Sliding invariants and classification of singular holomorphic foliations in the plane

By introducing a new invariant called the set of slidings, we give a complete strict classification of the class of germs of non-dicritical holomorphic foliations in the plan whose Camacho-Sad indices are not rational. Moreover, we will show that, in this class, the new invariant is finitely determined. Consequently, the finite determination of the class of isoholonomic non-dicritical foliations and absolutely dicritical foliations that have the same Dulac maps are proved.Comment: 21 pages, 2 figur

Commuting foliations

The aim of this paper is to extend the notion of commutativity of vector fields to the category of singular foliations, using Nambu structures, i.e. integrable multi-vector fields. We will classify the relationship between singular foliations and Nambu structures, and show some basic results about commuting Nambu structures.Comment: New version, with a completely new section which clarifies the relationship between singular foliations and Nambu structures. The size of the paper has doubled from 10 to 20 page

Formes normales de singularités de feuilletages et invariants de glissement

Les objets étudiés dans cette thèse sont les germes de feuilletages holomorphes singuliers dans le plan. Elle est divisée en deux parties: La première partie est consacrée à donner des formes normales formelles de feuilletages topologiquement quasi-homogènes dans des conditions génériques. Toute forme normale est donnée comme la somme de trois termes: un terme générique quasi-homogène initial, un terme hamiltonien et un terme radial. Nous montrons également que le nombre de coefficients libres dans les termes hamiltoniens sont conformes à la dimension de l'espace des modules de Mattei de déploiement. Dans la deuxième partie, par l'introduction d'un nouvel invariant appelé l'ensemble des glissements, nous donnons une classification stricte complète de la classe des germes de feuilletages holomorphes non dicritiques dont les indices de Camacho-Sad ne sont pas rationnels. Par ailleurs, nous allons montrer que, dans cette classe, le nouvel invariant est de détermination finie. Par conséquent, nous obtenons la détermination finie de la classe des feuilletages non dicritiques isoholonomiques et de feuilletages absolument dicritiques qui ont les même applications de Dulac.The objects studied in this thesis are the germs of singular holomorphic foliations in the plan. It is divided into two part: The first part is devoted to the construction of formal normal forms of topologically quasi-homogeneous foliations under generic conditions. Any such normal form is given as the sum of three terms: an initial generic quasi-homogeneous term, a hamiltonian term and a radial term. We also show that the number of free coefficients in the hamiltonian terms are consistent with the dimension of Mattei's moduli space of unfoldings. In the second part, by introducing a new invariant called the set of slidings, we give a complete strict classification of the class of germs of non-dicritical holomorphic foliations whose Camacho-Sad indices are not rational. Moreover, we will show that, in this class, the new invariant is finitely determined. Consequently, the finite determination of the class of isoholonomy non-dicritical foliations and absolutely dicritical foliations that have the same Dulac maps are proved