Proposition 16.3.21`4C5C62135747`

Proposition 16.3.21.4C5C62135747 Let $\varphi \in \mathrm{PSH}(X,\theta )$. Consider a model potential $\phi \in \mathrm{PSH}(X,\theta )_{>0}$. Assume that $\varphi \preceq \phi $. Then

\[ \sup _{\{ \phi \neq -\infty \} } (\varphi -\phi )=\sup _ X \varphi . \]

Proof. Observe that $\phi \leq 0$, so on the set $\{ \phi \neq -\infty \} $, we have

\[ \varphi -\phi \geq \varphi . \]

So the $\geq $ direction follows from corollary 5.15.21. Conversely, by assumption, we can find a constant $C>0$ so that

\[ \varphi -\sup _ X \varphi \leq \min \{ \phi +C,0 \} . \]

It follows that $\varphi -\sup _ X \varphi \leq P_{\theta }[\phi ]=\phi $. Therefore, the reverse inequality follows. $\square$

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@misc{pome-tag-4C5C62135747,
  author = {{Pome contributors}},
  title = {Proposition 16.3.21},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/4C5C62135747/}
}

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Proposition 16.3.214C5C62135747

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Proposition 16.3.21`4C5C62135747`