Lemma 20.2.24.DDEF49346283 Let $\pi \colon Z\rightarrow X$ be a proper bimeromorphic morphism, where $Z$ is a connected Kähler manifold. Assume that $W$ (resp. $Y$) is an irreducible analytic subset of $Z$ (resp. $X$) such that the restriction $\Pi \colon W\rightarrow Y$ of $\pi $ is defined and is bimeromorphic, so that we have the following commutative diagram
Then for any $\varphi \in \mathrm{QPSH}(X)$ with $\nu (\varphi ,Y)=0$, we have
3B7501D970A6
20.2.25
\begin{equation} \widetilde{\Pi }^*\operatorname{Tr}_{Y}(\varphi )\sim _ P\operatorname{Tr}_ W(\pi ^*\varphi ). \end{equation}
Proof.
We first observe that by Zariski's main theorem, $\nu (\pi ^*\varphi ,W)=0$. So the right-hand side of (20.2.25) makes sense.
Step 1. Assume that $\varphi $ has analytic singularities. It suffices to apply example 20.2.15 to reformulate (20.2.25) as
\[ \widetilde{\Pi }^*(\varphi |_{\widetilde{Y}})\sim _ P(\pi ^*\varphi )|_{\widetilde{W}}. \]
In fact, the strict equality holds, which is nothing but the functoriality of pullbacks.
Step 2. Next we handle the general case. Choose a smooth closed real $(1,1)$-form $\theta $ such that $\theta _{\varphi }$ is a Kähler current. Take a quasi-equisingular approximation $(\varphi _ j)_ j$ of $\varphi $ in $\mathrm{PSH}(X,\theta )$. By corollary 18.1.15, $(\pi ^*\varphi _ j)_ j$ is a quasi-equisingular approximation of $\pi ^*\varphi $. From Step 1, we know that for each $j$,
\[ \widetilde{\Pi }^*\operatorname{Tr}_{Y}(\varphi _ j)\sim _ P\operatorname{Tr}_ W(\pi ^*\varphi _ j). \]
Letting $j\to \infty $, the right-hand side converges to $\operatorname{Tr}_ W(\pi ^*\varphi )$ by the definition of the trace operator. On the left-hand side, the $d_ S$-convergence of $\operatorname{Tr}_ Y(\varphi _ j)$ to $\operatorname{Tr}_ Y(\varphi )$, together with 16.8.26 and corollary 16.9.11, implies (20.2.25).
$\square$
Pome
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