Extension of plurisubharmonic functions in the Lelong class

Abstract

Let X be an algebraic subvariety of C n and X be its closure in P n . In their paper [2] Coman-Guedj-Zeriahi proved that any plurisubharmonic function with logarithmic growth on X extends to a plurisubharmonic function with logarithmic growth on C n when the germs (X, a) in P n 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 C n . Proof of the Theorem 1.2We need some lemmas to prove Theorem 1.2.Lemma 2.1. Let X be as in Theorem 1.2 and let a ∈ X \ X. If two irreducible components X i and X j of the germ (X, a) are not linked then (X, a) =X i ∪X j whereX i andX j are germs of subvarieties of X at a such thatX i ∩X j ∩C n = ∅.