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Zbl 0933.32043
Lins Neto, A.; Sad, P.; Azevedo Scárdua, Bruno
On topological rigidity of projective foliations. (English)
[J] Bull. Soc. Math. Fr. 126, No.3, 381-406 (1998). ISSN 0037-9484

This paper deals with the following situation. Let ${\cal X}(n)$ denote the space of holomorphic foliations on the complex projective plane ${\Bbb C}\text{P}^2$, of degree $n$, which leave invariant the line at infinity. Then a foliation ${\cal F}\in{\cal X}(n)$ is called topologically trivial in the class ${\cal X}(n)$ if any topologically trivial analytic deformation of ${\cal F}$ within this class is analytically trivial. Let $\text{Rig}(n)\subset{\cal X}(n)$ denote the subset of such topologically rigid foliations in ${\cal X}(n)$. \par A well-known and important theorem of {\it Y. S. Ilyashenko} establishes that $\text{Rig}(n)$ is residual in ${\cal X}(n)$ for any $n\geq 2$ [Proc. Int. Cong. Math., Helsinki 1978, Vol. 2, 821-826 (1980; Zbl 0434.34003)]. This result is significantly improved in the present paper in the following two ways: On the one hand, the authors show that $\text{Rig}(n)$ contains an open dense subset of ${\cal X}(n)$, and on the other hand, they use a weaker notion of topologically trivial deformations, and thus a stronger notion of topological rigidity. They also give some information about the non-rigid foliations, which is a description in certain cases.
[J.A.Álvarez López (Santiago de Compostela)]

MSC 2000:: *32S65 Singularities of holomorphic vector fields
57R30 Foliations; geometric theory

Keywords: foliation; rigidity; holonomy group; non solvable group of diffeomorphisms; lamination

Citations: Zbl 0434.34003

Cited in: Zbl 1237.32004

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