Hodge Structures of (p, k)-planes

A (p, k)-plane is an abelian cover of the projective plane with Galois group (Z/pZ) k . In this talk, we study the Hodge structure of such surfaces by analyzing their intermediate quotients. In particular, we classify all families for which the intermediate quotients of the last level - corresponding to the subgroups of G isomorphic to Z/pZ - are either rational surfaces or K3 surfaces, and at least one of them is a K3 surface. These assumptions allow us to obtain a strong control over the trascendental Hodge structure of the second cohomology of the surface and to investigate some properties, such as the Mumford-Tate conjecture. This is joint work with F. Fallucca and M. Penegini.