The structure of a class of finite ramified coverings and canonical forms of analytic matrix-functions in a neighborhood of a ramified turning point

摘要

Let X be the germ of a complex- or real-analytic manifold M at a point x_o∈ M, or the henselian germ of an algebraic manifold M over a field k of characteristic zero at a point x_o∈ M(k), D∈ X a divisor. Under some assumptions on D and its singularities we give a description of the structure, the singularities, and the divisor class group of all finite normal coverings of X ramified over D. Let g : X-- gl(n) be an analytic or a k-algebraic family, respectively, of semisimple matrices, the eigenvalues of which are ramified on D as functions of x∈ X. Put U = X -- D.