Foliations with algebraic limit sets. (English) Zbl 0769.57017

Camacho, C.; Lins Neto, Alcides; Sad, P.

Ann. Math. (2) 136, No. 2, 429-446 (1992).

This paper describes codimension one foliations on \(\mathbb{P}^ n(\mathbb{C})\). The limit set of a foliation \(\mathcal F\) is \({\mathcal L}({\mathcal F})=\overline{\bigcup_ L\lim(L)}\), where \(\lim(L) = \bigcap_{m \geq 1}\overline{L\setminus K_ m}\) for any leaf \(L\), with \(K_ m\) an exhausting sequence of compacta. The simplest foliations are those with algebraic limit set. Under the two extra conditions: (C1) the intersection of the singular set with the codimension one components \({\mathcal L}_ 1({\mathcal F})\) of \({\mathcal L}({\mathcal F})\) is nondicritical, and \({\mathcal L}_ 1({\mathcal F})\) contains all separatrices of its singularities, (C2) any irreducible component of \({\mathcal L}_ 1({\mathcal F})\) contains an attractor in its holonomy group, the authors show that \(\mathcal F\) is the pull-back of a linear foliation \({\mathcal L}: \lambda_ 1v du-\lambda_ 2u dv = 0\), \(\lambda_ 1 \neq 0 \neq \lambda_ 2\), on \(\mathbb{P}^ 2\) under a rational map.
The main part of the work consists in proving the theorem for \(n=2\), using a resolution of the singularities of \(\mathcal F\).

Reviewer: J.Stevens (Hamburg)

MSC:

57R30	Foliations in differential topology; geometric theory
32S65	Singularities of holomorphic vector fields and foliations
32S45	Modifications; resolution of singularities (complex-analytic aspects)

Keywords:

resolution of singularities; codimension one foliations on \(\mathbb{P}^ n(\mathbb{C})\); limit set of a foliation; attractor; holonomy group