18.3 The Kähler characterization649A8A26E78B

First suppose that $X$ is Kähler, that $\theta $ is a smooth closed big real $(1,1)$-form, and that $u\in \mathrm{PSH}(X,\theta )$ has positive non-pluripolar mass. Let $P_\theta [u]_{\mathcal I}$ be the multiplier-ideal envelope of definition 18.4.1 below, and let $d_ S$ be the singularity pseudometric of 16.9 The $d_ S$-pseudometric. The characterization of $\mathcal I$-good singularities in Theorem 18.1.2 states that the following three conditions are equivalent:

The last condition involves only envelopes. It therefore offers a starting point for the Hermitian problem even where the metric and the mass characterizations are unavailable.

The integral on the right of the middle condition is the multiplier-ideal volume of [SGPT]. The upper and lower volumes of definition 16.28.58, by contrast, vary the potential within its ordinary singularity type. The distinction matters in a nonclosed background, where even a bounded change of the potential can alter its total mass.