$K$-rings of wonderful varieties and matroids

TL;DR

This paper develops a combinatorial framework for the $K$-ring of wonderful varieties associated with hyperplane arrangements, extending to matroids and related moduli spaces, with applications to Euler characteristics.

Contribution

It introduces a combinatorial presentation of the $K$-ring for hyperplane arrangements and matroids, and establishes an isomorphism with the Chow ring satisfying a Riemann--Roch-type formula.

Findings

Provides a combinatorial formula for the $K$-ring of matroids.

Establishes an isomorphism between the $K$-ring and the Chow ring of the matroid.

Applies results to compute Euler characteristics of line bundles on wonderful varieties.

Abstract

We study the $K$-ring of the wonderful variety of a hyperplane arrangement and give a combinatorial presentation that depends only on the underlying matroid. We use this combinatorial presentation to define the $K$-ring of an arbitrary loopless matroid. We construct an exceptional isomorphism, with integer coefficients, to the Chow ring of the matroid that satisfies a Hirzebruch--Riemann--Roch-type formula, generalizing a recent construction of Berget, Eur, Spink, and Tseng for the permutohedral variety (the wonderful variety of a Boolean arrangement). As an application, we give combinatorial formulas for Euler characteristics of arbitrary line bundles on wonderful varieties. We give analogous constructions and results for augmented wonderful varieties, and for Deligne--Mumford--Knudsen moduli spaces of stable rational curves with marked points.