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

By Stefan Papadima, Alexander I. Suciu. 2014

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A central question in arrangement theory is to determine whether the characteristic polynomial Δq of the algebraic monodromy acting on the homology group Hq(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 Δ1 corresponding to certain eigenvalues of order a power of a prime p are equal to the Aomoto–Betti numbers βp(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 SL2(C) ‐representation varieties, which are governed by the Maurer–Cartan equation.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/53ac281ab188b2140e5c39bccfbed2b7c1d3d94b