In view of propositions 6.25, 6.27 and the fact that the restriction of the canonical metric to closures of orbits and to toric subvarieties is the canonical metric (corollaries 5.23 and 5.25), we are reduced to treat the case $Y=X_{\Sigma}$.

Thus we assume that $X_{\Sigma}$ has dimension $d$. We next prove that $X_{\Sigma}$ is integrable with respect to ${\overline{L}}_{0}^{{\operatorname{can}}},\dots,{\overline{L}}_{d}^{{\operatorname{can}}}$ and that the corresponding global height is zero. By a polarization argument, we can reduce to the case $\Psi_{0}=\dots=\Psi_{d}=\Psi$. The proof is done by induction on $d$. For short, write $L={\mathcal{O}}(D_{\Psi})$ and $s=s_{\Psi}$.

Let $d=0$. By equation (2.40), for each $v\in\mathfrak{M}_{\mathbb{K}}$,

$$\operatorname{h}_{{\overline{L}}^{v,{\operatorname{can}}}}(X_{\Sigma};s)=-\log\|s\|_{v,\Psi}=\Psi(0)=0.$$

Furthermore, $\operatorname{h}_{{\overline{L}}^{{\operatorname{can}}}}(X_{\Sigma};s)=\sum_{v}n_{v}\operatorname{h}_{{\overline{L}}^{v,{\operatorname{can}}}}(X_{\Sigma};s)=0$.

Now let $d\geq 1$. By the construction of local heights, for each $v\in\mathfrak{M}_{\mathbb{K}}$,

(6.36)		$\displaystyle\operatorname{h}_{{\overline{L}}^{v,{\operatorname{can}}}}(X_{\Sigma};s_{0},\dots,s_{d-1},s)=$	$\displaystyle\operatorname{h}_{{\overline{L}}^{v,{\operatorname{can}}}}(\operatorname{div}(s);s_{0},\dots,s_{d-1})$
		$\displaystyle-\int_{X_{\Sigma}^{v,{\text{\rm an}}}}\log\|s\|_{v,\Psi}c_{1}({\overline{L}}^{v,{\operatorname{can}}})^{d}\wedge\delta_{X_{\Sigma}}.$

As shown in (6.14), the last term in the equality above vanishes. Hence

$$\operatorname{h}_{{\overline{L}}^{v,{\operatorname{can}}}}(X_{\Sigma};s_{0},\dots,s_{d-1},s)=\operatorname{h}_{{\overline{L}}^{v,{\operatorname{can}}}}(\operatorname{div}(s);s_{0},\dots,s_{d-1}).$$

The divisor $\operatorname{div}(s)$ is a linear combination of subvarieties of the form $V(\tau)$, $\tau\in\Sigma^{1}$, and the restriction of the canonical metric to these varieties coincides with their canonical metrics. With the inductive hypothesis, this shows that $X_{\Sigma}$ is integrable with respect to ${\overline{L}}^{{\operatorname{can}}}$. Adding up the resulting equalities over all places,

$$\operatorname{h}_{{\overline{L}}^{{\operatorname{can}}}}(X_{\Sigma};s_{0},\dots,s_{d-1},s)=\operatorname{h}_{{\overline{L}}^{{\operatorname{can}}}}(\operatorname{div}(s);s_{0},\dots,s_{d-1}).$$

Using again the inductive hypothesis, $\operatorname{h}_{{\overline{L}}^{{\operatorname{can}}}}(X_{\Sigma};s_{0},\dots,s_{d-1},s)\in\operatorname{def}(\mathbb{K}^{\times})$.

We now prove the statements of the theorem. Again by a polarization argument, we can also reduce to the case when ${\overline{L}}_{0}=\dots={\overline{L}}_{d}={\overline{L}}$. By the definition of approachable adelic toric metrics, $X_{\Sigma}$ is also integrable with respect to ${\overline{L}}$. Furthermore,

$$\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})=\operatorname{h}_{{\overline{L}}}(X_{\Sigma};s_{0},\dots,s_{d})-\operatorname{h}_{{\overline{L}}^{{\operatorname{can}}}}(X_{\Sigma};s_{0},\dots,s_{d})$$

for any choice of sections $s_{i}$ intersecting $X_{\Sigma}$ properly. Hence, the classes of $\operatorname{h}_{{\overline{L}}}(X_{\Sigma})$ and of $\operatorname{h}_{{\overline{L}}}(X_{\Sigma};s_{0},\dots,s_{d})$ agree up to $\operatorname{def}(\mathbb{K}^{\times})$. But the latter is the global height of $X_{\Sigma}$ with respect to ${\overline{L}}$, hence the second statement. ∎

This follows readily from Corollary 6.21 and Proposition 6.35. ∎

By the definition of adelic field, ${\operatorname{val}}_{\mathbb{K}_{v}}(p)=0$ for almost all $v\in\mathfrak{M}_{\mathbb{K}}$. Therefore, the integrability of $Y$ follows as in the proof of Proposition 6.35.

Let $\Sigma$ be the complete regular fan of $N_{\mathbb{R}}$ induced by $H$ and $\Sigma_{\Delta^{r}}$, and let $X_{\Sigma}$ be the associated toric variety. Write $\varphi=\varphi_{H,p}$ for short. The fact that $H(N)$ is saturated implies that $\varphi$ has degree 1 and so $Y=\varphi_{*}X_{\Sigma}$. By the functoriality of the global height (Theorem 2.57(2)),

$$\operatorname{h}_{{\overline{{\mathcal{O}}(1)}}^{{\operatorname{can}}}}(Y)=\operatorname{h}_{\varphi^{*}({\overline{{\mathcal{O}}(1)}}^{{\operatorname{can}}})}(X_{\Sigma}).$$

Let $v\in\mathfrak{M}_{\mathbb{K}}$. Using the results in Example 6.31, it follows from Theorem 6.37 that

$$\operatorname{h}_{\varphi^{\ast}({\overline{{\mathcal{O}}}}(1)^{{\operatorname{can}}})}(X_{\Sigma})=\left[(n+1)!\sum_{v}\int_{{\overline{\Delta}}}n_{v}{\overline{\vartheta}}_{v}\,\text{\rm d}\operatorname{vol}_{M}\right].$$

where ${\overline{\Delta}}=\operatorname{conv}(0,m_{1}-m_{0},\dots,m_{r}-m_{0})\subset M_{\mathbb{R}}$ and ${\overline{\vartheta}}_{v}$ is the function parameterizing the upper envelope of the extended polytope

$$\operatorname{conv}\left((0,0),(m_{1}-m_{0},\log|p_{1}/p_{0}|_{v}),\dots,(m_{r}-m_{0},\log|p_{r}/p_{0}|_{v})\right)\subset M_{\mathbb{R}}\times\mathbb{R}.$$

We have that ${\overline{\Delta}}=\Delta-m_{0}$ and ${\overline{\vartheta}}_{v}=\tau_{-m_{0}}\vartheta_{v}-\log|p_{0}|_{v}$. Hence,

$$\int_{{\overline{\Delta}}}{\overline{\vartheta}}_{v}\,\text{\rm d}\operatorname{vol}_{M}=\int_{\Delta}\vartheta_{v}\,\text{\rm d}\operatorname{vol}_{M}-\log|p_{0}|_{v}\operatorname{vol}_{M}(\Delta).$$

Since $\sum_{v}n_{v}\log|p_{0}|_{v}\in\operatorname{def}(\mathbb{K}^{\times})$, we deduce the result. ∎