(i) The only thing to show is that $P_{\theta}(u)$ is $\theta$-psh. Since $P_{\theta}(u)\leq u$ and $u$ is continuous, it follows that the usc regularization satisfies $P_{\theta}(u)^{*}\leq u$. Now, $P_{\theta}(u)^{*}$ is $\theta$-psh by Theorem 7.9, and is hence a competitor in the definition of $P_{\theta}(u)$. Thus $P_{\theta}(u)=P_{\theta}(u)^{*}$ is indeed $\theta$-psh.

(ii) is trivial.

(iii) follows from the fact that given $\varphi\in\PSH(X,\theta)$, $\varphi^{\prime}\in\PSH(X,\theta^{\prime})$ with $\varphi\leq u$ and $\varphi^{\prime}\leq u^{\prime}$, $t\varphi+(1-t)\varphi^{\prime}$ belongs to $\PSH(X,t\theta+(1-t)\theta^{\prime})$ and is dominated by $tu+(1-t)u^{\prime}$.

(iv) and (v) are seen similarly.

(vi) is a formal consequence of (ii) and (iv).

(vii) By Proposition 5.2 we may assume after perhaps passing to a higher model that there exists a model function $v$ determined on $\mathcal{X}$ such that $\theta+dd^{c}v$ is $\mathcal{X}$-positive, i.e. determined by an ample class in $N^{1}(\mathcal{X}/S)$. As a consequence, there exists an open neighborhood $V\subset N^{1}(\mathcal{X}/S)$ of $\theta$ such that $\tau+dd^{c}v$ is $\mathcal{X}$-positive for all $\tau\in V$.

We claim that $P_{\tau}(u)$ is uniformly bounded on $X$ for $\tau\in V$. Indeed for each $\tau\in V$ we have $\mathbf{R}\subset\PSH(X,\tau+dd^{c}v)$, hence $P_{\tau+dd^{c}v}(u-v)\geq\inf_{X}u-\sup_{X}|v|$. By (v) it follows that

which proves the claim.

Now for each $x\in X$ the function $\tau\mapsto P_{\tau}(u)(x)$ is concave on $V$, hence locally Lipschitz continuous on $V$, with local Lipschitz constant only depending on $\sup_{\tau\in V}|P_{\tau}(u)(x)|$, which is in turn bounded independently of $x\in X$, and the result follows. ∎

The inequality $\widetilde{P}_{\theta}(u)\leq P_{\theta}(u)$ is trivial. To prove that equality holds on $X^{\mathrm{qm}}$, pick $\varepsilon>0$ and $x\in\emb_{\mathcal{X}}(\Delta_{\mathcal{X}})$ for some SNC model $\mathcal{X}$ on which $\theta$ is determined. By construction, there exists $\psi\in\PSH(X,\theta)$ such that $\psi\leq 0$ and $\psi(x)\geq P_{\theta}(u)(x)-\varepsilon$. By the definition of $\PSH(X,\theta)$, there then exists a $\theta$-psh model function $\varphi$ such that $|\varphi-(\psi-\varepsilon)|\leq\varepsilon$ on $\emb_{\mathcal{X}}(\Delta_{\mathcal{X}})$. Thus $\varphi\leq\varphi\circ p_{\mathcal{X}}\leq\psi\circ p_{\mathcal{X}}\leq 0$ on $X$ and $\varphi(x)\geq\psi(x)-2\varepsilon\geq P_{\theta}(u)(x)-3\varepsilon$. We conclude that $\widetilde{P}_{\theta}(u)=P_{\theta}(u)$ on $X^{\mathrm{qm}}$. ∎

We shall reduce the statement to a geometric assertion that can be proved using asymptotic multiplier ideals.

First, we may assume that $u\in\mathcal{D}(X)$, thanks to Corollary 2.3 and (vi) of Proposition 8.2.

Second, we can reduce to the case when $\theta\in N^{1}(\mathcal{X}/S)_{\mathbf{Q}}$ is a rational class, using (vii) of Proposition 8.2.

Third, we may further reduce to the case $u=0$, after replacing $\theta$ with $\theta+dd^{c}u$, using (v) of Proposition 8.2.

After scaling, we may finally assume that $\theta$ is the curvature form of a model metric determined by a line bunle $\mathcal{L}$ on some model $\mathcal{X}$. Now we conclude the proof using the following result. ∎

For $m\gg 1$, $m\mathcal{L}|_{X}$ is globally generated, which shows that the ideal sheaf $\mathfrak{a}_{m}$ is vertical. Since $\mathcal{O}_{\mathcal{X}}(m\mathcal{L})\otimes\mathfrak{a}_{m}$ is globally generated by the definition of $\mathfrak{a}_{m}$, it follows that $\varphi_{m}\in\mathcal{D}(X)$ is $\theta$-psh by Lemma 5.6. Note that $\mathfrak{a}_{m}\cdot\mathfrak{a}_{l}\subset\mathfrak{a}_{m+l}$ for all $m,l$. This yields the super-additivity property $m\varphi_{m}+l\varphi_{l}\leq(m+l)\varphi_{m+l}$. As a consequence, the pointwise limit $\lim_{m}\varphi_{m}$ exists and coincides with $\sup_{m}\varphi_{m}$.

Step 1. Let us first prove that $P_{\theta}(0)=\sup_{m}\varphi_{m}$ on $X^{\mathrm{qm}}$. This is similar to Step 2 of Theorem 5.11. Since $\varphi_{m}$ is $\theta$-psh and $\varphi_{m}\leq 0$ for all $m$, we have $\sup_{m}\varphi_{m}\leq P_{\theta}(0)$ on $X$. To see that equality holds on $X^{\mathrm{qm}}$, pick $\varepsilon>0$ and $x\in\emb_{\mathcal{X}^{\prime}}(\Delta_{\mathcal{X}^{\prime}})$ for some SNC model $\mathcal{X}^{\prime}$ dominating $\mathcal{X}$. By Lemma 8.4 there exists a $\theta$-psh model function $\varphi$ such that $\varphi\leq 0$ and $\varphi(x)\geq P_{\theta}(0)(x)-\varepsilon$. Replacing $\mathcal{X}^{\prime}$ by a higher model, we may assume that $\varphi=\varphi_{D}$ is determined by some divisor $D\in\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{Q}}$. Invoking Proposition 5.2 we may also assume that there exists $D^{\prime}\in\Div_{0}(\mathcal{X}^{\prime})_{\mathbf{Q}}$ with $-\varepsilon\leq\varphi_{D^{\prime}}\leq 0$ on $X$ and $\pi^{*}\mathcal{L}+D+D^{\prime}$ ample. Since $D+D^{\prime}\leq 0$ we then have

Now the left-hand side is globally generated for some $m$, and we conclude that

hence

Step 2. Introduce for each $m\in\mathbf{N}$ the asymptotic multiplier ideal $\mathfrak{b}_{m}=\mathcal{J}(\mathfrak{a}_{\bullet}^{m})\subset\mathcal{O}_{\mathcal{X}}$ associated to the graded sequence $\mathfrak{a}_{\bullet}$. We refer to Appendix B for the definition and the proof of the fundamental properties of multiplier ideals in our present setting. We shall use the following results. First we have the elementary inclusion $\mathfrak{a}_{m}\subset\mathfrak{b}_{m}$ for all $m$. Second, the subadditivity property (cf. Theorem B.7) implies $\mathfrak{b}_{ml}\subset\mathfrak{b}^{l}_{m}$ for any $l,m$. We infer that $\mathfrak{a}_{ml}\subset\mathfrak{b}_{ml}\subset\mathfrak{b}_{m}^{l}$ for any $m,l$ and hence

on $X^{\mathrm{qm}}$ for all $m$, where the last equality follows from the first step.

Since both $P_{\theta}(0)$ and $\varphi_{m}$ remain unchanged when $\mathcal{X}$ is replaced with a higher model, we may assume that there exists an effective divisor $E\in\Div_{0}(\mathcal{X})_{\mathbf{Q}}$ such that $\mathcal{A}:=\mathcal{L}-E$ is ample on $\mathcal{X}$. By the uniform global generation property of multiplier ideals (Theorem B.8) we may then choose $m_{0}\in\mathbf{N}$ such that $\mathcal{O}_{\mathcal{X}}(m\mathcal{L}+m_{0}\mathcal{A})\otimes\mathfrak{b}_{m}$ is globally generated for all $m$. Since $\mathcal{O}_{\mathcal{X}}(m\mathcal{L}+m_{0}\mathcal{A})$ injects in $\mathcal{O}_{\mathcal{X}}((m+m_{0})\mathcal{L})$ by multiplying with the canonical section of $\mathcal{O}_{\mathcal{X}}(m_{0}E)$, it follows that

Replacing $m$ with $m-m_{0}$ and using (8.1) we infer $(m-m_{0})P_{\theta}(0)\leq m\varphi_{m}+m_{0}\varphi_{E}$, so that

on $X^{\mathrm{qm}}$ for $m\gg 1$. As $\varphi_{m}$, $P_{\theta}(0)$ and $\varphi_{E}$ are all $\theta$-psh, Proposition 7.6 shows that this inquality extends to all of $X$. Now $\varphi_{E}$ is bounded and $\varphi_{m}$ is uniformly bounded, as follows from $\varphi_{1}\leq\varphi_{m}\leq 0$, so $\varphi_{m}$ converges uniformly on $X$ to $P_{\theta}(0)$, as was to be shown. ∎

We may assume $\varphi=P_{\theta}(v)$, in which case the result follows from Theorem 8.3. ∎

By Theorem 8.7 there exists a decreasing net $(\varphi_{j})$ of $\theta$-psh model functions converging pointwise to $\varphi$. For each $\varepsilon>0$ the compact set $X$ is the increasing union of the open sets $\{\varphi_{j}<\varphi+\varepsilon\}$, hence $\varphi_{j}<\varphi+\varepsilon$ for some $j$ (Dini’s lemma). It follows that $\varphi$ lies in the closure of $\mathcal{D}(X)\cap\PSH(X,\theta)$ in $C^{0}(X)$ with respect to the topology of uniform convergence. Since the latter is defined by a norm, the result follows. ∎

(i)$\Longrightarrow$(ii). Let $u\in C^{0}(X)$. By (i) there exists a decreasing net $(\varphi_{j})$ of $\theta$-psh model functions converging pointwise to $P_{\theta}(u)$. Since $P_{\theta}(u)\leq u$, we see that the compact set $X$ is for each $\varepsilon>0$ the increasing union of the open sets $\{\varphi_{j}<u+\varepsilon\}$, hence $\varphi_{j}<u+\varepsilon$ for some $j$. Since $\varphi_{j}-\varepsilon$ is $\theta$-psh and dominated by $u$, we get $\varphi_{j}-\varepsilon\leq P_{\theta}(u)$ by definition of the envelope, which proves (ii).

(ii)$\Longrightarrow$(iii). Since the set of $\varphi\in\mathcal{D}(X)\cap\PSH(X,\omega)$ such that $\varphi\leq u$ is stable by max, (ii) shows that we can construct an increasing family $\varphi_{j}\in\mathcal{D}(X)\cap\PSH(X,\omega)$ converging pointwise to $P_{\theta}(u)$. But $P_{\theta}(u)-\varphi_{j}$ is usc for each $j$, and Dini’s lemma therefore shows that the convergence is uniform on $X$.

(iii)$\Longrightarrow$(i). Let $\varphi$ be a $\theta$-psh function. We first claim that for each $x\in X$ we have

Indeed, given $\varepsilon>0$ there exists $u\in C^{0}(X)$ such that $u\geq\varphi$ and $u(x)\leq\varphi(x)+\varepsilon$, simply because $\varphi$ is usc. Since $\varphi$ is $\theta$-psh we have $\varphi\leq P_{\theta}(u)$. By (iii) may then find $\psi\in\mathcal{D}(X)\cap\PSH(X,\omega)$ such that $P_{\theta}(u)\leq\psi\leq P_{\theta}(u)+\varepsilon$. We thus have $\psi\geq\varphi$ and $\psi(x)\leq\varphi(x)+2\varepsilon$, and the claim follows.

Now consider the set $I$ of all $\psi\in\mathcal{D}(X)\cap\PSH(X,\omega)$ such that $\psi>\varphi$ on $X$. Note that this condition implies that $\psi\geq\varphi+\varepsilon$ for some $\varepsilon>0$, since $\varphi-\psi$ is usc. We claim that $I$ is a directed set, which will conclude the proof. To see this, let $\psi_{1},\psi_{2}\in I$ and choose $\varepsilon>0$ such that $\min\{\psi_{1},\psi_{2}\}\geq\varphi+3\varepsilon$. We then also have $P(\min\{\psi_{1},\psi_{2}\})\geq\varphi+3\varepsilon$. By (iii) we find $\psi_{3}\in\mathcal{D}(X)\cap\PSH(X,\omega)$ such that $|\psi_{3}-P(\min\{\psi_{1},\psi_{2}\})|\leq\varepsilon$. Then $\varphi+\varepsilon\leq\psi_{3}-\varepsilon\leq\min\{\psi_{1},\psi_{2}\}$, which concludes the proof. ∎