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Singularities that are inaccessible by geometry. (Singularités non abordables par la géométrie.) (French) Zbl 0940.32013
Summary: This paper is devoted to local analytic objects (i.e. germs of vector fields or diffeomorphisms) in any dimension, with special emphasis on the interplay between the two main difficulties: small denominators and resonance. We introduce an arborification technique, which is well suited for tackling diophantine small denominators and we recall the definition of resurgent functions and monomials, which are essential in any resonant context. We show how a single equation, the so-called Bridge Equation, not only yields all holomorphic invariants (i.e. all analytic invariants depending holomorphically on the object) but also most intrinsic properties of local objects, such as: sectorial normalization, criteria for the existence of invariant analytic varieties, etc.

32S65 Singularities of holomorphic vector fields and foliations
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