Milnor fibrations of arrangements with trivial algebraic monodromy

Abstract

Each complex hyperplane arrangement gives rise to a Milnor fibration of its complement. Although the Betti numbers of the Milnor fiber $F$ can be expressed in terms of the jump loci for rank 1 local systems on the complement, explicit formulas are still lacking in full generality, even for $b_1(F)$. We study here the "generic" case (in which $b_1(F)$ is as small as possible), and look deeper into the algebraic topology of such Milnor fibrations with trivial algebraic monodromy. Our main focus is on the cohomology jump loci and the lower central series quotients of ${\pi }_1(F)$. In the process, we produce a pair of arrangements for which the respective Milnor fibers have the same Betti numbers, yet non-isomorphic fundamental groups: the difference is picked by the higher-depth characteristic varieties and by the Schur multipliers of the second nilpotent quotients.

arXiv comment52 pages; to appear in a special issue of Revue Roumaine de Mathématiques Pures et Appliquées

References in corpus (37)

• Characteristic varieties of arrangements
• On the fundamental group of the complement of a complex hyperplane arrangement
• Topology and geometry of cohomology jump loci
• The Braid Monodromy of Plane Algebraic Curves and Hyperplane Arrangements
• Fundamental groups of line arrangements: Enumerative aspects
• Chen Lie algebras
• The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy
• The spectral sequence of an equivariant chain complex and homology with local coefficients
• Toric complexes and Artin kernels
• Translated tori in the characteristic varieties of complex hyperplane arrangements
• Multinets, parallel connections, and Milnor fibrations of arrangements
• Algebraic invariants for Bestvina-Brady groups
• Bieri-Neumann-Strebel-Renz invariants and homology jumping loci
• Hyperplane arrangements and Milnor fibrations
• On the monodromy action on Milnor fibers of graphic arrangements
• Homology of iterated semidirect products of free groups
• Hall invariants, homology of subgroups, and characteristic varieties
• When does the associated graded Lie algebra of an arrangement group decompose?
• Formality properties of finitely generated groups and Lie algebras
• Fundamental groups, Alexander invariants, and cohomology jumping loci
• Torsion in Milnor fiber homology
• Characteristic varieties of quasi-projective manifolds and orbifolds
• Chen ranks and resonance
• Local systems on complements of arrangements of smooth, complex algebraic hypersurfaces
• Characteristic varieties and Betti numbers of free abelian covers
• Non-formality of Milnor fibers of line arrangements
• Combinatorial covers and vanishing of cohomology
• Abelian duality and propagation of resonance
• Double coverings of arrangement complements and $2$-torsion in Milnor fiber homology
• Some analogs of Zariski's Theorem on nodal line arrangements
• Reduced resonance schemes and Chen ranks
• On the monodromy of Milnor fibers of hyperplane arrangements
• Homology, lower central series, and hyperplane arrangements
• Formality and finiteness in rational homotopy theory
• Cohomology, Bocksteins, and resonance varieties in characteristic 2
• Line arrangements and direct sums of free groups
• Alexander invariants and cohomology jump loci in group extensions

Cited by in corpus (1)

• Highly connected non-formal Milnor fibers via polyhedral products