# Piecewise Laurent Polynomials and (Operational) Equivariant K-theory of Toric Varieties

# Piecewise Laurent Polynomials and (Operational) Equivariant K-theory of Toric Varieties

For a smooth compact toric variety $X$, results of Bifet-de Concini-Procesi and Brion show that the equivariant cohomology of $X$ is identified with the ring of piecewise polynomials on the associated fan. In 2006, Payne extended this to arbitrary toric varieties, identifying the ring of piecewise polynomials with the operational equivariant Chow cohomology of $X$. It turns out that a similar story holds for K-theory: when $X$ is smooth and compact, Brion-Vergne and Vezzosi-Vistoli show that the equivariant K-theory of algebraic vector bundles on $X$ can be identified with the ring of "piecewise Laurent polynomials" on the associated fan. On the other hand, the bivariant machinery of Fulton-MacPherson can be applied to construct an "operational" equivariant K-theory for singular toric varieties. In this talk, I will describe ongoing joint work with Sam Payne: for an arbitrary toric variety $X$, we show that the ring of piecewise Laurent polynomials on the fan is identif ied with the operational equivariant K-theory of $X$. The proof requires us to develop some foundational aspects of operational K-theory, as well as the usual equivariant K-theory of coherent sheaves. Our point of view leads to the curious result that the abstract operational theory is tractable and computable on varieties where the usual K-theory (of algebraic vector bundles) is completely unknown.