Definition 22.1.50C0E37055D126

Definition 22.1.50.C0E37055D126 Let $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )$. Let $Y\subseteq X$ be an irreducible analytic subset. We say that the trace operator of $\Gamma $ along $Y$ is well-defined if

for all sufficiently negative $\tau $ and every $\omega \in \textup{Käh}(X)$. We define

Here $\omega $ is any Kähler form on $X$; this number is independent of the choice of $\omega $ by the compatibility of the transition maps.

In this case, we define $\operatorname{Tr}_ Y(\Gamma )\in \mathrm{PSH}^{\mathrm{NA}}(\widetilde{Y},\theta |_{\widetilde{Y}})$³ as the unique element such that for any $\omega \in \textup{Käh}(\widetilde{Y})$, the component

is defined as follows:

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   We let

   284D786E2B4E 22.1.52

   \begin{equation} \left( \operatorname{Tr}_ Y(\Gamma )^{\theta |_{\widetilde{Y}}+\omega } \right)_{\max }=\left(\operatorname{Tr}_ Y(\Gamma )\right)_{\max }; \end{equation}
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   for each $\tau \in \mathbb {R}$ less than $\left(\operatorname{Tr}_ Y(\Gamma )\right)_{\max }$, we define

   \[ \operatorname{Tr}_ Y(\Gamma )^{\theta |_{\widetilde{Y}}+\omega }_{\tau }\coloneqq P_{\theta |_{\widetilde{Y}}+\omega }\left[ \operatorname{Tr}_ Y^{\theta +\tilde{\omega }}\left(\Gamma ^{\theta +\tilde{\omega }}_{\tau }\right) \right]_{\mathcal{I}}, \]

   where $\tilde{\omega }$ is an arbitrary Kähler form on $X$ such that $\omega \geq \tilde{\omega }|_{\widetilde{Y}}$.