On the extension of K\"ahler currents on compact K\"ahler manifolds: holomorphic retraction case

In the present paper, we show that given a compact K"ahler manifold $(X,\omega )$ with a K"ahler metric $\omega$, and a complex submanifold $V\subset X$ of positive dimension, if $V$ has a holomorphic retraction structure in $X$, then any quasi-plurisubharmonic function $\phi$ on $V$ such that $\omega|_V+\sqrt{-1}\partial\bar\partial\varphi\geq \varepsilon\omega|_V$with$\varepsilon>0$ can be extended to a quasi-plurisubharmonic function $\mathrm{\Phi }$ on $X$, such that $\omega +\sqrt{-1}\mathrm{\partial }\bar{\mathrm{\partial }}\mathrm{\Phi }\ge {\epsilon }^{\mathrm{\prime }}\omega$ for some ${\epsilon }^{\mathrm{\prime }}>0$. This is an improvement of results in \cite{WZ20}. Examples satisfying the assumption that there exists a holomorphic retraction structure contain product manifolds, thus contains many compact K"ahler manifolds which are not necessarily projective.