Proposition 17.15.22.88665EBBD399 Let $(\Gamma ^ i)_{i\in I}$ be an increasing net in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.21). Then $\operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i}$ as defined in definition 17.15.20 lies in $\mathrm{TC}(X,\theta )_{>0}$. Moreover, if $\Gamma ^ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ for all $i\in I$, then $\operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i}$ lies in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ as well.
Moreover, we have
75480BC1C778
17.15.23
\begin{equation} \left( \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right)_{-\infty }=\operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i_{-\infty }. \end{equation}
Proof.
Fix $\tau <\sup _ i\Gamma ^ i_{\max }$. Choose $i_0$ such that $\tau <\Gamma ^{i_0}_{\max }$. On the tail $i\geq i_0$, the potentials $\Gamma ^ i_\tau $ form an increasing net of model potentials of positive mass, uniformly bounded above by $0$. Hence proposition 16.4.9 shows that
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i_\tau = \operatorname *{sup}_{i\geq i_0}{}\! {}^{*}\Gamma ^ i_\tau \]
is model. Its mass is positive by monotonicity, since it dominates $\Gamma ^{i_0}_\tau $. The decreasingness and concavity in $\tau $ follow from the directedness of the net and the corresponding properties of the curves $\Gamma ^ i$. Thus $\operatorname *{sup}_{}{}\! {}^{*}_ i\Gamma ^ i$ is a test curve. If all $\Gamma ^ i$ are $\mathcal I$-model, the same tail argument and proposition 16.6.30 show that the supremum is $\mathcal I$-model.
It remains to argue (17.15.23). Fix any $i_0\in I$. Since the curves are decreasing and are extended as in 17.12.3, the value at $-\infty $ can be computed by taking the supremum over rational $\tau <\Gamma ^{i_0}_{\max }$.
By proposition 5.14.5, there is a pluripolar set $Z\subseteq X$ such that for any $x\in X\setminus Z$,
\[ \left( \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right)_{-\infty }(x)=\sup _{\mathbb {Q}\ni \tau < \Gamma ^{i_0}_{\max }}\left(\operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i_{\tau }\right)(x)=\sup _{\mathbb {Q}\ni \tau < \Gamma ^{i_0}_{\max },i\in I}\Gamma ^ i_{\tau }(x)=\sup _{i\in I}\Gamma ^ i_{-\infty }(x). \]
So they are equal everywhere by proposition 5.14.6.
$\square$
Pome
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