Seiberg-Witten vanishing theorem for $S^1$-manifolds with fixed points

Abstract

In this paper we show that the Seiberg--Witten invariant is zero for all smooth 4--manifolds with $b_{+}>1$ which admit circle actions that have at least one fixed point. Furthermore, we show that all symplectic 4--manifolds which admit circle actions with fixed points are rational or ruled, and thus admit a symplectic circle action.

Includes new theorem classifying symplectic 4-manifolds with circle actions having fixed points. Minor updates to existing proofs to reflect the new theorem. 7 pages, 2 figures