Proposition 2.35 . [02JI]

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Proposition 2.35.

With the previous notation, let $K^{\prime}$ be a finite extension of $K$ . Set $(X^{\prime},Y^{\prime})=(X,Y)\times\operatorname{Spec}(K^{\prime})$ and let $\varphi\colon{X^{\prime}}^{{\text{\rm an}}}\to X^{{\text{\rm an}}}$ be the induced map. Let $\varphi^{\ast}{\overline{L}}_{i}$ , $i=0,\dots,d-1$ , be the line bundles with algebraic metrics on $X^{\prime}$ obtained by base change. Then

\varphi_{\ast}\left(\operatorname{c}_{1}(\varphi^{\ast}{\overline{L}}_{0})\land\dots\land\operatorname{c}_{1}(\varphi^{\ast}{\overline{L}}_{d-1})\land\delta_{{Y^{\prime}}}\right)=\operatorname{c}_{1}({\overline{L}}_{0})\land\dots\land\operatorname{c}_{1}({\overline{L}}_{d-1})\land\delta_{Y}.