Pome89E4AF40A761For a potential $a$ with analytic singularities, the multiplier-ideal order controls the ordinary order: by lemma 5.38.11, every quasi-plurisubharmonic $v$ with $v\preceq _{\mathcal I}a$ satisfies $v\preceq a$. The ideal envelope of $a$ is then a competitor for its ordinary envelope, and the two envelopes coincide.
Proposition 18.4.8.EB95710FF4D7 Let $a\in \mathrm{PSH}(X,\theta )$ have analytic singularities. Then
In particular the common envelope has analytic singularities with the same coefficient and ideal as $a$, up to a bounded remainder.
Proof. Set $q=P_\theta [a]_{\mathcal I}$. Projection gives $q\sim _{\mathcal I}a$, so lemma 5.38.11 gives $q\preceq a$. Since $q\leq 0$, it is a competitor for $P_\theta [a]$. Together with proposition 18.4.4, this yields
Hence the two envelopes are equal and ordinarily equivalent to $a$. Since they differ from $a$ by a bounded function, they inherit its analytic presentation. $\square$
The same argument applies to the potential of a countable divisorial current of finite mass, with lemma 5.38.5 in place of lemma 5.38.11.
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 $v\in \mathrm{PSH}(Z,\Omega )$,
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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