We start by recalling the construction of $\Delta_{\mathscr{X}}$ and $p_{\mathscr{X}}$ from [BFJ16a]. Note that, in loc. cit. the residue field $k$ is of characteristic zero, but once we assume that the model $\mathscr{X}$ is an SNC model, using the results of [MN15, § 3.1] it is possible to extend the presentation of [BFJ16a] to the case of positive and mixed characteristic.

Let $\Div_{0}(\mathscr{X})$ be the group of vertical Cartier divisors on $\mathscr{X}$. Denote $\Div_{0}(\mathscr{X})_{\mathbb{R}}=\Div_{0}(\mathscr{X})\otimes\mathbb{R}$ and let $\Div_{0}(\mathscr{X})_{\mathbb{R}}^{\ast}$ be the dual. As explained in 2.2, each $D\in\Div_{0}(\mathscr{X})$ determines a model function $\varphi_{D}$. The map $D\mapsto\varphi_{D}$ is linear in $D$ and can be extended by linearity to a map $\Div_{0}(\mathscr{X})_{\mathbb{R}}\to C^{0}(X^{{\mathrm{an}}})$.

There is a map $\ev_{\mathscr{X}}\colon{X^{{\mathrm{an}}}}\to\Div_{0}(\mathscr{X})_{\mathbb{R}}^{\ast}$ determined by

Let $D_{1},\dots,D_{\ell}$ be the components of the special fiber $\mathscr{X}_{s}$. Each $D_{i}$, $i=1,\dots,\ell$, determines a divisorial point $x_{i}\in X^{{\mathrm{an}}}$ and we denote by $e_{i}=\ev_{\mathscr{X}}(x_{i})$. For each $J\subset\{1,\dots,\ell\}$ we write $D_{J}=\bigcap_{j\in J}D_{j}$ and $\sigma_{J}=\conv(e_{j},j\in J)$. Then the abstract skeleton of $\mathscr{X}$ is

By [BFJ16a, Thm. 3.1], the image of $\ev_{\mathscr{X}}$ is $\Delta^{\abs}_{\mathscr{X}}$ and there exists a unique function $\emb_{\mathscr{X}}\colon\Delta^{\abs}_{\mathscr{X}}\to X^{{\mathrm{an}}}$ such that

1. (i)

   $\ev_{\mathscr{X}}\circ\emb_{\mathscr{X}}=\Id_{\Delta^{\abs}_{\mathscr{X}}}$;

2. (ii)

   for each $s\in\Delta_{\mathscr{X}}^{\abs}$, if $s\in\rint(\sigma_{J})$, then ${\mathrm{red}}(\emb_{\mathscr{X}}(s))=\xi_{D_{J}}$, where $\xi_{D_{J}}$ is the generic point of $D_{J}$.

Then the skeleton and the retraction are given by

We now go back to the regular toric case. In particular, $X$ is a toric smooth projective variety over $K$ and $\mathscr{X}$ is a toric projective SNC model. Then all the divisors of $\Div_{0}(\mathscr{X})$ are toric divisors. Therefore, for $D\in\Div_{0}(\mathscr{X})_{\mathbb{R}}$, the function $\varphi_{D}$ is invariant under the action of the compact torus $\mathbb{S}=\Val_{K}^{-1}(0)$. The restriction of $\varphi_{D}$ to $\mathbb{T}^{{\mathrm{an}}}$ factorizes as

where $\phi_{D}$ is the function from [BPS14, Def. 4.3.6] corresponding to the trivial line bundle $\mathcal{O}_{X}$ with the metric determined by $D$ and the section $1$.

We now define $\ev_{\Pi}\colon N_{\mathbb{R}}\to\Div_{0}(\mathscr{X})^{\ast}$ by

By construction, the restriction of $\ev_{\Pi}$ to each polyhedron $\Lambda\in\Pi$ is affine. Moreover, using (A.7) and (A.8) we deduce that

As before let $D_{1},\dots,D_{\ell}$ be the components of the special fiber $\mathscr{X}_{s}$ and $x_{i}$ the divisorial point determined by $D_{i}$. Then the set of vertices of $\Pi$ is $\Pi^{0}=\{u_{1},\dots,u_{\ell}\}$, where $u_{i}=\Val_{K}(x_{i})$. Therefore $\ev_{\Pi}(u_{i})=e_{i}$. Since $\ev_{\Pi}$ is affine in each polyhedron of $\Pi$ we deduce that the image of $\ev_{\Pi}$ is $\Delta^{\abs}_{\mathscr{X}}$ and that $\ev_{\Pi}$ determines a homeomorphism $\Delta_{\Pi}\to\Delta^{\abs}_{\mathscr{X}}$. We define $\emb_{\Pi}\colon\Delta^{\abs}_{\Pi}\to N_{\mathbb{R}}$ as the composition of the inverse of this homeomorphism with the inclusion $\Delta_{\Pi}\hookrightarrow N_{\mathbb{R}}$. Using equation (A.9) and Lemma A.1 one can check that $\zeta_{K}\circ\emb_{\Pi}$ satisfies the conditions (1) and (2) that characterize $\emb_{\mathscr{X}}$. Therefore

We next claim that $p_{\Pi}=\emb_{\Pi}\circ\ev_{\Pi}.$ Indeed, for every $D\in\Div_{0}(\mathscr{X})$, since $D$ is a model of the trivial vector bundle, we know that $\rec(\phi_{D})$ is the zero function. Therefore, writing any $u\in\Lambda\in\Pi$ is as in (A.6), one can show that

This implies that $\ev_{\Pi}=\ev_{\Pi}\circ\,p_{\Pi}$. By construction $\emb_{\Pi}\circ\ev_{\Pi}$ is the identity in the image of $p_{\Pi}$. Therefore

Using equations (A.11) (A.10) and (A.9) we deduce that

and

concluding the proof. ∎