Work

Singular sets of holonomy maps for algebraic foliations

Year: 2013

Type: article

Abstract: In this article we investigate the natural domain of definition of a holonomy map associated to a singular holomorphic foliation of the complex projective plane. We prove that germs of holonomy between algebraic curves can have large sets of singularities for the analytic continuation. In the Riccati context we provide examples with natural boundary and maximal sets of singularities. In the generic case we provide examples having at least a Cantor set of singularities and even a nonempty open set of singularities. The examples provided are based on the presence of sufficiently rich contracting dynamics in the holonomy pseudogroup of the foliation. This gives answers to some questions and conjectures of Loray and Ilyashenko, which follow-up on an approach to the associated ODE’s developed notably by Painlevé.

Source: Journal of the European Mathematical Society

DOI: 10.4171/jems/386

Authors: Gabriel Calsamiglia, Bertrand Deroin, Sidney Frankel, Adolfo Guillot

Institutions: Universidade Federal Fluminense, Université Paris-Sud, Rensselaer Polytechnic Institute, Universidad Nacional Autónoma de México

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze

Funders: Agence Nationale de la Recherche, Conselho Nacional de Desenvolvimento Científico e Tecnológico, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Max-Planck-Institut für Mathematik in den Naturwissenschaften

Awards: Uniformisation de familles de variétés complexes, Extension des théories de Teichmüller-Thurston