
123(3)
7.4 CF-Perturbations of a Good Coordinate System
126(9)
7.4.1 Embedding of Kuranishi Charts and CF-Perturbations
126(4)
7.4.2 CF-Perturbations on Good Coordinate Systems
130(1)
7.4.3 Extension of a Good Coordinate System and Relative Version of the Existence Theorem of CF-Perturbations
131(4)
7.5 Partition of Unity Associated to a Good Coordinate System
135(4)
7.6 Differential Forms on a Good Coordinate System and a Kuranishi Structure
139(1)
7.7 Integration Along the Fiber (pushout) on a Good Coordinate System
140(7)
8 Stokes' Formula
147(12)
8.1 Boundaries and Corners II
147(6)
8.2 Stokes' Formula for a Good Coordinate System
153(2)
8.3 Well-Definedness of Virtual Fundamental Cycle
155(4)
9 From Good Coordinate Systems to Kuranishi Structures and Back with CF-Perturbations
159(18)
9.1 CF-Perturbations and Embedding of Kuranishi Structures
159(4)
9.2 Integration Along the Fiber (pushout) for Kuranishi Structures
163(2)
9.3 Composition of GK-and KG-Embeddings: Proof of Definition-Lemma 5.17
165(7)
9.4 GG-Embedding and Integration: Proof of Proposition 9.16
172(1)
9.5 CF-Perturbations of Correspondences
173(1)
9.6 Stokes' Formula for a Kuranishi Structure
174(1)
9.7 Uniformity of CF-Perturbations on a Kuranishi Structure
175(2)
10 Composition Formula of Smooth Correspondences
177(16)
10.1 Direct Product and CF-Perturbation
177(2)
10.2 Fiber Product and CF-Perturbation
179(4)
10.3 Composition of Smooth Correspondences
183(2)
10.4 Composition Formula
185(8)
11 Construction of Good Coordinate Systems
193(28)
11.1 Construction of Good Coordinate Systems: The Absolute Case
193(12)
11.2 Construction of Good Coordinate Systems: When Thickening Is Given
205(6)
11.3 KG-Embeddings and Compatible Perturbations
211(5)
11.4 Extension of Good Coordinate Systems: The Relative Case
216(5)
12 Construction of CF-Perturbations
221(18)
12.1 Construction of CF-Perturbations on a Single Chart
221(8)
12.2 Sheaf CFK of CF-Perturbations on Hetero-Dimensional Compactum
229(10)
13 Construction of Multivalued Perturbations
239(14)
13.1 Sheaf MVK: of Multivalued Perturbations
239(3)
13.2 Construction of Multivalued Perturbations
242(6)
13.3 Extending Multivalued Perturbations from One Chart to Another: Remarks
248(5)
14 Zero-and One-Dimensional Cases via Multivalued Perturbation
253(24)
14.1 Virtual Fundamental Chain for a Good Coordinate System with Multivalued Perturbation
253(5)
14.2 Virtual Fundamental Chain of O-Dimensional K-Space with Multivalued Perturbation
258(3)
14.3 A Simple Morse Theory on a Space with a Good Coordinate System
261(5)
14.4 Denseness of the Set of Morse Functions on an Orbifold
266(6)
14.5 Similarity and Difference Between CF-Perturbation and Multivalued Perturbation: Remarks
272(5)
Part II System of K-Spaces and Smooth Correspondences
15 Introduction to Part II
277(36)
15.1 Outline of the Story of Linear K-Systems
279(14)
15.1.1 Floer Cohomology of Periodic Hamiltonian Systems
279(2)
15.1.2 Periodic Hamiltonian Systems and Axiom of Linear K-Systems
281(2)
15.1.3 Construction of Floer Cochain Complex
283(2)
15.1.4 Comer Compatibility Conditions
285(3)
15.1.5 Well-Definedness of Floer Cohomology and Morphism of Linear K-Systems
288(2)
15.1.6 Identity Morphism
290(2)
15.1.7 Homotopy Limit
292(1)
15.1.8 Story in the Case with Rational Coefficients
293(1)
15.2 Outline of the Story of Tree-Like K-Systems
293(12)
15.2.1 Moduli Space of Pseudo-holomorphic Disks: Review
293(1)
15.2.2 Axiom of Tree-Like K-System and the Construction of the Filtered A∞ Algebra
294(2)
15.2.3 Bifurcation Method and Pseudo-isotopy
296(2)
15.2.4 Bifurcation Method and Self-Gluing
298(7)
15.3 Discussion Deferred to the Appendices
305(8)
15.3.1 Orbifolds and Covering Space of Orbifolds/K-Spaces
305(1)
15.3.2 Admissibility of Orbifolds and of Kuranishi Structures
305(5)
15.3.3 Stratified Submersion
310(1)
15.3.4 Integration Along the Fiber and Local System
311(2)
16 Linear K-Systems: Floer Cohomology I -- Statement
313(32)
16.1 Axiom of Linear K-Systems
313(6)
16.2 Floer Cohomology Associated to a Linear K-System
319(5)
16.3 Morphism of Linear K-Systems
324(4)
16.4 Homotopy and Higher Homotopy of Morphisms of Linear K-Systems
328(6)
16.5 Composition of Morphisms of Linear K-Systems
334(3)
16.6 Inductive System of Linear K-Systems
337(8)
17 Extension of a Kuranishi Structure and Its Perturbation from Boundary to Its Neighborhood
345(54)
17.1 Introduction to Chap. 17
345(3)
17.2 Outer Collaring on One Chart
348(8)
17.3 Outer Collaring and Embedding
356(4)
17.4 Outer Collaring of Kuranishi Structures
360(4)
17.5 Collared Kuranishi Structure
364(8)
17.6 Products of Collared Kuranishi Structures
372(1)
17.7 Extension of Collared Kuranishi Structures
373(15)
17.7.1 Statement
374(2)
17.7.2 Extension Theorem for a Single Collared Kuranishi Chart
376(2)
17.7.3 Construction of Kuranishi Chart U+
378(9)
17.7.4 Completion of the Proof of Lemma 17.60
387(1)
17.7.5 Proof of Proposition 17.58
388(1)
17.8 Extension of Collared CF-Perturbations
388(2)
17.9 Extension of Kuranishi Structures and CF-Perturbations from a Neighborhood of a Compact Set
390(6)
17.10 Conclusion of Chap. 17
396(3)
18 Corner Smoothing and Composition of Morphisms
399(60)
18.1 Why Corner Smoothing?
399(1)
18.2 Introduction to Chap. 18
400(2)
18.3 Partial Outer Collaring of Cornered K-Spaces
402(7)
18.4 In Which Sense Is Corner Smoothing Canonical?
409(2)
18.5 Corner Smoothing of (0, ∞)k
411(3)
18.6 Corner Smoothing of Collared Orbifolds and of Kuranishi Structures
414(4)
18.7 Composition of Morphisms of Linear K-Systems
418(7)
18.8 Associativity of the Composition
425(3)
18.9 Parametrized Version of Morphism: Composition and Gluing
428(7)
18.9.1 Compositions of Parametrized Morphisms
428(2)
18.9.2 Gluing Parametrized Morphisms
430(5)
18.10 Identity Morphism
435(16)
18.11 Geometric Origin of the Definition of the Identity Morphism
451(8)
18.11.1 Interpolation Space of the Identity Morphism
451(2)
18.11.2 Identification of the Interpolation Space of the Identity Morphism with Direct Product
453(1)
18.11.3 Interpolation Space of the Homotopy
454(3)
18.11.4 Identification of the Interpolation Space of the Homotopy with Direct Product
457(2)
19 Linear K-Systems: Floer Cohomology II - Proof
459(54)
19.1 Construction of Cochain Complexes
459(6)
19.2 Construction of Cochain Maps
465(3)
19.3 Proof of Theorem 16.9 (1) and Theorem 16.39 (1)
468(5)
19.4 Composition of Morphisms and of Induced Cochain Maps
473(5)
19.5 Construction of Homotopy
478(5)
19.6 Proof of Theorem 16.9 (2)(except (f)), Theorem 16.31(1) and Theorem 16.39 (2)(except (e))? (3)
483(8)
19.7 Construction of Higher Homotopy
491(5)
19.8 Proof of Theorem 16.39 (2)(e), (4)--(6) and Theorem 16.9 (2)(f)
496(17)
20 Linear K-Systems: Floer Cohomology III -- Morse Case by Multisection
513(6)
20.1 Extension of a Multisection from Boundary to Its Neighborhood
514(1)
20.2 Completion of the Proof of Theorem 20.2
514(5)
21 Tree-Like K-Systems: A∞ Structure I -- Statement
519(20)
21.1 Axiom of Tree-Like K-Systems: A∞ Correspondence
519(11)
21.2 Filtered A∞ Algebra and its Pseudo-Isotopy
530(4)
21.3 Statement of the Results
534(5)
22 Tree-Like K-Systems: A∞ Structure II -- Proof
539(24)
22.1 Existence of CF-Perturbations
539(7)
22.2 Algebraic Lemmas: Promotion Lemmas via Pseudo-Isotopy
546(5)
22.3 Pointwiseness of Parametrized Family of Smooth Correspondences
551(1)
22.4 Proof of Theorem 21.35
552(11)
Part III Appendices
23 Orbifolds and Orbibundles by Local Coordinates
563(14)
23.1 Orbifolds and Embeddings Between Them
564(6)
23.2 Vector Bundles on Orbifolds
570(7)
24 Covering Space of Effective Orbifolds and K-Spaces
577(12)
24.1 Covering Space of an Orbifold
577(2)
24.2 Covering Space of a K-Space
579(3)
24.3 Covering Spaces Associated to the Comer Structure Stratification
582(4)
24.4 Finite Group Action on a K-Space
586(3)
25 Admissible Kuranishi Structures
589(16)
25.1 Admissible Orbifolds
589(7)
25.2 Admissible Vector Bundles
596(5)
25.3 Admissibility of the Moduli Spaces of Pseudo-holomorphic Curves
601(4)
26 Stratified Submersion to a Manifold with Corners
605(10)
27 Local System and Smooth Correspondence in de Rham Theory with Twisted Coefficients
615(4)
28 Composition of KG-and GG-Embeddings: Proof of Lemma 3.34
619(2)
29 Global Quotients and Orbifolds
621(4)
References 625(6)
Index 631
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