  We extend the vanishing theorem for the Seiberg-Witten invariants of a
manifold with positive scalar curvature to the case when the curvature is
allowed to be negative on a set of small volume. (The precise curvature bounds
are described in the paper.) The idea is to combine the method of `semigroup
domination' with the Weitzenbock formula. The same curvature hypothesis implies
the vanishing of the Seiberg-Witten invariant of any finite covering space. We
also show that, in high dimensions, a spin manifold with mostly positive scalar
curvature in fact admits a metric of positive scalar curvature.
