Rigidity of certain holomorphic foliations
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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.References
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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
- 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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2240672