Complexes, duality and Chern classes of logarithmic forms along hyperplane arrangements

TL;DR

This paper explores dualities and complexes of logarithmic forms in hyperplane arrangements, generalizing classical formulas to relate Poincare polynomials and Chern classes in projective spaces.

Contribution

It introduces new duality results and extends the Borel-Serre and Mustata-Schenck formulas to broader classes of arrangements and sheaves.

Findings

Generalized Borel-Serre formula for sheaves on projective space.

Extended Mustata-Schenck formula relating Poincare and Chern polynomials.

Established dualities for logarithmic forms in hyperplane arrangements.

Abstract

We describe dualities and complexes of logarithmic forms and differentials for central affine and corresponding projective arrangements. We generalize the Borel-Serre formula from vector bundles to sheaves on projective d-space with locally free resolutions of length one. Combining these results we present a generalization of a formula due to Mustata and Schenck, relating the Poincare polynomial of an arrangement in projective 3-space (or a locally tame arrangement in projective d-space with zero-dimensional non-free locus) to the total Chern polynomial of its sheaf of logarithmic 1-forms.