Codimension one holomorphic foliations on PCn\mathbb P^n_{\mathbb C}: problems in complex geometry

After a short review on foliations, we prove that a codimension 1 holomorphic foliation on $\mathbb P^3_{\mathbb C}$ with simple singularities is given by a closed rational 1-form. The proof uses Hironaka-Matsumura prolongation theorem of formal objects

Pinceaux linéaires de feuilletages sur CP(3) et conjecture de Brunella

The general element of a pencil of Cod 1 foliation in CP(3) either has an invariant surface or contains a subfoliation by algebraic curves

Feuilletages en droites, équations des eikonales et aures équations différentielles

Differential Equation and Singularities. 60th years of J.M.Aroca, Felipe Cano Torres éditeurInternational audienceWe study complex vector fields with straight lines trajectories. Some applications to rational solutions of ikonals equations are give

Feuilletages et actions de groupes sur les espaces projectifs

A holomorphic foliation $\mathscr{F}$ on a compact complex manifold $M$ is said to be an $\mathscr{L}$-foliation if there exists an action of a complex Lie group $G$ such that the generic leaf of $\mathscr{F}$ coincides with the generic orbit of $G$. We study $\mathscr{L}$-foliations of codimension one, in particular in projective space, in the spirit of classical invariant theory, but here the invariants are sometimes transcendantal ones. We give a bestiary of examples and general properties. Some classification results are obtained in low dimensions