Math. Ann. 279, 501-515 (1988) Am O Springer-Verlag 1988 On Analytic Abelian Coverings Alexandru Dimca Department of Mathematics, INCREST, Bd. P~cii 220, RO-79622 Bucharest, Romania O. Introduction 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. fis 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 COmponents, we can associate to a Galois covering f (as soon as Yis nonsingular in codimension 1) a ramification multidegree fl=(dl ..... d,,). Here di>l is the ramification degree of f at some point x sitting over a point y e D~ which is smooth on both Y and D. More precisely, this means that the germ f: (X, x)~(Y, y) is equivalent (see for instance [7, Sect. 2]) to the germ fo:(C",0)~(C",0), f0 (ul .... , u.) = (u~', u2 ..... u.). In particular x r X, in~. We denote by A C(Y, D, d) the set (of isomorphism classes) of abelian coverings f: (X, 0)~(y, 0) having D as a branching set and d as the corresponding ramification multidegree (where f is isomorphic to a covering f' : (X', 0)~(Y, 0) if there is an isomorphism h : (X, 0)~(X', 0) such that f' o h =f). The aim of this paper is to show that the set AC(Y, D, d) can be described quite precisely under certain assumptions on Y and D. The assumptions are given in Sect. 1, together with a lot of examples of germs (Y, 0) which satisfy them for any branching set (D, 0) = (Y, 0). These examples are based on results of Brieskorn, Grothendieek, Hamm and Kato.