Kuranishi Structures and Virtual Fundamental Chains [electronic resource] / by Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono.

Fukaya, Kenji, author., Author,
Singapore : Springer Singapore : Imprint: Springer, 2020.
1 online resource (XV, 638 pages) : 149 illustrations, 34 illustrations in color.
1st ed. 2020.
Mathematics and Statistics (SpringerNature-11649)
Springer monographs in mathematics 1439-7382
Springer Monographs in Mathematics, 1439-7382
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Geometry, Differential.
Geometry, Hyperbolic.
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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. .
2.Notations and conventions
3.Kuranishi structure and good coordinate system
4.Fiber product of Kuranishi structures
5.Thickening of a Kuranishi structure
6.Multivalued perturbation
7.CF-perturbation and integration along the fiber (pushout)
8.Stokes' formula
9.From good coordinate system to Kuranishi structure and back with CF-perturbations
10.Composition formula of smooth correspondences
11.Construction of good coordinate system
12.Construction of CF-perturbations
13.Construction of multivalued perturbations
14.Zero and one dimensional cases via multivalued perturbation
15.Introduction to Part 2
16.Linear K-system: Floer cohomology I: statement
17.Extension of Kuranishi structure and its perturbation from boundary to its neighborhood
18.Smoothing corners and composition of morphisms
19.Linear K-system: Floer cohomology II: proof
20.Linear K-system: Floer cohomology III: Morse case by multisection
21.Tree-like K-system: A1 structure I: statement
22.Tree-like K-system: A1 structure II: proof
23. Orbifold and orbibundle by local coordinate
24.Covering space of effective orbifold and K-space
25.Admissible Kuranishi structure
26.Stratified submersion to a manifold with corners
27.Local system and smooth correspondence in de Rham theory with twisted coefficients
28.Composition of KG and GG embeddings: Proof of Lemma 3.34
29.Global quotient and orbifold. .
Oh, Yong-Geun. author., Author,
Ohta, Hiroshi. author., Author,
Ono, Kaoru. author., Author,
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10.1007/978-981-15-5562-6 doi
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