Geodesic
Singular sets of holonomy maps for algebraic foliations
Journal of the European Mathematical Society, Tome 15 (2013) no. 3, pp. 1067-1099

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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.
DOI : 10.4171/jems/386
Classification : 34-XX, 32-XX, 00-XX
Keywords: Holomorphic foliation, holonomy map, analytic extension 
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     title = {Singular sets of holonomy maps for algebraic foliations},
     journal = {Journal of the European Mathematical Society},
     pages = {1067--1099},
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Gabriel Calsamiglia; Bertrand Deroin; Sidney Frankel; Adolfo Guillot. Singular sets of holonomy maps for algebraic foliations. Journal of the European Mathematical Society, Tome 15 (2013) no. 3, pp. 1067-1099. doi: 10.4171/jems/386

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