On the extension of quasiplurisubharmonic functions

TL;DR

This paper proves that under certain conditions on a compact Kähler manifold, any plurisubharmonic function defined on an analytic subvariety can be extended to the whole manifold, generalizing previous results on line bundle metrics.

Contribution

It extends the class of Kähler manifolds where plurisubharmonic functions on subvarieties can be globally extended, including transcendental classes.

Findings

Extension of plurisubharmonic functions on subvarieties in specific Kähler manifolds.

Generalization of previous results on singular metrics of ample line bundles.

Application to transcendental Kähler classes in the Neron-Severi space.

Abstract

Let $(V,\omega)$ be a compact K\"ahler manifold such that $V$ admits a cover by Zariski-open Stein sets with the property that $\omega$ has a strictly plurisubharmonic exhaustive potential on each element of the cover. If $X\subset V$ is an analytic subvariety, we prove that any $\omega|_X$-plurisubharmonic function on $X$ extends to a $\omega$-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 K\"ahler class in the real Neron-Severi space $NS_{\mathbb R}(V)$ has this extension property.