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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 b2+ 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
Mathematical Subject Classification
Primary: 53C07
Secondary: 52C15
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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