Theorem 5.18.182D610C4DD6AD

Theorem 5.18.18.2D610C4DD6ADLet $X$ be a connected compact Kähler manifold and $Y\subseteq X$ be a connected closed submanifold. Take a Kähler form $\omega $ on $X$ and $\varphi \in \mathrm{PSH}(Y,\omega |_ Y)$ such that $\omega |_ Y+\mathrm{dd}^{\mathrm{c}}\varphi $ is a Kähler current and $\mathrm{e}^{\varphi }$ is a Hölder continuous function on $Y$. Then there exists $\tilde{\varphi }\in \mathrm{PSH}(X,\omega )$ satisfying

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   $\tilde{\varphi }|_ Y=\varphi $;

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   $\omega _{\tilde{\varphi }}$ is a Kähler current.

In addition, if $\varphi $ has analytic singularities, then so does $\tilde{\varphi }$.