The isotropy group of a foliation: the local case

TL;DR

This paper studies the structure of the group of biholomorphisms preserving a holomorphic foliation near a singularity, focusing on the quotient groups and their formal counterparts, especially for codimension one cases.

Contribution

It introduces and analyzes the quotient groups of biholomorphisms preserving a foliation, providing new insights into their structure in the local singular setting.

Findings

Characterization of the groups $Iso(a)$ and $Fix(a)$

Analysis of the quotient groups $Iso(a)/Fix(a)$ and $\, ext{formal groups}$

Special focus on codimension one foliations

Abstract

Given a holomorphic singular foliation $\fa$ of $(\C^n,0)$ we define $Iso(\fa)$ as the group of germs of biholomorphisms on $(\C^n,0)$ preserving $\fa$: $Iso(\fa)=\{\Phi\in Diff(\C^n,0)\,|\,\Phi^*(\fa)=\fa\}$. The normal subgroup of $Iso(\fa)$, of biholomorphisms sending each leaf of $\fa$ into itself, will be denoted as $Fix(\fa)$. The corresponding groups of formal biholomorphisms will be denoted as $\wh{Iso}(\fa)$ and $\wh{Fix}(\fa)$, respectively. The purpose of this paper will be to study the quotients $Iso(\fa)/Fix(\fa)$ and $\wh{Fix}(\fa)/\wh{Fix}(\fa)$, mainly in the case of codimension one foliation.