J-holomorphic curves and symplectic topology
- McDuff, Dusa, 1945-
- 2nd ed.
- Providence, R.I. : American Mathematical Society, c2012.
- Physical description
- xiv, 726 p. ; 26 cm.
- Colloquium publications (American Mathematical Society) ; v. 52.
QA1 .A5225 V.52 2ND ED
- Unknown QA1 .A5225 V.52 2ND ED
- Salamon, D. (Dietmar)
- Includes bibliographical references and index.
- Publisher's Summary
- 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 rigourous detail. In particular, the book contains complete proofs of Gromov's compactness theorem for spheres, of the gluing theorem for spheres, and of the associatively 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.
(source: Nielsen Book Data)
- Symplectic and contact topology.
- Symplectic manifolds.
- Pseudoholomorphic curves.
- Differential geometry -- Symplectic geometry, contact geometry -- Symplectic manifolds, general.
- Differential geometry -- Symplectic geometry, contact geometry -- Gromov-Witten invariants, quantum cohomology, Frobenius manifolds.
- Differential geometry -- Symplectic geometry, contact geometry -- Global theory of symplectic and contact manifolds.
- Manifolds and cell complexes -- Differential topology -- Symplectic and contact topology.
- Dynamical systems and ergodic theory -- Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems -- General theory, relations with symplectic geometry and topology.
- Several complex variables and analytic spaces -- Complex manifolds -- Pseudoholomorphic curves.
- Differential geometry -- Symplectic geometry, contact geometry -- Lagrangian submanifolds; Maslov index.
- Differential geometry -- Symplectic geometry, contact geometry -- Floer homology and cohomology, symplectic aspects.
- Global analysis, analysis on manifolds -- Partial differential equations on manifolds; differential operators -- Elliptic equations on manifolds, general theory.
- Algebraic geometry -- Projective and enumerative geometry -- Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants.
- Publication date
- Dusa McDuff, Dietmar Salamon.
- American Mathematical Society colloquium publications ; v. 52