Okounkov bodies and the Kähler geometry of projective manifolds · arXivDesk
1510.00510Oct 2, 201513 pages. New version introduces the notion of Okounkov domain and considers Kähler embeddings of domains with their standard Kähler structure
Okounkov bodies and the Kähler geometry of projective manifolds
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
Nearby in the stack
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