On the extensions of K\"ahler currents on compact K\"{a}hler manifolds

Abstract

Let $(X,\omega)$ be a compact K\"{a}hler manifold with a K\"{a}hler form $\omega$ 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 $\varphi$ be a strictly $\omega|_V$-psh function on $V$. In this paper, we prove that there is a strictly $\omega$-psh function $\Phi$ on $X$, such that $\Phi|_V=\varphi$. This result gives a partial answer to an open problem raised by Collins-Tosatti and Dinew-Guedj-Zeriahi, for the case of K\"{a}hler currents. We also discuss possible extensions of K\"ahler currents in a big class.