Cyclic group actions on 4-manifolds

Abstract

Let X be a closed, oriented Riemannian 4-manifold. Suppose that a cyclic group Z p (p is prime) acts on X by an orientation preserving isometry with an embedded Riemann surface ∑ as fixed point set. We study the representation of Z p on the Spin c-bundles and the Z p-invariant moduli space of the solutions of the Seiberg-Witten equations for a Spin c-structure ξ → X. When the Z p action on the determinant bundle det ξ ≡ L acts non-trivially on the restriction L

∑ over the fixed point set ∑, we consider α-twisted solutions of the Seiberg-Witten equations over a Spin c-structure ξ′ on the quotient manifold X/Z p ≡ X′, α ∈ (0, 1). We relate the Z p-invariant moduli space for the Spin c-structure ξ on X and the α-twisted moduli space for the Spin c-structure ξ′ on X′. From this we induce a one-to-one correspondence between these moduli spaces and calculate the dimension of the α-twisted moduli space. When Z p acts trivially on L

∑, we prove that there is a one-to-one correspondence between the Z p-invariant moduli space M(ξ) Zp and the moduli space M(ξ″) where ξ″ is a Spin c-structure on X' associated to the quotient bundle L/Z p → X′. When p = 2, we apply the above constructions to a Kähler surface X with b 2 +(X) > 3 and H 2(X; Z) has no 2-torsion on which an anti-holomorphic involution acts with fixed point set ∑, a Lagrangian surface with genus greater than 0 and [∑] ∈ 2H 2(X; Z). If K X 2 > 0 or K X 2 = 0 and the genus g(∑) > 1, we have a vanishing theorem for Seiberg Witten invariant of the quotient manifold X'. When K X 2 = 0 and the genus g(∑) = 1, if there is a Z 2-equivariant Spin c-structure ξ on X whose virtual dimension of the Seiberg-Witten moduli space is zero then there is a Spin c-structure ξ″ on X′ such that the Seiberg-Witten invariant is ± 1.