2023/10/03 by Mannino, Eline
#05B35 #14C17 (Primary) 14C15 #52C35 (Secondary) #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2310.01956
We define Chern numbers of a matroid. These numbers are obtained when intersecting appropriate matroid Chern-Schwartz-MacPherson cycles defined by López de Medrano, Rincón, and Shaw. We prove that when a matroid arises from a complex hyperplane arrangement the Chern numbers of the matroid correspond to the Chern numbers of the log cotangent bundle. A matroid of rank 3 has two Chern numbers. We prove that they are positive and that their ratio is bounded by 3, which is analogous to the Bogomolov-Miyaoka-Yau inequality. If the matroid is orientable, we generalize a result of Eterović, Figuera, and Urzúa to prove that the ratio is bounded above by 5/2. Finally, we give a formula for the Chern numbers of the uniform matroid of any rank.