13.28.20 Convergence for all obstacles70267E3AB91D

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. $\square$

Proof. The classifications of ordinary and ideal models in lemma 13.28.1, lemma 20.3.20 show that $P_\theta [u]=P_\theta [u]_{\mathcal I}$ if and only if $\mu _ u=D_ u$. Step 1 of the proof of proposition 13.28.22 shows that this is equivalent to equality of the envelopes at every smooth obstacle. Membership of $u$ in $\mathcal E_{\mathrm{env}}(Y,\theta ,q)$ implies the latter by proposition 18.12.2.

Conversely, assume that $\mu _ u=D_ u$. By lemma 20.3.20, $\mu _ q=D_ u$. Ordinary equivalence preserves polar curvature by corollary 5.27.15, so every potential equivalent to $q$ has non-pluripolar mass $A-D_ u(Y)$, by Stokes' theorem. Hence

If $D_ u(Y)<A$, then $P_\theta [u]=P_\theta [u]_{\mathcal I}$, and proposition 13.24.2 puts $u$ in $\mathcal E_{\mathrm{env}}(Y,\theta ,q)$.

If $D_ u(Y)=A$, the positive curvature measures $T_ u$ and $T_ q$ both dominate $D_ u$ and both have total mass $A$. Hence $T_ u=T_ q=D_ u$. The distribution $u-q$ is harmonic, and therefore constant on the compact connected curve. Thus $u=q+C$ for a constant $C$, and proposition 16.28.62 gives $u\in \mathcal E_{\mathrm{env}}(Y,\theta ,q)$ also in this case. $\square$

Proof. Use the diffuse polar probability measures $\sigma _ j$ concentrating at a point $x$ and the models $p_ j=p_{c\sigma _ j}$ from the proof of proposition 13.28.19, with $0<c<1$. Fix a smooth obstacle $F$ and put $q_{y,F}=P_{\theta ,F}[q_{c\delta _ y}]$. Rotations preserving $\theta $ and sending $x$ to $y$ can be chosen to depend continuously on $y$ near $x$; they transport the obstacle $F$ to its composition with the rotation. Since envelopes below two obstacles differ by at most the uniform distance between the obstacles, continuity of rotations in $L^1$ gives $q_{y,F}\to q_{x,F}$ in $L^1$ as $y\to x$.

The uniform $L^1$ bound near $x$ makes the average $w_{j,F}=\int q_{y,F}\, \mathrm d\sigma _ j(y)$ well defined. It lies below $F$, its curvature current dominates $c\sigma _ j$, and it converges to $q_{x,F}$ in $L^1$. The description of envelopes below obstacles in Step 1 of the proof of proposition 13.28.22 gives $w_{j,F}\leq P_{\theta ,F}[p_ j]\leq F$. These bounds and normalized compactness give $L^1$ precompactness. Every cluster point of the envelopes has curvature at least $c\delta _ x$, so it is at most $q_{x,F}$, and the lower bound gives the reverse inequality. Hence $P_{\theta ,F}[p_ j]\to q_{x,F}$ for every $F$. The ideal envelopes of the $p_ j$ below $F$ all equal $P_\theta (F)$, since their multiplier ideals are trivial, and their atomic parts vanish. Thus $[p_ j]$ is a $d^{\mathrm{all}}_\theta $-Cauchy sequence. Since $F_1=0$, a limit would also be a $d^{\mathrm{ot}}_\theta $-limit, which proposition 13.28.19 excludes.

For the strict comparison with polar total variation, use the rotated diffuse models $\widetilde p_ j=p\circ R_ j^{-1}$ from the end of 13.28.15. For every smooth $F$,

The same obstacle bound and the $L^1$ continuity of rotations give convergence to $P_{\theta ,F}[p]$. All atomic parts vanish, so $d^{\mathrm{all}}_\theta ([\widetilde p_ j],[p])\to 0$, whereas the polar total-variation distance remains $2c$. Proposition 13.28.22 gives the implication in the other direction.

Finally, the sequence in proposition 18.11.2 converges for the zero obstacle but not for some smooth obstacle. By Step 1 of the proof of proposition 13.28.22, replacing its members by their ordinary models does not change any obstacle envelope, so $d^{\mathrm{all}}_\theta $ is strictly stronger than $\tau _{\theta ,0}$ already on ordinary models. $\square$