Every Hermitian form dominates all sufficiently small multiples of a fixed Hermitian form. A single decreasing sequence of components therefore determines the limit of the whole system. Projecting the resulting curve back gives the greatest realized system below the original one, and strict inequality means that recovery fails.
Theorem 16.29.9.E94DE568F212 Fix a Hermitian form $h$ on $X$, and let $j$ range over the positive integers. For every $\boldsymbol \Gamma \in \mathscr S_{a,c}(\theta )$ and every $\tau <c$, the sequence $\Gamma ^{h/j}_\tau $ decreases, and its limit
\[ (\mathsf R\boldsymbol \Gamma )_\tau =\lim _{j\to \infty }\Gamma ^{h/j}_\tau \]
is $\theta $-plurisubharmonic, in particular not identically $-\infty $. These limits define an element $\mathsf R\boldsymbol \Gamma $ of $\mathscr T_{a,c}(\theta )$, which does not depend on $h$. The maps $\mathsf E$ and $\mathsf R$ satisfy
\[ \mathsf R\mathsf E=\mathrm{id}, \qquad \mathsf E\mathsf R(\boldsymbol \Gamma ) \leq \boldsymbol \Gamma . \]
Both maps preserve order. More precisely,
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16.29.10
\begin{equation} \mathsf E(\Gamma )\leq \boldsymbol \Gamma \quad \Longleftrightarrow \quad \Gamma \leq \mathsf R\boldsymbol \Gamma . \end{equation}
Thus $\mathsf E\mathsf R(\boldsymbol \Gamma )$ is the greatest system below $\boldsymbol \Gamma $ that is realized in the original background.
Proof.
Step 1: the limit. Fix $\tau <c$ and put $q_ j=\Gamma ^{h/j}_\tau $. By compatibility, $q_ j=P_{\theta +h/j}[q_{j+1}]_{\mathcal I}$, and $q_{j+1}\leq 0$ is a competitor for this envelope; hence $q_ j\geq q_{j+1}$. The functions $q_ j$ have supremum zero and a common smooth lower curvature bound. By theorem 5.25.2, corollary 5.25.3, their decreasing limit $q$ is not identically $-\infty $ and has supremum zero. Passing to the limit in the distributional inequalities
\[ \theta +h/j+\mathrm{dd}^{\mathrm{c}}q_ j\geq 0 \]
gives $q\in \mathrm{PSH}(X,\theta )$.
The defining family of an ideal envelope grows when the background increases. Together with $q\leq q_ j$ and order preservation, this gives
\[ q\leq P_\theta [q]_{\mathcal I} \leq P_{\theta +h/j}[q]_{\mathcal I} \leq P_{\theta +h/j}[q_ j]_{\mathcal I}=q_ j. \]
Taking the infimum over $j$ proves $P_\theta [q]_{\mathcal I}=q$. Concavity, decreasingness and constancy on the left half-line pass to the pointwise limits, so these limits form a curve in $\mathscr T_{a,c}(\theta )$.
Step 2: independence of $h$. For every $\eta >0$ there is $j$ with $h/j\leq \eta $, and compatibility then gives $\Gamma ^\eta _\tau \geq \Gamma ^{h/j}_\tau $. Consequently
\[ (\mathsf R\boldsymbol \Gamma )_\tau =\inf _{\eta \in \mathscr H(X)}\Gamma ^\eta _\tau , \]
which proves that $\mathsf R\boldsymbol \Gamma $ does not depend on $h$. This formula also shows that $\mathsf R$ preserves order.
Step 3: the two compositions. Write $q=(\mathsf R\boldsymbol \Gamma )_\tau $. Since $q\leq \Gamma ^\eta _\tau $ and $\Gamma ^\eta _\tau $ is fixed by its envelope, order preservation gives $P_{\theta +\eta }[q]_{\mathcal I}\leq \Gamma ^\eta _\tau $. This proves $\mathsf E\mathsf R\leq \mathrm{id}$.
Now start with a curve $\Gamma \in \mathscr T_{a,c}(\theta )$. Its slice $\Gamma _\tau $ is a competitor in every larger background, so
\[ \Gamma _\tau \leq (\mathsf R\mathsf E\Gamma )_\tau \leq P_{\theta +\eta }[\Gamma _\tau ]_{\mathcal I} \quad (\eta >0). \]
The two outer terms have the same multiplier ideals, so the two ideal inclusions given by these inequalities show that the middle term has those ideals as well. Since both the middle term and $\Gamma _\tau $ are $P_\theta [\cdot ]_{\mathcal I}$-fixed, they are equal. This proves $\mathsf R\mathsf E=\mathrm{id}$.
Step 4: the adjunction. By the transition proposition, $\mathsf E$ preserves order. If $\mathsf E(\Gamma )\leq \boldsymbol \Gamma $, applying $\mathsf R$ gives $\Gamma =\mathsf R\mathsf E(\Gamma )\leq \mathsf R\boldsymbol \Gamma $. Conversely, if $\Gamma \leq \mathsf R\boldsymbol \Gamma $, applying $\mathsf E$ and using $\mathsf E\mathsf R\leq \mathrm{id}$ gives $\mathsf E(\Gamma )\leq \boldsymbol \Gamma $. This proves (16.29.10), and the final maximality assertion follows from it.
$\square$
Proposition 16.29.11.93A1136C4F72 A system $\boldsymbol \Gamma \in \mathscr S_{a,c}(\theta )$ is realized in $\theta $ if and only if, for every $\tau <c$, its limiting slice $(\mathsf R\boldsymbol \Gamma )_\tau $ is multiplier-ideal equivalent to one, and hence every, slice $\Gamma ^\eta _\tau $.
Proof.
For a realized system, this follows from $\mathsf R\mathsf E=\mathrm{id}$ and the preservation of multiplier ideals by each projection. Conversely, the assumed equivalence and the fact that $\Gamma ^\eta _\tau $ is fixed by its envelope in the larger background give
\[ P_{\theta +\eta }[(\mathsf R\boldsymbol \Gamma )_\tau ]_{\mathcal I} =\Gamma ^\eta _\tau . \]
Thus the system equals $\mathsf E\mathsf R(\boldsymbol \Gamma )$.
$\square$
By theorem 5.37.3, multiplier-ideal equivalence means equality of all Lelong numbers on all local modifications. Recovery therefore holds exactly when the limiting slices keep these Lelong numbers. They are the same for all components of a system, whereas the limiting slice can have larger values: in the ruled-surface example of subsection 16.29.15, one of them increases from zero to one. For big backgrounds on manifolds with the bounded mass property, theorem 13.21.7 excludes such an increase when the initial slices have positive limiting lower volume.
When $X$ is Kähler, the Kähler forms are cofinal among the Hermitian forms: a sufficiently small positive multiple of a fixed Kähler form lies below any given Hermitian form. The compatibility condition therefore determines a system by its components indexed by Kähler forms. Conversely, every compatible family indexed by Kähler forms extends uniquely to a system: for $\eta >0$, choose a Kähler form $\kappa \leq \eta $ and set $\Gamma ^\eta =P_{\theta +\eta }[\Gamma ^\kappa ]_{\mathcal I}$. Two choices of $\kappa $ give the same result, as one sees by passing to a common smaller Kähler form and using the transition identity.
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