Irreducible holonomy groups and Riccati foliations in higher complex dimension
Journal of Singularities, Tome 19 (2019), pp. 177-197
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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.
@article{10_5427_jsing_2019_19j,
author = {V. Le\'on and M. Martelo, and B. Sc\'ardua},
title = {Irreducible holonomy groups and {Riccati} foliations in higher complex dimension},
journal = {Journal of Singularities},
pages = {177--197},
publisher = {mathdoc},
volume = {19},
year = {2019},
doi = {10.5427/jsing.2019.19j},
url = {http://geodesic.mathdoc.fr/articles/10.5427/jsing.2019.19j/}
}
TY - JOUR AU - V. León AU - M. Martelo, AU - B. Scárdua TI - Irreducible holonomy groups and Riccati foliations in higher complex dimension JO - Journal of Singularities PY - 2019 SP - 177 EP - 197 VL - 19 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.5427/jsing.2019.19j/ DO - 10.5427/jsing.2019.19j ID - 10_5427_jsing_2019_19j ER -
%0 Journal Article %A V. León %A M. Martelo, %A B. Scárdua %T Irreducible holonomy groups and Riccati foliations in higher complex dimension %J Journal of Singularities %D 2019 %P 177-197 %V 19 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.5427/jsing.2019.19j/ %R 10.5427/jsing.2019.19j %F 10_5427_jsing_2019_19j
V. León; M. Martelo,; B. Scárdua. Irreducible holonomy groups and Riccati foliations in higher complex dimension. Journal of Singularities, Tome 19 (2019), pp. 177-197. doi: 10.5427/jsing.2019.19j
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