Moscow Mathematical Journal

Volume 21, Issue 2, April–June 2021 pp. 271–286.

Integrable Deformations of Foliations: a Generalization of Ilyashenko's Result

Authors: Dominique Cerveau (1) and Bruno Scárdua (2)
Author institution:(1) Université de Rennes / CNRS-IRMAR-UMR 6625, F 35000-Rennes, France
(2) Inst. Matemática, Universidade Federal do Rio de Janeiro. 68530, Rio de Janeiro-RJ, 21.945-970 BRAZIL

Summary:

We study analytic deformations of holomorphic differential 1-forms. The initial 1-form is exact homogeneous and the deformation is by polynomial integrable 1-forms. We investigate under which conditions the elements of the deformation are still exact or, more generally, exhibit a first integral. Our results are related to natural extensions of classical results of Ilyashenko on limit cycles of perturbations of hamiltonian systems in two complex variables.

2020 Math. Subj. Class. Primary: 37F75, 57R30; Secondary: 32M25, 32S65.

Keywords: Holomorphic foliation, integrable form, deformation.

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