Proposition 16.3.10.E58386609911 Suppose that $\theta _1,\ldots ,\theta _ n$ are smooth closed real $(1,1)$-forms on $X$. Let $\varphi _ i\in \mathrm{PSH}(X,\theta _ i)$ for each $i=1,\ldots ,n$. Then
0F7C5A9FEE2D
16.3.11
\begin{equation} \int _ X \theta _{1,P_{\theta _1}[\varphi _1]}\wedge \cdots \wedge \theta _{n,P_{\theta _ n}[\varphi _ n]}=\int _ X \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n}. \end{equation}
Proof.
For each $C\in \mathbb {Z}_{>0}$ and each $i=1,\ldots ,n$, we have
\[ (\varphi _ i+C)\land V_{\theta _ i}\sim \varphi _ i. \]
It follows from the monotonicity theorem theorem 13.13.11 that
\[ \int _ X \theta _{1,(\varphi _1+C)\land V_{\theta _1}}\wedge \cdots \wedge \theta _{n,(\varphi _ n+C)\land V_{\theta _ n}}=\int _ X \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n}. \]
So (16.3.11) follows from (16.3.7) and corollary 13.13.12.
$\square$
Pome
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