PomeB7F265C09335The trace envelope of subsection 20.3.23 imposes the trace inequalities along all divisors over $Y$, including those that do not appear on a given model. It therefore gives a single potential on $Y$, in the original background $\theta |_ Y$, whose pulled-back curvature current dominates the trace current on every model.
Proposition 20.4.8.C592843AB1E3 There exists $v\in \mathrm{PSH}(Y,\theta |_ Y)$ such that, for every smooth compact modification $\pi :Z\to Y$,
and
Proof. Take for $v$ the trace envelope $\mathcal T_{\theta |_ Y}(u;Y)$ of (20.3.25). By theorem 20.3.27, it belongs to $\mathrm{PSH}(Y,\theta |_ Y)$ and satisfies every defining trace inequality. On a fixed model $Z$, lemma 5.38.1 therefore allows us to subtract every finite partial sum of $S_ Z(u;Y)$ from $\theta _ Z+\mathrm{dd}^{\mathrm{c}}(v\circ \pi )$ while preserving positivity. Passing to the limit of these partial sums, constructed in theorem 20.4.2, preserves positivity and proves the second assertion. $\square$
The potential in the proof is canonical: by theorem 20.3.30, $v\circ \pi $ is the trace envelope on $Z$ in the background $\theta _ Z$ for the same trace data. Its generic Lelong numbers may nevertheless exceed the prescribed trace numbers. Exact realization of the trace data requires the reverse inequality, which section 20.3 establishes for analytic singularities, for Kähler $Y$ in sufficiently large closed backgrounds, and for pullbacks from curves. By contrast, the ordinary singularity type attached to $S_ Z$ by corollary 5.38.6 records only the divisorial part on the particular model $Z$.
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