Proposition 13.28.22E7DAF71126F6

Proof. Step 1: envelopes below an obstacle. For any smooth $F$, translating the competitors by $F$ gives

and likewise for the ideal envelope. The background $\theta +\mathrm{dd}^{\mathrm{c}}F$ on the right is a smooth real form of area $A$, though it need not be positive. We claim that these two envelopes are the models of $\mu _ u$ and of $D_ u$, respectively, in that background, in the sense of proposition 13.29.1. Set $\mu =\mu _ u$ and let $w_\mu $ be its Green potential, and put $r=P_{\theta +\mathrm{dd}^{\mathrm{c}}F}[w_\mu -F]$. By the gauge bounds of proposition 18.5.1, $r+F$ differs from $P_\theta [w_\mu ]=p_\mu \sim w_\mu $ by a bounded function, so $r\sim w_\mu -F$. Hence $r=P_{\theta +\mathrm{dd}^{\mathrm{c}}F}[r]$ is an ordinary model, and its polar measure is $\mu $. Moreover $u-F\preceq r$, and both have mass $A-\mu (Y)$. If this mass is positive, corollary 13.19.8, applied in the big background $\theta +\mathrm{dd}^{\mathrm{c}}F$, shows that $P_{\theta +\mathrm{dd}^{\mathrm{c}}F}[u-F]=r$. If it is zero, the curvature measures of $u-F$ and $w_\mu -F$ both equal $\mu $, so these functions differ by a constant, and again $P_{\theta +\mathrm{dd}^{\mathrm{c}}F}[u-F]=r$. The assertion for the ideal envelope follows in the same way from ideal projection and the classification of atomic models. It also follows that the envelope $P_{\theta ,F}[u]$ is the upper envelope of all $v\leq F$ with $\theta +\mathrm{dd}^{\mathrm{c}}v\geq \mu $.

Step 2: the uniformity. By proposition 13.29.1, polar total variation gives $L^1$ convergence of these models for every fixed $F$. The same assertion for atomic measures gives continuity of every ideal-envelope coordinate with respect to the total variation of the atomic parts. The controlled ideal relations are also implied, uniformly, by atomic total variation: for fixed $t>0$ and $a>1$, the bound $\| D-E\| _{\mathrm{TV}}<(a-1)/(at)$ guarantees both inclusions in (18.8.3), by the formula for multiplier ideals on a curve. Indeed, if $tE(\{ x\} )\geq 1$, the bound gives $aD(\{ x\} )>E(\{ x\} )$; otherwise the target ideal is the unit ideal. Interchange $D$ and $E$ for the other inclusion. Each divisorial coordinate is the atomic total variation. Therefore the envelope terms in the series, together with the atomic term, generate exactly the uniformity of $\tau ^{\mathrm{all}}_\theta $: finitely many coordinates control the sum up to its uniformly small tail, and the sum controls each fixed coordinate.

Step 3: zero distance and analytic closure. By Step 1 with $F=0$, distance zero forces equal polar measures; the classification at every obstacle proves the converse. The same classification shows that the ordinary and ideal envelopes agree at every $F$ if and only if $\mu _ u=D_ u$. Hence theorem 18.12.3 identifies the closure of the analytic classes.

Step 4: the sets $\mathcal C_ s$. For a $d^{\mathrm{all}}_\theta $-Cauchy sequence in $\mathcal C_ s$, the atomic measures converge in total variation to a measure of mass at most $s$, which is again atomic. The convergence of models in Step 2 gives convergence of every obstacle coordinate, and the summable tail then gives convergence for $d^{\mathrm{all}}_\theta $. Conversely, convergence for $d^{\mathrm{all}}_\theta $ implies convergence of the atomic parts, because of the first summand. This proves completeness and the equality of topologies on $\mathcal C_ s$. Finally, polar total variation controls the atomic part and all ordinary-envelope coordinates, which proves the last assertion. $\square$