Algebraic cobordism rings of wonderful varieties and matroids

Abstract

We give two combinatorial presentations for the algebraic cobordism ring $\Omega^*(M)$ of the toric variety of the Bergman fan of any loopless matroid $M$. As a consequence of our presentations, we obtain an $\Omega^*(\mathrm{pt})$-algebra isomorphism $\Omega^*(M) \simeq CH^*(M) \otimes_{\mathbb{Z}} \Omega^*(\mathrm{pt})$, where $CH^*(M)$ is the Chow ring of $M$ and $\Omega^*(\mathrm{pt})$ is the algebraic cobordism ring of the point. This isomorphism generalizes, in part, the exceptional integral isomorphism between the Chow ring and $K$-ring of a matroid, studied in the recent works of Berget--Eur--Spink--Tseng and Larson--Li--Payne--Proudfoot. For a complex hyperplane arrangement $\mathcal{H}$, we prove that the algebraic cobordism ring of the wonderful variety $W_\mathcal{H}$ of $\mathcal{H}$ and the algebraic cobordism ring of the toric variety of the matroid underlying $\mathcal{H}$ are isomorphic, and that both rings coincide with the complex cobordism ring of $W_\mathcal{H}$.