On Threefolds Isogenous to a Product of Curves

Abstract

A threefold isogenous to a product of curves $X$ is a quotient of a product of three compact Riemann surfaces of genus at least two by the free action of a finite group. In this paper we study these threefolds under the assumption that the group acts diagonally on the product. We show that the classification of these threefolds is a finite problem, present an algorithm to classify them for a fixed value of $\chi ({\mathcal{O}}_X)$ and explain a method to determine their Hodge numbers. Running an implementation of the algorithm we achieve the full classification of threefolds isogenous to a product of curves with $\chi ({\mathcal{O}}_X)=-1$, under the assumption that the group acts faithfully on each factor.

arXiv comment18 pages, 1 table; v2: improved exposition, results unchanged; v3: final version, improved exposition, results unchanged