Transcendental Okounkov bodies

TL;DR

The paper establishes a transcendental Okounkov body theory for big (and, by limit, pseudoeffective) $\,(1,1)\-$classes on compact Kähler manifolds, proving a volume identity that realizes $\mathrm{vol}(\xi)$ as a real-volume of a convex body: $\mathrm{vol}_{\mathbb{R}^n}(\Delta(\xi))=\frac{1}{n!}\mathrm{vol}(\xi)$. The authors introduce and leverage partial Okounkov bodies, an extension theorem for Kähler currents, and a flag-lifting framework to carry the construction through birational models; they also connect transcendental Okounkov bodies to toric degenerations via moment bodies, proving $\Delta^{\mu}(\xi)=\Delta(\xi)$ for big classes and showing that toric degenerations densely approximate the transcendental body. This work generalizes the classical line-bundle case and provides a convex-geometry realization of transcendental volumes, with potential implications for transcendental Morse inequalities and stability questions. The approach blends complex-analytic techniques (currents with analytic singularities, restricted volumes) with algebro-geometric ideas (Okounkov bodies, lifting of flags, toric degenerations).

Abstract

We show that the volume of transcendental big $(1,1)$-classes on compact Kähler manifolds can be realized by convex bodies, thus answering questions of Lazarsfeld-Mustaţă 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 Kähler currents, as well as the bimeromorphic behavior of currents with analytic singularities. We also establish a connection between transcendental Okounkov bodies and toric degenerations.

Theorems & Definitions (95)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Definition 2.2
• Remark 2.3
• Lemma 2.4
• Lemma 2.5