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)

 
 

 

Rigidity of certain holomorphic foliations


Author: David Marín
Journal: Proc. Amer. Math. Soc. 134 (2006), 3595-3603
MSC (2000): Primary 32G34, 34Mxx, 57R30
DOI: https://doi.org/10.1090/S0002-9939-06-08418-8
Published electronically: June 8, 2006
MathSciNet review: 2240672
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: There is a well-known rigidity theorem of Y. Ilyashenko for (singular) holomorphic foliations in $ \mathbb{CP}^2$ and also the extension given by Gómez-Mont and Ortíz-Bobadilla (1989). Here we present a different generalization of the result of Ilyashenko: some cohomological and (generic) dynamical conditions on a foliation $ \mathcal{F}$ on a fibred complex surface imply the d-rigidity of $ \mathcal{F}$, i.e. any topologically trivial deformation of $ \mathcal{F}$ is also analytically trivial. We particularize this result to the case of ruled surfaces. In this context, the foliations not verifying the cohomological hypothesis above were completely classified in an earlier work by X. Gómez-Mont (1989). Hence we obtain a (generic) characterization of non-d-rigid foliations in ruled surfaces. We point out that the widest class of them are Riccati foliations.


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

  • 1. M. Belliart, I. Liousse, and F. Loray.
    Sur l'existence de points fixes attractifs pour les sous-groupes de $ \mathrm{Aut}(\mathbb{C},0)$.
    C.R. Acad. Sci. Paris, Série I, 324:443-446, 1997. MR 1440964 (98c:58134)
  • 2. Marco Brunella.
    Feuilletages holomorphes sur les surfaces complexes compactes.
    Ann. Sci. École Norm. Sup. (4), 30(5):569-594, 1997. MR 1474805 (98i:32051)
  • 3. Marco Brunella.
    Birational geometry of foliations.
    Monografías de Matemática. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2000. MR 1948251 (2004g:14018)
  • 4. Dominique Cerveau and Paulo Sad.
    Problèmes de modules pour les formes différentielles singulières dans le plan complexe.
    Comment. Math. Helv., 61(2):222-253, 1986. MR 0856088 (88f:58124)
  • 5. Raymond Gérard and Antoinette Sec.
    Feuilletages de Painlevé.
    Bull. Soc. Math. France, 100:47-72, 1972. MR 0306552 (46:5675)
  • 6. Étienne Ghys.
    À propos d'un théorème de J.-P. Jouanolou concernant les feuilles fermées des feuilletages holomorphes.
    Rend. Circ. Mat. Palermo (2), 49(1):175-180, 2000. MR 1753461 (2001i:32048)
  • 7. X. Gómez-Mont and L. Ortíz-Bobadilla.
    Sistemas dinámicos holomorfos en superficies.
    Sociedad Matemática Mexicana, México City, 1989. MR 1304495 (95j:32046)
  • 8. Xavier Gómez-Mont.
    Holomorphic foliations in ruled surfaces.
    Trans. Amer. Math. Soc., 312(1):179-201, 1989. MR 0983870 (91e:32027)
  • 9. Xavier Gómez-Mont.
    Unfoldings of holomorphic foliations.
    Publ. Mat., 33(3):501-515, 1989. MR 1038486 (91d:32026)
  • 10. Yu. S. Ilyashenko.
    Topology of phase portraits of analytic differential equations on a complex projective plane.
    Trudy Sem. Petrovsk., (4):83-136, 1978. MR 0524528 (84k:58164)
  • 11. J. P. Jouanolou.
    Hypersurfaces solutions d'une équation de Pfaff analytique.
    Math. Ann., 232(3):239-245, 1978. MR 0481129 (58:1274)
  • 12. David Marín and Marcel Nicolau.
    Riccati foliations on ruled surfaces.
    In preparation, 2005.
  • 13. Jean-François Mattei.
    Modules de feuilletages holomorphes singuliers. I. Équisingularité.
    Invent. Math., 103(2):297-325, 1991. MR 1085109 (92f:32056)
  • 14. Jean-François Mattei and Marcel Nicolau.
    Equisingular unfoldings of foliations by curves.
    Astérisque, 222(6):285-302, 1994.
    Complex analytic methods in dynamical systems (Rio de Janeiro, 1992). MR 1285392 (96c:32040)
  • 15. Isao Nakai.
    Separatrices for nonsolvable dynamics on $ (\mathbf{C},0)$.
    Ann. Inst. Fourier (Grenoble), 44(2):569-599, 1994. MR 1296744 (95j:58124)
  • 16. M. P. Painlevé.
    Leçons sur la théorie analytique des équations différentielles.
    Librairie scientifique A. Hermann, 1897.
    Leçons professées a Stockholm, 1895.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32G34, 34Mxx, 57R30

Retrieve articles in all journals with MSC (2000): 32G34, 34Mxx, 57R30


Additional Information

David Marín
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Barcelona, Spain
Email: davidmp@mat.uab.es

DOI: https://doi.org/10.1090/S0002-9939-06-08418-8
Keywords: Singular holomorphic foliation, complex fibred surface, deformation, unfolding, holonomy, rigidity
Received by editor(s): April 15, 2005
Received by editor(s) in revised form: June 29, 2005
Published electronically: June 8, 2006
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society