Holonomy and equivalence of analytic foliations

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.