Tangent classes for matroid building sets

Abstract

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 \(\G\) be a building set containing the top flat \(E\). We define a tangent class \(T_{M,\G}\) in the \(K\)-ring \(K(M,\G)\), which extends the tangent bundle class of the de Concini--Procesi wonderful model from realizable matroids to arbitrary matroids with building sets. The class \(T_{M,\G}\) satisfies a matroidal Hirzebruch--Riemann--Roch package. More precisely, its Hirzebruch class specializes to the Todd class and computes the Chow polynomial of \((M,\G)\). 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 \(T_{M,\G}\), including a Miyaoka--Yau type inequality with respect to the hyperplane class.

29 pages