Hodge structures of K3 type of bidouble covers of rational surfaces

A bidouble cover between complex algebraic varieties is a flat G:= (ℤ/2ℤ)2-Galois cover X → Y. In this situation, there exist three intermediate quotients Y1,Y2 and Y3 which correspond to the three subgroups ℤ/2ℤ ≤ G. We consider the following situation: Y will be a rational surface and Yi will be either a surface with pg = 0 or a K3 surface. The motivation for these assumptions is to have a strong control on the weight 2 Hodge-structure of the covering surface X, which allows us to study the infinitesimal Torelli property, the Chow groups and Chow motive, and the Tate and Mumford–Tate conjectures for X. In particular, we classify all the bidouble covers X → Y satisfying the conditions above whenever Y is minimal; we obtain surfaces X with pg(X) = 1, 2, 3. We also introduce another construction, called iterated bidouble cover, which allows us to obtain surfaces with higher value of pg for which we still have a strong control on the weight 2 Hodge-structure.