Theorem 13.25.5CBA354870D02

Proof. Suppose first that $p=P_\theta [p]_{\mathcal I}$. By theorem 13.24.3, there are analytic ordinary models $r_ j$ in background $\theta $ with $d_\theta (r_ j,p)\to 0$ and a common positive lower-volume bound. Corollary 13.25.3 and projection invariance in background $\eta _ j$ give $d_{\eta _ j}(r_ j,p)\to 0$, so $a_ j=r_ j$ works. One can also use the ordinary models $a_ j=P_{\eta _ j}[r_ j]$, since $P_{\eta _ j}[r_ j]\sim r_ j$ by proposition 18.4.8.

Conversely, suppose that such $a_ j$ exist. Normalize them below zero, and put $m_ j=\max (a_ j,p)$ and $\delta =\underline{\operatorname {vol}}_\theta (p)>0$. Inequality (13.22.2) and proposition 13.25.2 give

Indeed, the upper volume of $p$ in background $\eta _ j$ tends to its upper volume in background $\theta $, and the differences of $\overline{\operatorname {vol}}_{\eta _ j}(a_ j)$ and $\overline{\operatorname {vol}}_{\eta _ j}(m_ j)$ from $\overline{\operatorname {vol}}_{\eta _ j}(p)$ are bounded in absolute value by $d_{\eta _ j}(a_ j,p)$.

Enlarge the errors $\varepsilon _ j$ to positive rational numbers tending to zero, and put

As in the proof of theorem 13.24.3, $\theta +\mathrm{dd}^{\mathrm{c}}b_ j\geq 0$ and $b_ j$ has analytic singularities. Choose $A>0$ with $\theta \leq Ah$, and set

Then $\widehat\theta _ j\geq \eta _ j$ and $c_ j\widehat\theta _ j\geq \theta $. Since $r_ j\sim b_ j$ and $\rho \leq 0$, the potential $r_ j/c_ j$ is a nonpositive $\widehat\theta _ j$-plurisubharmonic function more singular than $a_ j$, hence more singular than $m_ j$. Therefore $r_ j\leq c_ j\widehat m_ j$. Also $p\leq \widehat m_ j\leq c_ j\widehat m_ j$.

The lower volume of $m_ j$ in background $\eta _ j$ is at least $\delta $. For large $j$, proposition 13.25.2 applies with $\beta =\eta _ j$, $\gamma =\widehat\theta _ j$ and $\varepsilon =(1+A/\kappa )\varepsilon _ j$. Consequently

By theorem 13.19.5, $\widehat m_ j$ has the same upper volume in background $\widehat\theta _ j$ as $m_ j$. On the other hand,

Addition, applied to $c_ ja_ j\in \mathrm{PSH}(X,c_ j\eta _ j)$ and $(1-c_ j)\rho \in \mathrm{PSH}(X,(1-c_ j)(\theta -\kappa h))$, and background monotonicity therefore bound the upper volume of $b_ j$ below by $c_ j^ n\overline{\operatorname {vol}}_{\eta _ j}(a_ j)$. Since $r_ j\sim b_ j$, the two comparisons yield

Here the last inequality uses $\theta \leq c_ j\widehat\theta _ j$ and $\max (r_ j,p)\leq c_ j\widehat m_ j$. Both outer terms tend to $\overline{\operatorname {vol}}_\theta (p)$, which proves $d_\theta (r_ j,p)\to 0$. By proposition 18.4.8, $r_ j=P_\theta [b_ j]_{\mathcal I}$ is an ideal model, so theorem 13.24.5 gives $p=P_\theta [p]_{\mathcal I}$.

For large $j$, the upper-volume deficit from $r_ j$ to $\max (r_ j,p)$ is small, and lemma 13.18.1 with $b=2$ gives a convex lower bound for $r_ j$ whose first term is $\frac12\max (r_ j,p)$. Volume addition gives $\underline{\operatorname {vol}}_\theta (r_ j)\geq 2^{-n}\delta $. Thus proposition 13.22.4, theorem 13.23.3 give the asserted $L^1$ and capacity convergence of the models $r_ j$. $\square$