Irreducible holonomy groups and Riccati foliations in higher complex dimension

Journal of Singularities
volume 19 (2019), 177-197

Received: 6 July 2019. Received in revised form: 12 November 2019.

DOI: 10.5427/jsing.2019.19j

Add a reference to this article to your citeulike library.

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. We studied the case of complex dimension one in an earlier work where finiteness is proved for irreducible groups under certain arithmetic hypothesis on the linear part. In dimension n ≥ 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.

2000 Mathematical Subject Classification:

Primary 37F75, 57R30; Secondary 32M25, 32S65.

Author(s) information: