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\(J\)-holomorphic curves and symplectic topology. (English) Zbl 1064.53051
Colloquium Publications. American Mathematical Society 52. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3485-1/hbk). xii, 669 p. (2004).
The theory of \(J\)-holomorphic curves was founded by M. Gromov around 1985. Its applications include many key results in symplectic topology and it was one of the main inspirations for the creation of Floer homology. It provides a natural context in which one can define Gromov-Witten invariants and quantum cohomology, which form the so-called A-side of the mirror symmetry conjectures. Insights from physics have themselves inspired many fascinating developments, for example the till now little understood connections between the theory of integrable systems and Gromov-Witten invariants. The book under review is a wonderful work about this topic.
The material is divided into twelve chapters (namely: 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 homology), five appendices (namely: Fredholm theory, elliptic regularity, the Riemann-Roch theory, stable curves of genus zero, singularities and intersections) and a bibliography with 314 titles.
Written by two top specialists on this topic, it will be useful to all those interested in this fascinating domain. Last but not least the graphical conditions offered by the American Mathematical Society are excellent.

53D05 Symplectic manifolds (general theory)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
58J05 Elliptic equations on manifolds, general theory
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry