The theory of $J$-holomorphic curves has been of great importance since its introduction by Gromov in 1985. In mathematics, its 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. The main goal of this book is to establish 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 second edition clarifies various arguments, corrects several mistakes in the first edition, includes some additional results in Chapter 10 and Appendices C and D, and updates the references to recent developments.

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lgrsnf/J-holomorphic Curves and Symplectic Topology.pdf

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J-Holomorphic Curves and Symplectic Topology (American Mathematical Society)

J-holomorphic Curves and Symplectic Topology : Second Edition

McDuff, Dusa; Salamon, Dietmar

Colloquium publications (American Mathematical Society), 2nd ed, Providence, R.I, 2012

Colloquium Publications, Second edition, Providence, 2012

United States, United States of America

0

lg2443798

{"last_page":726,"publisher":"American Mathematical Society","series":"Colloquium Publications 52"}

Includes bibliographical references and index.

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 Homology -- Appendix A : Fredholm Theory -- Appendix B : Elliptic Regularity -- Appendix C : The Riemann-roch Theorem -- Appendix D : Stable Curves Of Genus Zero -- Appendix E : Singularities And Intersections (written With Laurent Lazzarini). Dusa Mcduff, Dietmar Salamon. Includes Bibliographical References (p. 695-709) And Index.

2019-11-30