We find a bounded solution of the non-homogeneous Monge-Ampère equation under very weak assumptions on its right hand side.

It is described how both plurisubharmonicity and convexity of functions can be characterized in terms of simple to work with classes of holomorphic martingales, namely a class of driftless Itô processes satisfying a skew-symmetry property and a family of linear modifications of Brownian motion parametrized by a compact set.

Let $A_{\zeta }=\Omega -\overline{\rho \left(\zeta \right)·\Omega }$ be a family of generalized annuli over a domain U. We show that the logarithm of the Bergman kernel $K_{\zeta }\left(z\right)$ of $A_{\zeta }$ is plurisubharmonic provided ρ ∈ PSH(U). It is remarkable that $A_{\zeta }$ is non-pseudoconvex when the dimension of $A_{\zeta }$ is larger than one. For standard annuli in ℂ, we obtain an interesting formula for $\partial ²log{K}_{\zeta }/\partial \zeta \partial \zeta ̅$, as well as its boundary behavior.

In the moduli space ${ℳ}_d$ of degree $d$ rational maps, the bifurcation locus is the support of a closed $(1,1)$ positive current $T_{\mathrm{bif}}$ which is called the bifurcation current. This current gives rise to a measure ${\mu }_{\mathrm{bif}}:={\left(T_{\mathrm{bif}}\right)}^{2d-2}$ whose support is the seat of strong bifurcations. Our main result says that $\mathrm{supp}\left({\mu }_{\mathrm{bif}}\right)$ has maximal Hausdorff dimension $2(2d-2)$. As a consequence, the set of degree $d$ rational maps having $(2d-2)$ distinct neutral cycles is dense in a set of full Hausdorff dimension.

The aim of the paper is to investigate subextensions with boundary values of certain plurisubharmonic functions without changing the Monge-Ampère measures. From the results obtained, we deduce that if a given sequence is convergent in $C_{n-1}$-capacity then the sequence of the Monge-Ampère measures of subextensions is weakly*-convergent. As an application, we investigate the Dirichlet problem for a nonnegative measure μ in the class ℱ(Ω,g) without the assumption that μ vanishes on all pluripolar sets.

We prove that subextension of certain plurisubharmonic functions is always possible without increasing the total Monge-Ampère mass.

Let $D$ be a pseudoconvex domain in ${ℂ}_t^k\times {ℂ}_z^n$ and let $\phi$ be a plurisubharmonic function in $D$. For each $t$ we consider the $n$-dimensional slice of $D$, $D_t=\{z;(t,z)\in D\}$, let ${\phi }^t$ be the restriction of $\phi$ to $D_t$ and denote by $K_t(z,\zeta )$ the Bergman kernel of $D_t$ with the weight function ${\phi }^t$. Generalizing a recent result of Maitani and Yamaguchi (corresponding to $n=1$ and $\phi =0$) we prove that $log{K}_t(z,z)$ is a plurisubharmonic function in $D$. We also generalize an earlier results of Yamaguchi concerning the Robin function and discuss similar results in the setting...

Let $\mathrm{\Sigma }$ be the boundary of the unit ball $\mathrm{\Omega }$ of ${𝐂}^n$. A set of second order linear partial differential operators, tangential to $\mathrm{\Sigma }$, is explicitly given in such a way that, for $n\ge 3$, the corresponding PDE caractherize the trace of the solution of the pluriharmonic problem (either “in the large” or “local”), relative to $\mathrm{\Omega }$.

We try to find a geometric interpretation of the wedge product of positive closed laminar currents in C2. We say such a wedge product is geometric if it is given by intersecting the disks filling up the currents. Uniformly laminar currents do always intersect geometrically in this sense. We also introduce a class of strongly approximable laminar currents, natural from the dynamical point of view, and prove that such currents intersect geometrically provided they have continuous potentials.

0. Introduction. Nous donnons ici une étude systématique des systèmes doublement orthogonaux "de Bergman" et leurs applications à certains aspects de l'analyse pluricomplexe: espaces de fonctions holomorphes, fonctions séparément analytiques. C'est en quelque sorte un article de synthèse. On y trouve cependant des démonstrations détaillées qui n'ont paru nulle part ailleurs.