On the integrability of holomorphic vector fields

Leonardo Câmara; Bruno Scárdua

doi:10.3934/dcds.2009.25.481

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2009, Volume 25, Issue 2: 481-493. Doi: 10.3934/dcds.2009.25.481

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On the integrability of holomorphic vector fields

Leonardo Câmara^1, and
Bruno Scárdua^2,

1.
Departamento de Matemática, Universidade Federal do Espírito Santo, Av. Fernando Ferrari 514, 29075-910, Vitória, ES
2.
Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21.945-970, Rio de Janeiro, RJ

Received: January 31, 2008

Revised: December 31, 2008

Published: May 2009

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Abstract

We determine topological and algebraic conditions for a germ of holomorphic foliation $\mathcal{F}_X$ induced by a generic vector field $X$ on $(\mathbb{C}^{3},0)$ to have a holomorphic first integral, i.e., a germ of holomorphic map $F$ : $(\mathbb{C}^{3},0)\rightarrow(\mathbb{C}^{2},0)$ such that the leaves of $\mathcal{F}_X$ are contained in the level curves of $F$.

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Mathematics Subject Classification: Primary: 37F75; Secondary: 32H50, 32S65.

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