Plurisubharmonic functions with logarithmic singularities

Bedford, E.; Taylor, B. A.

Annales de l'Institut Fourier, Tome 38 (1988) no. 4, pp. 133-171.

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

$${\displaystyle {\rho }_u\left(z\right)=\underset{\lambda \to \infty }{lim\; sup}u\left(\lambda z\right)-log\left(\lambda z\right)}$$

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

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

$${\displaystyle {\rho }_u\left(z\right)=\underset{\lambda \to \infty }{lim\; sup}u\left(\lambda z\right)-log\left(\lambda z\right)}$$

defined on ${\mathbf{P}}^{n-1}$, the hyperplane at infinity. We study the classes $L_{+}$, and (respectively) $L_p$ of plurisubharmonic functions which have the form $u=log(1+|z\left|\right)+O\left(1\right)$ 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 ${\mathbf{C}}^n$ with the Robin function on ${\mathbf{P}}^{n-1}$. 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 $\{\Psi =-\infty \}$ for some $\Psi \in L_{\rho }$, and so the polar propagator $E^{*}$, given as the intersection of the sets $\{\Psi =-\infty \}$ containing $E$, is polar. Ir $A$ is an algebraic hypersurface which is disjoint from $E$, then $E^{*}$ cannot contain $A$.

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