On analytic abelian coverings

Abstract

An analytic map germ f : (X, 0)~(Y, 0) is called a covering if it has a representative f which is a (global) covering as in [2, p. 41] i.e. f i s a finite surjective proper map: Let f : (X, 0 ) ~ (Y, 0) be an analytic covering of n-dimensional complex analytic germs. The total space (X, 0) is always assumed to be normal, while the conditions on the base space (Y, 0) are discussed in detail in Sect. 1. We freely identify a map or space germ with a convenient representative of it. The group of covering transformations G(f) of the covering f consists of all isomorphisms h : (X, 0)--.(X, 0) such that fo h =f . We call f a Galois covering if the group G(f) acts transitively on the fibers of f . If moreover the group G ( f ) is abelian, we call f an abelian covering. A branching set for the covering f is a pure 1-codimensional germ (D, 0) c (Y, 0) such that if we put Yo = Y ~ D and Xo = f 1 (Yo), then the induced map fo : Xo ~ Yo is an unramified covering. If D=i=~, Di is the decomposition in irreducible