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 $\Delta_q$ of the algebraic monodromy acting on the homology group $H_q(F(\mathcal{A}),\mathbb{C})$ of the Milnor fiber of a complex hyperplane arrangement $\mathcal{A}$ is determined by the intersection lattice $L(\mathcal{A})$. Under simple combinatorial conditions, we show that the multiplicities of the factors of $\Delta_1$ corresponding to certain eigenvalues of order a power of a prime $p$ are equal to the Aomoto--Betti numbers $\beta_p(\mathcal{A})$, which in turn are extracted from $L(\mathcal{A})$. When $\mathcal{A}$ 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 $\mathcal{A}$ to resonance varieties in positive characteristic. Using modular invariants of nets, we find a new realizability obstruction (over $\mathbb{C}$) for matroids, and we estimate the number of essential components in the first complex resonance variety of $\mathcal{A}$. Our approach also reveals a rather unexpected connection of modular resonance with the geometry of ${\rm SL}_2(\mathbb{C})$-representation varieties, which are governed by the Maurer--Cartan equation.