Since any ${K^{\circ}}$-model of $X$ is dominated by a projective ${K^{\circ}}$-model of $X$ [Gub03, Proposition 10.5], we may assume that ${\mathscr{X}}$ is projective. Let $n\coloneqq\dim(X)$. Hence ${\mathscr{X}}$ is irreducible of dimension $n+1$. We choose a closed curve $Y$ in the special fibre ${\mathscr{X}}_{s}$. Then we have to show that $\deg_{\mathscr{L}}(Y)\geq 0$. We follow the strategy of [Goo69] to use the blow up $\pi:{\mathscr{X}}^{\prime}\rightarrow{\mathscr{X}}$ in $Y$ (as suggested in [BFJ16, Remark 5.13]). Then $E:=\pi^{-1}(Y)$ is an effective Cartier divisor on ${\mathscr{X}}^{\prime}$ which is vertical. Moreover, ${\mathscr{X}}^{\prime}$ is projective and hence we have a very ample invertible sheaf ${\mathscr{H}}^{\prime}$ on ${\mathscr{X}}^{\prime}$. Since $E$ is an $n$-dimensional projective variety mapping onto $Y$, it follows from using generic hyperplane sections, the fibre theorem [Har77, Exercise II.3.22] and the fact that ${\mathscr{X}}$ is irreducible of dimension $n+1$ that $\pi_{*}(({\mathscr{H}}^{\prime})^{n-1}.E)$ is a positive multiple of $Y$. By projection formula, it is enough to show

(5.5.1)

$$\deg_{{\mathscr{L}}^{\prime}}(({\mathscr{H}}^{\prime})^{n-1}.E)\geq 0$$

for ${\mathscr{L}}^{\prime}:=\pi^{*}({\mathscr{L}})$. We may assume that $Y\subset V(\mathfrak{a}_{m})$ for every $m$ as otherwise there is a global section $s_{m}$ of ${\mathscr{L}}^{\otimes m}$ such that $s_{m}|_{Y}\neq 0$ and hence

$$\deg_{\mathscr{L}}(Y)=\deg({\rm div}(s_{m}|_{Y}))\geq 0$$

would make the claim obvious. The crucial new idea is now to consider the family of blow ups $\psi_{m}:{\mathscr{X}}_{m}\rightarrow{\mathscr{X}}^{\prime}$ of ${\mathscr{X}}_{m}$ in the closed subscheme $\pi^{-1}(V(\mathfrak{a}_{m}))$ for all integers $m\geq 1$. Replacing ${\mathscr{X}}_{m}$ by its normalization, we can assume that ${\mathscr{X}}_{m}$ is normal. Let $\pi_{m}=\pi\circ\psi_{m}$. We set $D_{m}\coloneqq\pi_{m}^{-1}(\mathfrak{a}_{m})$. It is an effective Cartier divisor on ${\mathscr{X}}_{m}$, and we denote by $s_{-D_{m}}$ the canonical meromorphic section of $\mathcal{O}(-D_{m})$. Note that all these models have generic fibre $X$. We conclude that $E_{m}:=\pi_{m}^{-1}(Y)=\psi_{m}^{-1}(E)$ is an effective Cartier divisor. Note that ${\mathscr{H}}_{m}:=\psi_{m}^{*}({\mathscr{H}}^{\prime})$ is generated by global sections. We conclude from refined intersection theory that

(5.5.2)

$${\rm cl}(Z)=c_{1}({\mathscr{H}}_{m})^{n-1}.{\rm cyc}(E_{m})\in{\rm CH}_{1}(E_{m})$$

for an effective $1$-dimensional cycle $Z$ of ${\mathscr{X}}_{m}$ with support over $Y$. We consider the invertible sheaf ${\mathscr{L}}_{m}:=\pi_{m}^{*}({\mathscr{L}}^{\otimes m})\cong\psi_{m}^{*}({\mathscr{L}}^{\prime\otimes m})$ of ${\mathscr{X}}_{m}$. We claim that

(5.5.3)

$$\deg_{{\mathscr{L}}_{m}}(Z)\geq\deg_{{\mathcal{O}}(D_{m})}(Z).$$

To prove this, let $Z_{m}$ be any irreducible component of $Z$. We choose $\zeta_{m}\in Z_{m}$ and let $\zeta:=\pi_{m}(\zeta_{m})$. We note first that the stalk of ${\mathscr{L}}_{m}(-D_{m})$ at $\zeta_{m}$ is generated by global sections. Indeed, it follows from the definitions that there is a global section $s_{m}$ of ${\mathscr{L}}^{\otimes m}$ and an invertible section $\ell_{m}$ of ${\mathscr{L}}^{\otimes m}$ at $\zeta$ such that $\pi_{m}^{*}(s_{m}/\ell_{m})$ is an equation of the Cartier divisor $D_{m}$ at $\zeta_{m}$. It follows from the definition of the base ideal $\mathfrak{a}_{m}$ that $t_{m}:=\pi_{m}^{*}(s_{m})\otimes s_{-D_{m}}$ is a global section of ${\mathscr{L}}_{m}(-D_{m})$ and the choice of $s_{m}$ yields that $t_{m}$ generates the stalk at $\zeta_{m}$. We deduce that the restriction of $t_{m}$ to $Z_{m}$ is a global section which is not identically zero and hence

$$\deg_{{\mathscr{L}}_{m}}(Z_{m})=\deg_{{\mathcal{O}}(D_{m})}(Z_{m})+\deg({\rm div}(t_{m}|_{Z_{m}}))\geq\deg_{{\mathcal{O}}(D_{m})}(Z_{m})$$

proving (5.5.3). By projection formula and (5.5.2), we have

$$m\deg_{{\mathscr{L}}^{\prime}}(c_{1}({\mathscr{H}}^{\prime})^{n-1}.E)=\deg_{{\mathscr{L}}_{m}}(Z)$$

and hence (5.5.3) leads to

$$m\deg_{{\mathscr{L}}^{\prime}}(c_{1}({\mathscr{H}}^{\prime})^{n-1}.E)\geq\deg_{{\mathcal{O}}(D_{m})}(Z)=\deg_{{\mathcal{O}}(D_{m})}(c_{1}({\mathscr{H}}_{m})^{n-1}.{\rm cyc}(E_{m})).$$

Commutativity of intersection product shows

(5.5.4)

$$m\deg_{{\mathscr{L}}^{\prime}}(c_{1}({\mathscr{H}}^{\prime})^{n-1}.E)\geq\deg(c_{1}({\mathscr{H}}_{m})^{n-1}.E_{m}.{\rm cyc}(D_{m})).$$

The intersection product can be computed on the model ${\mathscr{X}}_{m}$ over the valuation ring ${K^{\circ}}$ using the intersection theory with Cartier divisors from [Gub98] (see also [GS15b, Section 2] for the normal case). We have

$${\rm cyc}(D_{m})=\sum_{W}\mu_{W}W,$$

where $W$ ranges over all irreducible components of the special fibre of ${\mathscr{X}}_{m}$. Using [BPS14, Proposition 1.3.3] there is a unique point $\xi_{W}$ of the analytification ${X^{\rm an}}$ of the generic fibre of ${\mathscr{X}}_{m}$ with reduction equal to the generic point of $W$ (see [Ber90, Proposition 2.4.4] and [Gub07b, 2.5, 2.6]) and the multiplicities $\mu_{W}$ are given by

$$\mu_{W}=-\log\|s_{D_{m}}(\xi_{W})\|.$$

We insert this in (5.5.4) and use again projection formula to get

(5.5.5)

$$m\deg_{{\mathscr{L}}^{\prime}}(c_{1}({\mathscr{H}}^{\prime})^{n-1}.E)\geq\sum_{V}\sum_{W:\psi_{m}(W)=V}\mu_{W}[W:V]\deg(c_{1}({\mathscr{H}}^{\prime})^{n-1}.E.V),$$

where $V$ ranges over all irreducible components of $({\mathscr{X}}^{\prime})_{s}$ and $W$ ranges over the irreducible components of $({\mathscr{X}}_{m})_{s}$ with $\psi_{m}(W)=V$. Here, $[W:V]$ is the degree of the induced map $W\rightarrow V$. Note that $\psi_{m}(\xi_{W})$ is a divisorial point of ${X^{\rm an}}$ which reduces to the generic point of $V$ in the model ${\mathscr{X}}^{\prime}$. We conclude that there are only finitely many possibilities for $\psi_{m}(\xi_{W})$ independently of the choice of $m$.

We choose ${\varepsilon}>0$ small. By the above finiteness, there is a sufficiently large $m$ such that

$$0\leq-\frac{1}{m}\log|\mathfrak{a}_{m}|(\psi_{m}(\xi_{W}))\leq{\varepsilon}$$

for all $W$ as above. We conclude that

$$0\leq\mu_{W}=-\log\|s_{D_{m}}(\xi_{W})\|=-\log|\mathfrak{a}_{m}|(\xi_{V})\leq m{\varepsilon}$$

for all $V$ and $W$ as above with $\psi_{m}(W)=V$. Let $-R$ be the minimum of the finitely many intersection numbers $\deg(({\mathscr{H}}^{\prime})^{n-1}.E.V)$ and $0$. Then (5.5.5) leads to

$$\deg_{{\mathscr{L}}^{\prime}}(c_{1}({\mathscr{H}}^{\prime})^{n-1}.E)\geq-R{\varepsilon}\sum_{V}\sum_{W:\psi_{m}(W)=V}[W:V].$$

By projection formula for $\psi_{m}$ applied to the Cartier divisor ${\rm div}(\rho)$ on ${\mathscr{X}}^{\prime}$ for any non-zero $\rho$ in the maximal ideal of $R$,

we deduce easily that

$$\sum_{W:\psi_{m}(W)=V}m_{W}[W:V]=m_{V}$$

for the multiplicity $m_{V}$ (resp. $m_{W}$) of $({\mathscr{X}}^{\prime})_{s}$ (resp. $({\mathscr{X}}_{m})_{s}$) in $V$ (resp. $W$). We conclude that

$$\deg_{{\mathscr{L}}^{\prime}}(c_{1}({\mathscr{H}}^{\prime})^{n-1}.E)\geq-R{\varepsilon}\sum_{V}m_{V}.$$

The numbers $R$ and $m_{V}$ are independent of ${\varepsilon}$. This proves (5.5.1) and hence the claim. ∎

Since we may check semipositivity after a base extension (see Lemma 3.3), we may replace $K$ by a finite field extension of $K$. Then, using Lemma 3.6, we may assume that $X$ is a variety.

Let $\varphi$ be a model function on $X$ which is the pointwise limit of $\theta$-psh functions. Replacing $\theta$ by $\theta+dd^{c}\varphi$, we may assume that $\varphi=0$. Then the existence of a $\theta$-psh function yields that $\theta$ is semipositive and hence $\{\theta\}$ is nef (see 4.8). Let ${\mathscr{X}}$ be a ${K^{\circ}}$-model of $X$ such that $\theta$ is determined on ${\mathscr{X}}$. Then the restriction of $\theta_{\mathscr{X}}$ to $X$ is nef.

We may replace ${\mathscr{X}}$ by a generically finite covering ${\mathscr{X}}^{\prime}$ for any ${K^{\circ}}$-model ${\mathscr{X}}^{\prime}$ with generic fibre $X^{\prime}$. This does not change convergence of metrics and semipositivity. It is here, where we use that pointwise convergence holds on ${X^{\rm an}}$. By [dJ96, Theorem 4.5], up to replacing $K$ by a finite field extension, we may assume that ${\mathscr{X}}$ is SNC (see 5.1). The proof of Proposition 4.13 shows that $N^{1}({\mathscr{X}}/S)$ is a finite dimensional ${\mathbb{R}}$-vector space as we can see it as a subspace of $N^{1}({\mathscr{X}}_{s})$. We have also seen that the ample cone in $N^{1}({\mathscr{X}}/S)$ is the intersection of $N^{1}({\mathscr{X}}/S)$ with the ample cone in $N^{1}({\mathscr{X}}_{s})$ and hence it is open in $N^{1}({\mathscr{X}}/S)$. We conclude that there are ${\mathscr{H}}_{1},\dots,{\mathscr{H}}_{n}$ ample line bundles on ${\mathscr{X}}$ such that their numerical classes $\alpha_{j}$ form a basis of $N^{1}({\mathscr{X}}/S)$. Then there are $\lambda_{j}\in{\mathbb{R}}$ such that

$$c_{1}({\mathscr{L}}):=\sum_{j}\lambda_{j}c_{1}({\mathscr{H}}_{j})\in{\rm Pic}({\mathscr{X}})_{\mathbb{R}}$$

represents $\theta$. Let ${\varepsilon}_{j}$ be small positive numbers such that the numbers $\lambda_{j}+{\varepsilon}_{j}$ are rational. We consider the ${\mathbb{Q}}$-line bundle

$${\mathscr{L}}_{\varepsilon}:=\bigotimes_{j}{\mathscr{H}}_{j}^{\otimes(\lambda_{j}+{\varepsilon}_{j})}$$

on ${\mathscr{X}}$ and let $L_{\varepsilon}:={\mathscr{L}}_{\varepsilon}$. Since $\{\theta\}$ is nef and ${\varepsilon}_{j}>0$, it follows that $L_{\varepsilon}$ is ample. For any model function $\psi$ on $X$, we have

$$c_{1}(L_{\varepsilon},e^{-\psi}{\|\hskip 4.30554pt\|}_{{\mathscr{L}}_{\varepsilon}})=dd^{c}\psi+\theta+\sum_{j}{\varepsilon}_{j}\alpha_{j}.$$

We conclude that a $\theta$-psh model function $\psi$ yields a semipositive model metric $e^{-\psi}{\|\hskip 4.30554pt\|}_{{\mathscr{L}}_{\varepsilon}}$. Since $0$ is the pointwise limit of $\theta$-psh model functions $\psi$, we deduce that ${\|\hskip 4.30554pt\|}_{{\mathscr{L}}_{\varepsilon}}$ is the pointwise limit of semipositive model metrics on $L_{\varepsilon}$. It follows from Proposition 5.2 that ${\|\hskip 4.30554pt\|}_{{\mathscr{L}}_{\varepsilon}}$ is semipositive. This means that ${\mathscr{L}}_{\varepsilon}$ is nef.

By definition of nef and using $N^{1}({\mathscr{X}}/S)\subset N^{1}({\mathscr{X}}_{s})$, we see that the cone in $N^{1}({\mathscr{X}}/S)$ of nef classes is the intersection of $N^{1}({\mathscr{X}}/S)$ with the nef cone in $N^{1}({\mathscr{X}}_{s})$. In particular, the cone of nef classes is closed in $N^{1}({\mathscr{X}}/S)$. Using ${\varepsilon}=({\varepsilon}_{1},\dots,{\varepsilon}_{n})\to 0$, we deduce that ${\mathscr{L}}$ is nef. Since ${\mathscr{L}}$ represents $\theta$, we conclude that $\varphi=0$ is $\theta$-psh. ∎

Using 4.7, this is a special case of Theorem 5.6. ∎