On the extension of Kähler currents on compact Kähler manifolds: holomorphic retraction case

Abstract

In the present paper, we show that given a compact Kähler manifold $(X,\omega )$ with a Kähler 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 $\omega {|}_V+\sqrt{-1}\partial \bar{\partial }\phi \ge \epsilon \omega {|}_V$ with $\epsilon >0$ can be extended to a quasi-plurisubharmonic function $\Phi$ on $X$, such that $\omega +\sqrt{-1}\partial \bar{\partial }\Phi \ge {\epsilon }^{\prime }\omega$ for some ${\epsilon }^{\prime }>0$. 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 Kähler manifolds which are not necessarily projective.

Improvements of arXiv: 2002.11381. Comments welcome!