In the $G$-topology on ${X^{{\mathrm{an}}}}$ induced by the strictly $K$-affinoid domains, a $\mathbb{Z}$-model function is given locally by $-\log|\gamma|$ for a rational function $\gamma$ on $X$. Hence (a) follows from [GRW16, Prop. 5.6]. Property (b) was proven in [GH15, Prop. B.7] for any dimension.

To prove (c), we choose a subdivision of $S(\mathscr{X})$ as in Definition 3.4 for $G$. As in [BPR13, §3], this subdivison is the skeleton of a strictly semistable model $\mathscr{X}^{\prime}$ dominating $\mathscr{X}$ and with the same retraction $\tau$. Then (c) follows from (b). ∎

If we pass to the completion $\mathbb{C}_{K}$ of an algebraic closure of $K$, there is a strictly semistable model $\mathscr{X}^{\prime}$ dominating $\mathscr{X}$ such that $f$ is determined on $\mathscr{X}$. This is proven in [BL85, §7]. We note that the property $\theta$-psh holds if and only if the corresponding property holds after base change to $\mathbb{C}_{K}$. This is a consequence of the projection formula in algebraic intersection theory. Since the degree is invariant under base change, it follows from [Thu05, Prop. 2.2.21] that the left hand side of (3.1) is invariant under base change as well. We conclude that we may assume that $K$ is algebraically closed and that $\mathscr{X}=\mathscr{X}^{\prime}$, i.e. $f$ is determined on $\mathscr{X}$. Then (3.1) follows from the slope formula of Katz, Rabinoff, and Zureick-Brown [KRZB16, Thm. 2.6]. ∎

It follows from Proposition 3.5 that $\varphi\circ\tau$ is a model function. To check that $\varphi\circ\tau$ is $\theta$-psh, we may assume $K$ algebraically closed as we have seen in the proof of Proposition 3.7. Moreover, we have seen that there is a strictly semistable model $\mathscr{X}^{\prime}$ of $X$ dominating $\mathscr{X}$ such that $f$ is determined on $\mathscr{X}^{\prime}$. Then

and $\varphi\circ\tau$ is constant along edges of $\Delta^{\prime}$ which are not contained in $\Delta$. By Proposition 3.5, there is a piecewise linear function $F^{\prime}$ on $\Delta^{\prime}$ with $\varphi=F^{\prime}\circ\tau^{\prime}$ for the canonical retraction $\tau^{\prime}\colon{X^{{\mathrm{an}}}}\to\Delta^{\prime}$ such that $mF^{\prime}$ is integral $\Gamma$-affine for a non-zero $m\in\mathbb{N}$. Moreover, the function $F^{\prime}$ is affine on the edges of $\Delta^{\prime}$. Let $F$ be the restriction of $F^{\prime}$ to $\Delta$. The same arguments as in [BFJ16a, Prop. 5.7] show that the $\theta$-psh function $\varphi$ is a uniform limit of functions of the form $\frac{1}{m}\log|\mathfrak{a}|$ with non-zero $m\in\mathbb{N}$ and with a vertical fractional ideal sheaf $\mathfrak{a}$ on $\mathscr{X}$. By [Ber99, Thm. 5.2(ii)], we deduce that $\varphi\leq\varphi\circ\tau$. Using the terminology introduced in Proposition 3.7, for all $x\in\Delta$ and $v\in T_{x}(\Delta^{\prime})$ we obtain $\lambda_{x,\nu}(F^{\prime})=\lambda_{x,\nu}(F)$ if $v\in T_{x}(\Delta^{\prime})$ and $\lambda_{x,\nu}(F^{\prime})\leq 0$ if $\nu\in T_{x}(\Delta^{\prime})\setminus T_{x}(\Delta)$. This implies

Here this first inequality comes from Proposition 3.7, since $\varphi$ is $\theta$-psh. Applying Proposition 3.7 again, we conclude that $\varphi\circ\tau=F\circ\tau$ is $\theta$-psh. ∎

Step 1. By Proposition 2.9(iv), we have

Hence replacing $\theta$ by $\theta+dd^{c}f$ and $f$ by $0$, we can assume that $f=0$ by Proposition 3.5(b).

Step 2. Let $\Delta\coloneqq S(\mathscr{X})$ denote the skeleton of $\mathscr{X}$. By Propositions 3.5 and 3.8, we get that

for the set $\mathcal{A}$ of non-positive piecewise linear functions $F$ on $\Delta$ such that $mF$ is integral $\Gamma$-affine for some $m\in\mathbb{N}_{>0}$ and such that $F\circ\tau$ is $\theta$-psh. Note that the piecewise linear functions are not assumed to be affine on the edges of $\Delta$. Since $X$ is a smooth projective curve and the de Rham class $\{\theta\}$ is nef, it is clear that $\mathcal{A}$ is non-empty. We introduce the function $F_{0}\colon\Delta\to\mathbb{R}$ defined by

By (3.2), we get that ${P}_{\theta}(0)=F_{0}\circ\tau$. Hence we can reduce (i) to prove that $F_{0}$ is affine on each edge of $\Delta$.

Step 3. For $F\in\mathcal{A}$, let $L(F)\colon\Delta\to\mathbb{R}$ be the function which is affine on the edges of $\Delta$ and which agrees with $F$ on the set $V$ of vertices of $\Delta$. As $F\leq 0$, we deduce immediately $L(F)\leq 0$. Since $F\circ\tau$ is $\theta$-psh, Proposition 3.7 shows that $F$ is convex on each edge of $\Delta$ and hence $F\leq L(F)$.

By passing from $F$ to $L(F)$, the slopes do not decrease in the vertices and using that $F\circ\tau$ is $\theta$-psh, it follows from Proposition 3.7 that $L(F)\circ\tau$ is $\theta$-psh as well.

The slopes of the function $L(F)$ might be non-rational. However, we can approximate the slopes of $L(F)$ in a rational way at any vertex and thus for any $\varepsilon>0$ we find a piecewise linear function $L_{\varepsilon}(F)$ on $\Delta$ such that

1. (i)

   $L_{\varepsilon}(F)$ agrees with $F$ on $V$.

2. (ii)

   $L_{\varepsilon}(F)$ has rational slopes.

3. (iii)

   $L_{\varepsilon}(F)$ is convex on the edges of $\Delta$.

4. (iv)

   $L_{\varepsilon}(F)\geq F$.

5. (v)

   $\sup|L_{\varepsilon}(F)-L(F)|<\varepsilon$

We claim that $L_{\varepsilon}(F)\in\mathcal{A}$. It follows from (ii) that $mL_{\varepsilon}(F)$ is integral $\Gamma$-affine for some $m\in\mathbb{N}_{>0}$. Since $F$ and $L(F)$ agree on $V$, it is clear from (i) and (iii) that $L_{\varepsilon}(F)\leq L(F)\leq 0$. To show that $L_{\varepsilon}(F)\circ\tau$ is $\theta$-psh, we use the slope criterion from Proposition 3.7. Note that (3.1) is fulfilled in the interior of each edge of $\Delta$ by (iii). In a vertex of $\Delta$, the inequality (3.1) is satisfied by using the corresponding inequality for $F$, (i) and (iv). This proves $L_{\varepsilon}(F)\in\mathcal{A}$. As a consequence we find

Step 4. We claim $F_{0}=L(F_{0})$. First note that since the $\max$ of convex functions in convex, $F_{0}$ is convex on each edge of $\Delta$, thus $F_{0}\leq L(F_{0})$.

We pick an $\varepsilon>0$. For any $v\in V$, there is $f_{v}\in\mathcal{A}$ with $f_{v}(v)>L(F_{0})(v)-\varepsilon$. It follows from [GM16, Prop. 3.12] that the maximum of two $\theta$-psh model functions is again a $\theta$-psh model function. Using 2.5, we conclude that $\mathcal{A}$ is closed under the operation $\max$. Thus $L(\max\{f_{v}\mid v\in V\})\in\mathcal{A}$ is in $\varepsilon$-distance to $L(F_{0})$ at every vertex of $\Delta$ and hence at every point of $\Delta$. As $\varepsilon>0$ can be chosen arbitrarily small, (3.3) yields $L(F_{0})\leq F_{0}$ and hence we get Step 4. Note that Step 4 proves (i).

Step 5. In the case $\Gamma\subset\mathbb{Q}$, we have $L(F)\in\mathcal{A}$ for $F\in\mathcal{A}$. Indeed, we note that in this special case the edges have rational lengths and $F$ takes rational values at $V$. We deduce from Proposition 3.5 that $L(F)\circ\tau$ is a model function. Let $\mathcal{B}$ be the set of $F\in\mathcal{A}$ such that $F$ is affine on every edge of $\Delta$. For $F\in\mathcal{A}$ we thus have $L(F)\in\mathcal{B}$ which shows via (3.3) that we might restrict the $\sup$ to $\mathcal{B}$ in the definition of $F_{0}$. Recall that $V$ is the set of vertices of $\Delta$. Since the edge lengths of $\Delta$ are rational, the map $\Psi\colon\mathcal{B}\to\mathbb{R}^{V}$ defined by $\Psi(F)=(F(v))_{v\in V}$ identifies $\mathcal{B}$ with the rational points of a rational polyhedron in $\mathbb{R}^{V}$ defined by the linear inequalities of Proposition 3.7. Note that by affineness on the edges, we need to check the slope inequalities only at the vertices. If a rational linear form $\varphi$ is bounded from above on a rational polyhedron $P$, then $\varphi|_{P}$ achieves its maximum in a rational point. Hence there exists $G\in\mathcal{B}$ such that

We claim that $G=F_{0}$. Considering $F\in\mathcal{B}$, we get $\max(G,F)\in\mathcal{A}$ by Step 4 and hence $H^{\prime}\coloneqq L(\max(G,F))\in\mathcal{B}$. Hence $H^{\prime}\geq G$, $H^{\prime}\geq F$ and $H^{\prime}\in\mathcal{B}$. But by (3.4) we deduce that for $v\in V$ we have $H^{\prime}(v)=G(v)$. Since functions in $\mathcal{B}$ are determined by their values on $V$, we have $H^{\prime}=G$. Hence $G\geq F$, whence $G=F_{0}$. It follows that $P_{\theta}(0)=F_{0}\circ\tau=G\circ\tau$ is a $\theta$-psh function proving (ii). ∎

By Proposition 2.9 (viii) it is enough to prove the continuity of ${P}_{\theta}(u)$. By the semistable reduction theorem [BL85, §7], there is a finite field extension $K^{\prime}/K$ such that $X^{\prime}\coloneqq X\otimes_{K}K^{\prime}$ has a strictly semistable model $\mathscr{X}^{\prime}$ with $\theta^{\prime}\coloneqq q^{*}\theta$ determined on $\mathscr{X}^{\prime}$, where $q\colon X^{\prime}\to X$ is the canonical map. It follows from Proposition 3.10 that ${P}_{\theta^{\prime}}(u\circ q)$ is continuous. We know from Lemma 2.11 that

By [Ber90, Prop. 1.3.5], the topological space of ${X^{{\mathrm{an}}}}$ is the quotient of $(X^{\prime})^{\rm an}$ by the automorphism group of $K^{\prime}/K$. We conclude that ${P}_{\theta}(u)$ is continuous. ∎

By definition, we have ${P}(\|\ \|)=\|\ \|e^{-{P}_{\theta}(0)}$ for $\theta\coloneqq c_{1}(L,\|\ \|)$, hence the claim follows from Proposition 3.10. ∎

If $\deg(L)\leq 0$, then it is clear from the definition that $\vol(L,\|\ \|_{1},\|\ \|_{2})=0$. So we may assume that $L$ is ample. We need the energy $E(L,{\|\ \|}_{1},{\|\ \|}_{2})$ with respect to continuous semipositive metrics ${\|\ \|}_{1},{\|\ \|}_{2}$ on $L^{{\mathrm{an}}}$ introduced in [BGJKM16, Def. 2.4.4]. By Corollary 3.11, the envelopes ${P}({\|\ \|}_{1})$ and ${P}({\|\ \|}_{2})$ are semipositive model metrics on $L^{{\mathrm{an}}}$. In particular, they are continuous and hence it follows from [BGJKM16, Cor. 6.2.2] that

In the case of semipositive model metrics associated to line bundles $\mathscr{L}_{1},\mathscr{L}_{2}$ on a ${K^{\circ}}$-model $\mathscr{X}$, our assumption $\Gamma\subset\mathbb{Q}$ yields that the energy is defined as a $\mathbb{Q}$-linear combination of intersection numbers of the line bundles $\mathscr{L}_{1},\mathscr{L}_{2}$ on $\mathscr{X}$ proving the claim. ∎