Wednesday, February 15th, 2012
9:30am - 10:30am, in Science 3-113

David Anderson

University of Washington

Okounkov bodies: from algebraic to convex geometry

Abstract: Building on earlier work of Okounkov, in 2008 Kaveh, Khovanskii, Lazarsfeld, and Mustata showed how to construct a convex body in n-dimensional Euclidean space naturally associated to a line bundle on an n-dimensional algebraic variety, in such a way that the convex geometry of this body reflects algebro-geometric properties of the line bundle. This construction generalizes a well-understood correspondence between toric varieties and polytopes: when one starts with a toric variety and an equivariant line bundle, the associated convex body is the polytope arising from the yoga of toric geometry. After describing the history and construction of these so-called "Okounkov bodies" from an elementary point of view, I will explain how the toric correspondence can be made tighter: under the right conditions, the Okounkov body is a polytope, and the variety in question deforms to a toric variety with the same Okounkov body. The toric correspondence provides a remarkably useful bridge between several branches of mathematics, and we will see connections between geometry, algebra, combinatorics, and representation theory.