Extension of Plurisubharmonic Functions in the Lelong Class

Abstract

Let $X$ be an algebraic subvariety of ${\mathbb{C}}^n$ and $\bar{X}$ be its closure in ${\mathbb{P}}^n.$ In their paper \cite{CGZ} Coman-Guedj-Zeriahi proved that any plurisubharmonic function with logarithmic growth on $X$ extends to a plurisubharmonic function with logarithmic growth on ${\mathbb{C}}^n$ when the germs $(\bar{X},a)$ in ${\mathbb{P}}^n$ are irreducible for all $a\in \bar{X}\setminus X.$ In this paper we consider $X$ for which the germ $(\bar{X},a)$ is reducible for some $a\in \bar{X}\setminus X$ and we give a necessary and sufficient condition for $X$ so that any plurisubharmonic function with logarithmic growth on $X$ extends to a plurisubharmonic function with logarithmic growth on ${\mathbb{C}}^n.$

arXiv commentRevised version of the paper with the same title