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Article Dans Une Revue Publications mathematiques de l' IHES Année : 2005

A KAM phenomenon for singular holomorphic vector fields

Laurent Stolovitch

Résumé

Let $X$ be a germ of holomorphic vector field at the origin of ${\bf C}^n$ and vanishing there. We assume that $X$ is a "nondegenerate" good perturbation of a singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals. We show that $X$ admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection of a polydisc with an analytic set of the form ``resonant monomials = constants". Such a biholomorphism conjugates the restriction of $X$ to one of its invariant varieties to the restriction of a linear diagonal vector field to a toric variety. Moreover, we show that the set of "frequencies" defining the invariant sets is of positive measure.

Dates et versions

hal-00121995 , version 1 (23-12-2006)

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Citer

Laurent Stolovitch. A KAM phenomenon for singular holomorphic vector fields. Publications mathematiques de l' IHES, 2005, 102, pp.99-165. ⟨hal-00121995⟩
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