Groups of formal diffeomorphisms in several complex variables and closed one-forms

TL;DR

This paper investigates groups of formal diffeomorphisms in several complex variables, exploring their invariance properties and applications to holomorphic foliations and Liouvillian integration.

Contribution

It introduces new methods to connect group invariance with closed forms and vector fields, advancing the understanding of holomorphic foliation structures.

Findings

Existence of invariant vector fields and closed forms for certain groups

Application to constructing integrating factors for foliations

Insights into Liouvillian integrability and transverse structures

Abstract

We study groups of formal diffeomorphisms in several complex variables. For abelian, metabelian 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.