Let X be a germ of holomorphic vector field at the origin of Cn and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection...

The aim of this note is to give a clearer and more direct proof of the main result of another paper of the author. Moreover, we give some complementary results related to R-complete algebraic foliations with R a rational function of type ℂ*.

We study compact Kähler manifolds $X$ admitting nonvanishing holomorphic vector fields, extending the classical birational classification of projective varieties with tangent vector fields to a classification modulo deformation in the Kähler case, and biholomorphic in the projective case. We introduce and analyze a new class of $\mathrm{𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑖𝑎𝑙𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛𝑠}$, and show that they form a smooth subspace in the Kuranishi space of deformations of the complex structure of $X$. We extend Calabi’s theorem on the structure of compact Kähler...

We obtain a classification of codimension one holomorphic foliations on ${\mathbb{P}}^4$ with degenerate Gauss maps.

Nous considérons un germe de feuilletage holomorphe singulier non-dicritique $ℱ$ défini sur une boule fermée $\overline{𝔹}\subset {ℂ}^2$, satisfaisant des hypothèses génériques, de courbe de séparatrice $S$. Nous démontrons l’existence d’un voisinage ouvert $U$ de $S$ dans $\overline{𝔹}$ tel que, pour toute feuille $L$ de ${ℱ}_{\left|\right(U\setminus S)}$, l’inclusion naturelle $ı:L\hookrightarrow U\setminus S$ induit un monomorphisme ${ı}_{*}:{\pi }_1\left(L\right)\hookrightarrow {\pi }_1(U\setminus S)$ au niveau du groupe fondamental. Pour cela, nous introduisons la notion géométrique de « connexité feuilletée » avec laquelle nous réinterprétons la notion d’incompressibilité....

We give a necessary condition for a holomorphic vector field to induce an integrable osculating plane distribution and, using this condition, we give a characterization of such fields. We also give a generic classification for vector fields which have two invariant coordinate planes.

We prove the existence of minimal and rigid singular holomorphic foliations by curves on the projective space $ℂ{\mathbb{P}}^n$ for every dimension $n\ge 2$ and every degree $d\ge 2$. Precisely, we construct a foliation $ℱ$ which is induced by a homogeneous vector field of degree $d$, has a finite singular set and all the regular leaves are dense in the whole of $ℂ{\mathbb{P}}^n$. Moreover, $ℱ$ satisfies many additional properties expected from chaotic dynamics and is rigid in the following sense: if $ℱ$ is conjugate to another holomorphic foliation...

We give a complete topological classification of germs of holomorphic foliations in the plane under rather generic conditions. The key point is the introduction of a new topological invariant called monodromy representation. This monodromy contains all the relevant dynamical information, in particular the projective holonomy representations whose topological invariance was conjectured in the eighties by Cerveau and Sad and is proved here under mild hypotheses.

We present several characterizations and representations of semi-complete vector fields on the open unit balls in complex Euclidean and Hilbert spaces.

We study the simultaneous linearizability of $d$–actions (and the corresponding $d$-dimensional Lie algebras) defined by commuting singular vector fields in ${ℂ}^n$ fixing the origin with nontrivial Jordan blocks in the linear parts. We prove the analytic convergence of the formal linearizing transformations under a certain invariant geometric condition for the spectrum of $d$ vector fields generating a Lie algebra. If the condition fails and if we consider the situation where small denominators occur, then...

We study topology of leaves of $1$-dimensional singular holomorphic foliations of Stein manifolds. We prove that for a generic foliation all leaves, except for at most countably many, are contractible, the rest are topological cylinders. We show that a generic foliation is complex Kupka-Smale.

0n se donne une variété complexe $V$, compacte, de dimension complexe $n$, un champ de vecteurs $v$ holomorphe sur $V$, un fibré vecoriel $E$ de rang $r$ au dessus de $V$ et une $ℂ$-action ${\theta }_v$ sur $E$. Il est bien connu que si $v$ n’a pas de singularité, tous les nombres de Chern $c_I\left(E\right)⌢\left[V\right]$ sont nuls ($\left|I\right|=n$). Si $v$ a des singularités, Bott a démontré que ces nombres de Chern se localisent près de ces singularités donnant lieu à des résidus . Ces résidus ont été calculés d’abord par Bott dans le cas d’une singularité isolée non dégénérée,...

On peut construire facilement des exemples de connexions plates de rang $2$ sur ${\mathbb{P}}^2$ comme tirés en arrière de connexions sur ${\mathbb{P}}^1$. On donne un exemple de connexion qui ne peut être obtenue de cette manière. Cet exemple est construit à partir d’une solution algébrique de l’équation de Painlevé VI. On en déduit un feuilletage modulaire. La preuve de ce fait repose sur la classification des feuilletages sur les surfaces projectives par leurs dimensions de Kodaira, fruit du travail de Brunella, McQuillan et...

Let (F₁,..., Fₙ): ℂⁿ → ℂⁿ be a locally invertible polynomial map. We consider the canonical pull-back vector fields under this map, denoted by ∂/∂F₁,...,∂/∂Fₙ. Our main result is the following: if n-1 of the vector fields $\partial /\partial F_j$ have complete holomorphic flows along the typical fibers of the submersion $(F₁,...,F_{j-1},F_{j+1},...,Fₙ)$, then the inverse map exists. Several equivalent versions of this main hypothesis are given.

We investigate the interplay between invariant varieties of vector fields and the inflection locus of linear systems with respect to the vector field. Among the consequences of such investigation we obtain a computational criterion for the existence of rational first integrals of a given degree, bounds for the number of first integrals on families of vector fields, and a generalization of Darboux's criteria. We also provide a new proof of Gomez--Mont's result on foliations...