Let $\pi :Z\to Y$ be a modification of connected compact complex manifolds. Extend the restricted ideals to $Z$ by taking the ideals $\mathfrak a_ t(u;Y)\mathcal O_ Z$ generated by their pullbacks. For a divisor $F$ on a further modification $\rho :W\to Z$, their asymptotic order is $\tau _{Y,u}(F)$, computed on $\pi \circ \rho :W\to Y$. For a smooth real $(1,1)$-form $\Gamma $ on $Z$, write $\mathcal T^\pi _\Gamma (u;Y)$ for the envelope on $Z$ defined as in (20.3.25), with background $\Gamma $ and the inequalities $\nu (a\circ \rho ,F)\geq \tau _{Y,u}(F)$ for all divisors $F$ over $Z$. Thus the trace data still comes from the multiplier ideals of $u$ on $X$.
For the pulled-back background $\pi ^*\Omega $, every competitor descends to $Y$, and a common refinement of two models identifies the divisorial inequalities on them.
Theorem 20.3.30.7EFA5C202804 Suppose $\Omega $ is a smooth real $(1,1)$-form on $Y$ for which $T=\mathcal T_\Omega (u;Y)$ is defined, and let $h$ be a Hermitian form on $Z$. Then
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20.3.31
\begin{equation} \mathcal T^\pi _{\pi ^*\Omega }(u;Y)=T\circ \pi , \qquad \mathcal T^\pi _{\pi ^*\Omega +\varepsilon h}(u;Y) \searrow T\circ \pi \quad (\varepsilon \downarrow 0). \end{equation}
The convergence is pointwise and in $L^1(Z)$, and the limit is independent of $h$.
Let $\delta _ m\geq 0$ tend to zero and suppose, for all sufficiently large $m$, that
\[ b_ m:=(u_ m|_ Y)\circ \pi \in \mathrm{PSH}(Z,\pi ^*\Omega +\delta _ m h). \]
Then
\[ P_{\pi ^*\Omega +\delta _ m h}[b_ m] =P_{\pi ^*\Omega +\delta _ m h}[b_ m]_{\mathcal I} \longrightarrow T\circ \pi \quad \text{in }L^1(Z). \]
If $\delta _{2^ j}$ decreases, these envelopes decrease pointwise to $T\circ \pi $ along the dyadic indices.
Proof.
For any nonempty family of trace competitors, the argument with finite maxima and semicontinuity in Step 1 of the proof of theorem 20.3.27 shows that the envelope is its largest member. By proposition 5.43.18, every $\pi ^*\Omega $-plurisubharmonic function is uniquely of the form $a\circ \pi $ with $a\in \mathrm{PSH}(Y,\Omega )$, and since $\pi $ is surjective, $a\leq 0$ if and only if $a\circ \pi \leq 0$. If $a$ satisfies all trace inequalities over $Y$, then $a\circ \pi $ satisfies those over $Z$, since every modification of $Z$, composed with $\pi $, is a modification of $Y$.
Conversely, suppose that $a\circ \pi $ satisfies the inequalities over $Z$. Let $E$ be a prime divisor on a smooth compact modification $V\to Y$. Resolving the graph of the bimeromorphic map between $V$ and $Z$ gives a smooth compact modification $W$ dominating both. The map $W\to V$ is an isomorphism near the generic point of $E$, so the strict transform $F$ of $E$ has the same ideal orders and generic Lelong numbers as $E$, and the inequality at $F$ over $Z$ is exactly the required inequality at $E$. Thus the two families of competitors correspond under pullback, and the largest member on $Z$ is $T\circ \pi $. This proves the first identity.
For $\varepsilon >0$, the pullback $T\circ \pi $ is still a competitor for $\mathcal T^\pi _{\pi ^*\Omega +\varepsilon h}(u;Y)$. These envelopes lie between $T\circ \pi $ and zero and decrease as $\varepsilon \downarrow 0$, so their limit $S$ satisfies $S\geq T\circ \pi $ and is $\pi ^*\Omega $-plurisubharmonic. The $L^1$ convergence of pullbacks and semicontinuity of Lelong numbers preserve every trace inequality, so $S$ belongs to the unperturbed family. Maximality gives $S\leq T\circ \pi $, hence equality. Since the lower bound $T\circ \pi $ is integrable on $Z$, dominated convergence gives $L^1$ convergence.
For the final assertion, denote the analytic envelope in the statement by $q_ m$. The proof of theorem 20.3.27 applies to the global coherent ideals $\mathfrak a_ m(u;Y)\mathcal O_ Z$: resolving each of them and applying lemma 5.38.11 gives $T\circ \pi \preceq b_ m$. Hence
\[ T\circ \pi \leq q_ m\leq 0,\qquad \nu (q_ m\circ \rho ,F) =\frac1m\operatorname {ord}_ F\mathfrak a_ m(u;Y) \]
for every divisor $F$ over $Z$, where ordinary and ideal envelopes agree by proposition 18.4.8. Every $L^1$ cluster point is $\pi ^*\Omega $-plurisubharmonic and satisfies all trace inequalities, so the first identity in (20.3.31) identifies it with $T\circ \pi $, and compactness gives convergence of the whole sequence. Subadditivity orders the dyadic analytic types, decreasing dyadic backgrounds then order the envelopes, and their pointwise limit is the same $L^1$ limit.
$\square$
For $\Omega =\theta |_ Y$, the last hypothesis holds with $\delta _ m=C\epsilon _ m$, where the constant $C$ satisfies $\pi ^*(\omega |_ Y)\leq Ch$. Thus the restricted analytic envelopes converge to the same pulled-back trace envelope on every compact model. Formula (20.3.31) also gives compatibility under successive modifications.
For arbitrary changes of background, inclusion of the families of competitors orders the envelopes. Equality of their ordinary types would require a reverse bound on the singularities. For curve fibrations, the fixed divisorial potential of theorem 20.3.39 gives this bound in every sufficiently large background.
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