If $s$ and $s^{\prime}$ are two toric sections, then there is an element $m\in M$ such that $s^{\prime}=\chi^{m}s$. Since for any element $t\in\mathbb{S}^{{\text{\rm an}}}$ we have $|\chi^{m}(t)|=1$, if the function $\|s(p)\|$ is $\mathbb{S}^{{\text{\rm an}}}$-invariant, then the function $\|s^{\prime}(p)\|=\|\chi^{m}(p)s(p)\|$ is also $\mathbb{S}^{{\text{\rm an}}}$-invariant. ∎

Let $\|\cdot\|$ be a toric metric on $L^{{\text{\rm an}}}$. Since $s$ is a regular nowhere vanishing section on $X_{0}^{{\text{\rm an}}}$, $\psi_{\|\cdot\|}$ is a well defined continuous function on $N_{\mathbb{R}}$. Let $\{m_{\sigma}\}$ be a set of defining vectors of $\Psi$. For each cone $\sigma\in\Sigma$, the section $\chi^{m_{\sigma}}s$ is a regular nowhere vanishing section on $X_{\sigma}^{{\text{\rm an}}}$. Therefore $\log(\|(\chi^{m_{\sigma}}s)(p)\|)$ is a continuous function on $X_{\sigma}^{{\text{\rm an}}}$ that is $\mathbb{S}^{{\text{\rm an}}}$-invariant. So it defines a continuous function on $X_{\sigma}(\mathbb{R}_{\geq 0})$. By equation (5.4),

	$\displaystyle\psi_{\|\cdot\|}({\operatorname{val}}(p))-m_{\sigma}({\operatorname{val}}(p))$	$\displaystyle=\frac{1}{\lambda_{K}}\left(\log(\|s(p)\|)-\log(|\chi^{-m_{\sigma}}(p)|)\right)$
		$\displaystyle=\frac{1}{\lambda_{K}}\log(\|(\chi^{m_{\sigma}}s)(p)\|).$

Therefore $\psi_{\|\cdot\|}-m_{\sigma}$ extends to a continuous function on $N_{\sigma}\simeq X_{\sigma}(\mathbb{R}_{\geq 0})$. If we see that $\Psi-m_{\sigma}$ extends also to a continuous function on $N_{\sigma}$ we will be able to extend $\psi_{\|\cdot\|}-\Psi$ to a continuous function on $N_{\sigma}$ for every $\sigma\in\Sigma$ and therefore to $N_{\Sigma}$.

Let $\tau$ be a face of $\sigma$ and let $u\in N(\tau)_{\mathbb{R}}$. Let $W(\tau,U,p)$ be a neighbourhood of $u$ as in (5.6). By taking $U$ small enough and $p$ big enough we can assume that $W(\tau,U,p)\cap N_{\mathbb{R}}$ is contained in the set of cones that have $\tau$ as a face. Since $\Psi$ and $m_{\sigma}$ agree when restricted to $\sigma$ (hence when restricted to $\tau$) it follows that, if $w+t\in W(\tau,U,p)\cap N_{\mathbb{R}}$ with $w\in U$ and $t\in p+\tau$, then $(\Psi-m_{\sigma})(w+t)$ only depends on $w$ and not on $t$. Hence it can be extended to a continuous function on the whole $W(\tau,U,p)$. By moving $\tau$, $u$, $U$ and $p$ we see that it can be extended to a continuous function on $N_{\sigma}$.

Let now $\psi$ be a function on $N_{\mathbb{R}}$ such that $\psi-\Psi$ extends to a continuous function on $N_{\Sigma}$. We define a toric metric $\|\cdot\|_{\psi}$ on $L^{{\text{\rm an}}}$ over the set $X_{0}^{{\text{\rm an}}}$ by the formula

$$\|s(p)\|_{\psi}=\exp(\lambda_{K}\psi({\operatorname{val}}_{K}(p))).$$

Then, by the argument before, $\psi-m_{\sigma}$ extends to a continuous function on $N_{\sigma}$, which proves that $\|\cdot\|_{\psi}$ extends to a metric over $X_{\sigma}^{{\text{\rm an}}}$. Varying $\sigma\in\Sigma$ we obtain that $\|\cdot\|_{\psi}$ extends to a metric over $X_{\Sigma}^{{\text{\rm an}}}$. ∎

Since we are assuming that $\Sigma$ is complete, the space $N_{\Sigma}\simeq X_{\Sigma}(\mathbb{R}_{\geq 0})$ is compact. Thus the corollary follows from Proposition 5.16. ∎

This follows easily from the definitions. ∎

Let $s^{\prime}$ be another toric section of $L$. Then there is an element $m\in M$ such that $s^{\prime}=\chi^{m}s$. The corresponding virtual support function is $\Psi^{\prime}=\Psi-m$. Denote by $\|\cdot\|$ and $\|\cdot\|^{\prime}$ the metrics associated to $s,\Psi$ and to $s^{\prime},\Psi^{\prime}$ respectively. Then

$$\|s(p)\|^{\prime}=\|\chi^{-m}s^{\prime}(p)\|^{\prime}=\operatorname{e}^{\lambda_{K}(m+\Psi^{\prime})({\operatorname{val}}(p))}=\operatorname{e}^{\lambda_{K}\Psi({\operatorname{val}}(p))}=\|s(p)\|.$$

Thus both metrics agree. ∎

This follows easily from the definitions. ∎

The section $s^{\prime}$ is a nowhere vanishing section over $X_{\Sigma,\sigma}$. Therefore, the function $g_{{\overline{L}},s^{\prime}}\colon X^{{\text{\rm an}}}_{\Sigma,\sigma}\to\mathbb{R}$ of diagram (5.13) can be extended to a continuous function on $X_{\Sigma,\sigma}$ that we also denote $g_{{\overline{L}},s^{\prime}}$. By the definition of the inverse image of a metric, there is a commutative diagram

Then the result is a consequence of the definition of $\psi_{\iota^{\ast}{\overline{L}},\iota^{\ast}s^{\prime}}$ and of the commutativity of the diagram

that follows from Proposition 5.9. ∎

Choose a toric section $s$ of $L$ whose divisor meets $V(\sigma)$ properly. Let $\Psi$ be the corresponding virtual support function. The condition of proper intersection is equivalent to $\Psi|_{\sigma}=0$. Then $\Psi$ extends to a continuous function ${\overline{\Psi}}$ on $N_{\sigma}$ and the restriction of ${\overline{\Psi}}\to N(\sigma)$ is equal to $\Psi(\sigma)$. Hence the result follows from Proposition 5.22. ∎

First observe that the condition $l_{i}\in\mathbb{R}$ in the Archimedean case and $l_{i}\in\mathbb{Q}$ in the non-Archimedean case is equivalent to the condition $l_{i}\in\mathbb{Q}\,{\operatorname{val}}_{K}(K^{\times})$. Let $e>0$ be an integer such that $em_{i}\in M$ and $el_{i}\in{\operatorname{val}}_{K}(K^{\times})$ for $i=0,\dots,r$.

Consider the linear map $H\colon N_{\mathbb{R}}\to\mathbb{R}^{r}$ given by $H(u)=(em_{i}(u)-em_{0}(u))_{i=1,\dots,r}$ and the affine map $A=H+\boldsymbol{l}$ with $\boldsymbol{l}=(el_{i}-el_{0})_{i=1,\dots,r}$. By Lemma 3.79,

$$e\psi=A^{\ast}\Psi_{\Delta^{r}}+em_{0}+el_{0}.$$

We claim that, for each $\sigma\in\Sigma$ there exists $\sigma_{i_{0}}\in\Sigma_{\Delta^{r}}$ such that $H(\sigma)\subset\sigma_{i_{0}}$. Indeed, $\Psi(u)=\min_{i}\{m_{i}(u)\}$. Since $\Psi$ is a support function on $\Sigma$, for each $\sigma\in\Sigma$, there exists an $i_{0}$ such that $\Psi(u)=m_{i_{0}}(u)$ for all $u\in\sigma$. Writing $e_{0}^{\vee}=0$, this condition implies

$$\min_{0\leq i\leq r}\{e_{i}^{\vee}(H(u))\}=e_{i_{0}}^{\vee}(H(u))\quad\text{for all }u\in\sigma.$$

Hence, $H(\sigma)\subset\sigma_{i_{0}}$, where $\sigma_{i_{0}}\in\Sigma_{\Delta^{r}}$ is the cone $\{v|\min_{0\leq i\leq r}\{e_{i}^{\vee}(v)\}=e_{i_{0}}^{\vee}(v)\}$ and the claim is proved.

Therefore, we can apply Theorem 4.9 and given a point $p\in\mathbb{P}^{r}_{r}(K)$ such that ${\operatorname{val}}_{K}(p)=\boldsymbol{l}$, there is an equivariant map $\varphi_{p,H}\colon X_{\Sigma}\to\mathbb{P}^{r}$. By Example 4.44, there is an isomorphism $L^{\otimes e}\simeq\varphi_{p,H}^{*}\mathcal{O}(1)$ and $a\in K^{\times}$ with ${\operatorname{val}}_{K}(a)=l_{0}$ such that $(a^{-1}\chi^{-m_{0}}s)^{\otimes e}$ corresponds to $\varphi_{p,H}^{*}(s_{\Psi_{\Delta^{r}}})$.

Let ${\overline{L}}$ be the line bundle $L$ equipped with the metric induced by the above isomorphism and the canonical metric of $\mathcal{O}(1)^{{\text{\rm an}}}$. Then

$$\psi_{{\overline{L}},s}=\psi_{{\overline{L}},a^{-1}\chi^{-m_{0}}s}+m_{0}+l_{0}=\frac{1}{e}A^{\ast}\Psi_{\Delta^{r}}+m_{0}+l_{0}=\psi,$$

as stated. ∎

This follows readily from the previous result together with Example 2.32 in the Archimedean case and Example 2.25 in the non-Archimedean case and the fact that the inverse image of an approachable metric is also approachable. ∎