Title: Hirzebruch-Riemann-Roch in Combinatorial K-Theory
Abstract: The Hirzebruch-Riemann-Roch Theorem provides an intricate connection between the K-theory and the intersection theory of an algebraic variety. What data is needed to describe this connection explicity? For smooth projective toric varieties, it turns out that all the necessary information is contained in the Ehrhart polynomial. I will present work in progress with Matthew Dupraz and Leonid Monin in which we show that this can be generalized to any smooth complete variety whose numerical Chow ring is generated in degree 1. Namely, one can recover the numerical K-ring, the numerical Chow ring, the Chern character, and the Todd class only from the Snapper polynomial in a very satisfying way. Abstracting this situation, we can associate to any polynomial (that satisfies a certain condition), a K-ring, a Chow ring, a Chern character, and a Todd class such that Hirzebruch-Riemann-Roch holds. In particular, our framework produces a Hirzebruch-Riemann-Roch theorem for Bergman fans of arbitrary matroids. In my talk, I will also discuss which properties our abstract K-rings share with K-rings of varieties and how one can detect the existence of exceptional isomorphisms from the polynomial one starts with.