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Books > Science & Mathematics > Mathematics > Geometry > Differential & Riemannian geometry
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Kuranishi Structures and Virtual Fundamental Chains (Hardcover, 1st ed. 2020)
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Kuranishi Structures and Virtual Fundamental Chains (Hardcover, 1st ed. 2020)
Series: Springer Monographs in Mathematics
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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.
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