Seiberg-Witten invariants and real curves

Abstract

On a compact oriented four-manifold with an orientation preserving involution c, we count solutions of Seiberg-Witten equations, which are moreover symmetrical in relation to c, to construct "real" Seiberg-Witten invariants. Using Taubes' results, we prove that on a symplectic almost complex manifold with an antisymplectic and antiholomorphic involution, this invariants are not all trivial, and that the canonical bundle is represented by a real holomorphic curve.