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July  2018, 38(7): 3223-3237. doi: 10.3934/dcds.2018140

Dicritical nilpotent holomorphic foliations

Percy Fernández-Sánchez 1, , Jorge Mozo-Fernández 2,, and  Hernán Neciosup 3,

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

Received  April 2017 Revised  November 2017 Published  April 2018

Fund Project: 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).

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: Percy Fernández-Sánchez, Jorge Mozo-Fernández, Hernán Neciosup. Dicritical nilpotent holomorphic foliations. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3223-3237. doi: 10.3934/dcds.2018140
References:
[1]

C. Camacho, Dicritical singularities of holomorphic vector fields, Contemporary Mathematics, 269 (2001), 39-45.  doi: 10.1090/conm/269/04328.  Google Scholar

[2]

C. Camacho and A. Lins Neto, Teoria Geométrica das Folheações, IMPA, Projeto Euclides, 1979.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

D. Cerveau and J. -F. Mattei, Formes intégrables holomorphes singulières, Astérisque, 97 (1982), 193pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

V. Cossart, Desingularization in dimension 2, in Resolution of Surface Singularities, Lecture Notes in Mathematics, Springer-Verlag, 1101 (1984), 79-98. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Giraud, Desingularization in low dimension, in Resolution of Surface Singularities, Lecture Notes in Mathematics, Springer-Verlag, 1101 (1984), 50-78. Google Scholar

[13]

J. P. Jouanolou, Équations de Pfaff Algébriques, Lecture Notes in Mathematics, 708, Springer-Verlag, 1979.  Google Scholar

[14]

F. Loray, A preparation theorem for codimension one foliations, Ann. of Math., 163 (2006), 709-722.  doi: 10.4007/annals.2006.163.709.  Google Scholar

[15]

B. Malgrange, Frobenius avec singularités. I. Codimension un, Inst. Hautes Études Sci. Publ. Math., 46 (1976), 163-173.   Google Scholar

[16]

J. F. Mattei and R. Moussu, Holonomie et intégrales premiéres, Ann. Sci. École Normale Sup., 13 (1980), 469-523.   Google Scholar

[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.  Google Scholar

[18]

R. Meziani and P. Sad, Singularités nilpotentes et intégrales premières, Publ. Mat., 51 (2007), 143-161.  doi: 10.5565/PUBLMAT_51107_07.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

C. Camacho, Dicritical singularities of holomorphic vector fields, Contemporary Mathematics, 269 (2001), 39-45.  doi: 10.1090/conm/269/04328.  Google Scholar

[2]

C. Camacho and A. Lins Neto, Teoria Geométrica das Folheações, IMPA, Projeto Euclides, 1979.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

D. Cerveau and J. -F. Mattei, Formes intégrables holomorphes singulières, Astérisque, 97 (1982), 193pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

V. Cossart, Desingularization in dimension 2, in Resolution of Surface Singularities, Lecture Notes in Mathematics, Springer-Verlag, 1101 (1984), 79-98. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Giraud, Desingularization in low dimension, in Resolution of Surface Singularities, Lecture Notes in Mathematics, Springer-Verlag, 1101 (1984), 50-78. Google Scholar

[13]

J. P. Jouanolou, Équations de Pfaff Algébriques, Lecture Notes in Mathematics, 708, Springer-Verlag, 1979.  Google Scholar

[14]

F. Loray, A preparation theorem for codimension one foliations, Ann. of Math., 163 (2006), 709-722.  doi: 10.4007/annals.2006.163.709.  Google Scholar

[15]

B. Malgrange, Frobenius avec singularités. I. Codimension un, Inst. Hautes Études Sci. Publ. Math., 46 (1976), 163-173.   Google Scholar

[16]

J. F. Mattei and R. Moussu, Holonomie et intégrales premiéres, Ann. Sci. École Normale Sup., 13 (1980), 469-523.   Google Scholar

[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.  Google Scholar

[18]

R. Meziani and P. Sad, Singularités nilpotentes et intégrales premières, Publ. Mat., 51 (2007), 143-161.  doi: 10.5565/PUBLMAT_51107_07.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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