Irreducible holonomy groups and Riccati foliations in higher complex dimension

Abstract

We study groups of germs of complex diffeomorphisms having a property called irreducibility. The notion is motivated by a similar property of the fundamental group of the complement of an 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 on the singular or ramification set. The case of complex dimension one is studied in [7] where finiteness is proved for irreducible groups under certain arithmetic hypothesis on the linear part. In dimension $n \geq 2$ the picture changes since linear groups are not always abelian in dimension two or bigger. Nevertheless, we still obtain a finiteness result under some conditions in the linear part of the group, for instance if the linear part is abelian. Examples are given illustrating the role of our hypotheses. Applications are given to the framework of holomorphic foliations and analytic deformations of rational fibrations by Riccati foliations.