Germes de feuilletages pr\'esentables du plan complexe

TL;DR

This paper investigates the conditions under which certain complex plane foliations are incompressible, providing counterexamples and characterizations that refine previous results by Marin and Mattei.

Contribution

It demonstrates that the incompressibility hypothesis cannot be universally applied and characterizes foliations where Marin--Mattei's monodromy construction is valid.

Findings

Counterexamples of foliations where incompressibility fails

Proof that individual saddle-node foliations are incompressible

Characterization of foliations suitable for Marin--Mattei's monodromy

Abstract

Let F be a germ of a singular foliation of the complex plane. Assuming that F is a generalized curve D. Marin and J.-F. Mattei proved the incompressibility of the foliation in a neighborhood from which a finite set of analytic curves is removed. We show in the present work that this hypothesis cannot be eluded by building examples of foliations, reduced after one blow-up, for which the property does not hold. Even if we manage to prove that the individual saddle-node foliation is incompressible, their leaves not retracting tangentially on the boundary of the domain of definition forbids a generalization of Marin--Mattei's construction. We finally characterize those foliations for which the construction of Marin--Mattei's monodromy can be carried out.