Algebraic cobordism rings of wonderful varieties and matroids

Abstract

We give two combinatorial presentations for the algebraic cobordism ring ${\Omega }^{\ast }(M)$ of the toric variety of the Bergman fan of any loopless matroid $M$. As a consequence of our presentations, we obtain an ${\Omega }^{\ast }(\mathrm{pt})$-algebra isomorphism ${\Omega }^{\ast }(M)\simeq C{H}^{\ast }(M){\otimes }_{\mathbb{Z}}{\Omega }^{\ast }(\mathrm{pt})$, where $C{H}^{\ast }(M)$ is the Chow ring of $M$ and ${\Omega }^{\ast }(\mathrm{pt})$ 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 $W_{\mathcal{H}}$.

Minor revisions, updated Example 4.17