Theorem 18.12.326068B87D99A

Proof. Fix a smooth obstacle $F$. As in the preceding proof, transport of the obstacle and theorem 18.4.2 give $P_{\omega ,F}[u]_{\mathcal I}\sim _{\mathcal I}u$. For $a$ with analytic singularities, lemma 5.38.11 therefore makes this ideal envelope a competitor for the ordinary envelope, so $P_{\omega ,F}[a]=P_{\omega ,F}[a]_{\mathcal I}$.

Write $p=P_{\omega ,F}[u]_{\mathcal I}$ and $q_ j=P_{\omega ,F}[b_ j]_{\mathcal I}$. The dyadic construction gives $u\preceq _{\mathcal I}b_ j$, decreasing ordinary types, and convergence of every local divisorial Lelong number of $b_ j$ to that of $u$. The ideal envelope preserves the ideal order, as one sees by taking maxima with a translate of $b_ j$ below $F$, so $p\leq q_ j\leq F$. Since $q_ j=P_{\omega ,F}[b_ j]$ and the ordinary types decrease, the sequence $q_ j$ decreases. Its limit $q$ is bounded below by $p$, so it is $\omega $-plurisubharmonic, and $p\leq q\leq F$; the same bounds give convergence in $L^1$.

Convergence after pullback and semicontinuity of Lelong numbers, as in proposition 18.7.1, give $\nu _ E(q)\geq \nu _ E(u)$ for every local divisor $E$, and $p\leq q$ gives the reverse inequality. The Lelong-number criterion of theorem 5.37.3 yields $q\sim _{\mathcal I}u$. Since $q\leq F$, it is a competitor for $p$, which proves

The controlled ideal relations and all divisorial distances on fixed modifications converge by the proof of theorem 18.9.1; they do not depend on $F$. Consequently this one dyadic sequence converges in $\tau ^{\mathrm{all}}_\omega $ exactly when the stated envelope equalities hold.

For each $F$, equality of the two $L^1$ envelope coordinates is a closed condition. Analytic classes satisfy all these equalities, so the closure of the analytic classes under nets lies in $\mathcal G^{\mathrm{all}}_\omega $, and the dyadic sequence proves the converse. Finally, the uniform obstacle bound in the proof of proposition 18.12.1 extends equality from a dense family to every smooth obstacle. $\square$