Total Rigidity of Polynomial Foliations on the Complex Projective Plane

Yu. S. Ilyashenko

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Analysis and singularities. Part 2, Tome 259 (2007), pp. 64-76

Résumé

Polynomial foliations of the complex plane are topologically rigid. Roughly speaking, this means that the topological equivalence of two foliations implies their affine equivalence. There exist various nonequivalent formalizations of the notion of topological rigidity. Generic polynomial foliations of fixed degree have the so-called property of absolute rigidity, which is the weakest form of topological rigidity. This property was discovered by the author more than 30 years ago. The genericity conditions imposed at that time were very restrictive. Since then, this topic has been studied by Shcherbakov, Gómez-Mont, Nakai, Lins Neto–Sad–Scárdua, Loray–Rebelo, and others. They relaxed the genericity conditions and increased the dimension. The main conjecture in this field states that a generic polynomial foliation of the complex plane is topologically equivalent to only finitely many foliations. The main result of this paper is weaker than this conjecture but also makes it possible to compare topological types of distant foliations.