Product-Quotient Surfaces: Result and Problems

The following is an extended version of a talk given at the Kinosaki Symposium on Algebraic Geometry in October 2011. The aim is to give an overview of product-quotient surfaces, the results that have been proven so far in collaboration with several different authors as well as pointing out some of the the problems that are still open.Comment: Appeared in the proceedings of the Kinosaki Algebraic Geometry Symopsium held in October 201

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

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.Comment: 36 pages, article dedicated to the 60-th birthday of Gerard van der Gee