The Milnor Fibration of a Hyperplane Arrangement: from Modular Resonance to Algebraic Monodromy

A central question in arrangement theory is to determine whether the characteristic polynomial ∆q of the algebraic monodromy acting on the homology group HqpFpAq,Cq of the Milnor fiber of a complex hyperplane arrangement A is determined by the intersection lattice LpAq. Under simple combinatorial conditions, we show that the multiplicities of the factors of ∆1 corresponding to certain eigenvalues of order a power of a prime p are equal to the Aomoto–Betti numbers βppAq, which in turn are extracted from LpAq. 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 for matroids, and we 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 SL2pCq-representation varieties, which are governed by the Maurer–Cartan equation.