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2020 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Kuranishi Structures and Virtual Fundamental Chains

verfasst von: Prof. Kenji Fukaya, Prof. Yong-Geun Oh, Prof. Hiroshi Ohta, Prof. Kaoru Ono

Verlag: Springer Singapore

Buchreihe : Springer Monographs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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SUCHEN

Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
The technique of virtual fundamental cycles (and chains) was introduced in the year 1996 by several groups of mathematicians [FOn2, LiTi2, LiuTi, Ru2, Sie] to provide a differential geometric foundation on the study of moduli spaces of pseudo-holomorphic curves entering on the one hand in the Gromov–Witten theory and on the other in the study of the Arnold conjecture in symplectic geometry without the assumption of any kind of positivity on general compact symplectic manifolds.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 2. Notations and Conventions
Abstract
We use ‘hat’ such as \(\widehat {\mathcal U}\), \(\widehat {f}\), \(\widehat {\mathfrak S}\), \(\widehat {h}\) for an object defined on a Kuranishi structure \(\widehat {\mathcal U}\).
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

Abstract Theory of Kuranishi Structures, Fiber Products and Perturbations

Frontmatter
Chapter 3. Kuranishi Structures and Good Coordinate Systems
Abstract
In this chapter we define the notions of a Kuranishi structure and of a good coordinate system. We also study embedding between them, which describes a relation among those structures.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 4. Fiber Product of Kuranishi Structures
Abstract
Before studying fiber products we consider direct products. Let X i, i = 1, 2 be separable metrizable spaces, Z i ⊆ X i compact subsets, and \(\widehat {\mathcal U}_i\) Kuranishi structures of Z i ⊆ X i.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 5. Thickening of a Kuranishi Structure
Abstract
Let X be a paracompact metrizable space, and let \(\widehat {\mathcal U} = (\{\mathcal U_p\},\{\Phi _{pq}\})\) be a Kuranishi structure on it.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 6. Multivalued Perturbations
Abstract
We next define the notion of multivalued perturbations associated to a given good coordinate system.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 7. CF-Perturbations and Integration Along the Fiber (Pushout)
Abstract
As we mentioned in the Introduction, we study systems of K-spaces (K-systems) so that the boundary of each of its members is described by a fiber product of other members. We will obtain an algebraic structure on certain cochain complexes which realize the homology groups of certain spaces. They are the spaces over which we take fiber products between members of the system of K-spaces.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 8. Stokes’ Formula
Abstract
In this chapter, we state and prove Stokes’ formula. We first discuss the notion of a boundary or a corner of an orbifold and of a Kuranishi structure in more detail. The discussion below is a detailed version of [FOOO4, the last paragraph of page 762]. See also [Jo1, page 11]. [Jo3] gives a systematic account on this issue.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 9. From Good Coordinate Systems to Kuranishi Structures and Back with CF-Perturbations
Abstract
As we explained at the end of Chap. 4, it is more canonical to define the notion of fiber products of spaces with Kuranishi structure than to define that of fiber products of spaces with a good coordinate system. On the other hand, in Chap. 7, we gave the definition of CF-perturbation and of the pushout of differential forms via good coordinate systems. In this chapter, we describe how we go from a good coordinate system to a Kuranishi structure and back together with CF-perturbations on them, and prove in Theorem 9.14 that we can define the pushout via the Kuranishi structure itself in such a way that the outcome is independent of the auxiliary choice of good coordinate system.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 10. Composition Formula of Smooth Correspondences
Abstract
The purpose of this chapter is to prove [FOOO8, Lemma 12.15] = Theorem 10.21, where a fiber product of Kuranishi structures is used as a way to define the composition of smooth correspondences. For this purpose we work out the plan described in Sect. 6.​5 in the de Rham model.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 11. Construction of Good Coordinate Systems
Abstract
In this chapter we prove Theorem 3.​35 together with various addenda and variants.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 12. Construction of CF-Perturbations
Abstract
In this chapter, we prove Theorem 7.​51, that is, the existence of CF-perturbations with respect to which a given weakly submersive map becomes strongly submersive. We also prove its relative version, Proposition 7.​59. For this purpose we use the language of sheaf theory to prove Proposition 12.2 for a single Kuranishi chart and Theorem 12.24 for a general case.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 13. Construction of Multivalued Perturbations
Abstract
In this chapter, we discuss multivalued perturbations, especially their existence result, Theorem 6.​23. This result will be used in Chaps. 14 and 20. One of the advantages of using multivalued perturbations is that it enables us to work with \({\mathbb Q}\) coefficients. In the construction based on de Rham theory we can work only over \({\mathbb R}\) or \({\mathbb C}\). For many applications, it is enough to work over \({\mathbb R}\) or \({\mathbb C}\), for which we do not need to use the results of Chaps. 13 and 14. The discussion of this chapter is largely parallel to that of CF-perturbation given in Chap. 12.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 14. Zero-and One-Dimensional Cases via Multivalued Perturbation
Abstract
In Chaps. 7, 8, 9 and 10, we discussed smooth correspondence and defined virtual fundamental chains based on de Rham theory and CF-perturbations. In this chapter, we discuss another method based on multivalued perturbations. Here we restrict ourselves to the case when the dimension of K-spaces of our interest is 1, 0 or negative, and define a virtual fundamental chain over \({\mathbb Q}\) in the 0-dimensional case. In spite of this restriction, the argument of this chapter is enough for the purpose, for example, to prove all the results stated in [FOn2]. We recall that in [FOn2] we originally used a triangulation of the zero set of a multisection to define a virtual fundamental chain. In this chapter we present a different way from [FOn2].
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

System of K-Spaces and Smooth Correspondences

Frontmatter
Chapter 15. Introduction to Part II
Abstract
In Part I, we described the foundation of the theory of Kuranishi structures, good coordinate systems, and CF-perturbations (also multivalued perturbations), and we defined the integration along the fiber (pushout) of a strongly submersive map with respect to a CF-perturbation and also proved Stokes’ formula.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 16. Linear K-Systems: Floer Cohomology I – Statement
Abstract
We axiomatize the properties enjoyed by the system of moduli spaces of solutions of Floer’s equation.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 17. Extension of a Kuranishi Structure and Its Perturbation from Boundary to Its Neighborhood
Abstract
In Chap. 16 we formulated various versions of corner compatibility conditions. To prove the results stated in Chap. 16 (and which also appear elsewhere in this book and will appear in the future) we need to extend the Kuranishi structure given on the boundary ∂X satisfying corner compatibility conditions to one on X.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 18. Corner Smoothing and Composition of Morphisms
Abstract
The goal of this chapter is to define composition of morphisms of linear K-systems (Lemma-Definition 18.37) and to show that it is associative (Proposition 18.41). There are two key ingredients for the construction of composition of morphisms. One is ‘partial outer collaring’ and the other is ‘corner smoothing’.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 19. Linear K-Systems: Floer Cohomology II – Proof
Abstract
Let us start with a linear K-system or a partial linear K-system of energy cut level E 0 as in Definition 16.​6. We set E 0 = +∞ for the case of a linear K-system and E 0 to be the energy cut level for the case of a partial linear K-system.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 20. Linear K-Systems: Floer Cohomology III – Morse Case by Multisection
Abstract
In this chapter we provide the way we associate Floer cohomology with ground ring \({\mathbb Q}\) to a linear K-system when the critical submanifolds are 0-dimensional.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 21. Tree-Like K-Systems: A ∞ Structure I – Statement
Abstract
In Chaps. 21 and 22 we discuss construction of a filtered A ∞ structure associated to a relatively spin Lagrangian submanifold L of a symplectic manifold. This construction had been worked out in great detail in the book [FOOO3], [FOOO4] based on singular homology, before its de Rham version was given in [FOOO8, Section 12], [FOOO10], [Fuk4].
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 22. Tree-Like K-Systems: A ∞ Structure II – Proof
Abstract
In this chapter we prove Theorem 21.​35. The proof is mostly parallel to the proofs given in Chap. 19, and also similar to those given in [FOOO4, Subection 7.2], [Fuk4].
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono

Appendices

Frontmatter
Chapter 23. Orbifolds and Orbibundles by Local Coordinates
Abstract
In this chapter we describe the story of orbifolds as much as we need in this book. We restrict ourselves to effective orbifolds and regard only embeddings as morphisms. The category \(\mathcal {O}\mathcal {B}_{\mathrm {ef,em}}\) where objects are effective orbifolds and morphisms are embeddings among them is naturally a 1 category. Moreover it has the following property.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 24. Covering Space of Effective Orbifolds and K-Spaces
Abstract
We first define the notion of a covering space of an orbifold. Let U 1, U 2 be orbifolds and let π : U 1 → U 2 be a continuous map between their underlying topological spaces.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 25. Admissible Kuranishi Structures
Abstract
In this chapter we introduce the notion of an admissible Kuranishi structure. For this purpose we introduce the notion of an admissible orbifold, an admissible vector bundle, and various admissible objects associated to them, like an admissible section, and provide their fundamental properties.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 26. Stratified Submersion to a Manifold with Corners
Abstract
In Definition 3.​44 we defined the notions of a strongly smooth map and weakly submersive map from a K-space to a manifold without boundary or corners. In this chapter we give the corresponding definitions for the case when the target manifold P has a boundary or corners.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 27. Local System and Smooth Correspondence in de Rham Theory with Twisted Coefficients
Abstract
Let \(\mathcal L\) be a local system, i.e., a flat vector bundle, on a manifold M. We denote by \((\Omega ^{\bullet }(M;{\mathcal L})=\Gamma (M;\bigwedge ^{\bullet }T^*M \otimes {\mathcal L}), d=d_{\mathcal L})\) the de Rham complex with coefficients in \(\mathcal L\). We recall some basic operations on the de Rham complex with twisted coefficients.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 28. Composition of KG-and GG-Embeddings: Proof of Lemma 3.34
Abstract
Proof of Lemma 3.34
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Chapter 29. Global Quotients and Orbifolds
Abstract
In some of our earlier writings we assumed that a Kuranishi neighborhood is not only an orbifold but also is a quotient of a manifold by a finite group action globally. We explain that there is no difference between two formulations when we apply the theory. The contents of this chapter are not used anywhere else in this book. Our purpose is to clarify the relationship of the formulation of this book with that in the previous literature.
Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, Kaoru Ono
Backmatter
Metadaten
Titel
Kuranishi Structures and Virtual Fundamental Chains
verfasst von
Prof. Kenji Fukaya
Prof. Yong-Geun Oh
Prof. Hiroshi Ohta
Prof. Kaoru Ono
Copyright-Jahr
2020
Verlag
Springer Singapore
Electronic ISBN
978-981-15-5562-6
Print ISBN
978-981-15-5561-9
DOI
https://doi.org/10.1007/978-981-15-5562-6