18.2 Multiplier-ideal envelopes and I-goodness530E72454328

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Throughout this chapter, $X$ is a connected compact complex manifold of dimension $n$ and $\theta $ is a smooth real $(1,1)$-form, which need not be closed. We fix a smooth positive probability volume form $\mathrm{d}V$ on $X$ and take $L^1$ norms with respect to it. For $u\in \mathrm{PSH}(X,\theta )$, $\mathrm{MA}_\theta (u)$ denotes the extended non-pluripolar Monge–Ampère measure $\mathrm{MA}^{\mathrm{ext}}_\theta (u)$ of theorem 13.14.3; its total mass may be infinite. We write $u\preceq v$ and $u\sim v$ for the ordinary singularity order and equivalence of definition 16.7.1, and $u\preceq _{\mathcal I}v$ and $u\sim _{\mathcal I}v$ for inclusion and equality of the multiplier ideals $\mathcal I(ku)$ and $\mathcal I(kv)$ at every weight $k>0$, as in section 5.33. Finally, $P_\theta [u]$ and $P_{\theta ,F}[u]$ denote the ordinary singularity envelopes of definition 5.40.4, below zero and below a smooth obstacle $F$, and $V_\theta =P_\theta (0)$ is the potential with minimal singularities of section 5.40.

Multiplier ideals record weighted holomorphic integrability. Two potentials can have the same multiplier ideals at every weight although their ordinary envelopes differ, so passing from the ordinary singularity type of a potential to its multiplier-ideal class loses information. Analytic approximation makes this loss visible: along dyadic Demailly approximations, the envelopes converge to the greatest nonpositive representative of the multiplier-ideal class, which may exceed the ordinary envelope of the original potential.

In the Kähler theory, equality of these two envelopes is also characterized by a mass condition. In a nonclosed background the equality remains meaningful even when the potential has infinite mass, and we therefore study both envelopes along analytic approximations. Examples on curves show that this still does not give continuity of mass or completeness, because diffuse polar curvature can disappear in the limit. The same examples explain why the obstacle has to be taken into account when the background changes.