60 20 07 v 3 9 F eb 1 99 6 VIRTUAL MODULI CYCLES AND GW - INVARIANTS

The study of moduli spaces plays a fundamental role in our understanding geometry and topology of algebraic manifolds, or more generally, symplectic manifolds. One example is the Donaldson theory (and more recently the Seiberg-Witten invariants), which gives rise to differential invariants of 4-manifolds [Do]. When the underlying manifold is an algebraic surface, then it is the intersection theory on moduli spaces of vector bundles over this surface [Li, Mo]. More recently, inspired by the sigma model theory in mathematical physics ([W1], [W2]), quantum cohomology has been constructed on so called semi-positive symplectic manifolds, which include all algebraic manifolds of complex dimension less than 4, all Fano manifolds and Calabi-Yau spaces ([RT1]). The quantum cohomology uses the GW-invariants, which are the intersection numbers of certain induced homology classes on moduli spaces of rational curves in a given symplectic manifold. This is a generalization of classical enumerative geometry that counts the number of algebraic curves with appropriate constraints an a variety. In [RT2], general GW-invariants of higher genus (also see [Ru] for special cases) are constructed to establish a mathematical theory of the sigma model theory coupled with gravity on any semi-positive symplectic manifolds.