K-theory of toric varieties

Many geometric invariants of a normal toric variety X can be described in terms of its associated polyhedral fan F. Fulton and Sturmfels described the Chow ring of a complete toric variety using Minkowski weights, which are a ring of integer-valued functions on F satisfying a balancing condition. These weights have appeared in many contexts since their introduction, including in tropical geometry where they are central to tropical intersection theory.

Here, I will discuss a K-theoretic analogue of Minkowski weights which we call Grothendieck weights. These weights describe the "operational" K-theory of a complete toric variety, and their balancing condition can be expressed in terms of Ehrhart theory. We will see some of their properties, and connections they have to other invariants of toric varieties including the K-theories of vector bundles and coherent sheaves.