Step 1. For each $m\geq 0$ let $\mathfrak{a}_{m}\subset\mathcal{O}_{\mathcal{X}}$ be the base-ideal of $\mathcal{O}_{\mathcal{X}}(m\mathcal{L})$. We are going to show that $\tfrac{1}{m}\log|\mathfrak{a}_{m}|$ converges pointwise to $0$ on $X^{\mathrm{div}}$. Note that $\mathfrak{a}_{m}$ is vertical for $m\gg 1$ since $\mathcal{L}$ is ample on the generic fiber of $\mathcal{X}$. The sequence $\mathfrak{a}_{\bullet}=(\mathfrak{a}_{m})_{m\geq 0}$ is a graded sequence of ideals, i.e. we have $\mathfrak{a}_{m}\cdot\mathfrak{a}_{l}\subset\mathfrak{a}_{m+l}$ for all $m,l$. It follows that $(\log|\mathfrak{a}_{m}|)_{m}$ is a super-additive sequence, which implies that

(5.1)

$$\lim_{m\to\infty}\frac{1}{m}\log|\mathfrak{a}_{m}|=\sup_{m}\frac{1}{m}\log|\mathfrak{a}_{m}|\leq 0$$

pointwise on $X$. Pick a rational number $\varepsilon>0$ and $x\in X^{\mathrm{div}}$. Let $\theta$ be the curvature form of $h_{\mathcal{L}}$. Since $0$ is by assumption a pointwise limit of $\theta$-psh model functions, there exists a vertical blow-up $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ and $D\in\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{Q}}$ such that $\varphi_{D}$ is $\theta$-psh, $\varphi_{D}(x)\geq-\varepsilon$ and $\varphi_{D}(x_{E_{i}})\leq\varepsilon$ for each irreducible component $E_{i}$ of our given model $\mathcal{X}$. By Proposition 5.9 the latter condition yields $\varphi_{D}\leq\varepsilon$ on $X$, so that $D^{\prime}:=D+\varepsilon\mathcal{X}^{\prime}_{0}\in\Div_{0}(\mathcal{X}^{\prime})$ has $D^{\prime}\leq 0$ and satisfies $\varphi_{D^{\prime}}(x)\geq-2\varepsilon$. On the other hand, we may assume that $\mathcal{X}^{\prime}$ has been chosen high enough to apply Proposition 5.2 and get $D^{\prime\prime}\in\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{Q}}$ with $D^{\prime\prime}\leq 0$, $\varphi_{D^{\prime\prime}}\geq-\varepsilon$ on $X$ and $\pi^{*}\mathcal{L}+D^{\prime}+D^{\prime\prime}$ ample. Since $D^{\prime}+D^{\prime\prime}\leq 0$ we then have

$$\mathcal{O}_{\mathcal{X}^{\prime}}(m\left(\pi^{*}\mathcal{L}+D^{\prime}+D^{\prime\prime}\right))\subset\mathcal{O}_{\mathcal{X}^{\prime}}(m\pi^{*}\mathcal{L}).$$

Now the left-hand side is globally generated for some $m$. Since $\pi_{*}\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}}$ we conclude that

$$\mathcal{O}_{\mathcal{X}^{\prime}}(m(D^{\prime}+D^{\prime\prime}))\subset\mathcal{O}_{\mathcal{X}^{\prime}}\cdot\mathfrak{a}_{m},$$

hence

$$-3\varepsilon\leq\varphi_{D^{\prime}+D^{\prime\prime}}(x)\leq\frac{1}{m}\log|\mathfrak{a}_{m}|(x).$$

We have thus shown that $\sup_{m}\tfrac{1}{m}\log|\mathfrak{a}_{m}|\geq 0$ at each $x\in X^{\mathrm{div}}$, which implies as desired that $\tfrac{1}{m}\log|\mathfrak{a}_{m}|$ converges to $0$ pointwise on $X^{\mathrm{div}}$ thanks to (5.1).

Step 2. Let us now show that $\mathcal{L}$ is nef. For each $c>0$ let $\mathcal{J}(\mathfrak{a}_{\bullet}^{c})\subset\mathcal{O}_{\mathcal{X}}$ be the multiplier ideal attached to the graded sequence $\mathfrak{a}_{\bullet}$ (cf. Appendix B). We have the elementary inclusion $\mathfrak{a}_{m}\subset\mathcal{J}(\mathfrak{a}_{\bullet}^{m})$ for all $m\in\mathbf{N}$, whereas the subadditivity property (cf. Theorem B.7) implies $\mathcal{J}(\mathfrak{a}_{\bullet}^{ml})\subset\mathcal{J}(\mathfrak{a}_{\bullet}^{m})^{l}$ for all $l,m\in\mathbf{N}$. We infer that $\mathfrak{a}_{ml}\subset\mathcal{J}(\mathfrak{a}_{\bullet}^{m})^{l}$ for any $m,l$ and hence

$$\sup_{l}\tfrac{1}{l}\log|\mathfrak{a}_{ml}|\leq\log|\mathcal{J}(\mathfrak{a}_{\bullet}^{m})|\leq 0.$$

By Step 2 we conclude that $\log|\mathcal{J}(\mathfrak{a}_{\bullet}^{m})|=0$, i.e. $\mathcal{J}(\mathfrak{a}_{\bullet}^{m})=\mathcal{O}_{\mathcal{X}}$ since multiplier ideals are integrally closed by definition. The uniform global generation property of multiplier ideals (Theorem B.8) now yields an ample line bundle $\mathcal{A}\in\Pic(\mathcal{X})$ independent of $m$ such that $m\mathcal{L}+\mathcal{A}$ is globally generated for all $m\in\mathbf{N}$. This immediately shows that $\mathcal{L}$ is nef. ∎

. Suppose that $\varphi\in\mathcal{D}(X)$ is a pointwise limit of $\theta$-psh model functions. Our goal is to show that $\varphi$ is $\theta$-psh. Upon replacing $\theta$ with $\theta+dd^{c}\varphi$ we may assume that $\varphi=0$. Note that the existence of at least one $\theta$-psh model function implies that $(\theta_{\mathcal{X}})|_{\mathcal{X}_{K}}$ is nef. As in Proposition 5.8 we can choose finitely many ample line bundles $\mathcal{A}_{i}\in\Pic(\mathcal{X})$ such that their numerical classes $\alpha_{i}\in N^{1}(\mathcal{X}/S)$ form a basis of $N^{1}(\mathcal{X}/S)$. There exists arbitrarily small positive numbers $\varepsilon=(\varepsilon_{i})$ such that $\theta_{\mathcal{X}}+\sum_{i}\varepsilon_{i}\alpha_{i}$ is a rational class, hence the class of a $\mathbf{Q}$-line bundle $\mathcal{L}_{\varepsilon}$ on $\mathcal{X}$ whose restriction to $\mathcal{X}_{K}$ is ample. Since $0$ is a pointwise limit of $\theta$-psh model functions and since $h_{\mathcal{L}_{\varepsilon}}e^{-\psi}$ is semipositive for each $\theta$-psh model function $\psi$, we may now apply Lemma 5.12 to conclude that $\mathcal{L}_{\varepsilon}$ is nef. It follows that $\theta_{\mathcal{X}}\in\Nef(\mathcal{X}/S)$ by closedness of the nef cone. ∎