Hodge structures of K3 type of bidouble covers of rational surfaces

A bidouble cover between complex algebraic varieties is a flat [Formula: see text]-Galois cover [Formula: see text]. In this situation, there exist three intermediate quotients [Formula: see text] and [Formula: see text] which correspond to the three subgroups [Formula: see text]. We consider the following situation: [Formula: see text] will be a rational surface and [Formula: see text] will be either a surface with [Formula: see text] 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 [Formula: see text], which allows us to study the infinitesimal Torelli property, the Chow groups and Chow motive, and the Tate and Mumford–Tate conjectures for [Formula: see text]. In particular, we classify all the bidouble covers [Formula: see text] satisfying the conditions above whenever [Formula: see text] is minimal; we obtain surfaces [Formula: see text] with [Formula: see text]. We also introduce another construction, called iterated bidouble cover, which allows us to obtain surfaces with higher value of [Formula: see text] for which we still have a strong control on the weight 2 Hodge-structure.