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A Darboux-type theorem for germs of holomorphic one-dimensional foliations

Published online by Cambridge University Press:  04 August 2014

CÉSAR CAMACHO  and
BRUNO SCÁRDUA
CÉSAR CAMACHO
Affiliation:
IMPA, Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro, 22460-320, Brazil
BRUNO SCÁRDUA
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21.945-970 Rio de Janeiro, Brazil email scardua@im.ufrj.br, scardua@impa.br
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Abstract

We show that a germ of a holomorphic one-dimensional foliation at a singularity in a space of dimension two admits a holomorphic first integral if and only if there are infinitely many closed leaves and a finite number of separatrices, with each separatrix having linearizable holonomy. Indeed, if there are infinitely many closed leaves and the set of separatrices is finite, then the foliation admits either a holomorphic first integral or a formal simple integrating factor of Darboux type.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 8 , December 2015 , pp. 2458 - 2473
Copyright
© Cambridge University Press, 2014 

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