Topology of Hopf surfaces

Abstract

By a Hopf surface $S$ , we shall mean a 2-dimensional compact complex manifold of which the universal covering is $W=C^{2}-\{0\}$ , where $C^{2}$ is the space of two complex variables $(z_{1}, z_{2})$ and $0$ is the origin $(0,0)$ . $S$ can be represented as a quotient space $W/G$ with a group $G$ generated by some biholomorphic transformations of $W$ whose action is properly discontinuous and free.