En résumé, on retiendra que seules les surfaces d’Inoue-Hirzebruch et les surfaces génériques admettent un feuilletage holomorphe. Sur les surfaces d’Inoue-Hirzebruch il existe exactement deux feuilletages et sur les surfaces génériques au plus un. Le lieu singulier de la réunion des courbes rationnelles coïncide avec le lieu singulier du feuilletage. Les courbes rationnelles sont des feuilles en dehors des points singuliers du feuilletage.

Dans cet article, nous étudions le groupoïde de Galois d’un germe de feuilletage holomorphe de codimension un. Nous associons à ce $𝒟$-groupoïde de Lie un invariant biméromorphe  : le rang transverse. Nous étudions en détails les relations entre cet invariant, l’existence de suites de Godbillon-Vey particulières et l’existence d’une intégrale première dans une extension fortement normale du corps différentiel des germes de fonctions méromorphes. Nous obtenons ainsi une généralisation d’un théorème...

For germs of singularities of holomorphic foliations in $({ℂ}^2,0)$ which are regular after one blowing-up we show that there exists a functional analytic invariant (the transverse structure to the exceptional divisor) and a finite number of numerical parameters that allow us to decide whether two such singularities are analytically equivalent. As a result we prove a formal-analytic rigidity theorem for this kind of singularities.

Let $ℱ$ be a holomorphic one-dimensional foliation on ${\mathbb{P}}^n$ such that the components of its singular locus $\Sigma$ are curves $C_i$ and points $p_j$. We determine the number of $p_j$, counted with multiplicities, in terms of invariants of $ℱ$ and $C_i$, assuming that $ℱ$ is special along the $C_i$. Allowing just one nonzero dimensional component on $\Sigma$, we also prove results on when the foliation happens to be determined by its singular locus.

Given a foliation F in an algebraic surface having a rational first integral a genus formula for the general solution is obtained. In the case S = P2 some new counter-examples to the classic formulation of the Poincaré problem are presented. If S is a rational surface and F has singularities of type (1, 1) or (1,-1) we prove that the general solution is a non-singular curve.

We prove that a foliation on $\mathbf{C}{P}^2$ with hyperbolic singularities and with “many" parabolic leaves (i.e. leaves without Green functions) is in fact a linear foliation. This is done in two steps: first we prove that there exists an algebraic leaf, using the technique of harmonic measures, then we show that the holonomy of this leaf is linearizable, from which the result follows easily.

Let $X$ be a smooth foliation with complex leaves and let $\mathcal{D}$ be the sheaf of germs of smooth functions, holomorphic along the leaves. We study the ringed space $\left(X,\mathcal{D}\right)$. In particular we concentrate on the following two themes: function theory for the algebra $\mathcal{D}\left(X\right)$ and cohomology with values in $\mathcal{D}$.

Si illustrano alcuni sviluppi della teoria delle foliazioni di Monge-Ampère e delle sue applicazioni alla classificazione delle varietà complesse non compatte.

We are interested on families of formal power series in $\text{(}{ℂ}^{},0\text{)}$ parameterized by ${ℂ}^n$ ($\widehat{f}={\sum }_{m=0}^{\infty }{P}_m(x_1,\cdots ,x_n)x^m$). If every $P_m$ is a polynomial whose degree is bounded by a linear function ($deg{P}_m\le Am+B$ for some $A\>0$ and $B\ge 0$) then the family is either convergent or the series $\widehat{f}(c_1,\cdots ,c_n,x)\notin ℂ\left\{x\right\}$ for all $(c_1,\cdots ,c_n)\in {ℂ}^n$ except a pluri-polar set. Generalizations of these results are provided for formal objects associated to germs of diffeomorphism (formal power series, formal meromorphic functions, etc.). We are interested on describing the nature of the set of parameters where...

Let $ℱ$ be a holomorphic foliation by curves on ${\mathbb{P}}^3$. We treat the case where the set $Sing\left(ℱ\right)$ consists of disjoint regular curves and some isolated points outside of them. In this situation, using Baum-Bott’s formula and Porteuos’theorem, we determine the number of isolated singularities, counted with multiplicities, in terms of the degree of $ℱ$, the multiplicity of $ℱ$ along the curves and the degree and genus of the curves.

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....

A formula of Matsuo Oka (1990) expresses the Milnor number of a germ of a complex analytic map with a generic principal part in terms of the Newton polyhedra of the components of the map. In this paper this formula is generalized to the case of the index of a 1-form on a local complete intersection singularity (Theorem 1.10, Corollaries 1.11, 4.1). In particular, the Newton polyhedron of a 1-form is defined (Definition 1.6). This also simplifies the Oka formula in some particular cases (Propositions...