Lemma 13.13.8F77E450F00FD

Proof. By definition of $\operatorname{nK}(\{ \theta \} )$, it suffices to show that if $\varphi \in \mathrm{PSH}(X,\theta )$ has analytic singularities and $\theta _{\varphi }$ is a Kähler current, then $V_{\theta }$ is locally bounded outside $\{ \varphi =-\infty \} $. After replacing $\varphi $ by $\varphi -\sup _ X\varphi $, the function $\varphi $ is a candidate in (13.13.7); hence $V_{\theta }\geq \varphi $ and $V_{\theta }\leq 0$. Since $\varphi $ is locally bounded outside its pole set, so is $V_{\theta }$. $\square$