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

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 $\phi$ be a strictly $\omega {\mathrm{\mid }}_V$-psh function on $V$. In this paper, we prove that there is a strictly $\omega$-psh function $\mathrm{\Phi }$ on $X$, such that $\mathrm{\Phi }{\mathrm{\mid }}_V=\phi$. 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.