Seiberg-Witten vanishing theorem for S1-manifolds with fixed points

We show that the Seiberg-Witten invariant is zero for all smooth 4-manifolds with b(+)> 1 that admit circle actions having at least one fixed point. We also show that all symplectic 4-manifolds that admit (possibly nonsymplectic) circle actions with fixed points are rational or ruled, and thus admit a symplectic circle action.