It is clear that $(1)\Rightarrow(2)\Rightarrow(3)$. The implication $(3)\Rightarrow(2)$ follows from the equality $P_{\theta}(f)-f=P_{\theta+dd^{c}f}(0)$. It remains to prove $(2)\Rightarrow(1)$. We may write a given $f\in C^{0}(X)$ as a uniform limit on $X$ of model functions $f_{j}$, and $P_{\theta}(f_{j})\to P_{\theta}(f)$ uniformly on $X$ thanks to the Lipschitz property of $P_{\theta}$, see Proposition 2.15. By Theorem 3.1 we thus have

in the weak topology of measures. Since $P_{\theta}(f_{j})-f_{j}\to P_{\theta}(f)-f$ uniformly on $X$ and the measures $(\theta+dd^{c}P_{\theta}(f_{j}))^{n}$ have uniformly bounded (in fact, constant) mass, it follows that

Let us prove the final assertion. Pick $\theta^{\prime}\in\mathcal{Z}^{1,1}(X)$ such that $\{\theta^{\prime}\}=\{\theta\}$ in $N^{1}(X)$. By the analogue of the $dd^{c}$-lemma proved in [BFJ11, Theorem 4.3] there exists $g\in\mathcal{D}(X)$ such that $\theta^{\prime}=\theta+dd^{c}g$. Observe that a function $\varphi$ is $\theta^{\prime}$-psh iff $\varphi+g$ is $\theta$-psh. As a consequence we get $P_{\theta^{\prime}}(f)-f=P_{\theta}(f+g)-(f+g)$, hence

for all $f\in C^{0}(X)$. ∎

Pick any regular model $\mathcal{X}$. Then the linear map $N^{1}(\mathcal{X}/S)\to N^{1}(X)$ is surjective hence open. It is thus enough to prove the following claim: let $\theta_{\mathcal{X}}\in N^{1}(\mathcal{X}/S)$ have ample image in $N^{1}(X)$, and assume that $\theta_{\mathcal{X}}$ is the limit of a sequence $\theta_{m,\mathcal{X}}\in N^{1}(\mathcal{X}/S)$. If the corresponding forms $\theta_{m}\in\mathcal{Z}^{1,1}(X)$ all satisfy the orthogonality property, then so does $\theta$.

Let $f\in C^{0}(X)$. By Proposition 2.15 we have $P_{\theta_{m}}(f)\to P_{\theta}(f)$ uniformly on $X$. We claim that

with uniformly bounded mass. Since $(P_{\theta_{m}}(f)-f)\to(P_{\theta}(f)-f)$ uniformly on $X$, we have as before

which concludes the proof.

To prove the claim, pick any model function $g\in\mathcal{D}(X)$, and fix $\varepsilon>0$. By Corollary 2.12, we can find a $\theta$-psh model function $\varphi$ such that $\sup|\varphi-P_{\theta}(f)|\leq\varepsilon$. We then have

Using integration by parts, the last term can be bounded as follows.

where $\omega$ is a fixed form such that $(\omega+dd^{c}g)$ is semipositive, and $C=2\{\omega\}\,\{\theta\}^{n-1}$. In a similar way, the first term is bounded from above by

for $m$ large enough. Finally $g$ and $\varphi$ being model functions, the second term tends to zero as $m\to\infty$, and we get $\limsup_{m}I_{m}\leq 3C\varepsilon$. We conclude by letting $\varepsilon\to 0$. ∎

For all $m\gg 1$ set $\varphi_{m}:=\tfrac{1}{m}\log|\mathfrak{a}_{m}|=\tfrac{1}{m}\varphi_{F_{m}}$. This is a $\theta$-psh model function, and [BFJ11, Theorem 8.5] states that $\varphi_{m}\to P_{\theta}(0)$ uniformly on $X$. Unravelling the definitions, we find

By Theorem 3.1 the right-hand side converges to $\int P_{\theta}(0)\left(\theta+dd^{c}P_{\theta}(0)\right)^{n}$, which proves the result. ∎

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.

∎

Let $\overline{K}/\overline{F}$ be algebraic closures. The groups of components of the Picard groups $\Pic(Y_{\overline{F}})$ and $\Pic(Y_{\overline{K}})$ are then isomorphic, so we have an isomorphism $N^{1}(Y_{\overline{F}})_{\mathbf{Q}}\simeq N^{1}(Y_{\overline{K}})_{\mathbf{Q}}$ by the result of [Mat57] recalled above (compare [MP11, Proposition 3.1]). This isomorphism is furthermore compatible with ample classes by the Nakai-Moishezon criterion for ampleness.

It is enough to show the surjectivity of $N^{1}(Y)_{\mathbf{Q}}\to N^{1}\left(Y_{K}\right)_{\mathbf{Q}}$. Let $\beta\in N^{1}(Y_{K})_{\mathbf{Q}}$. By the previous result, we find $L\in\Pic(Y_{\overline{F}})_{\mathbf{Q}}$ mapping to the lift of $\beta$ in $N^{1}(Y_{\overline{K}})_{\mathbf{Q}}$. Since $Y_{\overline{F}}$ is in particular reduced, $L$ can be represented by some $\bar{D}\in\Div(Y_{\overline{F}})_{\mathbf{Q}}$. The average of the Galois orbit of $\bar{D}$ is then $\mathrm{Gal}(\bar{F}/F)$-invariant, hence descends to $D\in\Div(Y)_{\mathbf{Q}}$ by [Car58, Proposition 11, §4.6]. By construction, the image of $D$ under the composition

coincides with the image of $\beta$. But $N^{1}(Y_{K})_{\mathbf{Q}}\to N^{1}(Y_{\overline{K}})_{\mathbf{Q}}$ is injective by the projection formula, and the result follows. ∎

Set $\mathcal{F}_{m}:=\pi_{*}\mathcal{O}_{\mathcal{X}}(m\mathfrak{L})$, and pick a very ample line bundle $H$ on $B$. By the Castelnuovo-Mumford criterion [Laz04, Theorem 1.8.5] it is enough to show the existence of $m_{0}\in\mathbf{N}$ such that

for all $m$ large and divisible.

Since $\mathfrak{L}$ is ample over the generic point of $B$, the $\mathcal{O}_{B}$-algebra $\bigoplus_{m\geq 0}\mathcal{F}_{m}$ is finitely generated at the generic point of $B$. After perhaps replacing $\mathfrak{L}$ by $d\mathfrak{L}$ for some $d\in\mathbf{N}$, we may further assume that the generators have degree $1$, so that $\mathcal{F}_{m}/\mathcal{F}_{1}^{m}$ has zero-dimensional support for all $m\geq 1$. As a consequence, the map

is surjective for all $m\geq 1$. Upon replacing $\mathfrak{X}$ with $\Proj_{B}\left(\bigoplus_{m\geq 0}\mathcal{F}_{1}^{m}\right)$ we are thus reduced to proving (A.4) when $\mathfrak{L}$ is $\pi$-ample, i.e. ample on all fibers of $\pi$. In that case we have $R^{q}\pi_{*}\mathcal{O}_{\mathfrak{X}}(m\mathfrak{L})=0$ for $m\gg 1$ and $q>0$ by Serre vanishing, and the degeneration of the Leray spectral sequence yields

which vanishes for all $m\gg 1$ if we choose $m_{0}$ such that $\mathfrak{L}+m_{0}\pi^{*}H$ is ample on $\mathfrak{Y}$. ∎