With the previous notation, let ${\overline{L}}$ be a toric line bundle on $X_{\Sigma}$ generated by global sections, equipped with an approachable toric metric. We put on $\varphi^{\ast}L$ the structure of toric line bundle of Remark 4.36. Choose a toric section $s$ of $L$ and let $\Psi$ be the associated support function.

1. (1)

   If $H$ is not injective, then $\operatorname{h}^{\operatorname{tor}}_{\varphi^{\ast}{\overline{L}}}(X_{\Sigma_{1}})=0$.

2. (2)

   If $H$ is injective, then $\operatorname{h}^{\operatorname{tor}}_{\varphi^{\ast}{\overline{L}}}(X_{\Sigma_{1}})=[Q:H(N_{1})]\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y_{Q})$. Moreover

   (6.26)

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