Complements and higher resonance varieties of hyperplane arrangements

Abstract

Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where they are known, up to a sign, as Whitney numbers of the first kind, and appear as the coefficients of chromatic, or characteristic, polynomials. We show that certain combinations, some nonlinear, of these Betti numbers satisfy Schur positivity. At the same time, we study the higher degree resonance varieties of the arrangement. We draw some consequences, using homological algebra results and vector bundles techniques, of the fact that all resonance varieties are determinantal.

Erratum to Thm 1.1 added. Main results on deeper propagation, dimensional bounds for resonance varieties, and positivity results related to Betti numbers are unchanged

References in corpus (4)

• Critical points and resonance of hyperplane arrangements
• On combinatorial invariance of the cohomology of Milnor fiber of arrangements and Catalan equation over function field
• Inequalities for the Hodge numbers of irregular compact Kaehler manifolds
• Derivative complex, BGG correspondence, and numerical inequalities for compact Kähler manifolds