Homologically trivial integrable deformations of germs of holomorphic functions

Victor León and Bruno Scárdua

Journal of Singularities
volume 24 (2022), 119-125

Received: 16 November 2021.

DOI: 10.5427/jsing.2022.24d

Add a reference to this article to your citeulike library.

Abstract:

We study analytic deformations by integrable 1-forms a germ of a holomorphic function at the origin of the complex affine space in dimension three or higher. We prove that, under some mild nondegeneracy conditions on the function germ, the existence of a simple normal form for the deformation is equivalent to a homological condition: the annihilation of the deformed one-form in the first homology group of the non-singular fibers of the function germ. In many cases this implies the existence of a holomorphic first integral for the deformation.

2000 Mathematical Subject Classification:

Primary 37F75, 57R30; Secondary 32M25, 32S65

Key words and phrases:

integrable 1-form; holomorphic germ; first integral; first homology group

Author(s) information:

Victor León
ILACVN - CICN
Universidade Federal da Integração Latino-Americana
Parque tecnológico de Itaipu
Foz do Iguaçu-PR, 85867-970 - Brazil
email: victor.leon@unila.edu.br

Bruno Scárdua
Instituto de Matemática
Universidade Federal do Rio de Janeiro
CP. 68530
Rio de Janeiro-RJ, 21945-970 - Brazil
email: bruno.scardua@gmail.com