On the structure of metabelian Galois coverings of complex algebraic varieties

We describe the general structure of metabelian branched covers of smooth complex algebraic varieties. The main question that we consider is the following: given a smooth algebraic variety $Y$, which “building data” on $Y$ determine a metabelian cover $f:X\to Y$? We prove a structure theorem that answers this question completely. In order to achieve this, we extend the structure theorem for abelian covers of smooth varieties to the case where the total space and the base are normal varieties. For such abelian covers, we also provide a canonical bundle formula. For a metabelian cover $f:X\to Y$, we obtain a formula which relates the canonical bundle of $X$ to that of $Y$. Finally, we apply our results to the special case of metacyclic covers where we determine their basic invariants as well as their function fields.