Theorem 13.28.4.74646DD62F50 For every $u,v\in \mathrm{PSH}(Y,\theta )$,
FB9F740F1579
13.28.5
\begin{equation} d_\theta (u,v)=\| \mu _ u-\mu _ v\| _{\mathrm{TV}}. \end{equation}
In particular, $d_\theta $ is a metric on the set of ordinary models on $Y$. For positive atomic measures $D,E$ of mass at most $A$, let $q_ D,q_ E$ be the ideal models of lemma 20.3.20, with polar curvatures $D$ and $E$. Then
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13.28.6
\begin{equation} d_\theta (q_ D,q_ E)=\| D-E\| _{\mathrm{TV}}. \end{equation}
Proof.
By (5.27.16), the polar curvature of $\max (u,v)$ is the minimum $\mu _ u\wedge \mu _ v$ of the two measures. The formula $U=A-\mu (Y)$ therefore gives
\[ \begin{split} d_\theta (u,v)& =2U(\max (u,v))-U(u)-U(v)\\ & =\mu _ u(Y)+\mu _ v(Y)-2(\mu _ u\wedge \mu _ v)(Y)\\ & =\| \mu _ u-\mu _ v\| _{\mathrm{TV}}. \end{split} \]
The classification in lemma 13.28.1 shows that $d_\theta $ separates ordinary models, and the triangle inequality follows from that of the total variation norm. By lemma 20.3.20, the ideal models are exactly the models whose polar measures are atomic, which gives (13.28.6).
$\square$
Pome
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