Proposition 16.8.22 (Monotonicity theorem).82DCD8E3B1BA Let $\theta _1,\ldots ,\theta _ n$ be closed real smooth $(1,1)$-forms on $X$. Let $\varphi _ i,\psi _ i\in \mathrm{PSH}(X,\theta _ i)$ for $i=1,\ldots ,n$. Assume that $\varphi _ i\preceq _ P \psi _ i$ for each $i$. Then
\[ \int _ X \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n}\leq \int _ X \theta _{1,\psi _1}\wedge \cdots \wedge \theta _{n,\psi _ n}. \]
In other words, the non-pluripolar mass is a decreasing quantity with respect to the $P$-singularity types of the potentials.
Proof.
Fix a Kähler form $\omega $ on $X$. For each $i=1,\ldots ,n$, since $\varphi _ i\preceq _ P\psi _ i$, we have
\[ P_{\theta _ i+\epsilon \omega }[\varphi _ i]\leq P_{\theta _ i+\epsilon \omega }[\psi _ i] \]
for all $\epsilon >0$. Therefore, by proposition 16.3.10 and our previous monotonicity theorem theorem 13.13.11, we have
\[ \int _ X (\theta _1+\epsilon \omega )_{\varphi _1}\wedge \cdots \wedge (\theta _ n+\epsilon \omega )_{\varphi _ n}\leq \int _ X (\theta _1+\epsilon \omega )_{\psi _1}\wedge \cdots \wedge (\theta _ n+\epsilon \omega )_{\psi _ n}. \]
Letting $\epsilon \to 0+$ and applying the multi-linearity of the non-pluripolar products proposition 13.11.4 6, we find the desired inequality.
$\square$
Pome
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