On Threefolds Isogenous to a Product of Curves

TL;DR

This paper classifies threefolds isogenous to a product of curves with a fixed Euler characteristic, providing an algorithm for classification and methods to compute their Hodge numbers, under certain group action assumptions.

Contribution

It introduces a finite classification approach for these threefolds with fixed invariants and presents an algorithm to classify them when the group acts diagonally and faithfully.

Findings

Complete classification of threefolds with $oldsymbol{oxed{ ext{chi}(oldsymbol{oxed{ ext{O}_X)}}=-1}$ under specific group actions.

Development of an algorithm to classify threefolds isogenous to a product of curves for fixed $oldsymbol{oxed{ ext{chi}(oldsymbol{oxed{ ext{O}_X)}})}}$.

Method to determine Hodge numbers of these threefolds.

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.