Definition 17.12.1C8399F09D86D

Definition 17.12.1.C8399F09D86D A test curve $\Gamma $ in $\mathrm{PSH}(X,\theta )$ consists of a real number $\Gamma _{\max }$ together with a map $(-\infty ,\Gamma _{\max })\rightarrow \mathrm{PSH}(X,\theta )$ denoted by $\tau \mapsto \Gamma _{\tau }$ satisfying the following conditions:

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   The map $\tau \mapsto \Gamma _{\tau }$ is concave and decreasing;

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   $P_{\theta }[\Gamma _{\tau }]=\Gamma _{\tau }$;

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   the potential

   E3D67CAF2F52 17.12.2

   \begin{equation} \Gamma _{-\infty }\coloneqq \operatorname *{sup}_{\tau <\Gamma _{\max }}{}\! {}^{*}\Gamma _{\tau } \end{equation}

   satisfies

   \[ \int _ X \left(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma _{-\infty }\right)^ n>0. \]

Let $\phi \in \mathrm{PSH}(X,\theta )_{>0}$ be a model potential. The set of test curves $\Gamma $ with $\Gamma _{-\infty }=\phi $ is denoted by $\mathrm{TC}(X,\theta ;\phi )$.

The union of all $\mathrm{TC}(X,\theta ;\phi )$'s for various model potentials $\phi \in \mathrm{PSH}(X,\theta )_{>0}$ is denoted by $\mathrm{TC}(X,\theta )_{>0}$.