Cohomologies of tautological bundles of matroids

TL;DR

This paper computes the cohomologies of tautological bundles associated with matroids, revealing their dependence solely on the matroid, and applies these results to the geometry of hyperplane arrangement complements.

Contribution

It provides explicit cohomology calculations for tautological bundles of matroids and demonstrates their independence from realizations, with applications to hyperplane arrangement compactifications.

Findings

Cohomologies depend only on the matroid of the realization.

Vanishing of higher cohomologies for the log canonical bundle of certain compactifications.

Application to the moduli space of pointed rational curves.

Abstract

Tautological bundles of realizations of matroids were introduced in [BEST23] as a unifying geometric model for studying matroids. We compute the cohomologies of exterior and symmetric powers of these vector bundles, and show that they depend only on the matroid of the realization. As an application, we show that the log canonical bundle of a wonderful compactification of a hyperplane arrangement complement, in particular the moduli space of pointed rational curves, has vanishing higher cohomologies.