Lemma 18.28.9A86A369D4CBF

Proof. Both $\geq $ directions in (18.28.10) are trivial.

Conversely, let $\eta $ be a candidate in (18.28.8). Let $S\subseteq X$ be a pluripolar set so that $\eta |_{K\setminus S}\leq f|_{K\setminus S}$.

We now claim that we can choose $\chi \in \mathrm{PSH}(X,\theta )_{>0}$ with $\chi |_ S\equiv -\infty $ and $\chi \leq \eta $.

Assume this for the moment. For any $\delta \in (0,1)$, we have

Hence, $(1-\delta )\eta +\delta \chi $ is dominated by the right-hand side of (18.28.10). Letting $\delta \to 0+$, we conclude that $\eta $ is dominated by the right-hand side of (18.28.10) and hence, (18.28.10) follows.

It remains to construct $\chi $. Since $\varphi $ has positive mass, the class $\{ \theta \} $ is big. Fix a Kähler form $\omega _0$ on $X$ and choose $\rho \in \mathrm{PSH}(X,\theta )$ with $\theta _{\rho }\geq 2\epsilon _0\omega _0$ for some $\epsilon _0>0$. By theorem 5.13.20, there is a quasi-psh function $q$ such that $q|_ S\equiv -\infty $. Choose $A>0$ so that $A\omega _0+\mathrm{dd}^{\mathrm{c}}q\geq 0$. For $a>0$ small enough, $\rho +aq\in \mathrm{PSH}(X,\theta )$ and still has pole $-\infty $ on $S$. Replacing $\rho $ by $\rho +aq$, we may therefore assume that $\rho |_ S\equiv -\infty $. Since $\int _ X \theta _{\eta }^ n>0$, fixing such a $\rho $ and taking $\epsilon \in (0,1)$ small enough, we have

Thus, by proposition 16.3.25 and theorem 16.3.35, we have

It suffices to define this function as $\chi $. $\square$