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 $\mathcal{D}$.

Nous présentons quelques résultats au sujet des groupes engendrés par trois involutions antiholomorphes dans le cadre du plan hyperbolique complexe ${\mathbf{H}}_{ℂ}^2$.

Let $D$ be a bounded symmetric domain in ${ℂ}^2$ and $\Gamma \subset {\mathrm{Aut}}^{0}D$ an irreducible arithmetic lattice which operates freely on $D$. We prove that the cusp–compactification $\bar{X}$ of $X=D/\Gamma$ is hyperbolic.

We give a description of bounded pseudoconvex Reinhardt domains, which are complete for the Carathéodory inner distance.

We study an adaptation to the logarithmic case of the Kobayashi-Eisenman pseudo-volume form, or rather an adaptation of its variant defined by Claire Voisin, for which she replaces holomorphic maps by holomorphic $K$-correspondences. We define an intrinsic logarithmic pseudo-volume form ${\Phi }_{X,D}$ for every pair $(X,D)$ consisting of a complex manifold $X$ and a normal crossing Weil divisor $D$ on $X$, the positive part of which is reduced. We then prove that ${\Phi }_{X,D}$ is generically non-degenerate when $X$ is projective and $K_X+D$...

Some known localization results for hyperconvexity, tautness or $k$-completeness of bounded domains in ${ℂ}^n$ are extended to unbounded open sets in ${𝒞}^n$.

In 1981 J. Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate.In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2: in a logarithmic surface with logarithmic irregularity $2$ and logarithmic Kodaira dimension $2$, any...

For low order jets, it is known how to construct meromorphic frames on the space of the so-called vertical $k$-jets$J_{\mathsf{\text{vert}}}^k\left(𝒳\right)$ of the universal hypersurface$$𝒳\subset {\mathbb{P}}^{n+1}\times {\mathbb{P}}^{\frac{(n+1+d)!}{\left(\right(n+1)!d!)}-1}$$parametrizing all projective hypersurfaces $X\subset {\mathbb{P}}^{n+1}\left(ℂ\right)$ of degree $d$. In 2004, for $k=n$, Siu announced that there exist two constants $c_n\ge 1$ and $c_n^{\prime }\ge 1$ such that the twisted tangent bundle$$T_{J_{\mathsf{\text{vert}}}^n\left(𝒳\right)}\otimes {𝒪}_{{\mathbb{P}}^{n+1}}\left(c_n\right)\otimes {𝒪}_{{\mathbb{P}}^{\frac{(n+1+d)!}{\left(\right(n+1)!d!)}-1}}\left(c_n^{\prime }\right)$$is generated at every point by its global sections. In the present article, we establish this property outside a certain exceptional algebraic subset $\Sigma \subset J_{\mathsf{\text{vert}}}^n\left(𝒳\right)$ defined by the vanishing of certain Wronskians,...