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 ${\mathrm{SL}}_2(\mathbb{C})$-representation varieties, which are governed by the Maurer--Cartan equation.

50 pages, 5 figures; v2: results strengthened, proofs simplified; v3: accepted for publication in the Proceedings of the London Mathematical Society

References in corpus (6)

• On the monodromy action on Milnor fibers of graphic arrangements
• Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves
• The line geometry of resonance varieties
• Flat connections and resonance varieties: from rank one to higher ranks
• On combinatorial invariance of the cohomology of Milnor fiber of arrangements and Catalan equation over function field
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Cited by in corpus (20)

• Around the tangent cone theorem
• Double coverings of arrangement complements and $2$-torsion in Milnor fiber homology
• Homology graph of real arrangements and monodromy of Milnor Fiber
• On the Milnor monodromy of the irreducible complex reflection arrangements
• Naturality properties and comparison results for topological and infinitesimal embedded jump loci
• Rank two topological and infinitesimal embedded jump loci of quasi-projective manifolds
• Milnor fibrations of arrangements with trivial algebraic monodromy
• Cohomology, Bocksteins, and resonance varieties in characteristic 2
• On the topology of the Milnor fibration of a hyperplane arrangement
• Efficiency and complexity of hyperplane arrangements
• Length of Perverse Sheaves on Hyperplane Arrangements
• Resonant bands, Aomoto complex, and real 4-nets
• Gysin morphisms for non-transversal hyperplane sections with an application to line arrangements
• Some computations on the characteristic variety of a line arrangement
• Topological computation of the first Milnor fiber cohomology of hyperplane arrangements
• The homology groups of finite cyclic covering of line arrangement complement
• Configurations of points and topology of real line arrangements
• On two pencils of cubic curves
• On Betti Numbers of Milnor Fiber of Hyperplane Arrangements
• On algebraic surfaces associated to line arrangements