Theorem 20.3.307EFA5C202804

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

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. $\square$