Smooth finite abelian uniformizations of projective spaces and Calabi-Yau orbifolds (Q2466358)

Let \(f\colon Y \to {\mathbb P}^n\) be an abelian cover, namely \(Y\) is a normal complex variety such that there exists a finite abelian group \(G\) that acts faithfully on \(Y\), the quotient \(Y/G\) is isomorphic to \({\mathbb P}^n\) and \(f\) is the quotient map. Such a map \(f\) is determined by the components of the branch locus together with some combinatorial data related to the \(G\)-action above each component (an algebraic treatment of the general theory of abelian covers can be found in [Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 14, No. 5, 695--705 (2007; Zbl 0721.14009)]). Here the author determines necessary and sufficient conditions on a collection of irreducible hypersurfaces of \({\mathbb P}^n\) and the corresponding combinatorial data for the existence of a smooth \(G\)-cover of \(f\colon Y \to {\mathbb P}^n\) associated with it. In addition, he studies in detail the case in which \(Y\) is a Calabi--Yau manifold, listing all the possibilities for small values of \(n\).