PomeTheorem 5.19.10 (Boucksom).847815EFB4FE Assume that $\alpha $ is big. There is a Kähler current $T\in \alpha $ with analytic singularities, such that the polar locus of $T$ is exactly $\operatorname{nK}(\alpha )$. In particular, $\operatorname{nK}(\alpha )$ is a proper Zariski closed subset of $X$.
Theorem 5.19.10 (Boucksom)847815EFB4FE
Proof. The big-class part of the theorem on positivity loci constructs a single Kähler current with analytic singularities whose pole set is $E_{\mathrm{nK}}(\alpha)$. This set is the same as $\operatorname{nK}(\alpha)$ here, by the defining intersection of singular loci of Kähler currents in $\alpha$. The same theorem proves that this pole set is a proper analytic subset. Thus its constructed current has all the required properties, including equality, rather than just containment, of the polar locus.
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