Let us fix an ample class $\alpha\in N^{1}(X)$. By Lemma A.2 we may assume $\alpha\in N^{1}(X)_{\mathbf{Q}}$.

Since $X$ is algebraizable, we can find a smooth projective curve $B$ over the residue field $k$ such that $F=k(B)$; a closed point $0\in B$ and a regular parameter $t\in\mathcal{O}_{B,0}$ inducing an isomorphism $S\simeq\spec\widehat{\mathcal{O}}_{B,0}$; and a smooth projective variety $Y$ over $F$ such that $X=Y_{K}$.

By Lemma A.5 below we may then choose an ample $\mathbf{Q}$-line bundle $L\in\Pic(Y)_{\mathbf{Q}}$ mapping to $\alpha$ in $N^{1}_{\mathbf{Q}}(X)$. We can also find a normal, flat and projective $B$-scheme $\mathfrak{Y}$ having $Y$ as its generic fiber and such that $L\in\Pic(Y)_{\mathbf{Q}}$ extends to $\mathfrak{L}\in\Pic(\mathfrak{Y})_{\mathbf{Q}}$. The latter is therefore ample on the generic fiber of the structure morphism $\pi:\mathfrak{Y}\to B$, hence in particular $\pi$-big. Since the natural morphism $\mathcal{X}:=\mathfrak{Y}\times_{B}S\to\mathfrak{Y}$ is regular, $\mathcal{X}$ is normal, as well as flat and projective over $S$, hence a model of $X$ according to our definition. The $\mathbf{Q}$-line bundle $\mathfrak{L}$ induces $\mathcal{L}\in\Pic(\mathcal{X})_{\mathbf{Q}}$.

The curvature form $\theta\in\mathcal{Z}^{1,1}(X)$ of the model metric defined by $\mathcal{L}$ has $\alpha$ as its de Rham class. Our goal is to show that (A.1) holds for each $f\in\mathcal{D}(X)$. We may in fact assume that $f=0$. Indeed let $\mathcal{X}^{\prime}$ be a determination of $f$, which may be taken to dominate $\mathfrak{Y}$. The model $\mathcal{X}^{\prime}$ is then the blow-up of $\mathcal{X}$ along a vertical ideal sheaf $\mathfrak{a}$. Since $(t^{m})\subset\mathfrak{a}$ for some $m\in\mathbf{N}$, $\mathfrak{a}$ comes from an ideal sheaf on $\mathfrak{Y}$, and the blow-up $\mathfrak{Y}^{\prime}$ of $\mathfrak{Y}$ along this ideal satisfies $\mathfrak{Y}^{\prime}\times_{B}S=\mathcal{X}^{\prime}$ since blow-ups commute with flat base change. Replacing $\mathfrak{Y}$ with $\mathfrak{Y}^{\prime}$, we may thus assume that $\mathcal{X}$ is a determination of $f$, so that there exists a vertical $\mathbf{Q}$-divisor $E\in\Div_{0}(\mathcal{X})$ such that $f=f_{E}$. Since $E$ is vertical, it also comes from $\mathfrak{Y}$. Replacing $\mathfrak{L}$ with $\mathfrak{L}+E$ reduces us as desired to the case $f=0$.

After perhaps passing to a multiple, we may further assume that $\mathfrak{L}\in\Pic(\mathfrak{Y})$. According to Lemma A.3, we are to show that the approximate Zariski decompositions of $\mathcal{L}$ are asymptotically orthogonal.

Denote by $\mathfrak{a}_{m}$ the base-ideal of $m\mathcal{L}$ on $\mathcal{X}$, and let $\mathfrak{b}_{m}\subset\mathcal{O}_{\mathfrak{Y}}$ be the relative base-ideal of $m\mathfrak{L}$ on $\mathfrak{Y}/B$. By flat base change we have $\mathfrak{a}_{m}=\mathfrak{b}_{m}\cdot\mathcal{O}_{\mathcal{X}}$. Let $\rho_{m}:\mathfrak{Y}_{m}\to\mathfrak{Y}$ be the normalized blow-up of $\mathfrak{Y}$, and let $G_{m}$ be the effective Cartier divisor of $\mathfrak{Y}_{m}$ such that $\mathcal{O}_{\mathfrak{Y}_{m}}(-G_{m})=\mathfrak{b}_{m}\cdot\mathcal{O}_{\mathfrak{Y}_{m}}$. Note that $G_{m}$ is supported on finitely many fibers over $B$ for $m\gg 1$, since $m\mathfrak{L}$ is $\pi$-ample. Observe also that $G_{m}$ pulls back to the similarly defined divisor $F_{m}$ on $\mathcal{X}_{m}:=\mathfrak{Y}_{m}\times_{B}S$. Finally set $\mathfrak{M}_{m}:=\rho_{m}^{*}(m\mathfrak{L})-G_{m}$, which pulls back to $\mathcal{M}_{m}$ on $\mathcal{X}_{m}$. Once again by flat base change, it is enough to show that

We are going to prove this by reducing to the absolute case of a big line bundle $\mathfrak{D}$ on $\mathfrak{Y}$. By Lemma A.6 below we may choose an ample line bundle $H\in\Pic(B)$ such that the sheaves

are globally generated over $B$ for all $m\gg 1$ sufficiently divisible. Since $\mathfrak{L}$ is $\pi$-big, we may assume (after perhaps replacing $H$ with a large enough multiple) that $\mathfrak{D}:=\mathfrak{L}+\pi^{*}H$ is a big line bundle on the projective $k$-variety $\mathfrak{Y}$.

The relative base-ideal $\mathfrak{b}_{m}$ of $m\mathfrak{L}$ coincides with the relative base-ideal of $m\mathfrak{D}$ since $m\mathfrak{L}$ and $m\mathfrak{D}$ are $\pi$-linearly equivalent by construction. The fact that

is globally generated therefore shows that $\mathfrak{b}_{m}$ is also the (absolute) base-ideal of $m\mathfrak{D}$. As a consequence we get that $\mathfrak{P}_{m}:=\rho_{m}^{*}(m\mathfrak{D})-G_{m}$ is the (absolute) base-point free part of $m\mathfrak{D}$, and we infer from [BDPP04, Theorem 4.1] that $\left(\tfrac{1}{m}\mathfrak{P}_{m}\right)^{n}\cdot\left(\tfrac{1}{m}G_{m}\right)\to 0$. But $\mathfrak{P}_{m}=\mathfrak{M}_{m}+(\pi\circ\rho_{m})^{*}(mH)$ implies $\mathfrak{M}_{m}^{n}\cdot G_{m}=\mathfrak{P}_{m}^{n}\cdot G_{m}$ since $G_{m}$ is supported on finitely many fibers over $B$, and the result follows.

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