Gromov-Witten invariants and quantum cohomology

This article is based on the lectures given by the author at the University of Adelaide in 1998. In Section 1, we define Gromov-Witten invariants of symplectic manifolds as intersection pairings on the moduli space of pseudoholomorphic curves. The invariants are computed in various examples. We also study the quantum product structure on the cohomology groups and its associativity. In Section 2, we introduce relative Gromov-Witten invariants when there is a symplectic submanifold of (real) codimension two. Symplectic cutting can be regarded as the result of degeneration along such a submanifold. The Gromov-Witten invariants of the total space can be expressed in terms of the relative invariants of the symplectic cuts. In Section 3, we discuss the question of naturality of quantum cohomology, i.e., finding morphisms between symplectic manifolds that induce homomorphisms of quantum cohomology. After a review of Mori's program of minimal models in three (complex) dimensions, we state various results and conjectures regarding naturality and in particular in relation to mirror symmetry.