Okounkov bodies and the Kähler geometry of projective manifolds

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 Kähler form on $D$ extends to a Kähler 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 Kähler 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.

arXiv comment13 pages. New version introduces the notion of Okounkov domain and considers Kähler embeddings of domains with their standard Kähler structure