Feuilletages holomorphes de codimension 1: une \'etude locale dans le cas dicritique

TL;DR

This paper analyzes the singularities of dicritical holomorphic foliations of small multiplicity in three dimensions and links their deformations to Liouvillian integrability issues.

Contribution

It provides a detailed local study of singularities in dicritical holomorphic foliations and relates deformations to integrability problems.

Findings

Characterization of singularities in small multiplicity cases

Connection between deformations and Liouvillian integrability

Insights into the structure of dicritical holomorphic foliations

Abstract

Nous d\'ecrivons les singularit\'es de feuilletages holomorphes dicritiques de petite multiplicit\'e en dimension $3$. En particulier nous relions l'existence de d\'eformations et de d\'eploiements non triviaux \`a des probl\`emes d'int\'egrabilit\'e liouvillienne. We describe the singularities of dicritical holomorphic foliations of small multiplicity in dimension $3$. In particular we connect the existence of non trivial deformations and deployments to problems of liouvillian integrability.