Tangent classes for matroid building sets

Significant combinatorial constraints and structure on matroids can come from notions in algebraic geometry, even without the matroids themselves being representable. Let $M$ be a loopless matroid on a finite ground set $E$, and let $\text{G}$ be a building set containing the top flat $E$. We define a tangent class $T_{M,\text{G}}$ in the $K$-ring $K(M,\text{G})$, which extends the tangent bundle class of the de Concini–Procesi wonderful model from realizable matroids to arbitrary matroids with building sets. The class $T_{M,\text{G}}$ satisfies a matroidal Hirzebruch–Riemann–Roch package. More precisely, its Hirzebruch class specializes to the Todd class and computes the Chow polynomial of $(M,\text{G})$. In the realizable case, these identities agree with the usual tangent-bundle computations on the corresponding wonderful model. As an application, we prove Chern-number inequalities for $T_{M,\text{G}}$, including a Miyaoka–Yau type inequality with respect to the hyperplane class.