PomeProposition 20.2.12.B293F74D2022 Let $\varphi \in \mathrm{QPSH}(X)$ such that $\nu (\varphi ,Y)=0$. Assume that $\varphi |_ Y\not\equiv -\infty $. Then
\[ \varphi |_{\widetilde{Y}}\preceq _ P \operatorname{Tr}_ Y(\varphi ). \]
Proposition 20.2.12B293F74D2022
Proof. Take a Kähler form $\omega $ such that $\omega _{\varphi }$ is a Kähler current. Let $(\varphi _ j)_{j>0}$ be a quasi-equisingular approximation of $\varphi $ in $\mathrm{PSH}(X,\omega )_{>0}$. We may assume that $\varphi _ j\leq 0$ for all $j\geq 1$.
Then
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\begin{equation} \varphi _ j|_{\widetilde{Y}}\leq P_{\omega |_{\widetilde{Y}}}\left[\varphi _ j|_{\widetilde{Y}}\right] \end{equation}
for all $j\geq 1$. In particular,
\[ \varphi |_{\widetilde{Y}}\leq \inf _{j\geq 1}P_{\omega |_{\widetilde{Y}}}\left[\varphi _ j|_{\widetilde{Y}}\right]. \]
Thanks to corollary 16.9.14,
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\begin{equation} \operatorname{Tr}_ Y(\varphi )\sim _ P \inf _{j\geq 1} P_{\omega |_{\widetilde{Y}}}[\varphi _ j|_{\widetilde{Y}}]. \end{equation}
We conclude our assertion. $\square$
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