Moscow Mathematical Journal

Volume 18, Issue 1, January–March 2018 pp. 163–179.

Stable Singularities and Stable Leaves of Holomorphic Foliations in Dimension Two

Authors: V. León (1) and B. Scárdua (2)
Author institution:(1) ILACVN – CICN, Universidade Federal da Integraçāo Latino-Americano, Parque tecnológico de Itaipu, Foz do Iguaçu-PR, 85867-970 – Brazil
(2) Instituto de Matemática – Universidade Federal do Rio de Janeiro, CP. 68530-Rio de Janeiro-RJ, 21945-970 – Brazil

Summary:

We consider germs of holomorphic foliations with an isolated singularity at the origin 0 ∈ ℂ². We introduce a notion of Lstability for the singularity, similar to Lyapunov stability. We prove that L-stability is equivalent to the existence of a holomorphic first integral, or the foliation is a real logarithmic foliation. A notion of L-stability is also naturally introduced for a leaf of a holomorphic foliation in a complex surface. We prove that the holonomy groups of L-stable leaves are abelian, of a suitable type. This implies the existence of local closed meromorphic 1-forms defining the foliation, in a neighborhood of compact L-stable leaves. Finally, we consider the case of foliations in the complex projective plane. We prove that a foliation on ℂP² admitting a L-stable invariant algebraic curve is the pull-back by some polynomial map of a suitable linear logarithmic foliation.

2010 Math. Subj. Class. Primary: 37F75, 57R30; Secondary: 32M25, 32S65.

Keywords: Holomorphic foliation; Lyapunov stability, singularity.

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