On the Extension of Quasiplurisubharmonic Functions

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 NS_ℝ(V) has this extension property.