PomeF128B6972881Let $U_ m$ be the ambient multiplier-ideal approximants used in theorem 20.3.27. After discarding finitely many dyadic indices, their restrictions belong to $\mathrm{PSH}(B,a\kappa )$. Put
Proposition 20.3.57.B0E6CF965FAB The potentials $v_ j$ are the full ordinary analytic models
They decrease to $\mathcal T_{\Omega _{a,b}}(u;Y)$ in $L^1(Y)$. For every smooth compact modification $\pi :Z\to Y$, their pullbacks converge in $L^1(Z)$, and for every prime divisor $D\subset Z$,
Proof. For $h\in \mathrm{PSH}(B,a\kappa )$, each finite obstacle envelope $P_{\Omega _{a,b}}(q^*h+A,0)$ is invariant under the elliptic action. Its full supremum descends to the base, where (20.3.50) gives the background condition. Surjectivity makes the obstacle inequalities equivalent on $Y$ and $B$. Hence
Letting $A\to +\infty $ and taking upper semicontinuous regularizations proves $P_{\Omega _{a,b}}[q^*h]=q^*P_{a\kappa }[h]$. Analytic type follows from proposition 18.4.8.
The base models decrease to $w_ a$ by theorem 20.3.27. Their pullbacks therefore decrease to $q^*w_ a$ and converge in $L^1(Y)$ by genuine decreasing quasi-plurisubharmonic convergence. The extracted-divisor formula (20.3.45) transfers convergence of all base divisorial Lelong numbers to every $D$ over $Y$; horizontal divisors have coefficient zero throughout. Finally, lemma 5.43.19 gives $L^1$ convergence on each compact modification. $\square$
These results answer the trace question for arbitrary genuine projective data pulled back to the Hopf pair $(H_{n+1},H_ n)$, including nonanalytic data. They neither require bounded mass on $H_ n$ nor supply convergence of extremal singularity volumes there. The unrestricted realization problem of 24.9.1 still concerns data without this projective pullback structure.
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