Let $X$ be a compact Kähler manifold and $\Delta$ be a $ℝ$-divisor with simple normal crossing support and coefficients between $1/2$ and $1$. Assuming that $K_X+\Delta$ is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on $X\setminus \mathrm{Supp}\left(\Delta \right)$ having mixed Poincaré and cone singularities according to the coefficients of $\Delta$. As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair $(X,\Delta )$.

Les fonctions plurisousharmoniques négatives dans un domaine Ω de ℂⁿ forment un cône convexe. Nous considérons les points extrémaux de ce cône, et donnons trois exemples. En particulier, nous traitons le cas de la fonction de Green pluricomplexe. Nous calculons celle du bidisque, lorsque les pôles se situent sur un axe. Nous montrons que cette fonction ne coïncide pas avec la fonction de Lempert correspondante. Cela donne un contre-exemple à une conjecture de Dan Coman.

We prove, among other results, that $min(u,v)$ is plurisubharmonic (psh) when $u,v$ belong to a family of functions in $\mathrm{psh}\left(D\right)\cap {\Lambda }_{\alpha }\left(D\right),$ where ${\Lambda }_{\alpha }\left(D\right)$ is the $\alpha$-Lipchitz functional space with $1<\alpha <2.$ Then we establish a new characterization of holomorphic functions defined on open sets of ${ℂ}^n.$

We generalize a theorem of Siciak on the polynomial approximation of the Lelong class to the setting of toric manifolds with an ample line bundle. We also characterize Lelong classes by means of a growth condition on toric manifolds with an ample line bundle and construct an example of a nonample line bundle for which Siciak's theorem does not hold.

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 $Y$.

We construct $d$-closed and $d{d}^c$-closed positive currents associated to a holomorphic map $\phi$ via cluster points of normalized weighted truncated image currents. They are constructed using analogues of the Ahlfors length-area inequality in higher dimensions. Such classes of currents are also referred to as Ahlfors currents. We give some applications to equidistribution problems in value distribution theory.

A compact set $K\subset {ℂ}^N$ satisfies Łojasiewicz-Siciak condition if it is polynomially convex and there exist constants B,β > 0 such that $V_K\left(z\right)\ge B{\left(dist(z,K)\right)}^{\beta }$ if dist(z,K) ≤ 1. (LS) Here $V_K$ denotes the pluricomplex Green function of the set K. We cite theorems where this condition is necessary in the assumptions and list known facts about sets satisfying inequality (LS).

The main result of this paper is the following: if a compact subset E of ${ℝ}^n$ is UPC in the direction of a vector $v\in S^{n-1}$ then E has the Markov property in the direction of v. We present a method which permits us to generalize as well as to improve an earlier result of Pawłucki and Pleśniak [PP1].

Consider the normed space $(\mathbb{P}\left({ℂ}^N\right),|\left|·\right|\left|\right)$ of all polynomials of N complex variables, where || || a norm is such that the mapping $L_g:(\mathbb{P}\left({ℂ}^N\right),\left|\right|·\left|\right|\left)\ni f\mapsto gf\in \right(\mathbb{P}\left({ℂ}^N\right),\left|\right|·\left|\right|)$ is continuous, with g being a fixed polynomial. It is shown that the Markov type inequality $|\partial /\partial z_j{P\left|\right|\le M\left(degP\right)}^m\left|\right|P\left|\right|$, j = 1,...,N, $P\in \mathbb{P}\left({ℂ}^N\right)$, with positive constants M and m is equivalent to the inequality $\left|\right|{\partial }^N/\partial z₁...\partial z_{N}P\left|\right|\le M^{\text{'}}{\left(degP\right)}^{m^{\text{'}}}\left|\right|P\left|\right|$, $P\in \mathbb{P}\left({ℂ}^N\right)$, with some positive constants M’ and m’. A similar equivalence result is obtained for derivatives of a fixed order k ≥ 2, which can be more specifically formulated in the language of normed algebras. In...

We study two known theorems regarding Hermitian matrices: Bellman's principle and Hadamard's theorem. Then we apply them to problems for the complex Monge-Ampère operator. We use Bellman's principle and the theory for plurisubharmonic functions of finite energy to prove a version of subadditivity for the complex Monge-Ampère operator. Then we show how Hadamard's theorem can be extended to polyradial plurisubharmonic functions.

In our earlier paper [CKZ], we proved that any plurisubharmonic function on a bounded hyperconvex domain in ${ℂ}^n$ with zero boundary values in a quite general sense, admits a plurisubharmonic subextension to a larger hyperconvex domain. Here we study important properties of its maximal subextension and give informations on its Monge-Ampère measure. More generally, given a quasi-plurisubharmonic function $\varphi$ on a given quasi-hyperconvex domain $D\subset X$ of a compact Kähler manifold $(X,\omega )$, with well defined Monge-Ampère...

Soit $K$ un compact polynomialement convexe de ${\mathbf{C}}^n$ et $V_K$ son “potentiel logarithmique extrémal” dans ${\mathbf{C}}^n$. Supposons que $K$ est régulier (i.e. $V_K$ continue) et soit $f$ une fonction holomorphe sur un voisinage de $K$. On construit alors une suite $\{{\mathbf{P}}_{\ell }{\}}_{\ell \ge 1}$ de polynôme de $n$ variables complexes avec deg$({\mathbf{P}}_{)}\le \ell$ pour $\ell \ge 1$, telle que l’erreur d’approximation ${\max }_{z\in K}|f\left(z\right)-P_{\ell }\left(z\right)|$ soit contrôlée de façon assez précise en fonction du “pseudorayon de convergence” de $f$ par rapport à $K$ et du degré de convergence $\ell$. Ce résultat est ensuite utilisé pour étendre...

Soit $X$ un schéma projectif intègre défini sur un corps de nombres $F$ ; soit $L$ un fibré en droites ample sur $X$ muni d’une métrique adélique semi-positive au sens de Zhang. Les résultats principaux de cet article sont :(1)Une formule qui calcule les hauteurs locales (relativement à $L$) d’un diviseur de Cartier sur $X$ comme des « mesures de Mahler » généralisées, c’est-à-dire les intégrales de fonctions de Green pour $D$ contre des mesures associées à $L$ ;(2)Un théorème d’équidistribution des points de « petite »...