Proposition 13.28.244805035AEC54

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. $\square$