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

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. Using the above results under some quite general assumptions on g and D we construct an irreducible nonsingular variety U(c), a finite etale morphism a(c) : U(c) → U, and a morphism u(c) : U(c) → GL(n) (all in the same category as X and g), such that t(c)(x) = u(c)(x)g(x)u(c)(x)(−1) is a diagonal matrix, for all x ∈ U(c). This construction gives, among other things, an extension in a refined form (on the level of U(c)-sections) of the classical one-parameter Perturbation Theory of matrices to the case of many parameters, ramified eigenvalues, not necessarily hermitian matrices, etc. We also prove the stable triviality of the eigenbundles of g on U and vanishing of their Chern classes.