We investigate the quadratic homogeneous holomorphic vector fields on ${\mathbf{C}}^n$ that are semicomplete, this is, those whose solutions are single-valued in their maximal definition domain. To a generic quadratic vector field we rationally associate some complex numbers that turn out to be integers in the semicomplete case, thus showing that the linear equivalence classes of semicomplete vector fields are contained in some sort of lattice in the space of linear equivalence classes of quadratic ones. We prove...

Let $ℱ:L\to TS$ be a foliation on a complex, smooth and irreducible projective surface $S$, assume $ℱ$ admits a holomorphic first integral $f:S\to {\mathbb{P}}^1$. If $h^0(S,{𝒪}_S(-n{𝒦}_S\left)\right)\>0$ for some $n\ge 1$ we prove the inequality: $(2n-1)(g-1)\le h^1(S,{ℒ}^{\text{'}-1}(-(n-1)K_S\left)\right)+h^0(S,{ℒ}^{\text{'}})+1$. If $S$ is rational we prove that the direct image sheaves of the co-normal sheaf of $ℱ$ under $f$ are locally free; and give some information on the nature of their decomposition as direct sum of invertible sheaves.

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...

Dans un article précédent [Singularité des flots holomorphes, Ann. Inst. Fourier, Grenoble, 46-2 (1996), 411-428], le deuxième auteur démontrait, en particulier, qu’un champ de vecteurs holomorphe complet sur une surface complexe ne peut posséder une singularité isolée dont le deuxième jet est nul. Nous nous proposons ici de donner une description précise des champs de vecteurs holomorphes complets sur les surfaces complexes qui possèdent une singularité isolée dont le premier jet est nul. Dans...

This paper presents a classification of plane dicritical nilpotent singularities, i.e. singularities which have nilpotent linear part and infinitely many separatrices. In particular the existence of meromorphic first integrals is discussed. The same ideas are applied to other kind of dicritical singularities.

We study germs of singular holomorphic vector fields at the origin of ${ℂ}^n$ of which the linear part is $1$-resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some “sectorial domains” which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms....

We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the “small divisors” are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth $\alpha$-Gevrey vector field with an hyperbolic linear part admits a smooth $\beta$-Gevrey transformation to a smooth $\beta$-Gevrey normal form. The Gevrey order $\beta$ depends on...

We show that the singular holomorphic foliations induced by dominant quasi-homogeneous rational maps fill out irreducible components of the space ${ℱ}_q(r,d)$ of singular foliations of codimension $q$ and degree $d$ on the complex projective space ${\mathbb{P}}^r$, when $1\le q\le r-2$. We study the geometry of these irreducible components. In particular we prove that they are all rational varieties and we compute their projective degrees in several cases.

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.

On démontre l’énoncé classique du théorème de décomposition de la polaire générique dans le contexte maximal des feuilletages courbes généralisées à modèle logarithmique non résonnant. On montre aussi la propriété d’éloignement des séparatrices pour le feuilletage polaire.

On démontre qu'une feuille transcendante d'un feuilletage analytique sur une surface fibrée doit intersecter toute courbe algébrique non invariante et non contenue dans une réunion de fibres de la fibration; comme application on montre qu'une équation différentielle algébrique qui possède une solution locale avec une singularité essentielle n'a pas de ramification mobile, ce qui généralise les théorèmes de Malmquist et Yosida.

Dans cet article nous étudions les feuilletages holomorphes réduits en dimension complexe 2. Plus précisément, nous caractérisons par leur espace de module analytique, ceux qui sont transversalement projectifs en dehors d'un sous-ensemble analytique propre. Ceci entraî ne que cette classe de feuilletages est obtenue par pull-back d'équations de Riccati. Nous montrons enfin que cette dernière propriété peut être mise en défaut dans le cas non réduit.

We prove that the algebraic multiplicity of a holomorphic vector field at an isolated singularity is invariant by $C^1$ equivalences.

We confirm a conjecture of Bernstein–Lunts which predicts that the characteristic variety of a generic polynomial vector field has no homogeneous involutive subvarieties besides the zero section and subvarieties of fibers over singular points.

Let $X$ be a normal projective variety, and let $A$ be an ample Cartier divisor on $X$. Suppose that $X$ is not the projective space. We prove that the twisted cotangent sheaf ${\Omega }_X\otimes A$ is generically nef with respect to the polarisation $A$. As an application we prove a Kobayashi-Ochiai theorem for foliations: if $ℱ⊊T_X$ is a foliation such that $detℱ\equiv i_{ℱ}A$, then $i_{ℱ}$ is at most the rank of $ℱ$.