Integrating factors for groups of formal complex diffeomorphisms

TL;DR

This paper explores the classification and integrability of groups of formal complex diffeomorphisms in multiple variables, linking algebraic properties with invariant structures to aid in understanding holomorphic foliations.

Contribution

It extends the dictionary between algebraic properties and integrability from one variable to several complex variables, focusing on invariant vector fields and forms.

Findings

Existence of formal invariant vector fields for abelian and solvable groups.

Construction of closed differential forms invariant under group actions.

Application to holomorphic foliations with singularities.

Abstract

We study groups of formal or germs of analytic diffeomorphisms in several complex variables. Such groups are related to the study of the transverse structure and dynamics of Holomorphic foliations, via the notion of holonomy group of a leaf of a foliation. For dimension one, there is a well-established dictionary relating analytic/formal classification of the group, with its algebraic properties (finiteness, commutativity, solvability, ...). Such system of equivalences also characterizes the existence of suitable {\it integrating factors}, i.e., invariant vector fields and one-forms associated to the group. In this paper we search the basic lines of such dictionary for the case of several complex variables groups. For abelian, metabelian, solvable or nilpotent groups we investigate the existence of suitable formal vector fields and closed differential forms which exhibit an invariance property under the group action. Our results are applicable in the construction of suitable integrating factors for holomorphic foliations with singularities. We believe they are a starting point in the study of the connection between Liouvillian integration and transverse structures of holomorphic foliations with singularities in the case of arbitrary codimension.