Cyclic group actions on Spin$^c$ bundles

Hongxia Li, Ximin Liu

Let $L$ be a complex line bundle over a closed, oriented Riemannian 4-manifold $X$ with $c_1(L)=\omega_2(TX)\mod2$. Let $\mathbf Z_p$ be a cyclic group of order $p$ ($p$ is prime) that acts on $X$ as orientation preserving isometry with an oriented, connected, compact 2-dimensional submanifold $\Sigma$ as a fixed point set and on $L$ such that the projection $L\to X$ is a $\mathbf Z_p$-map. In this paper, we investigate the action of $\mathbf Z_p$ on the Seiberg-Witten equations, and obtain a relation of the dimension of the moduli space of the quotient bundle and its pull-back bundle. Also, we discuss the Seiberg-Witten invariant of the quotient bundle when $X$ is a Kähler manifold.