Pseudoholomorphic Curves and Seiberg-Witten Invariants

Abstract

Let X be a smooth closed four manifold. The Seiberg-Witten equations yield an important invariant of the diffeomorphism type of X. In many cases these invariants are closely related to the geometry of X. For example, when X has a metric of positive scalar curvature, they vanish. When X is Kahler, they can be interpreted holomorphically as a signed count of divisors. Here we discuss similar results for the case where X is a closed symplectic 4-manifold. In particular we prove an existence result for embedded symplectic surfaces in particular homology classes when associated line bundles have non-zero Seiberg-Witten invariants. Together with a non-vanishing result, these give strong constraints on the topology of X. We treat both the cases b2+ > 1 and b2+ = 1. Using these results we prove classification theorems for topologically simple symplectic 4-manifolds. In particular we give a detailed proof of the symplectic rigidity of CP2.