Let $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ be the normalization of the blow-up of $\mathcal{X}$ along $\mathfrak{a}$ and let $D\in\Div_{0}(\mathcal{X}^{\prime})$ be the vertical Cartier divisor such that $\mathfrak{a}\cdot\mathcal{O}_{\mathcal{X}^{\prime}}=\mathcal{O}_{\mathcal{X}^{\prime}}(D)$. The assumption implies that $\pi^{*}\mathcal{L}\otimes\mathcal{O}_{\mathcal{X}^{\prime}}(D)$ is also generated by its global sections, so that $\mathcal{L}+D$ is nef. The result follows since the model function $\log|\mathfrak{a}|$ is determined on $\mathcal{X}^{\prime}$ by $D$. ∎

Let $\pi:\mathcal{X}^{\prime}\to\mathcal{X}$ be a vertical blow-up such that $\varphi=\varphi_{D}$ for some $D\in\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{Q}}$. Since $\theta$ is determined by $\theta_{\mathcal{X}}\in N^{1}(\mathcal{X}/S)$, the assumption that $\varphi$ is $\theta$-psh implies that $D$ is $\pi$-nef. By Lemma 1.4 and Kleiman’s criterion [Kle66], we may find a vertical $\pi$-ample $\mathbf{Q}$-divisor $A\in\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{Q}}$ arbitrarily close to $D$. It is then clear that $\varphi_{A}$ is uniformly close to $\varphi=\varphi_{D}$ on $X$ (see the proof of Corollary 2.4). Since $A$ is $\pi$-ample we may find $m\gg 1$ such that $\mathcal{O}_{\mathcal{X}^{\prime}}(mA)$ is $\pi$-globally generated. If we set $\mathfrak{a}:=\pi_{*}\mathcal{O}_{\mathcal{X}^{\prime}}(mA)$ we then have $\varphi_{A}=\tfrac{1}{m}\log|\mathfrak{a}|$, which concludes the proof. ∎

Convexity is clear from the definition. To prove stability under maxima, let $\varphi_{1},\varphi_{2}\in\mathcal{D}(X)$ be $\theta$-psh, pick a common determination $\mathcal{X}$ of $\theta$ and the $\varphi_{i}$’s and let $D_{i}\in\Div_{0}(\mathcal{X})_{\mathbf{Q}}$ be a representative of $\varphi_{i}$ for $i=1,2$.

Since the ample cone of $N^{1}(\mathcal{X}/S)$ is open, we may find ample line bundles $\mathcal{A}_{1},\dots,\mathcal{A}_{r}\in\Pic(\mathcal{X})$ whose numerical classes $\alpha_{1},\dots,\alpha_{r}$ form a basis of $N^{1}(\mathcal{X}/S)$. We may thus find $t_{1},\dots,t_{r}\in\mathbf{R}$ such that $\mathcal{L}:=\sum_{j}t_{j}\mathcal{A}_{j}$ is a representative of $\theta$ in $\Pic(\mathcal{X})_{\mathbf{R}}$. Let $\varepsilon_{1},\dots,\varepsilon_{r}>0$ be (small) positive numbers such that $t_{j}+\varepsilon_{j}\in\mathbf{Q}$ for each $i$ and set $\mathcal{L}_{\varepsilon}:=\sum_{j}(t_{j}+\varepsilon_{j})\mathcal{A}_{j}$. Since $\varphi,\varphi^{\prime}$ are $\theta$-psh it follows that $\mathcal{L}_{\varepsilon}+D_{i}$ is an ample $\mathbf{Q}$-divisor on $\mathcal{X}$ for $i=1,2$. We may thus find a positive integer $m$ such that $m\mathcal{L}_{\varepsilon}\in\Pic(\mathcal{X})$, $mD_{i}\in\Div_{0}(\mathcal{X})$ and both sheaves $\mathcal{O}_{\mathcal{X}}\left(m\left(\mathcal{L}_{\varepsilon}+D_{i}\right)\right)$, $i=1,2$ are generated by their global sections on $\mathcal{X}$. If we introduce the vertical fractional ideal sheaf

$$\mathfrak{a}_{m}:=\mathcal{O}_{\mathcal{X}}(mD_{1})+\mathcal{O}_{\mathcal{X}}(mD_{2})$$

then it follows that $\mathcal{O}_{\mathcal{X}}(m\mathcal{L}_{\varepsilon})\otimes\mathfrak{a}_{m}$ is also generated by its global sections. By Lemma 5.6, $\log|\mathfrak{a}_{m}|=m\max\left\{\varphi_{1},\varphi_{2}\right\}$ is thus psh with respect to $m(\theta+\sum_{j}\varepsilon_{j}\alpha_{j})$, that is:

$$\theta+\sum_{j}\varepsilon_{j}\alpha_{j}+dd^{c}\max\{\varphi_{1},\varphi_{2}\}\geq 0.$$

Letting $\varepsilon_{j}\to 0$, we conclude as desired that $\theta+dd^{c}\max\{\varphi_{1},\varphi_{2}\}\geq 0$. ∎

This follows directly from Lemma 5.7 and Proposition 3.9. ∎

Let $\varphi\in\mathcal{D}(X)$. By Proposition 5.2 we may find a model $\mathcal{X}$ and a model function $\psi$ such that $\theta$, $\varphi$ and $\psi$ are all determined on $\mathcal{X}$ and such that $\theta+dd^{c}\psi$ is $\mathcal{X}$-positive. Since the closed $(1,1)$-form $dd^{c}\varphi$ is determined on $\mathcal{X}$ we may thus find a rational number $0<\varepsilon\ll 1$ such that $\theta+dd^{c}(\psi+\varepsilon\varphi)\geq 0$. It follows that $\varepsilon\varphi=(\psi+\varepsilon\varphi)-\psi$ is a difference of $\theta$-psh model functions, and the result follows. The case when $\theta$ is semipositive is proved in a similar way. ∎