Smooth double covers of K3 surfaces

In this paper we classify the topological invariants of the possible branch loci of a smooth double cover $f:X\rightarrow Y$ of a K3 surface $Y$. We describe some geometric properties of $X$ which depend on the properties of the branch locus. We give explicit examples of surfaces $X$ with Kodaira dimension 1 and 2 obtained as double cover of K3 surfaces and we describe some of them as bidouble cover of rational surfaces. Then, we classify the K3 surfaces which admit smooth double covers $X$ satisfying certain conditions; under these conditions the surface $X$ is of general type, $h^{1,0}(X)=0$ and $h^{2,0}(X)=2$. We discuss the variation of the Hodge structure of $H^2(X,\mathbb{Z})$ for some of these surfaces $X$.