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Proceedings of the American Mathematical Society
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
Published electronically: June 8, 2006
MathSciNet review: 2240672
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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.


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08418-8
PII: S 0002-9939(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.