16.7 Comparison of singularitiesDE52193AE570

DE52193AE570

A central theme of these constructions is that singularities of quasi-plurisubharmonic functions can be compared at several levels of precision. The preorder $\preceq $ of definition 16.7.1 compares potentials up to additive constants. It is the most concrete notion, but it is often too fine for global pluripotential theory: the $P$-envelope identifies potentials with the same model representative, while multiplier ideals identify singularities that have the same integrability behavior. This chapter makes these two coarser comparisons systematic.

We first introduce the $P$-preorder and the $\mathcal I$-preorder on $\mathrm{QPSH}(X)$. The $P$-preorder is defined by comparing $P$-envelopes, and its equivalence classes are the $P$-singularity types. The $\mathcal I$-preorder is defined by comparing multiplier ideals $\mathcal I(k\varphi )$ for all $k\in \mathbb Z_{>0}$. These two preorders are compatible with ordinary singularity comparison, but they capture different information: the former is governed by non-pluripolar mass, while the latter is governed by valuative and multiplier-ideal data.

The second main object of the chapter is the pseudometric $d_ S$. For positive-mass potentials in a fixed class, $d_ S$ is defined through the Darvas $d_1$-distance between the associated geodesic rays. Its zero locus is precisely $P$-equivalence. Thus $d_ S$ should be viewed as a convergence structure on $P$-singularity types. The chapter proves the basic properties of $d_ S$, including monotone convergence, sandwich criteria and compatibility with changing the background form.

The final results establish the continuity properties that will be used in the following constructions. We prove continuity of the $\mathcal I$-envelope, continuity of non-pluripolar mass and volume, and a general equisingularity criterion. These results allow one to pass from analytic approximations to limiting singularities while controlling both $P$-types and multiplier ideals. They are used repeatedly in the theory of $\mathcal I$-good singularities, trace operators, test curves, partial Okounkov bodies and non-Archimedean potentials.