Corollary 5.36.2.8F2184BBF4C4 Let $\theta $ be a smooth real $(1,1)$-form on $X$ and $u\in \mathrm{PSH}(X,\theta )$. The approximants $u_ m$ of theorem 5.36.1 for $\gamma =-\theta $ satisfy
\[ \theta +\mathrm{dd}^{\mathrm{c}}u_ m\geq -\epsilon _ m\omega . \]
In particular, they have analytic type $(1/m,\mathcal I(mu))$.
Proof.
Apply theorem 5.36.1 to $u$ with the lower bound $\gamma =-\theta $; the theorem does not require $\gamma $ to be closed. Adding $\theta $ to the curvature inequality gives the assertion. In the normalization of [DemAM11], the theorem is applied to the potential $u/2$, since $\frac{\mathrm{i}}{\pi }\partial \bar\partial (u/2)=\mathrm{dd}^{\mathrm{c}}u$.
$\square$
Pome
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