#### 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.

Notes on file formats
Clifford Henry Taubes

Department of Mathematics

Harvard University

Cambridge, MA 02138, USA

Email: chtaubes@abel.math.harvard.edu

GT home page

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