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

Taubes, Clifford Henry¹

¹ Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138, USA

Geometry & topology, Tome 3 (1999) no. 1, pp. 167-210.

Résumé

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.

Détail

DOI : 10.2140/gt.1999.3.167

Keywords: Four–manifold invariants, symplectic geometry

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