Combinatorial covers and vanishing of cohomology

TL;DR

This paper develops a spectral sequence approach to prove vanishing theorems for the cohomology of hyperplane arrangement complements, generalizing known results and exploring duality properties of spaces and groups.

Contribution

It introduces a new spectral sequence method to establish cohomology vanishing results for arrangements, extending previous theorems and considering broader local system conditions.

Findings

Proves vanishing of cohomology for complements of hyperplane arrangements.

Generalizes known vanishing theorems to new classes of local systems.

Connects cohomology vanishing with duality properties of spaces and groups.

Abstract

We use a Mayer-Vietoris-like spectral sequence to establish vanishing results for the cohomology of complements of linear and elliptic hyperplane arrangements, as part of a more general framework involving duality and abelian duality properties of spaces and groups. In the process, we consider cohomology of local systems with a general, Cohen-Macaulay-type condition. As a result, we recover known vanishing theorems for rank-1 local systems as well as group ring coefficients, and obtain new generalizations.