By Proposition 5.24, $\psi_{{\overline{L}}_{1},\varphi^{\ast}s}=A^{\ast}\psi_{{\overline{L}},s}$. By Proposition 3.46(3) we obtain that $\operatorname{stab}(A^{\ast}\psi_{{\overline{L}},s})=H^{\vee}(\Delta_{\Psi})$ and that

from which equation (6.28) follows.

To prove equation (6.29), we observe that, by the definition of $\operatorname{h}^{\operatorname{tor}}$,

where $\varphi^{\ast}({\overline{L}}^{{\operatorname{can}}})$ has the toric structure induced by $s$ and the metric induced by the canonical metric of $L^{{\text{\rm an}}}$. We remark here that this metric differs from the canonical metric of $\varphi^{\ast}L^{{\text{\rm an}}}$. Now equation (6.29) follows from equation (6.28) and the definition of the canonical metric. ∎