SEIBERG-WITTEN INVARIANTS AND (ANTI-)SYMPLECTIC INVOLUTIONS

Abstract

Let $X$ be a closed, symplectic 4-manifold. Suppose that there is either a symplectic or an anti-symplectic involution $\sigma : X\,{\to}\, X$ with a 2-dimensional compact, oriented submanifold $\Sigma$ as a fixed point set. If $\sigma$ is a symplectic involution then the quotient $X/\sigma$ with $b_2^+(X/\sigma)\,{\ge}\, 1$ is a symplectic 4-manifold. If $\sigma$ is an anti-symplectic involution and $\Sigma$ has genus greater than 1 representing non-trivial homology class, we prove a vanishing theorem on Seiberg-Witten invariants of the quotient $X/\sigma$ with $b_2^+(X/\sigma)\,{ >}\,1.$ If $\Sigma$ is a torus with self-intersection number 0, we get a relation between the Seiberg-Witten invariants on $X$ and those of $X/\sigma$ with $b_2^+(X), b_2^+(X/\sigma)\,{ >}\,2$ which was obtained in [ 21 ] when the genus $g(\Sigma)\,{ >}\,1$ and $\Sigma\cdot\Sigma\,{=}\,0$ . This work was supported by a Korea Research Foundation Grant (No KRF-2002-072-C00010).