Characteristic varieties of arrangements

Abstract

Abstract. The k th Fitting ideal of the Alexander invariant B of an arrangement A of n complex hyperplanes defines a characteristic subvariety, V k (A), of the algebraic torus (C * ) n . In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of V k (A). For any arrangement A, we show that the tangent cone at the identity of this variety coincides with R 1 k (A), one of the cohomology support loci of the Orlik-Solomon algebra. Using work of Arapura [1] and Libgober [18], we conclude that all positive-dimensional components of V k (A) are combinatorially determined, and that R 1 k (A) is the union of a subspace arrangement in C n , thereby resolving a conjecture of Falk [11]. We use these results to study the reflection arrangements associated to monomial groups.

The spectral sequence of an equivariant chain complex and homology with local coefficients

“…Part (2) generalizes [4, Theorem 1.2], valid only for complements of complex hyperplane arrangements, and k = C. Earlier results in this direction, also within the confines of arrangement theory, were obtained in [5] and [3]. The result in Part (2) was recently proved by M. Yoshinaga [35], under an additional condition, satisfied by arrangement complements, but not by arbitrary minimal CW-complexes.…”

Torsion in Milnor fiber homology

“…Consequently, as shown in [8] using properties of Fitting ideals, for q = 1 and d < n, the above proposition simplifies to:…”

Resonance of basis-conjugating automorphism groups

“…Permuting indices if need be, we may assume that a ∈ H 1 (PΣ n ; k) satisfies a 2,1 = 0 and a 3,4 = 0. We will show that this assumption implies that the map ψ a : I 2 → E 3 injects; hence a / ∈ R. Specifically, we will exhibit a subspace V ⊂ E 3 and a projection π : E 3 V so that the composition π • ψ a : I 2 → V is an isomorphism. Let V be the union of the sets {e 1,2 e 2,1 e 3,4 } ∪ {e 2,1 e i,j e j,i | 1 ≤ i < j ≤ n, {i, j} = {1, 2}} , {e 3,4 e 1,2 e k,1 , e 3,4 e 2,1 e 1,k , e 3,4 e 1,2 e 1,k | 3 ≤ k ≤ n} ,…”

“…By Theorem 1.1, the irreducible components of R 1 (PΣ n , k) are two-and three-dimensional. On the other hand, the results of [3] or [12] may be used to show that the irreducible components of the first resonance variety of H * (F n × · · · × F n ; k) are all n-dimensional.…”