Plurisubharmonic functions with logarithmic singularities
Annales de l'Institut Fourier, Volume 38 (1988) no. 4, p. 133-171

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

ρu(z)=lim supλu(λz)-log(λz)

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 -. 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=-} for some ΨL ρ , and so the polar propagator E * , given as the intersection of the sets {Ψ=-} containing E, is polar. Ir A is an algebraic hypersurface which is disjoint from E, then E * cannot contain A.

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

ρu(z)=lim supλu(λz)-log(λz)

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 -. 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 }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=-} pour une fonction uL ρ , donc que l’ensemble de propagation E * , l’intersection des ensembles {Ψ=-} contenant E, est polaire. Soit A une hypersurface algébrique, EA=, alors E * ne contient pas A.

@article{AIF_1988__38_4_133_0,
     author = {Bedford, E. and Taylor, B. A.},
     title = {Plurisubharmonic functions with logarithmic singularities},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {38},
     number = {4},
     year = {1988},
     pages = {133-171},
     doi = {10.5802/aif.1152},
     zbl = {0626.32022},
     mrnumber = {90f:32016},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1988__38_4_133_0}
}
Bedford, E.; Taylor, B. A. Plurisubharmonic functions with logarithmic singularities. Annales de l'Institut Fourier, Volume 38 (1988) no. 4, pp. 133-171. doi : 10.5802/aif.1152. http://www.numdam.org/item/AIF_1988__38_4_133_0/

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