Cyclic group actions on 4-manifolds

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 Sigma as fixed point set. We study the representation of Zp 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 xi --> X. When the Z(p) action on the determinant bundle det xi equivalent to L acts non-trivially on the restriction L\(Sigma) over the fixed point set Sigma, we consider alpha-twisted solutions of the Seiberg-Witten equations over a Spin(c)-structure xi' on the quotient manifold X/Z(p) - X', alpha is an element of (0, 1). We relate the Z(p)-invariant moduli space for the Spin(c)-structure xi on X and the alpha-twisted moduli space for the Spin(c)-structure xi' on X'. From this we induce a one-to-one correspondence between these moduli spaces and calculate the dimension of the alpha-twisted moduli space. When Z(p) acts trivially on L\(Sigma), we prove that there is a one-to-one correspondence between the Z(p)-invariant moduli space M(xi)(Zp) and the moduli space M(xi") where xi" 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 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 Sigma, a Lagrangian surface with genus greater than 0 and [Sigma] is an element of 2H(2)(X; Z). If K-x(2) > 0 or K-x(2) = 0 and the genus g(Sigma) > 1, we have a vanishing theorem for Seiberg-Witten invariant of the quotient manifold X'. When K-x(2) = 0 and the genus g(Sigma) = 1, if there is a Z(2)-equivariant Spin(c)-structure on xi whose virtual dimension of the Seiberg-Witten moduli space is zero then there is a Spin(c)-structure xi" on X' such that the Seiberg-Witten invariant is +/-1.