## Foliations with algebraic limit sets.(English)Zbl 0769.57017

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 $$\mathcal F$$.

### MSC:

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