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 Σ 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 detξ≡ L acts non-trivially on the restriction L|Σ over the fixed point set Σ, 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|_Σ, 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'. vskip0pt When p = 2, we apply the above constructions to a Kahler 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 Σ, a Lagrangian surface with genus greater than 0 and [Σ]∈2H ₂(H ;Z). If ^X_K 2 > 0 or K ²_X = 0 and the genus g(Σ)> 1, we have a vanishing theorem for Seiberg–Witten invariant of the quotient manifold X'. When K ²_X = 0 and the genus g(Σ)= 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.