In this paper we review the moduli theory of polarized CY manifolds. We briefly sketched Kodaira-Spencer-Kuranishi local deformation theory developed by the author and G. Tian. We also construct the Teichmüller space of polarized CY manifolds following the ideas of I. R. Shafarevich and I. I. Piatetski-Shapiro. We review the fundamental result of E. Viehweg about the existence of the course moduli space of polarized CY manifolds as a quasi-projective variety. Recently S. Donaldson computed the moment...

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,...

LVM and LVMB manifolds are a large family of non kähler manifolds. For instance, Hopf manifolds and Calabi-Eckmann manifolds can be seen as LVMB manifolds. The LVM manifolds have a natural action of a real torus and the quotient of this action is a polytope. This quotient allows us to relate closely LVM manifolds to the moment-angle manifolds studied by Buchstaber and Panov. Our aim is to generalize the polytope associated to a LVM manifold to the LVMB case and study the properties of this generalization....

We show that if X, Y are smooth, compact k-dimensional submanifolds of ℝⁿ and 2k+2 ≤ n, then each diffeomorphism ϕ: X → Y can be extended to a diffeomorphism Φ: ℝⁿ → ℝⁿ which is tame (to be defined in this paper). Moreover, if X, Y are real analytic manifolds and the mapping ϕ is analytic, then we can choose Φ to be also analytic. We extend this result to some interesting categories of closed (not necessarily compact) subsets of ℝⁿ, namely, to the category of Nash submanifolds...

The aim of the present paper is to study meromorphic extension spaces. The obtained results allow us to get the invariance of meromorphic extendibility under finite proper surjective holomorphic maps.

Nous mettons en perspective différentes méthodes de changement d’échelles et illustrons leur pertinence en mettant sur pieds des preuves simples et élémentaires de plusieurs théorèmes biens connus en analyse ou géométrie complexe. Les situations abordées sont variées et la plupart des théorèmes démontrés sont des classiques initialement obtenus entre la fin du xixe  et la seconde moitié du xxe  siècle.

We prove the existence of non-positively curved Kähler-Einstein metrics with cone singularities along a given simple normal crossing divisor of a compact Kähler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved Kähler-Einstein metrics with cone singularities. As an application we extend to this setting classical results of Lichnerowicz and Kobayashi on the parallelism and vanishing of appropriate holomorphic tensor fields.

Un des problèmes les plus intéressants de la géométrie différentielle complexe consiste à comprendre les classes de Kähler de variétés complexes admettant des métriques à courbure scalaire constante. La question de l’unicité a été récemment résolue par Donaldson, Mabuchi, Chen–Tian. Des liens forts avec la stabilité algébrique des variétés ont été mis en évidence. L’exposé s’attachera à exposer les idées nouvelles qui ont mené à ces résultats.