PomeProposition 22.1.53.E7009D90BCC6 Let $\pi \colon Y\rightarrow X$ be a proper bimeromorphic morphism from a compact Kähler manifold $Y$. Then all definitions in this section are invariant under pullback to $Y$.
Proposition 22.1.53E7009D90BCC6
Proof. We only prove the assertion for the trace operator, as the other proofs are similar.
We shall use the notations above. Observe that for any closed positive smooth $(1,1)$-form $\omega $ on $X$ with positive mass, we have
\[ \left(\widetilde{\Pi }^* \operatorname{Tr}_ Z(\Gamma )\right)_{\max }=\left( \operatorname{Tr}_ Z(\Gamma )\right)_{\max }=\sup \left\{ \tau <\Gamma _{\max }: \nu \left(\Gamma ^{\theta +\omega }_{\tau },Z\right)=0 \right\} , \]
and
\[ \begin{aligned} \left(\operatorname{Tr}_ W(\pi ^*\Gamma )\right)_{\max }=& \sup \left\{ \tau <\Gamma _{\max }: \nu \left((\pi ^*\Gamma )^{\pi ^*\theta +\pi ^*\omega }_{\tau },W\right)=0 \right\} \\ =& \sup \left\{ \tau <\Gamma _{\max }: \nu \left(\pi ^*\Gamma ^{\theta +\omega }_{\tau },W\right)=0 \right\} \\ =& \sup \left\{ \tau <\Gamma _{\max }: \nu \left(\Gamma ^{\theta +\omega }_{\tau },Z\right)=0 \right\} . \end{aligned} \]
Here we implicitly used proposition 22.1.17. Therefore,
\[ \left(\widetilde{\Pi }^* \operatorname{Tr}_ Z(\Gamma )\right)_{\max }=\left(\operatorname{Tr}_ W(\pi ^*\Gamma )\right)_{\max }. \]
Let $\tau \in \mathbb {R}$ be less than this common value. Take a Kähler form $\omega _{\widetilde{Z}}$ (resp. $\omega _{\widetilde{W}}$) on $\widetilde{Z}$ (resp. $\widetilde{W}$). Take a Kähler form $\omega _ Y$ on $Y$ (resp. $\omega _ X$ on $X$) such that
\[ \omega _{\widetilde{Z}}\geq \omega _ X|_{\widetilde{Z}},\quad \omega _{\widetilde{W}} \geq \omega _ Y|_{\widetilde{W}},\quad \omega _ Y \geq \pi ^*\omega _ X. \]
We want to show that, after projection to the common background $(\pi ^*\theta )|_{\widetilde{W}}+\omega _{\widetilde{W}}$,
\[ P_{(\pi ^*\theta )|_{\widetilde{W}}+\omega _{\widetilde{W}}}\left[\widetilde{\Pi }^*\left(\operatorname{Tr}_ Z(\Gamma )^{\theta |_{\widetilde{Z}}+\omega _{\widetilde{Z}}}_\tau \right)\right]_{\mathcal I} \sim _ P \left(\operatorname{Tr}_ W(\pi ^*\Gamma )\right)^{(\pi ^*\theta )|_{\widetilde{W}}+\omega _{\widetilde{W}}}_\tau . \]
Unfolding the definitions, we reduce to
\[ \widetilde{\Pi }^* \operatorname{Tr}_ Z^{\theta +\omega _ X}\left( \Gamma ^{\theta +\omega _ X}_{\tau } \right)\sim _ P \operatorname{Tr}_{W}^{\pi ^*\theta +\omega _ Y}\left( \left( \pi ^*\Gamma \right)^{\pi ^*\theta +\omega _ Y}_{\tau }\right). \]
Using proposition 20.2.21, this is equivalent to
\[ \widetilde{\Pi }^* \operatorname{Tr}_ Z\left( \Gamma ^{\theta +\omega _ X}_{\tau } \right)\sim _ P \operatorname{Tr}_{W}\left( \left( \pi ^*\Gamma \right)^{\pi ^*\theta +\pi ^*\omega _ X}_{\tau }\right). \]
This is a consequence of 20.2.24. $\square$
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