Transcendental Okounkov bodies

TL;DR

This paper demonstrates that the volume of transcendental big (1,1)-classes on compact Kähler manifolds can be represented by convex bodies, linking complex geometry with convex analysis and toric degenerations.

Contribution

It introduces a method to realize volumes of transcendental classes via convex bodies, extending Okounkov body theory beyond algebraic settings.

Findings

Volumes of transcendental classes are represented by convex bodies.

Established a connection between transcendental Okounkov bodies and toric degenerations.

Analyzed properties of Kähler currents and their extensions.

Abstract

We show that the volume of transcendental big $(1,1)$-classes on compact K\"ahler manifolds can be realized by convex bodies, thus answering questions of Lazarsfeld-Musta\c{t}\u{a} and Deng. In our approach we use an approximation process by partial Okounkov bodies together with properties of the restricted volume, and we study the extension of K\"ahler currents, as well as the bimeromorphic behavior of currents with analytic singularities. We also establish a connection between transcendental Okounkov bodies and toric degenerations.