Proposition 20.3.24787C2A9E0C8D

Proof. The restrictions $u_ m|_ Y$ have a common smooth lower bound for their curvature. After subtracting their suprema, normalized compactness gives a subsequence converging in $L^1$ to some $v$. The inequalities $(\theta +\epsilon _ m\omega )|_ Y+\mathrm{dd}^{\mathrm{c}}(u_ m|_ Y)\geq 0$ pass to the limit as $\epsilon _ m\to 0$, giving $\theta |_ Y+\mathrm{dd}^{\mathrm{c}}v\geq 0$.

For each fixed compact modification $\pi :Z\to Y$, lemma 5.43.19 gives $L^1$ convergence of the pullbacks along the same subsequence, and lemma 5.38.7 gives, at every point $z$ of a prime divisor $E\subset Z$,

Taking the generic value along $E$ proves the assertion. No diagonal extraction over divisors is needed, since the pullback lemma applies to each modification once the subsequence on $Y$ has been fixed. $\square$