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

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

1999-07-04

arxiv.org/abs/math/9907199v1

Geom. Topol. 3 (1999), 167-210

Mathematics

Symplectic Geometry

44 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol3/paper8.abs.html

Scientific paper

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.

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