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

Clifford Henry Taubes

doi:10.2140/gt.1999.3.167

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arXiv · math/9907199

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

Clifford Henry Taubes

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Abstract

A smooth, compact 4-manifold with a Riemannian metric and b^(2+) > 0 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.

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BibTeX RIS

Clifford Henry Taubes. 1999-07-04. Seiberg-Witten invariants and pseudo-holomorphic subvarieties for self-dual, harmonic 2-forms. https://doi.org/10.2140/gt.1999.3.167

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