David Witt Nyström: Okounkov bodies and embeddings of torus-invariant Kähler balls.

In the 90's Okounkov found a way to associate a convex body to any ample line bundle. These convex bodies are now known as Okounkov bodies, and they generalize the moment polytopes from toric geometry. In the first part of my talk I will describe Okounkov's construction.

 A toric moment polytope can be thought of as the image of a moment map, corresponding to a torus-invariant Kähler form in the first Chern class of the line bundle. In the second part of the talk I will describe how this is approximately true for a general Okounkov body. Namely, I will show how to embed a torus-invariant Kähler ball into X so that the Kähler form on the ball extends to a Kähler form on X lying in the first Chern class of the line bundle, and so that the image of the moment map of the ball approximates the Okounkov body. This is inspired by recent work of Kaveh proving the symplectic version of this result.