Dicritical nilpotent holomorphic foliations

Percy Fernández-Sánchez; Jorge Mozo-Fernández; Hernán Neciosup

doi:10.3934/dcds.2018140

Article Contents

2018, Volume 38, Issue 7: 3223-3237. Doi: 10.3934/dcds.2018140

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Dicritical nilpotent holomorphic foliations

1.
Dpto. Ciencias - Sección Matemáticas, Pontificia Universidad Católica del Perú, Av. Universitaria 1801, San Miguel, Lima 32, Peru
2.
Dpto. Álgebra, Análisis Matemático, Geometría y Topología, Facultad de Ciencias, Universidad de Valladolid, Paseo de Belén, 7; 47011 Valladolid, Spain
3.
Dpto. Ciencias - Sección Matemáticas, Pontificia Universidad Católica del Perú, Av. Universitaria 1801, San Miguel, Lima 32, Peru

* Corresponding author: Jorge Mozo-Fernández
* Corresponding author: Jorge Mozo-Fernández

Received: March 31, 2017

Revised: October 31, 2017

Published: June 2018

This work was funded by the Dirección de Gestión de la Investigación at the PUCP through grant DGI-2015-1-0045, and by the Ministerio de Economía y Competitividad from Spain, under Projects "Álgebra y Geometría en Dinámica Real y Compleja Ⅲ" (Ref.: MTM2013-46337-C2-1-P) and "Álgebra y geometr ía en sistemas dinámicos y foliaciones singulares" (Ref: MTM2016-77642-C2-1-P).

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Abstract

We study in this paper several properties concerning singularities of foliations in $ {\left( {{\mathbb{C}}^{3}}\rm{,}\bf{0} \right)}$ that are pull-back of dicritical foliations in $ {\left( {{\mathbb{C}}^{2}}\rm{,}\bf{0} \right)}$. Particularly, we will investigate the existence of first integrals (holomorphic and meromorphic) and the dicriticalness of such a foliation. In the study of meromorphic first integrals we follow the same method used by R. Meziani and P. Sad in dimension two. While the foliations we study are pull-back of foliations in $ {\left( {{\mathbb{C}}^{2}}\rm{,}\bf{0} \right)}$, the adaptations are not straightforward.

Keywords:

Holomorphic foliations,
dicritical foliations.

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

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References

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