Plurisubharmonic functions with logarithmic singularities

Bedford, E. ; Taylor, B. A.

Annales de l'Institut Fourier, Tome 38 (1988), p. 133-171 / Harvested from Numdam

Résumé

On associe à une fonction $u$, plurisousharmonique dans ${\bf C}^n$ de croissance logarithmique à l’infini, la fonction de Robin

ρu(z)=lim supλ→∞u(λz)-log(λz)

dans l’hyperplan ${\bf P}^{n-1}$ à l’infini. On étudie $L_+$, la classe des fonctions de la forme $u=\log (1+|z|)+O(1)$ et $L_p$, la classe des fonctions pour lesquelles la fonction $\rho _u$ n’est pas identiquement $-\infty $. On obtient une formule intégrale qui relie la mesure de Monge-Ampère sur l’espace ${\bf C}^n$ et la fonction de Robin. Sous titre d’application, on donne un critère sur les mesures de Monge-Ampère d’une suite de fonctions $\lbrace u_j\rbrace \subseteq L_+$ qui est nécessaire et suffisante pour la convergence des fonctions de Robin $\lbrace r_u\rbrace $. Par conséquent, on trouve qu’un ensemble polaire $E$ est contenu dans $\lbrace \Psi =-\infty \rbrace $ pour une fonction $u\in L_\rho $, donc que l’ensemble de propagation $E^*$, l’intersection des ensembles $\lbrace \Psi =-\infty \rbrace $ contenant $E$, est polaire. Soit $A$ une hypersurface algébrique, $E\cap A = \emptyset$, alors $E^*$ ne contient pas $A$.

Abstract

To a plurisubharmonic function $u$ on ${\bf C}^ n$ with logarithmic growth at infinity, we may associate the Robin function

ρu(z)=lim supλ→∞u(λz)-log(λz)

defined on ${\bf P}^{n-1}$, the hyperplane at infinity. We study the classes $L_+$, and (respectively) $L_p$ of plurisubharmonic functions which have the form $u=\log (1+|z|)+O(1)$ and (respectively) for which the function $\rho _u$ is not identically $-\infty $. We obtain an integral formula which connects the Monge-Ampère measure on the space ${\bf C}^n$ with the Robin function on ${\bf P}^{n-1}$. As an application we obtain a criterion on the convergence of the Monge-Ampère measures of a sequence of functions in $L_+$ which is equivalent to the convergence of the associated Robin functions. As a consequence, it is shown that a polar set $E$ is contained in $\lbrace \Psi = -\infty \rbrace $ for some $\Psi \in L_\rho $, and so the polar propagator $E^*$, given as the intersection of the sets $\lbrace \Psi = -\infty \rbrace $ containing $E$, is polar. Ir $A$ is an algebraic hypersurface which is disjoint from $E$, then $E^*$ cannot contain $A$.

Détail

Publié le : 1988-01-01

MR 90f:32016 | Zbl 0626.32022

DOI : https://doi.org/10.5802/aif.1152

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