Multipoint Okounkov bodies

TL;DR

This paper introduces a new method to construct multipoint Okounkov bodies from a big line bundle on a projective manifold, capturing key geometric features like volume, Seshadri constants, and Kähler packings.

Contribution

It provides a novel construction of multipoint Okounkov bodies that encodes important geometric invariants and extends the understanding of these bodies in toric manifolds and surfaces.

Findings

Constructed multipoint Okounkov bodies encode volume and Seshadri constants.

Application to toric manifolds and surfaces demonstrates the method's effectiveness.

Enables the study of Kähler packings centered at multiple points.

Abstract

Starting from the data of a big line bundle $L$ on a projective manifold $X$ with a choice of $N\geq 1$ different points on $X$ we provide a new construction of $N$ Okounkov bodies which encodes important geometric features of ($L\to X,p_{1},\dots,p_{N}$) such as the volume of $L$, the (moving) multipoint Seshadri constant of $L$ at $p_{1},\dots,p_{N}$, and the possibility to construct K\"ahler packings centered at $p_{1},\dots,p_{N}$. Toric manifolds and surfaces are examined in detail.