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

This issue Previous Article Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families Next Article Low Mach number limit for a model of accretion disk

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: April 2017

Revised: November 2017

Published: July 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).

Abstract / Introduction Full Text(HTML) Figure(1) Related Papers Cited by

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.

Citation:

Full Text(HTML)

Figure 1. The image of a closed leaf is closed

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	C. Camacho, Dicritical singularities of holomorphic vector fields, Contemporary Mathematics, 269 (2001), 39-45. doi: 10.1090/conm/269/04328.
[2]	C. Camacho and A. Lins Neto, Teoria Geométrica das Folheações, IMPA, Projeto Euclides, 1979.
[3]	F. Cano, Reduction of the singularities of codimension one holomorphic foliations in dimension three, Annals of Math, 160 (2004), 907-1011. doi: 10.4007/annals.2004.160.907.
[4]	F. Cano, M. Ravara-Vago and M. Soares, Local Brunella's alternative I. RICH foliations, Int. Math. Res. Not. IMRN, 9 (2015), 2525-2575. doi: 10.1093/imrn/rnu011.
[5]	F. Cano and D. Cerveau, Desingularization of non-dicritical holomorphic foliations and existence of separatrices, Acta Math., 169 (1992), 1-103. doi: 10.1007/BF02392757.
[6]	D. Cerveau and J. -F. Mattei, Formes intégrables holomorphes singulières, Astérisque, 97 (1982), 193pp.
[7]	D. Cerveau and R. Moussu, Groupes d'automorphismes de $ ({\mathbb{C}},0)$ et équations différentielles $ ydy+··· = 0$, Bull. Soc. Math. France, 116 (1988), 459-488. doi: 10.24033/bsmf.2108.
[8]	D. Cerveau and J. Mozo Fernández, Classification analytique des feuilletages singuliers réduits de codimension 1 en dimension $ n≥3$, Erg. Th. and Dyn. Systems, 22 (2002), 1041-1060. doi: 10.1017/S0143385702000561.
[9]	V. Cossart, Desingularization in dimension 2, in Resolution of Surface Singularities, Lecture Notes in Mathematics, Springer-Verlag, 1101 (1984), 79-98.
[10]	P. Fernández Sánchez and J. Mozo Fernández, Quasi-ordinary cuspidal foliations in $ ({\mathbb{C}}^3,0)$, J. Differential Equations, 226 (2006), 250-268. doi: 10.1016/j.jde.2005.09.006.
[11]	P. Fernández Sánchez, J. Mozo Fernández and H. Neciosup, On codimension one foliations with prescribed cuspidal separatrix, J. Differential Equations, 256 (2014), 1702-1717. doi: 10.1016/j.jde.2013.12.002.
[12]	J. Giraud, Desingularization in low dimension, in Resolution of Surface Singularities, Lecture Notes in Mathematics, Springer-Verlag, 1101 (1984), 50-78.
[13]	J. P. Jouanolou, Équations de Pfaff Algébriques, Lecture Notes in Mathematics, 708, Springer-Verlag, 1979.
[14]	F. Loray, A preparation theorem for codimension one foliations, Ann. of Math., 163 (2006), 709-722. doi: 10.4007/annals.2006.163.709.
[15]	B. Malgrange, Frobenius avec singularités. I. Codimension un, Inst. Hautes Études Sci. Publ. Math., 46 (1976), 163-173.
[16]	J. F. Mattei and R. Moussu, Holonomie et intégrales premiéres, Ann. Sci. École Normale Sup., 13 (1980), 469-523.
[17]	R. Meziani, Classification analytique d'équations différentielles $ ydy+··· = 0$ et espaces de modules, Bol. Soc. Brasil Mat., 27 (1996), 23-53. doi: 10.1007/BF01246703.
[18]	R. Meziani and P. Sad, Singularités nilpotentes et intégrales premières, Publ. Mat., 51 (2007), 143-161. doi: 10.5565/PUBLMAT_51107_07.
[19]	J. J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhäuser Progress in Mathematics, 1999. doi: 10.1007/978-3-0348-8718-2.
[20]	Holonomie évanescente des équations différentielles dégénérées transverses, in Singularities and Dynamical Systems, North-Holland, 103 (1985), 161-173. doi: 10.1016/S0304-0208(08)72123-6.