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

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The National Academy of Sciences

RESUMO

Let X be the germ of a complex- or real-analytic manifold M at a point xo ∈ M, or the henselian germ of an algebraic manifold M over a field k of characteristic zero at a point xo ∈ 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 Uc, a finite etale morphism ac : Uc → U, and a morphism uc : Uc → GL(n) (all in the same category as X and g), such that tc(x) = uc(x)g(x)uc(x)−1 is a diagonal matrix, for all x ∈ Uc. This construction gives, among other things, an extension in a refined form (on the level of Uc-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.

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