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

A smooth, compact 4–manifold with a Riemannian metric and $b^{2 +} \geq 1$ 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

References

Forward citations

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

Volume 3, issue 1 (1999)

Clifford Henry Taubes