13.26 Two-sided background perturbationsE6BBFE91C762

Proof. Put $\theta _0=\theta -\kappa h/2$ and $h_0=(1+\kappa /2)h$. Since $\theta _0+\mathrm{dd}^{\mathrm{c}}\rho \geq \kappa h/2$, the form $\theta _0$ is big, and $\rho $ is a strict analytic potential for it with the rational margin $\kappa /(2+\kappa )$ relative to $h_0$. Let $\varepsilon \leq \min (\kappa /2,1/2)$. Then $\theta _0\leq \beta \leq \theta _0+h_0$ and $\beta \leq \gamma \leq \beta +2\varepsilon h_0$, so proposition 13.25.2 applies with $\theta _0$, $h_0$, $\rho $ and a fixed Hermitian form $\Omega _0\geq \theta _0+2h_0$ in place of $\theta $, $h$, $\rho $ and $\Omega $. These choices do not depend on $\beta $ and $\gamma $. Since the volumes in background $\beta $ are at most $v_+(\Omega _0)$, the multiplicative estimate of the proposition implies the additive estimate, with a constant $C_\delta $ independent of $\beta $, $\gamma $ and $u$. Applying it to $u$, $v$ and $\max (u,v)$, the three-error calculation in the proof of corollary 13.25.3 and projection invariance in background $\gamma $ give the second assertion. $\square$

Proof. Write $c=c_\varepsilon $, $s=1-c$, $v=cp+s\rho $ and $T(p)=T^\rho _{\eta ,\varepsilon }(p)$. Since $s\rho \in \mathrm{PSH}(X,s(\theta -\kappa h))$, addition gives $\operatorname {vol}_{\theta -s\kappa h}(v)\geq c^ n\operatorname {vol}_\theta (p)$, and since $\theta -s\kappa h=\theta -\varepsilon h\leq \eta $, monotonicity gives the lower bound in (13.26.4) for $v$. In particular $v$ has positive lower volume in background $\eta $, so theorem 13.19.5 shows that $T(p)=P_\eta [v]$ is an ordinary model with the same two volumes as $v$.

Since $\varepsilon <\kappa /2$, we have $c\geq 1/2$, and the choice of $B$ gives

Since $\eta \leq \theta +\varepsilon h\leq c\widehat\theta _\varepsilon $, every competitor $w$ for $T(p)$, that is, every nonpositive $\eta $-plurisubharmonic $w\preceq v$, satisfies $w/c\in \mathrm{PSH}(X,\widehat\theta _\varepsilon )$ and $w/c\preceq p$, because $\rho \leq 0$. Taking envelopes, and using that $v\leq 0$ is itself a competitor, yields

Volume monotonicity, together with theorem 13.19.5 in the larger background $\widehat\theta _\varepsilon $, proves the upper bound in (13.26.4).

Now let $q\in \mathcal M_\delta (\theta )$ and $m=\max (p,q)$. The same pointwise bounds give

Consequently the upper volumes of $T(p)$, $T(q)$ and of their maximum lie between $c^ n$ times the respective upper volumes of $p$, $q$, $m$ in the backgrounds $\theta $ and $\widehat\theta _\varepsilon $. The lower volumes of $p$, $q$ and $m$ are at least $\delta $, and their upper volumes are at most $K$. By proposition 13.25.2 with $\beta =\theta $ and $\gamma =\widehat\theta _\varepsilon $, each of the three errors $\overline{\operatorname {vol}}_\eta (T(x))-c^ n\overline{\operatorname {vol}}_\theta (x)$, $x=p,q,m$ (with $T(m)$ read as $\max (T(p),T(q))$), is nonnegative and at most $c^ nK((1+B\varepsilon /\kappa _\delta )^ n-1)$. As in the proof of corollary 13.25.3, this proves (13.26.5).

For fixed $p$, corollary 13.21.4, lemma 5.29.19 give $P_{\widehat\theta _\varepsilon }[p]\to p$ in $L^1$ and in capacity along every decreasing sequence $\varepsilon \downarrow 0$. An arbitrary sequence $\varepsilon _ j\to 0$ is handled by comparing with the decreasing sequence $\sup _{k\geq j}\varepsilon _ k$, since the envelopes increase with the background and lie above $p$. Quasi-continuity and integrability give $cp+s\rho \to p$ in both senses. The bounds (13.26.6) prove the asserted limits.

To compare two choices along a sequence $\eta _ j$, denote their coefficients by $c_ j^{(i)}$ and their upper backgrounds by $\theta +B_ i\varepsilon _ j^{(i)}h$, $i=0,1$. Put $c_ j=\min _ i c_ j^{(i)}$ and $a_ j=\max _ i B_ i\varepsilon _ j^{(i)}$. Since the envelopes are nonpositive, both transported models lie below $c_ jP_{\theta +a_ jh}[p]$. Moreover $\eta _ j\leq c_ j(\theta +a_ jh)$, by the form inequality above for an index attaining the smaller coefficient. The upper volume of the maximum of the two transported models is therefore at most $c_ j^ n\overline{\operatorname {vol}}_{\theta +a_ jh}(p)$, which tends to $\overline{\operatorname {vol}}_\theta (p)$. Each of their individual upper volumes is at least $(c_ j^{(i)})^ n\overline{\operatorname {vol}}_\theta (p)$ and at most the upper volume of the maximum, so it has the same limit. Hence their $d_{\eta _ j}$-comparison tends to zero. $\square$

Proof. Suppose 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. The transports $a_ j=T^\rho _{\eta _ j,\varepsilon _ j}(r_ j)$ have analytic singularities: the convex coefficients are rational, and ordinary projection preserves the singularity type of an analytic potential by proposition 18.4.8. The estimate (13.26.5) gives $d_{\eta _ j}(a_ j,p_ j)\to 0$.

Conversely, assume that such analytic approximants exist, and put $\delta =\underline{\operatorname {vol}}_\theta (p)$. By theorem 13.26.3, the lower volume of $p_ j$ is at least $c_{\varepsilon _ j}^ n\delta \geq 2^{-n}\delta $. The upper-volume deficit from $a_ j$ to $\max (a_ j,p_ j)$ is at most $d_{\eta _ j}(a_ j,p_ j)$, which tends to zero. Applying lemma 13.18.1 with $b=2$ in background $\eta _ j$, and then addition, gives

for all sufficiently large $j$.

Put $\beta _ j=\theta +\varepsilon _ jh$. Lemma 13.26.1, with $\beta =\eta _ j$ and $\gamma =\beta _ j$, and projection invariance in background $\beta _ j$ imply $d_{\beta _ j}(a_ j,p_ j)\to 0$. Also $d_{\beta _ j}(p_ j,p)\to 0$. Indeed, (13.26.6) bounds $\max (p_ j,p)$ above by $c_{\varepsilon _ j}P_{\widehat\theta _{\varepsilon _ j}}[p]$, and $c_{\varepsilon _ j}\widehat\theta _{\varepsilon _ j}\geq \beta _ j$ by the form inequality in the proof of theorem 13.26.3. Hence the upper volume of this maximum in background $\beta _ j$ is at most $c_{\varepsilon _ j}^ n\overline{\operatorname {vol}}_{\widehat\theta _{\varepsilon _ j}}(p)$, which tends to $\overline{\operatorname {vol}}_\theta (p)$. The upper volumes of $p_ j$ and $p$ in background $\beta _ j$ lie between $c_{\varepsilon _ j}^ n\overline{\operatorname {vol}}_\theta (p)$ and this bound, so they have the same limit.

Project $a_ j$, $p_ j$ and $p$ to ordinary models in background $\beta _ j$. Their lower volumes have a common positive bound, and their upper volumes are bounded by $v_+(\Omega )$, since $\beta _ j\leq \Omega $. The three-point estimate (13.22.6) in background $\beta _ j$ therefore gives $d_{\beta _ j}(a_ j,p)\to 0$, with constants independent of $j$. Now theorem 13.25.5, applied with the backgrounds $\beta _ j$, shows that $p$ is an ideal model and gives the stated approximation in the original background.

For another choice of transport, theorem 13.26.3 shows that the two reference models in background $\eta _ j$ have volume comparison tending to zero. The lower-volume bound just proved and the same uniform three-point estimate transfer the convergence condition in both directions. This proves independence of the choice. $\square$