Proposition 13.14.16.1342466C71A9 Assume $v_+(\omega )<+\infty $. If $\theta $ is a smooth real $(1,1)$-form with $\theta \leq C\omega $ for some $C>0$, and $u\in \mathrm{PSH}(X,\theta )$, then
\[ \mathrm{MA}^{\mathrm{ext}}_\theta (u)(X)\leq C^ n v_+(\omega ). \]
Proof.
The comparison in theorem 13.14.3, applied with the Hermitian form $C\omega $, gives
\[ \mathbf1_{D_ t}\mathrm{MA}^{\mathrm{ext}}_\theta (u) \leq \mathbf1_{D_ t}(C\omega +\mathrm{dd}^{\mathrm{c}}u_ t)^ n. \]
Since $u_ t/C$ is bounded and $\omega $-plurisubharmonic, the right-hand side has mass at most $C^ n v_+(\omega )$. Let $t\to \infty $.
$\square$
Pome
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