Theorem 18.1.32.A320A575BBEB Let $\theta _1,\ldots ,\theta _ n$ be smooth closed real $(1,1)$-forms representing big cohomology classes. Suppose that $(\varphi _ j^ k)_{k\in I}$ are nets in $\mathrm{PSH}(X,\theta _ j)$ and $\varphi _ j\in \mathrm{PSH}(X,\theta _ j)$ for $j=1,\ldots ,n$. We assume that $\varphi _ j^ k\xrightarrow {d_ S}\varphi _ j$ for each $j=1,\ldots ,n$. Then
BCAEFDCBCBE3
18.1.33
\begin{equation} \lim _{k\in I}\operatorname{vol}\left( \theta _{1,\varphi _1^ k},\ldots , \theta _{n,\varphi _ n^ k}\right)=\operatorname{vol}\left( \theta _{1,\varphi _1},\ldots , \theta _{n,\varphi _ n}\right). \end{equation}
Proof.
Fix a Kähler form $\omega $. Then for any $\epsilon >0$, we have
\[ \lim _{k\in I}\operatorname{vol}\left( (\theta _1+\epsilon \omega )_{\varphi _1^ k},\ldots , (\theta _ n+\epsilon \omega )_{\varphi _ n^ k}\right)=\operatorname{vol}\left( (\theta _1+\epsilon \omega )_{\varphi _1},\ldots , (\theta _ n+\epsilon \omega )_{\varphi _ n}\right) \]
as a consequence of corollary 16.9.36, theorem 16.9.30 and theorem 16.9.49.
Now thanks to 18.1.31, there is $C>0$ so that for each $k\in I$,
\[ \begin{aligned} 0\leq & \operatorname{vol}\left( (\theta _1+\epsilon \omega )_{\varphi _1^ k},\ldots , (\theta _ n+\epsilon \omega )_{\varphi _ n^ k}\right)-\operatorname{vol}\left( \theta _{1,\varphi _1^ k},\ldots , \theta _{n,\varphi _ n^ k}\right)\leq C\epsilon ,\\ 0\leq & \operatorname{vol}\left( (\theta _1+\epsilon \omega )_{\varphi _1},\ldots , (\theta _ n+\epsilon \omega )_{\varphi _ n}\right)-\operatorname{vol}\left( \theta _{1,\varphi _1},\ldots , \theta _{n,\varphi _ n}\right)\leq C\epsilon . \end{aligned} \]
Therefore, (18.1.33) follows.
$\square$
Pome
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