Grothendieck classes and Chern classes of hyperplane arrangements

TL;DR

This paper explores the relationships between characteristic polynomials, Chern classes, and Grothendieck classes of hyperplane arrangements, providing new formulas and conjectures that connect algebraic, topological, and combinatorial properties.

Contribution

It introduces novel methods to compute characteristic polynomials from Grothendieck and Chern classes, and establishes new conjectures for free arrangements.

Findings

Characteristic polynomial can be derived from the Grothendieck class of the complement.

The total Chern class of the complement relates to the characteristic polynomial.

For free arrangements, the Chern class matches that of a bundle of differential forms.

Abstract

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; this has also been observed by Huh. 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 Mustata and Schenck. We conjecture that this relation holds for all free divisors. 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.