On the extension of Kähler currents on compact complex manifolds

Abstract

Let $(X,ω)$ be a compact Kähler manifold and let $V\subset X$ be a closed complex submanifold. Coman-Guedj-Zeriahi proposed the problem: is every $ω|_V$-plurisubharmonic function on $V$ the restriction of an $ω$-plurisubharmonic function on $X$? In this paper, we solve this problem affirmatively, even for a compact Hermitian manifold.