Grothendieck Classes and Chern Classes of Hyperplane Arrangements

We show that the characteristic polynomial of a hyperplane arrangement can be recovered from the class in the Grothendieck group of varieties of the complement of the arrangement. This gives a quick proof of a theorem of Orlik and Solomon relating the characteristic polynomial with the ranks of the cohomology of the complement of the arrangement. We also show that the characteristic polynomial can be computed from the total Chern class of the complement of the arrangement. In the case of free arrangements, we prove that this Chern class agrees with the Chern class of the dual of a bundle of differential forms with logarithmic poles along the hyperplanes in the arrangement; this follows from work of Mustaţǎ and Schenck. We conjecture that this relation holds for any locally quasi-homogeneous free divisor. We give an explicit relation between the characteristic polynomial of an arrangement and the Segre class of its singularity (‘Jacobian’) subscheme. This gives a variant of a recent result of Wakefield and Yoshinaga, and shows that the Segre class of the singularity subscheme of an arrangement together with the degree of the arrangement determine the ranks of the cohomology of its complement. We also discuss the positivity of the Chern classes of hyperplane arrangements: we give a combinatorial interpretation of this phenomenon, and we discuss the cases of generic and free arrangements.