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Lemma 17.12.6348ACDBCB24E

Lemma 17.12.6.348ACDBCB24E Assume that $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$. Then for each $\tau <\Gamma _{\max }$, we have

F4B7B1DBBE89 17.12.7
\begin{equation} \int _ X \left(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma _{\tau }\right)^ n>0. \end{equation}

In particular, $\Gamma _{\tau }$ is a model potential in $\mathrm{PSH}(X,\theta )_{>0}$.

Proof. Fix $\tau \in (-\infty ,\Gamma _{\max })$.

By assumption, $\Gamma _{-\infty }$ has positive mass. By corollary 13.13.12, we have

\[ \int _ X \theta _{\Gamma _{-\infty }}^ n=\lim _{\tau \to -\infty }\int _ X \theta _{\Gamma _{\tau }}^ n. \]

In particular, for a sufficiently small $\tau _0<\tau $, we have

\[ \int _ X \theta _{\Gamma _{\tau _0}}^ n>0. \]

Now take $\tau '\in (\tau , \Gamma _{\max })$ and $t\in (0,1)$ so that

\[ \tau =(1-t)\tau '+t\tau _0. \]

From the concavity of $\Gamma $, we find that

\[ \Gamma _{\tau }\geq (1-t)\Gamma _{\tau '}+t\Gamma _{\tau _0}. \]

By the monotonicity theorem theorem 13.13.11 and the multi-linearity of the non-pluripolar products proposition 13.11.4 6,

\[ \int _ X \theta _{\Gamma _{\tau }}^ n\geq \int _ X \theta _{(1-t)\Gamma _{\tau '}+t\Gamma _{\tau _0}}^ n\geq t^ n \int _ X \theta _{\Gamma _{\tau _0}}^ n>0 \]

and (17.12.7) follows. $\square$

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