Proposition 22.1.60.492A145E9067 Let
\[ (\Gamma ^ i)_{i\geq 1}\subseteq \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0},\quad \Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}. \]
Assume one of the following conditions holds:
577BE71DB8C5
The sequence $(\Gamma ^ i)_{i\geq 1}$ is decreasing and $\Gamma =\inf _{i\geq 1}\Gamma ^ i$. Assume that
E10A583B2E40
22.1.61
\begin{equation} \operatorname{vol}\Gamma =\lim _{i\to \infty }\operatorname{vol}\Gamma ^ i. \end{equation}
75E1B24CB49A
The sequence $(\Gamma ^ i)_{i\geq 1}$ is increasing and $\Gamma =\operatorname *{sup}_{i\geq 1}{}\! {}^{*}\Gamma ^ i$.
Then
636AFE1086C1
22.1.62
\begin{equation} \operatorname{DH}(\Gamma ^ i)\rightharpoonup \operatorname{DH}(\Gamma ). \end{equation}
Proof.
We may assume that $X$ admits a smooth flag $Y_{\bullet }$.
Assume 1. Let
\[ \eta \coloneqq \inf _{i\geq 1}\Gamma ^ i_{-\infty }. \]
Since $\Gamma _\tau \leq \Gamma ^ i_{-\infty }$ for every $i$ and every $\tau <\Gamma _{\max }$, we have
\[ \Gamma _{-\infty }\leq \eta . \]
By proposition 16.4.4, the decreasing limit $\eta $ is model and
\[ \int _ X\theta _\eta ^ n=\lim _{i\to \infty }\int _ X\theta _{\Gamma ^ i_{-\infty }}^ n=\lim _{i\to \infty }\operatorname{vol}\Gamma ^ i. \]
Using (22.1.61), this equals
\[ \operatorname{vol}\Gamma =\int _ X\theta _{\Gamma _{-\infty }}^ n. \]
Since $\Gamma _{-\infty }\leq \eta $ and both potentials are model, proposition 16.8.7 gives
\[ \Gamma _{-\infty }=\eta . \]
Thus
\[ \Gamma _{-\infty }=\inf _{i\geq 1}\Gamma ^ i_{-\infty }. \]
We want to derive (22.1.62) from 19.6.22. It boils down to prove the following: For any $\tau <\Gamma _{\max }$, we have
\[ \Delta _{Y_{\bullet }}(\theta ,\Gamma ^ i_{\tau })\xrightarrow {d_{\mathrm{Haus}}} \Delta _{Y_{\bullet }}(\theta ,\Gamma _{\tau }). \]
For fixed $\tau <\Gamma _{\max }$, the potentials $\Gamma ^ i_\tau $ decrease to $\Gamma _\tau $. By proposition 16.4.4, their masses converge to the mass of $\Gamma _\tau $; hence $\Gamma ^ i_\tau \xrightarrow {d_ S}\Gamma _\tau $ by corollary 16.9.14. Applying the continuity of the partial Okounkov body map, theorem 19.5.26 4, gives
\[ \Delta _{Y_{\bullet }}(\theta ,\Gamma ^ i_{\tau })\xrightarrow {d_{\mathrm{Haus}}}\Delta _{Y_{\bullet }}(\theta ,\Gamma _{\tau }). \]
The proof under the assumption 2 is similar. We only need to apply 19.6.23 instead of 19.6.22.
$\square$
Pome
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