Okounkov Bodies and the Kähler Geometry of Projective Manifolds

Paper in proceeding, 2023

Given a projective manifold X equipped with an ample line bundle L, we show how to embed certain torus-invariant domains D⊆ Cn 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 Δ (L), and the image of D under the standard moment map will approximate Δ (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.