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...

We prove fibration theorems on compact Kähler manifolds with conditions on first cohomology groups of fundamental groups with respect to unitary representations into Hilbert spaces. If the fundamental group T of compact Kähler manifold X violates Property (T) of Kazhdan’s, then $H^1(Gamma,\Phi )\ne 0$ for some unitary representation $\Phi$. By our earlier work there exists a $d$-closed holomorphic 1-form with coefficients twisted by some unitary representation ${\Phi }^{\text{'}}$, possibly non-isomorphic to $\Phi$. Taking norms we obtains a positive...

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.

The aim of this work is to study global $3$-webs with vanishing curvature. We wish to investigate degree $3$ foliations for which their dual web is flat. The main ingredient is the Legendre transform, which is an avatar of classical projective duality in the realm of differential equations. We find a characterization of degree $3$ foliations whose Legendre transform are webs with zero curvature.

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

We define the notion of CR equivalence for Levi-flat foliations and compare in a local setting these foliations to their linear parts. We study also the situation where the foliation has a first integral ; a condition is given so that this integral is the real part of a holomorphic function.