Irreducible holonomy groups and first integrals for holomorphic foliations

TL;DR

This paper investigates irreducible groups of complex diffeomorphisms, focusing on their finiteness and implications for holomorphic foliations, including the existence of first integrals under specific conditions.

Contribution

It introduces new finiteness results for irreducible groups and applies these findings to establish the existence of first integrals in holomorphic foliations.

Findings

Finiteness results for irreducible groups of germs of complex diffeomorphisms.

Conditions under which first integrals exist for holomorphic foliations.

Applications to the tangent cone analysis after blow-ups.

Abstract

We study groups of germs of complex diffeomorphisms having a property called irreducibility. The notion is motivated by the similar property of the fundamental group of the complement of na irreducible hypersurface in the complex projective space. Natural examples of such groups of germ maps are given by holonomy groups and monodromy groups of integrable systems (foliations) under certain conditions. We prove some finiteness results for these groups extending previous results in [CL]. Applications are given to the framework of germs of holomorphic foliations. We prove the existence of first integrals under certain irreducibility or more general conditions on the tangent cone of the foliation after a punctual blow-up.