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Proposition 20.4.8C592843AB1E3

Proposition 20.4.8.C592843AB1E3 There exists $v\in \mathrm{PSH}(Y,\theta |_ Y)$ such that, for every smooth compact modification $\pi :Z\to Y$,

\[ \nu (v\circ \pi ,E)\geq \tau _{Y,u}(E) \quad \hbox{for every prime divisor }E\subset Z, \]

and

\[ \pi ^*(\theta |_ Y)+\mathrm{dd}^{\mathrm{c}}(v\circ \pi )\geq S_ Z(u;Y). \]

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. $\square$

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