A \(J\)-holomorphic curve is a \((j,J)\)-holomorphic map \(u : \Sigma \to M\) from a Riemann surface \((\Sigma,j)\) to an almost complex manifold \((M,J)\). Following the ideas and methods of [M. Gromov, Invent. Math. 82, 307-347 (1985;Zbl 0592.53025)] the theory of \(J\)-holomorphic curves is one of the new techniques in searching of global results in symplectic geometry. The book is devoted mainly to establish the fundamental theorems in the subject and give a useful introduction to the methods and applications of the theory of \(J\)-holomorphic curves. The authors establish the foundational Fredholm theory and compactness results necessary in the basic constructions of the theory. The Gromov- Witten invariants are discussed, and in particular their existence and applications to quantum cohomology (complete proof of associativity of the quantum cup-product) and to the theory of symplectic manifolds which satisfy some positivity condition. The extensions of the theory of \(J\)- holomorphic curves to the Calabi-Yau manifolds and relation of this theory to the Floer homology is also discussed. In Appendix A, B there are presented some special techniques (e.g. gluing techniques for \(J\)- holomorphic spheres) and detailed proves concerning elliptic regularity.

Reviewer: St.Janeczko (Warszawa)