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

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