We return to the big background $\theta $ on $X$ and apply the monotone limits above to the envelope systems of section 16.29. Fix real numbers $a<c$. Recall that $\mathscr T_{a,c}(\theta )$ consists of the pointwise concave, decreasing families $\Gamma =(\Gamma _\tau )_{\tau <c}$ of ideal models in background $\theta $ that are constant for $\tau \leq a$ (definition 16.29.2). An envelope system $\boldsymbol \Gamma \in \mathscr S_{a,c}(\theta )$ is a compatible family of such curves $\Gamma ^\eta =(\Gamma ^\eta _\tau )_{\tau <c}\in \mathscr T_{a,c}(\theta +\eta )$, indexed by the Hermitian forms $\eta $ (definition 16.29.6). The map $\mathsf E$ projects a curve in background $\theta $ to a system, and $\mathsf R$ takes a system to its limiting curve $(\mathsf R\boldsymbol \Gamma )_\tau =\lim _ j\Gamma ^{h/j}_\tau $, for any Hermitian form $h$ (theorem 16.29.9). The system is recovered from its limit if $\boldsymbol \Gamma =\mathsf E\mathsf R\boldsymbol \Gamma $.
Compatibility fixes every local divisorial Lelong number of the slices before the perturbation tends to zero, and recovery means that these numbers do not increase in the limit. A positive limiting lower volume rules out such an increase. Concavity propagates positivity from the constant left part to every finite parameter, so a single condition on the initial slice suffices for the whole system.
Theorem 13.21.7.9EA752566EF2 Let $\boldsymbol \Gamma \in \mathscr S_{a,c}(\theta )$, fix a Hermitian form $h$, and put $\Gamma =\mathsf R\boldsymbol \Gamma $. Then
\[ \lim _{j\to \infty }\underline{\operatorname {vol}}_{\theta +h/j} (\Gamma ^{h/j}_ a) =\underline{\operatorname {vol}}_\theta (\Gamma _ a). \]
In particular this limit is independent of $h$. If it is positive, then $\boldsymbol \Gamma =\mathsf E(\Gamma )$, and for every $\tau <c$ both singularity volumes satisfy
\[ \operatorname {vol}_{\theta +h/j}(\Gamma ^{h/j}_\tau ) \searrow \operatorname {vol}_\theta (\Gamma _\tau ), \qquad \underline{\operatorname {vol}}_\theta (\Gamma _\tau )>0. \]
Proof.
Every ideal model is an ordinary model by corollary 18.4.4, and the retraction theorem gives $\Gamma ^{h/j}_\tau \searrow \Gamma _\tau $ in the decreasing backgrounds $\theta +h/j$. Thus theorem 13.21.2 proves the identity for the initial lower volumes, including the case where the limit is zero.
Suppose that this limit $\ell $ is positive. For $\tau \leq a$, the slice is $\Gamma _ a$. For $a<\tau <c$, choose $r\in (\tau ,c)$ and put $s=(\tau -a)/(r-a)$, so that $\tau =(1-s)a+sr$. Concavity gives
\[ (1-s)\Gamma _ a+s\Gamma _ r\leq \Gamma _\tau . \]
Homogeneity, proposition 13.17.2 and volume monotonicity imply
\[ \underline{\operatorname {vol}}_\theta (\Gamma _\tau ) \geq (1-s)^ n\ell >0. \]
Theorem 13.21.2 now gives both volume limits for every fixed $\tau <c$.
It remains to show that the slices keep their multiplier ideals in the limit. Fix $\tau <c$ and write $q=\Gamma _\tau $, $q_ j=\Gamma ^{h/j}_\tau $ and $\theta _ j=\theta +h/j$. By proposition 13.20.6, the upper volumes of the fixed potential $q$ in the backgrounds $\theta _ j$ converge to its upper volume in $\theta $, which is the limit of the upper volumes of the $q_ j$. Consequently
\[ D_ j=\overline{\operatorname {vol}}_{\theta _ j}(q_ j) -\overline{\operatorname {vol}}_{\theta _ j}(q)\longrightarrow 0, \qquad \underline{\operatorname {vol}}_{\theta _ j}(q_ j) \geq \underline{\operatorname {vol}}_\theta (q)>0. \]
After discarding finitely many indices, choose $b_ j>1$ with $b_ j\to +\infty $ and $b_ j^ nD_ j<\underline{\operatorname {vol}}_{\theta _ j}(q_ j)$. Since $q\leq q_ j$, the stretching lemma lemma 13.18.1 in background $\theta _ j$ gives $v_ j\in \mathrm{PSH}(X,\theta _ j)$ such that
\[ (1-b_ j^{-1})q_ j+b_ j^{-1}v_ j\leq q. \]
Choose a Hermitian form $\Omega \geq \theta +h$, so that all $v_ j$ are $\Omega $-plurisubharmonic. By lemma 5.26.2, there is $\alpha >0$ such that $\mathcal I(\alpha v_ j)=\mathcal O_ X$ for all $j$; additive normalization does not change these ideals. Let $E$ be a local divisor over a coordinate neighborhood $U$, with generic Lelong number $\nu _ E$ and log discrepancy $A_ U(E)$ as in section 5.38. Lemma 5.38.9, with $m=\alpha $, gives $\nu _ E(v_ j)\leq A_ U(E)/\alpha $. Taking Lelong numbers in the convex inequality therefore yields
\[ 0\leq \nu _ E(q)-\nu _ E(q_ j) \leq \frac{A_ U(E)}{\alpha b_ j}\longrightarrow 0. \]
Compatibility gives all components the same multiplier ideals, so $\nu _ E(q_ j)$ is independent of $j$, and hence equal to $\nu _ E(q)$. Blowing up a point on a local modification converts its pointwise Lelong number into such a divisorial number; on a curve the point itself is a divisor. Thus all Lelong numbers of $q$ and $q_ j$ on all local modifications agree, and theorem 5.37.3 gives $q\sim _{\mathcal I}q_ j$. This holds at every parameter $\tau <c$, so proposition 16.29.11 proves recovery.
$\square$
The lower bound $(1-s)^ n\ell $ may tend to zero as $\tau \uparrow c$; the theorem gives no positive lower bound that is uniform up to the cutoff. The hypothesis is invariant under a smooth change of background $\theta '=\theta +\mathrm{dd}^{\mathrm{c}}f$, $f\in C^\infty (X,\mathbb R)$, through the gauge bijection of proposition 16.29.13: the estimate there makes the transported initial slice ordinarily equivalent to $\Gamma _ a-f$, and the map $u\mapsto u-f$ preserves the curvature currents and hence both singularity volumes. The ruled-surface system of theorem 16.29.17, which is not recovered from its limit, has zero initial limiting lower volume, so the theorem does not apply to it.
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