Holonomy and equivalence of analytic foliations

TL;DR

This paper provides an analytic classification method for certain singular foliations by analyzing their holonomy, showing that conjugated holonomies imply the foliations are analytically equivalent under specific conditions.

Contribution

It establishes a link between the analytic classification of singular foliations and the conjugation of their holonomies, under certain hypotheses.

Findings

Holonomy conjugation determines foliation equivalence.

Analytic classification reduces to holonomy analysis.

Results apply to foliations generated by specific vector fields.

Abstract

The main goal of this paper is the analytic classification of the germs of singular foliations generated, up to an analytic change of coordinates, by the germs of vector fields of form the $x\partial_x+\sum_{i=1}^{n}a_i(x,\mathbf{z})\partial_{z_i}$, where $a_i(x,\mathbf{z})$ is a germ of analytic function with $a_i(x,0)=0$. We focus on the connection with the conjugation of the holonomies related to them. We prove, under some hypothesis, that these germs of singular foliations are analytically classified once their local holonomy along a given separatrix are analytically conjugated.