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

Abstract

A central question in arrangement theory is to determine whether the characteristic polynomial Δ q of the algebraic monodromy acting on the homology group H q ( F ( A ) , C ) of the Milnor fiber of a complex hyperplane arrangement A is determined by the intersection lattice L ( A ) . Under simple combinatorial conditions, we show that the multiplicities of the factors of Δ 1 corresponding to certain eigenvalues of order a power of a prime p are equal to the Aomoto–Betti numbers β p ( A ) , which in turn are extracted from L ( A ) . When A defines an arrangement of projective lines with only double and triple points, this leads to a combinatorial formula for the algebraic monodromy. To obtain these results, we relate nets on the underlying matroid of A to resonance varieties in positive characteristic. Using modular invariants of nets, we find a new realizability obstruction (over C) for matroids, and estimate the number of essential components in the first complex resonance variety of A. Our approach also reveals a rather unexpected connection of modular resonance with the geometry of SL 2 ( C ) -representation varieties, which are governed by the Maurer–Cartan equation.