Extension of Plurisubharmonic Functions in the Lelong Class

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Let X be an algebraic subvariety of Cⁿ and X be its closure in Pⁿ. In their paper CGZ Coman-Guedj-Zeriahi proved that any plurisubharmonic function with logarithmic growth on X extends to a plurisubharmonic function with logarithmic growth on Cⁿ when the germs (X,a) in Pⁿ are irreducible for all a∈ X X. In this paper we consider X for which the germ (X,a) is reducible for some a∈ X 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 Cⁿ.