On the cohomology of regular surfaces isogenous to a product of curves with $\chi(\mathcal{O}_S)=2$

TL;DR

This paper investigates the cohomology of regular surfaces isogenous to a product of curves with specific invariants, providing a structure theorem and identifying families with maximal Picard number.

Contribution

It presents a structure theorem for the cohomology of such surfaces with (}_S)=2, and discovers two families of general type surfaces with maximal Picard number.

Findings

Established a cohomology structure theorem for these surfaces.

Identified two families of surfaces with maximal Picard number.

Provided tools for studying the cohomology of surfaces isogenous to a product.

Abstract

Let $S$ be a surface isogenous to a product of curves of unmixed type. After presenting several results useful to study the cohomology of $S$ we prove a structure theorem for the cohomology of regular surfaces isogenous to a product of unmixed type with $\chi (\mathcal{O}_S)=2$. In particular we found two families of surfaces of general type with maximal Picard number.