Modules formels locaux de feuilletages holomorphes

TL;DR

This paper classifies formal invariants of certain holomorphic foliations and characterizes those with finite formal type, providing explicit criteria and showing their abundance within a specific class.

Contribution

It offers a complete list of formal invariants for a broad class of formal differential 1-forms and characterizes forms with finite formal type, including a combinatorial criterion.

Findings

Complete classification of formal invariants for formal differential 1-forms.

Characterization of 1-forms with finite formal type and semi-universal deformations.

Finite formal type forms form a dense open subset among second kind forms.

Abstract

We give a complete list of formal invariants for a large class of formal differential 1-forms $\w \in \Bbb C [[ x, y]]dx + \Bbb C [[ x, y]]dy$. \indent A $\hat{SL}$-equisingular deformation is an equireducible deformation which leaves invariant both the local formal types and the holonomy representation of the components of the exceptional divisor. We characterize the 1-forms with finite formal type (t.f.f), i.e. those which admit a semi-universal $\hat{SL}$-equisingular deformation, and we give an explicit combinatorial criterion of finiteness. \indent The set of 1-forms with finite formal type contains a dense open set (in the sense of Krull's topology)in the set of 1-forms of the second kind.