Corollary 13.20.3D51B82B18D30

Proof. Translate $u$ so that $u\leq V_\theta $. By bigness, there are $\rho \in \mathrm{PSH}(X,\theta )$ and $\delta >0$ with $\theta +\mathrm{dd}^{\mathrm{c}}\rho \geq \delta \omega $; after a translation, $\rho \leq V_\theta $. By lemma 13.20.1 with $r=V_\theta $, there is $b>1$ for which $h=Q_ b(u,V_\theta )$ is defined. Put

Since $b^{-1}h+(1-b^{-1})V_\theta \leq u$ and $\rho \leq V_\theta $, we have $w\leq u$; moreover $\theta +\mathrm{dd}^{\mathrm{c}}w\geq (1-b^{-1})\delta \omega $. Undoing the translation of $u$ by the same translation of $w$ gives the assertion with $\varepsilon =(1-b^{-1})\delta $. $\square$