Proposition 2.36 . [02JK]

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Proposition 2.36.

Let $\varphi\colon X^{\prime}\to X$ be a morphism of proper varieties over $K$ , $Y^{\prime}$ a $d$ -dimensional cycle of $X^{\prime}$ , and ${\overline{L}}_{i}=(L_{i},\|\cdot\|_{i})$ , $i=0,\dots,d-1$ , a collection of integrable metrized line bundles on $X$ . 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_{\varphi_{*}Y}.