Lemma 22.1.9.587EEDE88F4F Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$, let $\omega $ be a Kähler form on $X$, and let $0<t\leq 1$. Then for every $\tau <\Gamma _{\max }$, we have
\[ \begin{aligned} & \int _ X \left(\theta +t\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +t\omega }_{-\infty }\right)^ n-\int _ X \left(\theta +t\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +t\omega }_{\tau }\right)^ n\\ \leq {}& \int _ X \left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +\omega }_{-\infty }\right)^ n-\int _ X \left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +\omega }_{\tau }\right)^ n. \end{aligned} \]
Proof.
For $\sigma \in \{ -\infty ,\tau \} $, put
\[ S_ t^\sigma =\theta +t\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +t\omega }_{\sigma }. \]
By the transition compatibility and proposition 22.1.5, example 18.1.11, proposition 16.3.10, projecting $\Gamma ^{\theta +t\omega }_{\sigma }$ to the class $\theta +\omega $ does not change the total mass computed in the larger class. Thus
\[ \int _ X \left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +\omega }_{\sigma }\right)^ n = \int _ X \left(S_ t^\sigma +(1-t)\omega \right)^ n. \]
Expanding the right-hand side and subtracting the two identities for $\sigma =-\infty $ and $\sigma =\tau $, we get
\[ \begin{aligned} & \int _ X \left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +\omega }_{-\infty }\right)^ n-\int _ X \left(\theta +\omega +\mathrm{dd}^{\mathrm{c}}\Gamma ^{\theta +\omega }_{\tau }\right)^ n\\ ={}& \sum _{a=0}^ n \binom {n}{a}(1-t)^{n-a} \left(\int _ X (S_ t^{-\infty })^ a\wedge \omega ^{n-a}-\int _ X (S_ t^\tau )^ a\wedge \omega ^{n-a}\right). \end{aligned} \]
Since $\Gamma ^{\theta +t\omega }_{-\infty }\geq \Gamma ^{\theta +t\omega }_{\tau }$, each summand is non-negative by the monotonicity theorem for mixed non-pluripolar products. Keeping only the term $a=n$ gives the desired inequality.
$\square$
Pome
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