Geometry & Topology, Vol. 3 (1999) Paper no. 8, pages 167--210.

Seiberg--Witten Invariants and Pseudo-Holomorphic Subvarieties for Self-Dual, Harmonic 2--Forms

Clifford Henry Taubes

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.

Keywords. Four-manifold invariants, symplectic geometry

AMS subject classification. Primary: 53C07. Secondary: 52C15.

DOI: 10.2140/gt.1999.3.167

E-print: arXiv:math.SG/9907199

Submitted to GT on 26 July 1998. Paper accepted 8 May 1999. Paper published 4 July 1999.

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Clifford Henry Taubes
Department of Mathematics
Harvard University
Cambridge, MA 02138, USA

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