Pome

Corollary 13.24.1C72B68A062F4

Corollary 13.24.1.C72B68A062F4 Let $u\in \mathrm{PSH}(X,\theta )$, fix a Hermitian form $h$, and let $a_ j\in \mathrm{PSH}(X,\theta _ j)$, $\theta _ j=\theta +\epsilon _ jh$, be the dyadic analytic regularizations of $u$ of analytic type $(2^{-j},\mathcal I(2^ ju))$, with errors $\epsilon _ j\downarrow 0$, as in (18.7.5). Then

\[ \underline{\operatorname {vol}}_{\theta _ j}(a_ j) \searrow \mathcal V^-_{\mathcal I,\theta }(u). \]

If $\mathcal V^-_{\mathcal I,\theta }(u)>0$, then also

\[ \overline{\operatorname {vol}}_{\theta _ j}(a_ j) \searrow \mathcal V^+_{\mathcal I,\theta }(u). \]

Proof. By theorem 18.7.6, the analytic ordinary models $q_ j=P_{\theta _ j}[a_ j]$ decrease to $P_\theta [u]_{\mathcal I}$, and the backgrounds $\theta _ j$ decrease to $\theta $. Thus theorem 13.21.2 applies to the $q_ j$. Since $q_ j\sim a_ j$ by proposition 18.4.8, the volumes of $q_ j$ and $a_ j$ agree. $\square$

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