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

TL;DR

This paper proves that a specific family of minimal surfaces with certain invariants forms a connected component of the moduli space and introduces Inoue-type manifolds, extending the classification to a broader class.

Contribution

It establishes the connectedness of the Inoue surfaces' moduli component and introduces Inoue-type manifolds with a similar homotopy classification.

Findings

Inoue surfaces form a connected component of the moduli space.

Homotopically equivalent surfaces belong to the Inoue family.

Generalization to Inoue-type manifolds with similar properties.

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 type 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.