Proposition 16.6.22.10F9DDE25703 Let $\varphi \in \mathrm{PSH}(X,\theta )$. Assume that $\varphi $ has analytic singularities. Then
B621808C9E65
16.6.23
\begin{equation} \varphi \sim P_{\theta }[\varphi ] \sim P_{\theta }[\varphi ]_{\mathcal{I}}. \end{equation}
Hence, $P_{\theta }[\varphi ]$ and $P_{\theta }[\varphi ]_{\mathcal{I}}$ both have analytic singularities.
In particular,
D91D171A0296
16.6.24
\begin{equation} \int _ X \theta _{\varphi }^ n= \operatorname{vol}\theta _{\varphi }. \end{equation}
Proof.
First observe that (16.6.24) follows from (16.6.23) and theorem 13.13.11. It remains to establish (16.6.23).
In view of proposition 16.6.18, it suffices to show that
9169FBAAFD41
16.6.25
\begin{equation} P_{\theta }[\varphi ]_{\mathcal{I}}\preceq \varphi . \end{equation}
By proposition 16.6.10, proposition 16.4.1 and theorem 5.18.8, after passing to a log resolution we may assume that $\varphi $ has log singularities along an effective $\mathbb Q$-divisor $D$. This reduction does not change the assertion, since both the $P$- and the $\mathcal I$-envelope are compatible with bimeromorphic pull-back.
After multiplying $\theta $ and $\varphi $ by a sufficiently divisible positive integer, and then dividing the final comparison by the same integer, we may assume that $D$ is integral. In this case the Bergman-type quasi-equisingular approximants of $\varphi $ associated to the ideals $\mathcal I(2^ j\varphi )$ have the same singularity type as $\varphi $.
Since
\[ P_\theta [\varphi ]_{\mathcal I}\sim _{\mathcal I}\varphi \]
by proposition 16.6.11, we have
\[ \mathcal I(2^ jP_\theta [\varphi ]_{\mathcal I})=\mathcal I(2^ j\varphi ). \]
Thus the quasi-equisingular approximants $\eta _ j$ of $P_\theta [\varphi ]_{\mathcal I}$ and $\varphi _ j$ of $\varphi $ may be chosen with the same singularity type $(2^{-j},\mathcal I(2^ j\varphi ))$. Consequently $\eta _ j\sim \varphi _ j\sim \varphi $ for $j\gg 1$. Since $\eta _ j\searrow P_\theta [\varphi ]_{\mathcal I}$, choosing one such large $j$ gives
\[ P_\theta [\varphi ]_{\mathcal I}\leq \eta _ j\leq \varphi +C_ j, \]
and hence (16.6.25) holds.
$\square$
Pome
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