Pome293D12D027CCCorollary 13.21.5.293D12D027CC Suppose that $Y$ has the bounded mass property. Choose a Hermitian form $\Omega _0\geq \theta |_ Y$ on $Y$ and a Hermitian form $\Omega \geq 2\Omega _0$ on $Y$ with $v_-(\Omega )>0$. Then
for every prime divisor $E$ on every smooth compact modification $\pi :Z\to Y$.
Proof. Put $T=\mathcal T_\Omega (u;Y)$ and $r=\dim Y$. Normalize the majorant $v$ of proposition 20.3.24 by $\sup _ Yv=0$. It belongs to $\mathrm{PSH}(Y,\theta |_ Y)\subset \mathrm{PSH}(Y,\Omega /2)$ and is a competitor for $T$, so $v\leq T$. For every $w\in \mathrm{PSH}(Y,\Omega )$ with $w\sim T$, a constant translate of $v$ lies below $w$, so $P_{\Omega /2}(w)\not\equiv -\infty $. By lemma 16.28.69, applied on $Y$ with $a=1/2$,
Since $\Omega -\theta |_ Y\geq \Omega _0$ is Hermitian and the curvature errors of the approximants tend to zero, the restrictions $u_ m|_ Y$ belong to $\mathrm{PSH}(Y,\Omega )$ for all sufficiently large $m$. By theorem 20.3.27, their dyadic ordinary envelopes $q_ j=P_\Omega [u_{2^ j}|_ Y]$ decrease to $T$. Theorem 13.21.2, applied on $Y$ in the fixed background $\Omega $, and the analytic type of $u_{2^ j}|_ Y$ give
as required. $\square$
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