On the cohomology of surfaces with PG = q = 2 and maximal Albanese dimension

In this paper we study the cohomology of smooth projective complex surfaces S of general type with invariants pg = q = 2 and surjective Albanese morphism. We show that on a Hodge-theoretic level, the cohomology is described by the cohomology of the Albanese variety and a K3 surface X that we call the K3 partner of S. Furthermore, we show that in suitable cases we can geometrically construct the K3 partner X and an algebraic correspondence in S×X that relates the cohomology of S and X. Finally, we prove the Tate and Mumford–Tate conjectures for those surfaces S that lie in connected components of the Gieseker moduli space that contain a product-quotient or a mixed surface.