Theorem 18.1.42.763BE634A054 Let $(\varphi _ i^ j)_{j\in J}$ ($i=1,\ldots ,n$) be nets in $\mathrm{PSH}(X,\theta _ i)$, and $\varphi _ i\in \mathrm{PSH}(X,\theta _ i)$ ($i=1,\ldots ,n$). Assume that for each prime divisor $E$ over $X$, we have
\[ \lim _{j\in J} \nu \left(\varphi _ i^ j,E\right)=\nu \left(\varphi _ i,E\right),\quad i=1,\ldots ,n. \]
Then
\[ \varlimsup _{j\in J} \operatorname{vol}\left(\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1^ j, \dots ,\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n^ j \right) \leq \operatorname{vol}\left(\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1, \dots ,\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n\right). \]
Proof.
Step 1. We first assume that $\operatorname{vol}(\theta _ i+\mathrm{dd}^{\mathrm{c}}\varphi _ i^ j)>0$ and $\operatorname{vol}(\theta _ i+\mathrm{dd}^{\mathrm{c}}\varphi _ i)>0$ for all $i=1,\ldots ,n$ and $j\in J$.
Without loss of generality, we may assume that the $\varphi _ i^ j$'s and the $\varphi _ i$'s are $\mathcal{I}$-model for all $i=1,\ldots ,n$ and $j\in J$. Our assertion becomes
055370851A43
18.1.43
\begin{equation} \varlimsup _{j\in J} \int _ X \left(\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1^ j\right)\wedge \dots \wedge \left(\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n^ j \right) \leq \int _ X \left(\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1\right) \wedge \dots \wedge \left(\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n\right). \end{equation}
For each $j\in J$, define
\[ \psi _ i^ j\coloneqq \operatorname *{sup}_{k\geq j}{}\! {}^{*} \varphi _ i^ k,\quad i=1,\ldots ,n. \]
Observe that $\psi _ i^ j$ is $\mathcal{I}$-good thanks to proposition 18.1.19, since after the preceding reduction the tail family is bounded above by $0$. It follows from corollary 5.16.9 and our assumption that
\[ \lim _{j\in J}\nu \left(\psi _ i^ j,E\right)=\nu \left(\varphi _ i,E\right),\quad i=1,\ldots ,n. \]
Moreover, for each $j\in J$, the same formula gives $\varphi _ i\preceq _{\mathcal{I}}\psi _ i^ j$. Since $\varphi _ i$ is $\mathcal{I}$-model and $\psi _ i^ j$ is $\mathcal{I}$-good, proposition 16.8.17 gives
\[ \varphi _ i\leq P_{\theta _ i}\left[\psi _ i^ j\right]. \]
For each $i=1,\ldots ,n$, we define
\[ \psi _ i=\inf _{j\in J}P_{\theta _ i}\left[\psi _ i^ j\right]. \]
In particular, $\psi _ i\geq \varphi _ i$, so $\int _ X\theta _{i,\psi _ i}^ n>0$. Since $\psi _ i^ j$ is $\mathcal{I}$-good, $P_{\theta _ i}[\psi _ i^ j]$ is both model and $\mathcal{I}$-model. Due to proposition 16.6.29, $\psi _ i$ is $\mathcal{I}$-model. Thanks to proposition 16.4.4, we know
\[ \nu (\psi _ i,E)=\nu (\varphi _ i,E) \]
for any $i=1,\ldots ,n$ and any prime divisor $E$ over $X$. In other words, $\psi _ i\sim _{\mathcal{I}} \varphi _ i$ for $i=1,\ldots ,n$. But both $\varphi _ i$ and $\psi _ i$ are $\mathcal{I}$-good, therefore,
\[ \psi _ i\sim _{P} \varphi _ i,\quad i=1,\ldots ,n. \]
By proposition 16.8.22, we have
\[ \int _ X (\theta _1+\mathrm{dd}^{\mathrm{c}}\psi _1)\wedge \cdots \wedge (\theta _ n+\mathrm{dd}^{\mathrm{c}}\psi _ n)=\int _ X (\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1)\wedge \cdots \wedge (\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n). \]
Next by proposition 16.8.22 again,
\[ \begin{split} & \varlimsup _{j\in J} \int _ X \left(\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1^ j\right)\wedge \dots \wedge \left(\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n^ j \right)\\ \leq & \varlimsup _{j\in J} \int _ X \left(\theta _1+\mathrm{dd}^{\mathrm{c}}\psi _1^ j\right)\wedge \dots \wedge \left(\theta _ n+\mathrm{dd}^{\mathrm{c}}\psi _ n^ j \right). \end{split} \]
On the other hand, due to proposition 16.4.4 and corollary 16.9.14, for each $i=1,\ldots ,n$, we have
\[ \psi _ i^ j\xrightarrow {d_ S}\psi _ i. \]
We conclude from theorem 16.9.30 that
\[ \varlimsup _{j\in J} \int _ X \left(\theta _1+\mathrm{dd}^{\mathrm{c}}\psi _1^ j\right)\wedge \dots \wedge \left(\theta _ n+\mathrm{dd}^{\mathrm{c}}\psi _ n^ j \right)=\int _ X (\theta _1+\mathrm{dd}^{\mathrm{c}}\psi _1)\wedge \cdots \wedge (\theta _ n+\mathrm{dd}^{\mathrm{c}}\psi _ n). \]
Putting these equations together, (18.1.43) follows.
Step 2. Next we handle the general case.
Fix a Kähler form $\omega $ on $X$. For any $\epsilon \in (0,1]$, from Step 1, we know that
\[ \begin{split} & \varlimsup _{j\in J} \operatorname{vol}\left(\theta _1+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _1^ j, \dots ,\theta _ n+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _ n^ j \right) \\ \leq & \operatorname{vol}\left(\theta _1+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _1, \dots ,\theta _ n+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _ n\right). \end{split} \]
Using 18.1.31, we have
\[ \begin{aligned} & \varlimsup _{j\in J} \operatorname{vol}\left(\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1^ j, \dots ,\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n^ j \right)\\ \leq & \varlimsup _{j\in J} \operatorname{vol}\left(\theta _1+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _1^ j, \dots ,\theta _ n+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _ n^ j \right)\\ \leq & \operatorname{vol}\left(\theta _1+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _1, \dots ,\theta _ n+\epsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi _ n\right)\\ \leq & \operatorname{vol}\left(\theta _1+\mathrm{dd}^{\mathrm{c}}\varphi _1, \dots ,\theta _ n+\mathrm{dd}^{\mathrm{c}}\varphi _ n\right)+C\epsilon . \end{aligned} \]
But since $\epsilon $ is arbitrary, our assertion follows.
$\square$
Pome
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