Extension of plurisubharmonic functions in the Lelong class

Abstract

Let $X$ be an algebraic subvariety of $\mathbb{C}^{n}$ and $\overline{X}$ be its closure in $\mathbb{P}^{n}$. In their paper (J. Reine Angew. Math. 676 (2013), 33–49), Coman, Guedj and 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 $(\overline{X},a)$ in $\mathbb{P}^{n}$ are irreducible for all $a\in\overline{X}\setminus X$. In this paper we consider $X$ for which the germ $(\overline{X},a)$ is reducible for some $a\in\overline{X}\setminus X$ 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 $\mathbb{C}^{n}$.