Flats in the space of K\"ahler metrics and Okounkov bodies

TL;DR

This paper investigates conditions for flat subspaces in the space of Kähler metrics using Okounkov bodies and Legendre transforms, linking complex geometry with convex analysis.

Contribution

It introduces new criteria for flatness in the metric space of Kähler potentials based on Okounkov bodies and Legendre transforms.

Findings

Established sufficient conditions for flat subspaces in Kähler metric space.

Connected convex geometric tools with complex geometric structures.

Provided a framework for analyzing metric flatness via Okounkov bodies.

Abstract

We study sufficient conditions for the existence of flat subspaces in the space of continuous plurisubharmonic metrics on a polarised complex projective manifold, relying on the generalised Legendre transform to the Okounkov body defined by Witt Nystrom, and a result of Schwer--Lytchak.