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 σ : X → X with a 2-dimensional compact, oriented submanifold ∑ as a fixed point set. If σ is a symplectic involution then the quotient X/σ with b 2+(X/σ) ≥ 1 is a symplectic 4-manifold. If σ is an anti-symplectic involution and ∑ has genus greater than 1 representing non-trivial homology class, we prove a vanishing theorem on Seiberg-Witten invariants of the quotient X/σ with b2+(X/σ) > 1. If ∑ is a torus with self-intersection number 0, we get a relation between the Seiberg-Witten invariants on X and those of X/σ with b2+(X), b2+(X/σ) > 2 which was obtained in [21] when the genus g(∑) > 1 and ∑ · ∑ = 0.