Construction of foliations with prescribed separatrix

Abstract

We build a germ of singular foliation in C 2 with analytical class of separatrix and holonomy representations prescribed. Thanks to this construction, we study the link between moduli of a foliation and moduli of its separatrix.

Analytical and formal classifications of quasi-homogeneous foliations in (C2,0)

“…The main result is a realization kind Theorem 2.2, which provides an isoholonomic deformation of a foliation with a given deformation of its separatrix. This theorem specializes to a result of [5] if there is no parameter. 1 Since the arguments of [5] and these used in this section are widely similar, we are only going to detail the new difficulties coming with the introduction of a parameter.…”

“…This theorem specializes to a result of [5] if there is no parameter. 1 Since the arguments of [5] and these used in this section are widely similar, we are only going to detail the new difficulties coming with the introduction of a parameter. However, for the convenience of the reader we repeat here the relevant material from [5] thus making our exposition almost self-contained.…”

“…As already mentioned, we follow the proof performed in [5]. The most interest of the formalism introduced in [5] is the natural way it can be extended to the present context.…”

“…1 Since the arguments of [5] and these used in this section are widely similar, we are only going to detail the new difficulties coming with the introduction of a parameter. However, for the convenience of the reader we repeat here the relevant material from [5] thus making our exposition almost self-contained. Nevertheless, the reader more interested in the classification 1.1 can admit the main result of this section and begin its lecture Section 3.…”

Un problème de Riemann–Hilbert local non-linéaire pour les germes de feuilletages holomorphes singuliers en dimension deux

“…La preuve se trouve dans [2]. Nous en donnons ici un schéma simplifié : on établit d'abord le résultat sur le premier voisinage infinitésimal du diviseur ; puis, un cocycle de collage étant préparé, on effectue une double induction sur la longueur du processus de réduction de F et sur l'ordre du voisinage dans la filtration de M par les voisinages infinitésimaux de D. On y développe un algorithme de normalisation fondé sur deux éléments : l'équation cohomologique associée au résultat infinitésimal (2.1) et la formule de Campbell-Hausdorff.…”