PomeProposition 13.20.4.0729BE10D92E Assume in addition that $\theta \geq 0$. If $u\in \mathrm{PSH}(X,\theta )$ and $\underline{\operatorname {vol}}_\theta (u)>0$, then $P_{\varepsilon \theta }(u)\not\equiv -\infty $ for some $\varepsilon \in (0,1)$.
Proposition 13.20.40729BE10D92E
Proof. By corollary 13.20.3, choose $w\leq u$ and $\delta >0$ with $\theta +\mathrm{dd}^{\mathrm{c}}w\geq \delta \omega $. Choose $C>0$ with $\theta \leq C\omega $, and $0<t<\min (1,\delta /C)$. Then
\[ (1-t)\theta +\mathrm{dd}^{\mathrm{c}}w\geq (\delta -tC)\omega \geq 0. \]
Thus $w$ is a competitor for $P_{(1-t)\theta }(u)$, and $\varepsilon =1-t$ has the required property. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.