Corollary 18.7.9.A08D826CFAD4 Let $\omega $ be Hermitian and let $u\in \mathrm{PSH}(X,\omega )$. There are $b_ j\in \mathrm{PSH}(X,\omega )$ with analytic singularities and decreasing ordinary singularity types, such that $b_ j\to u$ in $L^1$ and
\[ P_\omega [b_ j]=P_\omega [b_ j]_{\mathcal I}\searrow P_\omega [u]_{\mathcal I}. \]
Consequently the classes of potentials with analytic singularities are dense in the whole metric space of ideal classes of proposition 18.6.1. The equality $P_\omega [u]=P_\omega [u]_{\mathcal I}$ is equivalent to convergence of these ordinary envelopes to $P_\omega [u]$ in $L^1$.
Proof.
Use $h=\omega $ in (18.7.5). Enlarge the error bounds to positive rational numbers $\epsilon _ j\downarrow 0$, and set
\[ c_ j=\frac1{1+\epsilon _ j},\qquad b_ j=c_ ja_ j. \]
Then $c_ j\uparrow 1$ is rational and
\[ \omega +\mathrm{dd}^{\mathrm{c}}b_ j =c_ j\bigl ((1+\epsilon _ j)\omega +\mathrm{dd}^{\mathrm{c}}a_ j\bigr )\geq 0. \]
The analytic coefficient of $b_ j$ is $c_ j/m_ j$. Moreover $b_ j\to u$ in $L^1$, since $a_ j\to u$ there and the sequence has bounded $L^1$ norm. Because $a_{j+1}\preceq a_ j$, $c_{j+1}\geq c_ j$, and each $a_ j$ is bounded above, one has $b_{j+1}\preceq b_ j$.
Put $q_ j=P_{(1+\epsilon _ j)\omega }[a_ j]=P_{(1+\epsilon _ j)\omega }[a_ j]_{\mathcal I}$. The preceding theorem gives $q_ j\searrow P_\omega [u]_{\mathcal I}$ pointwise and in $L^1$. Since $b_ j$ has analytic singularities, proposition 18.4.8 and the homogeneity in proposition 18.4.6 give
\[ r_ j:=P_\omega [b_ j]=P_\omega [b_ j]_{\mathcal I}=c_ jq_ j. \]
Here $q_ j\leq 0$ decreases and $c_ j\uparrow 1$, so $r_ j$ also decreases, with pointwise limit $P_\omega [u]_{\mathcal I}$. Moreover
\[ P_\omega [u]_{\mathcal I}\leq q_ j\leq r_ j\leq 0. \]
Since the lower bound is integrable, dominated convergence gives convergence in $L^1$.
The distance in proposition 18.6.1 from $[b_ j]_{\mathcal I}$ to $[u]_{\mathcal I}$ is exactly the $L^1$ distance from $r_ j$ to $P_\omega [u]_{\mathcal I}$, so it tends to zero. The last assertion follows because the limit of the $r_ j$ has been identified.
$\square$
Pome
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