K-theoretic balancing conditions and the Grothendieck group of a toric variety

Abstract

We introduce a ring of Z-valued functions on a complete fan Δ called Grothendieck weights to describe the ordinary operational K-theory of the associated toric variety X. These functions satisfy a K-theoretic analogue of the balancing condition for Minkowski weights, which is induced by a presentation of the Grothendieck group of X. We explicitly give a combinatorial presentation in low dimensions, and relate Grothendieck weights to other fan-based invariants such as piecewise exponential functions and Minkowski weights. As an application, we give an example of a projective toric surface X such that the forgetful map KT∘(X)→K∘(X) is not surjective.