Hyperplane Arrangements

Abstract

Let A = {H 1 , . . . , H l } be a hyperplane arrangement in C n and M be the complement of the union of hyperplanes in A, i.e., M = C n \ ∪ li=1 H i . The cohomology algebra H ∗ (M, C) has a complete combinatorial description. Let L be a local system on M and H ∗ (M, L) be the cohomology algebra with local coefficients. For [ω] ∈ H 1 (M, C), there is a chain complex: μ ω μ ω μ ω 0 → H 0 (M, C) → H 1 (M, C) → · · · → H n (M, C) → 0. The characteristic varieties of M are the jumping loci of the cohomology groups H ∗ (M, L). The resonance varieties of M are the jumping loci of the cohomology groups of the above complex. The aim of this thesis is to study some properties of these varieties.