19.1 The theory of Okounkov bodies2DF8DA44EB89

In this chapter, we apply our theory of singularities to the study of Okounkov bodies. We construct convex bodies $\Delta (L,\phi )$, the partial Okounkov bodies, attached to a big line bundle $L$ on a smooth projective variety $X$ together with a plurisubharmonic metric $\phi $ on $L$. These convex bodies provide a universal numerical invariant of the $\mathcal{I}$-equivalence class of $\phi $, and reduce many problems in pluripotential theory to problems in convex geometry, which are often more tractable.

We develop two parallel constructions: an algebraic one in section 19.4, where $\phi $ is encoded by the multiplier ideals $\mathcal{I}(k\phi )$, and a transcendental one in section 19.5, where $\phi $ is treated directly as an analytic object via valuations on closed positive currents. The two constructions agree when both apply, but each has its own natural domain of definition and its own range of applications.

Okounkov bodies associate convex bodies to graded linear series by recording successive vanishing orders along an admissible flag.

Let $X$ be an irreducible smooth projective variety of dimension $n$ and $L$ a big holomorphic line bundle on $X$. Given an admissible flag $X=Y_0\supseteq Y_1\supseteq \cdots \supseteq Y_ n$ on $X$ (definition 19.3.2), one attaches to $L$ a natural $n$-dimensional convex body $\Delta (L)\subseteq \mathbb {R}^ n$, generalizing the Newton polytope construction in toric geometry (definition 21.12.27). The convex body $\Delta (L)$ is the Okounkov body of $L$ with respect to the chosen flag. Concretely, the successive vanishing order along the flag defines a valuation $\nu \colon \mathbb {C}(X)^{\times }\to \mathbb {Z}^ n$, and

is a semigroup whose closed convex cone, intersected with the slice $\{ (x,1)\} $, is $\Delta (L)$. The construction shows that $\Delta (L)$ depends only on the numerical class of $L$, and the Okounkov bodies also show that the Okounkov bodies for all flags determine this class, so $\Delta (L)$ may be regarded as a universal numerical invariant of $L$.

The aim of this chapter is to perform the analogous construction at the level of singularity types of psh metrics. Setting

one immediately encounters the obstruction that $\Gamma (L,\phi )$ is in general not a semigroup, so the Lazarsfeld–Mustaţă and Kaveh–Khovanskii constructions break down. The remedy is the notion of an almost semigroup: $\Gamma (L,\phi )$ does satisfy a near-semigroup property (theorem 19.4.16), and the convex-body construction $\Gamma \mapsto \Delta (\Gamma )$ extends continuously from semigroups to almost semigroups (theorem 19.2.10). The partial Okounkov body of $(L,\phi )$ is then

and is a convex sub-body of $\Delta (L)$ — explaining the name partial. The universality of partial Okounkov bodies is captured by corollary 19.4.55: the $\mathcal{I}$-equivalence class of $\phi $ is determined by the Okounkov bodies of $(L,\phi )$ for sufficiently many flags.

The results in this chapter will be applied in section 21.13 to develop toric pluripotential theory on big line bundles.