18.5.3 Envelope equality and infinite mass`1A21A12C70DE`

For each $u\in \mathrm{PSH}(X,\theta )$, the equality

410A03897E45 18.5.4

\begin{equation} P_\theta [u]=P_\theta [u]_{\mathcal I} \end{equation}

is meaningful. We call it envelope goodness in the fixed background. Its relation to mass differs from the Kähler characterization: it can hold when the potential has infinite mass while its envelope has finite mass.

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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@misc{pome-tag-1A21A12C70DE,
  author = {{Pome contributors}},
  title = {18.5.3 Envelope equality and infinite mass},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/1A21A12C70DE/}
}

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18.5.3 Envelope equality and infinite mass1A21A12C70DE

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18.5.3 Envelope equality and infinite mass`1A21A12C70DE`