PomeDefinition 20.2.10.5239E9ADD0FF Let $\varphi \in \mathrm{QPSH}(X)$ such that $\nu (\varphi ,Y)=0$. We write $\operatorname{Tr}_{Y}(\varphi )$ for any trace operator of $\varphi $ along $Y$.
Given a closed smooth real $(1,1)$-form $\theta $ on $X$. When $\operatorname{Tr}_ Y(\varphi )$ can be chosen to lie in $\mathrm{PSH}(\widetilde{Y},\theta |_{\widetilde{Y}})_{>0}$, we write
\[ \operatorname{Tr}_ Y^{\theta }(\varphi )\coloneqq P_{\theta |_{\widetilde{Y}}}\bigl [ \operatorname{Tr}_ Y(\varphi ) \bigr ]=P_{\theta |_{\widetilde{Y}}}\bigl [ \operatorname{Tr}_ Y(\varphi ) \bigr ]_{\mathcal{I}}. \]
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