PomeProposition 17.12.16.7AF1F81E928C Let $\Gamma $ be a test curve in $\mathrm{PSH}(X,\theta )$. For each $x\in X$, the map $\mathbb {R} \ni \tau \mapsto \Gamma _{\tau }(x)$ is a closed concave function. Moreover, the map is proper as long as $\Gamma _{\Gamma _{\max }}(x)\neq -\infty $.
Proposition 17.12.167AF1F81E928C
Proof. We prove closedness. Fix $x\in X$. Assume that $\Gamma _{\tau }(x)\neq -\infty $ for some $\tau \in \mathbb {R}$. We only need to argue the upper semicontinuity of $\tau \mapsto \Gamma _{\tau }(x)$. The upper semicontinuity is clear at $\tau \geq \Gamma _{\max }$, so we are reduced to prove the following:
E703D3F2D79D
17.12.17
\begin{equation} \Gamma _{\tau }=\inf _{\tau '<\tau }\Gamma _{\tau '} \end{equation}
for any $\tau <\Gamma _{\max }$. Take $\tau ''\in (\tau ,\Gamma _{\max })$. Outside the polar locus of $\Gamma _{\tau ''}$, we know that (17.12.17) holds by continuity of real-valued concave functions. So (17.12.17) holds everywhere by proposition 5.14.6.
The final assertion follows from properness of a finite closed concave function on its effective domain. $\square$
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