On simultaneous uniformization and local nonuniformizability
Comptes Rendus. Mathématique, Volume 334 (2002) no. 6, pp. 489-494.

We prove existence of a one-dimensional holomorphic foliation (with isolated irremovable singularities) tangent to a rational vector field on appropriate affine algebraic surface of dimension 2 such that the family of leaves intersecting arbitrary given cross-section does not admit a uniformization holomorphic in the parameter by a family of simply connected domains in ¯. We show that such a foliation can be chosen transversally affine, having a Liouvillian first integral, with dense and hyperbolic leaves and an attracting cycle. This extends the author's result [4] giving a negative answer to Ilyashenko's simultaneous uniformization conjecture and answers negatively to the local version of this conjecture recently proposed by Shcherbakov.

On montre l'existence d'un feuilletage holomorphe de dimension un (à singularités isolées non effaçables) sur une surface algébrique affine lisse appropriée de dimension 2 qui est tangent à un champ vectoriel rationnel, et tel qu'aucune famille de feuilles intersectant une section transverse n'admet d'uniformisation holomorphe paramétrée par une famille d'ouverts simplement connexes de ¯. On montre, qu'un tel feuilletage peut être choisi transversalement affin, ayant une intégrale première de type Liouville, toutes les feuilles hyperboliques et denses et un cycle attractif. Cela étend le résultat précédent de l'auteur (donnant la réponse négative à la conjecture d'Ilyachenko sur l'uniformisation simultanée) et répond négativement à une version locale de cette conjecture proposée récemment par Chtcherbakov.

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Accepted:
Published online:
DOI: 10.1016/S1631-073X(02)02268-9
Glutsyuk, Alexey 1

1 CNRS, Unité de mathématiques pures et appliquées, MR, École normale supérieure de Lyon, 46, allée d'Italie, 69364 Lyon cedex 07, France
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Glutsyuk, Alexey. On simultaneous uniformization and local nonuniformizability. Comptes Rendus. Mathématique, Volume 334 (2002) no. 6, pp. 489-494. doi : 10.1016/S1631-073X(02)02268-9. http://www.numdam.org/articles/10.1016/S1631-073X(02)02268-9/

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