J-holomorphic curves and symplectic invariants

Abstract

Given an almost complex structure J on a manifold M, a map f from a Riemann surface to M is called a pseudoholomorphic (or J-holomorphic) curve if at each point p of the surface, the ordinary differential is a complex linear map with respect to the complex structures j(p) and J(f(p)) on tangent spaces. In these lectures, we present the basic results on the theory of pseudoholomorphic curves in almost complex manifolds when the almost complex structure is tame by a symplectic form. Both local and global results are described. We then apply this theory to the construction of symplectic invariants of spaces and of diffeomorphisms of symplectic manifolds. We conclude with a brief description of the quantum cohomology of a symplectic manifold, which can be viewed as an application of the theory of pseudoholomorphic 2-spheres.