Kuranishi Structures and Virtual Fundamental Chains

The package of Gromov's pseudo-holomorphic curves is a powerful tool in global symplectic geometry, and the Kuranishi structure was introduced in the mid-1990s to apply this machinery on general symplectic manifolds. This book provides a detailed explanation of the theory of Kuranishi structures, including the definition of K-spaces, maps to standard manifolds, fiber products, differential forms, partitions of unity, and the notion of CF-perturbations. It also discusses the system of K-spaces and the process of going from "geometry" to "homological algebra", and an axiomatic approach is taken to construct the associated algebraic structures.

\n Format: Hardback
\n Length: 638 pages
\n Publication date: 17 October 2020
\n Publisher: Springer Verlag, Singapore
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The package of Gromov's pseudo-holomorphic curves is a fundamental tool in global symplectic geometry and its applications, including mirror symmetry and Hamiltonian dynamics. In the mid-1990s, two of the authors of the present volume introduced the Kuranishi structure to apply this machinery on general symplectic manifolds without assuming any specific restrictions. This structure was further amplified by the authors in their monograph Lagrangian Intersection Floer Theory and in numerous other publications. 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 a 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's 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.

In conclusion, the theory of Kuranishi structures is a powerful tool in global symplectic geometry and its applications. The authors of the present book have provided a detailed and self-contained explanation of this theory, making it accessible to a wide range of readers. The book will be of great interest to researchers and students in the field.

\n Weight: 1126g\n
Dimension: 166 x 243 x 43 (mm)\n
ISBN-13: 9789811555619\n
Edition number: 1st ed. 2020\n