On the extensions of Kähler currents on compact Kähler manifolds

Abstract

Let $(X,ω)$ be a compact Kähler manifold with a Kähler form $ω$ of complex dimension $n$, and $V\subset X$ is a compact complex submanifold of positive dimension $k<n$. Suppose that $V$ can be embedded in $X$ as a zero section of a holomorphic vector bundle or rank $n-k$ over $V$. Let $φ$ be a strictly $ω|_V$-psh function on $V$. In this paper, we prove that there is a strictly $ω$-psh function $Φ$ on $X$, such that $Φ|_V=φ$. This result gives a partial answer to an open problem raised by Collins-Tosatti and Dinew-Guedj-Zeriahi, for the case of Kähler currents. We also discuss possible extensions of Kähler currents in a big class.