Resonant bands, Aomoto complex, and real 4-nets

Abstract

The resonant band is a useful notion for the computation of the nontrivial monodromy eigenspaces of the Milnor fiber of a real line arrangement. In this article, we develop the resonant band description for the cohomology of the Aomoto complex. As an application, we prove Date

Double coverings of arrangement complements and 2-torsion in Milnor fiber homology

“…, which is called the Aomoto complex. This complex plays crucial role in the computation of twisted cohomology groups [1,12,13,19,25,31,34,38].…”

“…For a given ω ∈ A 1 R (A), Orlik-Solomon algebra determines the cochain complex (A • R (A), ω ∧ −), which is called the Aomoto complex. This complex plays crucial role in the computation of twisted cohomology groups [1,12,13,19,25,31,34,38].…”

Some computations on the characteristic variety of a line arrangement

“…When n = 4, this gives another description of the 1-dimensional translated component found in [22]. For n = 8, the translated 1-dimensional component contains the two points P 8,2 , P 8,6 (in contrast with [24], p, 45-46; see remark 3.6).…”

“…(1) In [24], pages 45-46, the case n = 8 is considered and the authors claim that the six points P 8,k , k = 1, . .…”

Homology graph of real arrangements and monodromy of Milnor Fiber

“…where each β q,d is the multiplicity of an eigenvalue of h q with order d, and ϕ d is the cyclotomic polynomial of degree d. The computation of the eigenspaces of the monodromy operators, i.e. the cyclic modules [C[t, t −1 ]/ϕ d ] β q,d appearing in (1), is a difficult question which has been intensively studied the last decades and approached by different techniques such as nonresonant conditions for local systems (see for intance [6,7]), multinets ( [16,30]), minimality of the complement( [28,32,34,36]), graphs ( [1], [27]), and also mixed Hodge structure ( [3,4,5,11,12,13,15]). Many progress have been done for braid arrangements ( [29]), graphic arrangements ( [20]) and real line arrangements ( [33,35]).…”