Nel caso di una varietà di Banach complessa $X$, si costruisce una regolarizzata della metrica infinitesimale di Kobayashi. Se ne deduce una distanza integrata di Kobayashi e, se $X$ è iperbolica, si mostra che questa distanza è uguale alla distanza di Kobayashi.

Given a positive closed (1,1)-current $T$ defined on the regular locus of a projective variety $X$ with bounded mass near the singular part of $X$ and $Y$ an irreducible algebraic subset of $X$, we present uniform estimates for the locus inside $Y$ where the Lelong numbers of $T$ are larger than the generic Lelong number of $T$ along $Y$.

Let $(M,J,\omega )$ be a manifold with an almost complex structure $J$ tamed by a symplectic form $\omega$. We suppose that $M$ has the complex dimension two, is Levi-convex and with bounded geometry. We prove that a real two-sphere with two elliptic points, embedded into the boundary of $M$ can be foliated by the boundaries of pseudoholomorphic discs.

We consider a compact almost complex manifold $(M,J,\omega )$ with smooth Levi convex boundary $\partial M$ and a symplectic tame form $\omega$. Suppose that $S^2$ is a real two-sphere, containing complex elliptic and hyperbolic points and generically embedded into $\partial M$. We prove a result on filling $S^2$ by holomorphic discs.

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....