Plurisubharmonic functions with logarithmic singularities

Bedford, E.; Taylor, B. A.

doi:10.5802/aif.1152

Bedford, E. ; Taylor, B. A.

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

Résumé
Abstract

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

ρ_{u} (z) = \underset{λ \to \infty}{lim sup} u (λ z) - log (λ z)

dans l’hyperplan $P^{n - 1}$ 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 $ρ_{u}$ n’est pas identiquement $- \infty$ . On obtient une formule intégrale qui relie la mesure de Monge-Ampère sur l’espace $C^{n}$ 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 ${r_{u}}$ . Par conséquent, on trouve qu’un ensemble polaire $E$ est contenu dans ${Ψ = - \infty}$ pour une fonction $u \in L_{ρ}$ , donc que l’ensemble de propagation $E^{*}$ , l’intersection des ensembles ${Ψ = - \infty}$ contenant $E$ , est polaire. Soit $A$ une hypersurface algébrique, $E \cap A = \emptyset$ , alors $E^{*}$ ne contient pas $A$ .

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

ρ_{u} (z) = \underset{λ \to \infty}{lim sup} u (λ z) - log (λ z)

defined on $P^{n - 1}$ , 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 $ρ_{u}$ is not identically $- \infty$ . We obtain an integral formula which connects the Monge-Ampère measure on the space $C^{n}$ with the Robin function on $P^{n - 1}$ . 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 ${Ψ = - \infty}$ for some $Ψ \in L_{ρ}$ , and so the polar propagator $E^{*}$ , given as the intersection of the sets ${Ψ = - \infty}$ containing $E$ , is polar. Ir $A$ is an algebraic hypersurface which is disjoint from $E$ , then $E^{*}$ cannot contain $A$ .

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.1152

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     author = {Bedford, E. and Taylor, B. A.},
     title = {Plurisubharmonic functions with logarithmic singularities},
     journal = {Annales de l'Institut Fourier},
     pages = {133--171},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {38},
     number = {4},
     year = {1988},
     doi = {10.5802/aif.1152},
     mrnumber = {90f:32016},
     zbl = {0626.32022},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1152/}
}

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Bedford, E.; Taylor, B. A. Plurisubharmonic functions with logarithmic singularities. Annales de l'Institut Fourier, Tome 38 (1988) no. 4, pp. 133-171. doi : 10.5802/aif.1152. https://www.numdam.org/articles/10.5802/aif.1152/

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