Okounkov bodies and the K\"ahler geometry of projective manifolds

TL;DR

This paper demonstrates how to embed certain domains into projective manifolds using Okounkov bodies, allowing approximation of the manifold's K"ahler volume through Euclidean domains, with applications to ellipsoids and big line bundles.

Contribution

It introduces a method to embed torus-invariant domains into projective manifolds via Okounkov bodies, linking Euclidean geometry with K"ahler geometry.

Findings

Euclidean K"ahler forms extend to the manifold in the first Chern class of L

Volumes of embedded domains approximate the K"ahler volume of X

Results apply to ellipsoids and big line bundles

Abstract

Given a projective manifold $X$ equipped with an ample line bundle $L$, we show how to embed certain torus-invariant domains $D \subseteq\mathbb{C}^n$ into $X$ so that the Euclidean K\"ahler form on $D$ extends to a K\"ahler form on X lying in the first Chern class of $L$. This is done using Okounkov bodies $\Delta(L)$, and the image of $D$ under the standard moment map will approximate $\Delta(L)$. This means that the volume of $D$ can be made to approximate the K\"ahler volume of $X$ arbitrarily well. As a special case we can let $D$ be an ellipsoid. We also have similar results when $L$ is just big.