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A smooth, compact 4-manifold with a Riemannian metric and b^(2+) > 0 has anon-trivial, closed, self-dual 2-form. If the metric is generic, then the zeroset of this form is a disjoint union of circles. On the complement of this zeroset, the symplectic form and the metric define an almost complex structure; andthe latter can be used to define pseudo-holomorphic submanifolds andsubvarieties. The main theorem in this paper asserts that if the 4-manifold hasa non zero Seiberg-Witten invariant, then the zero set of any given self-dualharmonic 2-form is the boundary of a pseudo-holomorphic subvariety in itscomplement.

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