PomeCorollary 10.29.7.CBF3E42884D1 Let $X$ be an Oeljeklaus–Toma manifold with $t=1$, let $\eta $ be a smooth real $(1,1)$-form on $X$, and let $u\in \mathrm{PSH}(X,\eta )$. For every $c>0$, the set $\{ x:\nu (u,x)\geq c\} $ is finite. Consequently, for every $k>0$ the multiplier ideal $\mathcal I(ku)$ is trivial outside a finite set, and $\mathcal I(ku)$ has finite colength.
Corollary 10.29.7CBF3E42884D1
Proof. By Siu's theorem (5.34.2), the set $E_ c=\{ \nu (u,\cdot )\geq c\} $ is a compact analytic subset of $X$. It is a proper subset, because $\nu (u,x)=0$ outside the pluripolar set $\{ u=-\infty \} $, which has Lebesgue measure zero. Hence $E_ c$ is finite by the theorem of Ornea and Verbitsky. By Skoda's integrability criterion, recalled in [SGPT], $\mathcal I(ku)_ x=\mathcal O_{X,x}$ whenever $k\nu (u,x)<1$, so $\mathcal I(ku)$ is trivial outside the finite set $E_{1/k}$. A coherent ideal sheaf that is trivial outside a finite set has finite colength. $\square$
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