Proof. [02WK]

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Proof.

By Corollary 5.25, the inverse image of the canonical metric by a toric morphism is the canonical metric. Thus (1) and the first statement of (2) follow from 2.46 (2).

By Proposition 5.24 and Theorem 6.6 we deduce

\operatorname{h}^{\operatorname{tor}}_{\varphi^{\ast}{\overline{L}}}(X_{\Sigma_{1}})=(d+1)!\lambda_{K}\int_{\Delta_{\Psi\circ H}}(H^{\ast}\psi_{\|\cdot\|})^{\vee}\,\text{\rm d}\operatorname{vol}_{M_{1}}\\ =(d+1)!\int_{H^{\vee}(\Delta_{\Psi})}H^{\vee}_{\ast}(\vartheta_{\|\cdot\|})\,\text{\rm d}\operatorname{vol}_{M_{1}},

proving the result. ∎