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

TL;DR

This paper proves that under a holomorphic retraction condition, certain K"ahler currents on a submanifold can be extended to the whole manifold, broadening the class of manifolds where such extensions are possible.

Contribution

It establishes extension results for K"ahler currents on submanifolds with holomorphic retraction, including non-projective compact K"ahler manifolds, improving previous results.

Findings

Extension of K"ahler currents under holomorphic retraction

Applicable to non-projective compact K"ahler manifolds

Generalizes previous extension theorems

Abstract

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 $\varphi$ 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 $\Phi$ on $X$, such that $\omega+\sqrt{-1}\partial\bar\partial \Phi\geq \varepsilon'\omega$ for some $\varepsilon'>0$. 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.