Corollary 18.4.9`B5BE9775FBF5`

Corollary 18.4.9.B5BE9775FBF5 Let $S$ be a closed positive divisorial current of finite mass on a connected compact complex manifold $Z$, and choose a smooth residual decomposition $S=\beta +\mathrm{dd}^{\mathrm{c}}v$ as in corollary 5.38.6. For every Hermitian form $\Omega $ with $v\in \mathrm{PSH}(Z,\Omega )$,

\[ P_\Omega [v]=P_\Omega [v]_{\mathcal I}\sim v. \]

Proof. Set $q=P_\Omega [v]_{\mathcal I}$. Projection gives $q\sim _{\mathcal I}v$, and lemma 5.38.5 gives $q\preceq v$. Thus $q$ is a competitor for $P_\Omega [v]$. Together with proposition 18.4.4, this yields $v-\sup _ Zv\leq P_\Omega [v]=q\preceq v$. $\square$

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@misc{pome-tag-B5BE9775FBF5,
  author = {{Pome contributors}},
  title = {Corollary 18.4.9},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/B5BE9775FBF5/}
}

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Corollary 18.4.9B5BE9775FBF5

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Corollary 18.4.9`B5BE9775FBF5`