Proposition 5.17.9`5F9E17A436E9`

Proposition 5.17.9.5F9E17A436E9 Fix a closed smooth real $(1,1)$-form $\theta $ on $X$. Let $\varphi \in \mathrm{PSH}(X,\theta )$, and $B\Subset B' \subseteq X$ be two holomorphic balls. Assume that $\varphi $ is locally bounded on $B'$. Then there is a unique $\psi \in \mathrm{PSH}(X,\theta )$ so that $\psi $ is bounded on an open neighborhood of $\overline{B}$, and

\[ \begin{cases} \left( \theta |_ B+\mathrm{dd}^{\mathrm{c}}\psi |_ B \right)^ n = 0,\\ \psi = \varphi \quad \textup{on } X\setminus B. \end{cases} \]

Moreover, $\psi \geq \varphi $.

Proof. By the Dolbeault–Grothendieck lemma, after possibly shrinking $B'$, we may write $\theta =\mathrm{dd}^{\mathrm{c}}g$ for some bounded function $g\in C^{\infty }(B')$. We can then use $g$ to translate $\theta $-psh functions to psh functions, and our assertion follows immediately from proposition 5.14.21. $\square$

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@misc{pome-tag-5F9E17A436E9,
  author = {{Pome contributors}},
  title = {Proposition 5.17.9},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/5F9E17A436E9/}
}

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Proposition 5.17.95F9E17A436E9

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Proposition 5.17.9`5F9E17A436E9`