Proposition 5.63 . [02V6]

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Proposition 5.63.

Let $X_{\Sigma}\simeq\mathbb{P}^{1}_{K}$ be a one-dimensional toric variety over $K$ . Let $L\simeq\mathcal{O}(D_{\Psi})$ be a toric line bundle and let $\|\cdot\|$ be an algebraic metric defined by a semi-stable model and let $\|\cdot\|_{\mathbb{S}}$ be the associated toric metric. Then

c_{1}(L,\|\cdot\|_{\mathbb{S}})\land\delta_{X_{\Sigma}}=(\theta_{\Sigma})_{\ast}(\rho_{\Sigma})_{\ast}\left(c_{1}(L,\|\cdot\|)\land\delta_{X_{\Sigma}}\right).