Inoue type manifolds and Inoue surfaces: a connected component of the moduli space of surfaces with $K^2=7,p_g=0$

Abstract

We show that a family of minimal surfaces of general type with $p_g=0,K^2=7$, constructed by Inoue in 1994, is indeed a connected component of the moduli space: indeed that any surface which is homotopically equivalent to an Inoue surface belongs to the Inoue family.

The ideas used in order to show this result motivate us to give a new definition of varieties, which we propose to call Inoue type manifolds: these are obtained as quotients $\hat{X}/G$, where $\hat{X}$ is an ample divisor in a $K(\Gamma ,1)$ projective manifold $Z$, and $G$ is a finite group acting freely on $\hat{X}$ . For these types of manifolds we prove a similar theorem to the above, even if weaker, that manifolds homotopically equivalent to Inoue type manifolds are again Inoue type manifolds.