PomeTheorem 19.4.30.30EEF3A985B7 The Okounkov body map
\[ \Delta _{\nu }(\theta ,\bullet ):\left( \mathrm{PSH}(X,\theta )_{>0},d_ S\right)\rightarrow \left(\mathcal{K}_ n,d_{\mathrm{Haus}}\right) \]
is continuous.
Theorem 19.4.3030EEF3A985B7
Proof. Let $\varphi _ j\to \varphi $ be a $d_ S$-convergent sequence in $\mathrm{PSH}(X,\theta )_{>0}$. We want to show that
6AAC6A3D90A2
19.4.31
\begin{equation} \Delta _{\nu }(\theta ,\varphi _ j)\xrightarrow {d_{\mathrm{Haus}}}\Delta _{\nu }(\theta ,\varphi ). \end{equation}
Replacing the potentials by their $P$-envelopes does not change their partial Okounkov bodies, since $P$-equivalent potentials are $\mathcal{I}$-equivalent, nor does it change their $d_ S$-classes. Thus we may assume that all $\varphi _ j$'s and $\varphi $ are model potentials.
By proposition 16.9.17, after passing to a subsequence we can sandwich $\varphi _ j$ between a decreasing sequence and an increasing sequence, both converging to $\varphi $ in $d_ S$. Using proposition 19.4.28 and theorem 19.7.4, it is enough to prove (19.4.31) when $(\varphi _ j)_ j$ is monotone. By theorem 16.9.49, replacing $\varphi _ j$ and $\varphi $ by their $\mathcal{I}$-envelopes preserves this reduction, so we may further assume that the $\varphi _ j$'s and $\varphi $ are $\mathcal{I}$-model. In both cases, we claim that
\[ \Gamma (\theta ,\varphi _ j)\xrightarrow {d_{\mathrm{sg}}} \Gamma (\theta ,\varphi ) \]
as $j\to \infty $. In fact, using theorem 18.1.46, we can compute
\[ \begin{aligned} d_{\mathrm{sg}}\left(\Gamma (\theta ,\varphi _ j), \Gamma (\theta ,\varphi )\right) = & \varlimsup _{k\to \infty } k^{-n}\left| h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi _ j)\right)-h^0\left(X,L^ k\otimes \mathcal{I}(k\varphi )\right) \right|\\ =& \frac{1}{n!} \left| \operatorname{vol}\theta _{\varphi _ j}-\operatorname{vol}\theta _{\varphi }\right|, \end{aligned} \]
which converges to $0$ by theorem 16.9.57. $\square$
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