On the Extension of Quasiplurisubharmonic Functions

Abstract

Let (V, ω) be a compact Kähler manifold such that V admits a cover by Zariski-open Stein sets with the property that ω has a strictly plurisubharmonic exhaustive potential on each element of the cover. If X ⊂ V is an analytic subvariety, we prove that any ω∣x-plurisubharmonic function on X extends to a ω-plurisubharmonic function on V.This result generalizes a previous result of ours on the extension of singular metrics of ample line bundles. It allows one to show that any transcendental Kähler class in the real Neron—Severi space NSℝ(V) has this extension property.