Plurisubharmonic functions with logarithmic singularities

Bedford, E., and Taylor, B. A.. "Plurisubharmonic functions with logarithmic singularities." Annales de l'institut Fourier 38.4 (1988): 133-171. <http://eudml.org/doc/74812>.

@article{Bedford1988,
abstract = {To a plurisubharmonic function $u$ on $\{\bf C\}^ n$ with logarithmic growth at infinity, we may associate the Robin function\begin\{\}\rho \_u(z) = \displaystyle \limsup \_\{\lambda \rightarrow \infty \} u(\lambda z) - \log (\lambda z)\end\{\}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 $A$.},
author = {Bedford, E., Taylor, B. A.},
journal = {Annales de l'institut Fourier},
keywords = {complex Monge-Ampère operator; Green function; plurisubharmonic functions with logarithmic singularities; pluripolar set},
language = {eng},
number = {4},
pages = {133-171},
publisher = {Association des Annales de l'Institut Fourier},
title = {Plurisubharmonic functions with logarithmic singularities},
url = {http://eudml.org/doc/74812},
volume = {38},
year = {1988},
}

TY - JOUR
AU - Bedford, E.
AU - Taylor, B. A.
TI - Plurisubharmonic functions with logarithmic singularities
JO - Annales de l'institut Fourier
PY - 1988
PB - Association des Annales de l'Institut Fourier
VL - 38
IS - 4
SP - 133
EP - 171
AB - To a plurisubharmonic function $u$ on ${\bf C}^ n$ with logarithmic growth at infinity, we may associate the Robin function\begin{}\rho _u(z) = \displaystyle \limsup _{\lambda \rightarrow \infty } u(\lambda z) - \log (\lambda z)\end{}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 $A$.
LA - eng
KW - complex Monge-Ampère operator; Green function; plurisubharmonic functions with logarithmic singularities; pluripolar set
UR - http://eudml.org/doc/74812
ER -