Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Isolated invariant curves of a foliation


Authors: Nuria Corral and Percy Fernández-Sánchez
Journal: Proc. Amer. Math. Soc. 134 (2006), 1125-1132
MSC (2000): Primary 32S65; Secondary 37F75
DOI: https://doi.org/10.1090/S0002-9939-05-08044-5
Published electronically: August 29, 2005
MathSciNet review: 2196047
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We bound the equisingularity type of the set of isolated separatrices of a holomorphic foliation $\mathcal F$ of $({\mathbb C}^2,0)$ in terms of the Milnor number of $\mathcal F$. This result gives a bound for the degree of an algebraic invariant curve $C \subset {\mathbb P}^{2}_{\mathbb C}$ of a foliation $\mathcal G$ of ${\mathbb P}^{2}_{\mathbb C}$ in terms of the degree of $\mathcal G$, provided that all the branches of $C$ are isolated separatrices.


References [Enhancements On Off] (What's this?)

  • 1. M. Berthier; R. Meziani; P. Sad: On the classification of nilpotent singularities. Bull. Sci. Math., 123 (1999), 351-370. MR 1703202 (2000j:32052)
  • 2. E. Brieskorn; H. Knörrer: Plane algebraic curves. Birkhäuser, 1986. MR 886476 (88a:14001)
  • 3. M. Brunella: Some remarks on indices of holomorphic vector fields. Publ. Mat. 41 (1997), 527-544. MR 1485502 (98m:32052)
  • 4. M. Brunella: Birational geometry of foliations. Monografias de Matemática, IMPA, Rio de Janeiro, 2000. MR 1948251 (2004g:14018)
  • 5. C. Camacho; A. Lins Neto; P. Sad: Topological invariants and equidesingularization for holomorphic vector fields. J. Differential Geom., 20 (1984), no. 1, 143-174. MR 0772129 (86d:58080)
  • 6. F. Cano; D. Cerveau: Desingularization of nondicritical holomorphic foliations and existence of separatrices. Acta Math., 169 (1992), 1-103. MR 1179013 (93k:32069)
  • 7. F. Cano: Reduction of the singularities of codimension one singular foliations in dimension three. Annals of Math. 160, 3 (2004), 907-1011.
  • 8. M. M. Carnicer: The Poincaré Problem in the nondicritical case. Annals of Math., 140 (1994), 289-294. MR 1298714 (95k:32031)
  • 9. D. Cerveau; A. Lins Neto: Holomorphic foliations in $\mathbb{C} {\mathbb P}(2)$ having an invariant algebraic curve. Ann. Inst. Fourier, 41, 4 (1991), 883-903. MR 1150571 (93b:32050)
  • 10. D. Cerveau; R. Moussu: Groupes d'automorphismes de $(\mathbb{C} ,0)$ et équations différentielles $ydy + \ldots =0$. Bull. Soc. Math. France 116 (1998), 459-488. MR 1005391 (90m:58192)
  • 11. X. Gómez-Mont; J. Seade; A. Verjovsky: The index of a holomorphic flow with an isolated singularity. Math. Ann. 291 (1991), 737-751. MR 1135541 (93d:32066)
  • 12. F. Loray: A preparation theorem for codimension one foliations. Preprint: arXiv:math DG/0402411. To appear in Annals of Math.
  • 13. J. F. Mattei; R. Moussu: Holonomie et intégrales premières. Ann. Sci. Ecole Norm. Sup. (4) 13 (1980), 469-523. MR 0608290 (83b:58005)
  • 14. R. Meziani: Classification analytique d'équations différentielles $ydy + \ldots =0$ et espace de modules. Bol. Soc. Bras. Mat., 27, 1 (1996), 23-53. MR 1393026 (97g:32039)
  • 15. H. Poincaré: Sur l'intégration algébrique des équations différentielles du premier ordre et du premier degré. Rendiconti del Circolo Matematico di Palermo, 5 (1891), 161-191.
  • 16. A. Seidenberg: Reduction of singularities of the differential equation $Ady = B dx$. Amer. J. Math., 90 (1968), 248-269. MR 0220710 (36:3762)
  • 17. F. Takens: Singularities of vector fields. Inst. Hautes Études Sci. Publ. Math. 43 (1974), 47-100. MR 0339292 (49:4052)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32S65, 37F75

Retrieve articles in all journals with MSC (2000): 32S65, 37F75


Additional Information

Nuria Corral
Affiliation: Équipe de Géométrie et Dynamique, Institut de Mathématiques de Jussieu, 175, Université Paris 7, Rue du Chevaleret, 75013 Paris, France
Address at time of publication: Departamento Matemática Aplicada I, Universidad de Vigo, E.U.E.T. Forestal, Campus A Xunqueira, 36005 Pontevedra, España
Email: ncorral@uvigo.es

Percy Fernández-Sánchez
Affiliation: Instituto de Matemática y Ciencias Afines, Universidad de Ingeniería, Pontificia Universidad Católica del Perú, Casa de las Trece Monedas, Jr. Ancash, 536 Lima 1, Perú
Email: pefernan@pucp.edu.pe

DOI: https://doi.org/10.1090/S0002-9939-05-08044-5
Keywords: Singular foliation, isolated separatrix, algebraic invariant curve
Received by editor(s): July 22, 2004
Received by editor(s) in revised form: September 24, 2004, and November 2, 2004
Published electronically: August 29, 2005
Additional Notes: The first author was supported by a Postdoctoral Grant of the Spanish Ministry of Education and partially supported by the research project BFM 2001-2010 and by the Junta de Castilla y León (VA123/04). The second author was supported by a Postdoctoral Grant of the AECI, Spanish Ministry of Foreign Affairs and partially supported by the research project by Concytec (n. 823-2001), Perú
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society