On topological rigidity of projective foliations. (English) Zbl 0933.32043

Lins Neto, A.; Sad, P.; Azevedo Scárdua, Bruno

Bull. Soc. Math. Fr. 126, No. 3, 381-406 (1998).

This paper deals with the following situation. Let \({\mathcal X}(n)\) denote the space of holomorphic foliations on the complex projective plane \({\mathbb C}\text{P}^2\), of degree \(n\), which leave invariant the line at infinity. Then a foliation \({\mathcal F}\in{\mathcal X}(n)\) is called topologically trivial in the class \({\mathcal X}(n)\) if any topologically trivial analytic deformation of \({\mathcal F}\) within this class is analytically trivial. Let \(\text{Rig}(n)\subset{\mathcal X}(n)\) denote the subset of such topologically rigid foliations in \({\mathcal X}(n)\).
A well-known and important theorem of Y. S. Ilyashenko establishes that \(\text{Rig}(n)\) is residual in \({\mathcal X}(n)\) 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 \({\mathcal X}(n)\), 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.

Reviewer: J.A.Álvarez López (Santiago de Compostela)

MSC:

32S65	Singularities of holomorphic vector fields and foliations
57R30	Foliations in differential topology; geometric theory

Keywords:

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

Citations:

Zbl 0434.34003

References:

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