Volume 3, issue 1 (1999)

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

Clifford Henry Taubes

Geometry & Topology 3 (1999) 167–210
 arXiv: math.SG/9907199
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.

Keywords
Four–manifold invariants, symplectic geometry
Primary: 53C07
Secondary: 52C15
Publication
Received: 26 July 1998
Accepted: 8 May 1999
Published: 4 July 1999
Proposed: Robion Kirby
Seconded: Gang Tian, Tomasz Mrowka
Authors
 Clifford Henry Taubes Department of Mathematics Harvard University Cambridge Massachusetts 02138 USA