Proposition 18.5.5`A2602BC06784`

Proposition 18.5.5.A2602BC06784 If a Hermitian metric $\omega $ fails the bounded mass property of definition 13.14.14, there exists $U\in \mathrm{PSH}(X,\omega )$ such that

\[ P_\omega [U]=P_\omega [U]_{\mathcal I}=0, \qquad U\in \mathcal E_{\mathrm{env}}(X,\omega ,0), \qquad \mathrm{MA}^{\mathrm{ext}}_\omega (U)(X)=+\infty , \]

where $\mathcal E_{\mathrm{env}}(X,\omega ,0)$ is the relative envelope class of definition 16.28.61.

Proof. Take the potential $U$ of proposition 16.28.72, which has the stated ordinary envelope, class membership and infinite mass. Since $P_\omega [U]=V_\omega =0$, proposition 18.4.4 gives $P_\omega [U]_{\mathcal I}=0$. $\square$

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BibTeX citation

@misc{pome-tag-A2602BC06784,
  author = {{Pome contributors}},
  title = {Proposition 18.5.5},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/A2602BC06784/}
}

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Proposition 18.5.5A2602BC06784

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Proposition 18.5.5`A2602BC06784`