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

Jiafu Ning; Zhiwei Wang; Xiangyu Zhou

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arXiv · 2105.08224

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

Jiafu Ning·Zhiwei Wang·Xiangyu Zhou

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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.

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Jiafu Ning, Zhiwei Wang, Xiangyu Zhou. 2021-05-18. On the extension of K\"ahler currents on compact K\"ahler manifolds: holomorphic retraction case. https://arxiv.org/abs/2105.08224

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