Complexes, duality and Chern classes of logarithmic forms along 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 $\mathbb{P}^d$ with locally free resolutions of length one. Combining these results we present a generalization of a formula due to Mustaţă and Schenck, relating the Poincaré polynomial of an arrangement in $\mathbb{P}^3$ (or a locally tame arrangement in $\mathbb{P}^d$ with zero-dimensional non-free locus) to the total Chern polynomial of its sheaf of logarithmic 1-forms.