16.1 The envelope operators2AC9E2799656

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This chapter develops the envelope formalism that will be used in the following constructions. There are two parallel constructions. The first is the $P$-envelope, which records the singularity type relevant to non-pluripolar Monge–Ampère mass. The second is the $\mathcal I$-envelope, which records the singularity type detected by multiplier ideal sheaves. The distinction between these two envelopes is one of the main organizing principles of the book.

The starting point is the rooftop operator. Given two singular potentials, the rooftop construction produces the largest potential lying below both, and it provides a flexible way to compare singularities. From this we define the $P$-envelope $P_\theta [\varphi ]$. Its fixed points are the model potentials. Model potentials are the natural representatives of $P$-singularity types: they preserve non-pluripolar mass and satisfy strong contact-set properties. The first part of the chapter establishes the basic algebra of these envelopes, their behavior under monotone limits, and the domination and contact principles needed later.

We then introduce the Monge–Ampère energy relative to a model potential. This leads to the finite-energy classes $\mathcal E^1(X,\theta ;\phi )$ and to stability results for rooftop envelopes. These results will be used in the metric theory of singularities and in the study of geodesic rays.

The second part of the chapter develops the $\mathcal I$-envelope $P_\theta [\varphi ]_{\mathcal I}$. This envelope is defined by imposing multiplier ideal conditions rather than ordinary singularity-type conditions. It is therefore sensitive to valuative data in a way that the ordinary $P$-envelope is not. We prove its basic functorial properties, its independence of the chosen representative of the cohomology class, its bimeromorphic behavior, and its compatibility with analytic singularities.

The comparison between $P_\theta [\varphi ]$ and $P_\theta [\varphi ]_{\mathcal I}$ is delicate. They agree in important cases, for instance for analytic singularities, but they differ in general. Later chapters use this comparison to define $\mathcal I$-good singularities, trace operators, partial Okounkov bodies and non-Archimedean potentials. Thus the technical results of this chapter should be regarded as the basic calculus of singularity types used in the sequel.