Degeneration of Orlik–Solomon algebras and Milnor fibers of complex line arrangements

Abstract

We give a vanishing theorem for the monodromy eigenspaces of the Milnor fibers of complex line arrangements. By applying the modular bound of the local system cohomology groups given by Papadima-Suciu, the result is deduced from the vanishing of the cohomology of certain Aomoto complex over finite fields. In order to prove this, we introduce degeneration homomorphisms of Orlik-Solomon algebras.

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

“…Assuming the contrary, we will use Lemma 9.6 to derive a contradiction. Take a = 1 in Lemma 9.2, and denote by N the associated (3,9)-net on M (3). Write each class in the form…”

“…Let X = {v, v , v } be a size 3 subset of F 2 3 . It is easy to see that X is dependent if and only if X = {(i, 0), (i, 1), (i, 2)}, or X = {(0, j), (1, j), (2, j)}, or X = {(i, gi) | i ∈ F 3 }, for some g ∈ Σ 3 . Now assume that X = {(i, j), (i , j ), (i , j )} is independent.…”

A vanishing result for the first twisted cohomology of affine varieties and applications to line arrangements

“…Then at each point p ∈ C, any line H j passing through p is not in the tangent cone of the germ (C, p), since otherwise the germs (C, p) and (H j , p) are not separated after one blow-up. This implies the first two claims in (1). Moreover, if there is a line H j containing only one point p of T =1 , then the multiplicity of C at p has to be equal to d, which is possible only if C is a line, since C is irreducible.…”

“…The irreducible curve C has degree d = a + b by Proposition 3.3 (1). Hence, if a > b, the line L determined by p 1 and p 3 satisfies (L, C) ≥ 2a > d, and hence C = L, which is clearly not possible (L intersects the arrangement in points not in T =1 ).…”

“…complex line arrangement, see [1]. All these results, as well as [2], were motivated by Libgober's result in [11] saying that for the local systems L related to the Milnor fiber monodromy, one has such a vanishing as soon as there is a line in the arrangement A containing no bad point for L. We obtain as a direct application of Proposition 4.3 and using previous results by Mȃcinic, Papadima and Popescu in [14] a complete description on the Milnor fiber monodromy of the monomial arrangements a.k.a.…”