Anti-symplectic involutions with lagrangian fixed loci and their quotients

Abstract

We study the lagrangian embedding as a fixed point set of anti-symplectic involution τ on a symplectic 4-manifold X. Suppose the fixed loci of τ are the disjoint union of smooth Riemann surfaces Xτ = ∪̇σi; then each component becomes a lagrangian submanifold. Furthermore, if one of the components is a Riemann surface of genus g ≥ 2, then its quotient has vanishing Seiberg-Witten invariants. We will discuss some examples which allow an anti-symplectic involution with lagrangian fixed loci.