J-holomorphic curves and symplectic topology
Author(s)
Bibliographic Information
J-holomorphic curves and symplectic topology
（Colloquium publications / American Mathematical Society, v. 52）
American Mathematical Society, c2004
Available at 45 libraries
Note
Includes bibliographical references (p. 643-653) and index
Description and Table of Contents
Description
The theory of $J$-holomorphic curves has been of great importance since its introduction by Gromov in 1985. Its mathematical applications include many key results in symplectic topology. It was also one of the main inspirations for the creation of Floer homology. In mathematical physics, it provides a natural context in which to define Gromov-Witten invariants and quantum cohomology-two important ingredients of the mirror symmetry conjecture. This book establishes the fundamental theorems of the subject in full and rigorous detail. In particular, the book contains complete proofs of Gromov's compactness theorem for spheres, of the gluing theorem for spheres, and of the associativity of quantum multiplication in the semipositive case. The book can also serve as an introduction to current work in symplectic topology: There are two long chapters on applications, one concentrating on classical results in symplectic topology and the other concerned with quantum cohomology. The last chapter sketches some recent developments in Floer theory.
The five appendices of the book provide necessary background related to the classical theory of linear elliptic operators, Fredholm theory, Sobolev spaces, as well as a discussion of the moduli space of genus zero stable curves and a proof of the positivity of intersections of $J$-holomorphic curves in four dimensional manifolds. The book is suitable for graduate students and researchers interested in symplectic geometry and its applications, especially in the theory of Gromov-Witten invariants.
Table of Contents
Introduction $J$-holomorphic curves Moduli spaces and transversality Compactness Stable maps Moduli spaces of stable maps Gromov-Witten invariants Hamiltonian perturbations Applications in symplectic topology Gluing Quantum cohomology Floer cohomology Fredholm theory Elliptic regularity The Riemann-Roch theorem Stable curves of genus zero Singularities and intersections (written with Laurent Lazzarini) Bibliography List of symbols Index.
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