A Remark on the Deformation Equivalence Classes of Hopf Surfaces

Abstract

Let $S$ be a compact complex surface. With the exception of some complex surfaces, it is known that there are only finitely many deformation types of complex surfaces with the same homotopy type as $S$. Let $\mathcal{H}(S)$ be the set of deformation equivalence classes of complex surfaces homotopy equivalent to $S$. We evaluate $\# \mathcal{H}(S)$ when $S$ is a Hopf surface. As a corollary, we construct a sequence of Hopf surfaces $(S_n)$ such that although $\# \mathcal{H}(S_n)$ is finite for all $n$, the sequence $(\# \mathcal{H}(S_n))$ is unbounded.