Seiberg–Witten invariants and pseudo-holomorphic subvarieties for self-dual, harmonic 2–forms

Abstract

A smooth, compact 4–manifold with a Riemannian metric and $b^{2+}\ge 1$ has a non-trivial, closed, self-dual 2–form. If the metric is generic, then the zero set of this form is a disjoint union of circles. On the complement of this zero set, the symplectic form and the metric define an almost complex structure; and the latter can be used to define pseudo-holomorphic submanifolds and subvarieties. The main theorem in this paper asserts that if the 4–manifold has a non zero Seiberg–Witten invariant, then the zero set of any given self-dual harmonic 2–form is the boundary of a pseudo-holomorphic subvariety in its complement.