Finite Galois covers, cohomology jump loci, formality properties, and multinets

Abstract

We explore the relation between cohomology jump loci in a finite Galois cover, formality properties and algebraic monodromy action. We show that the jump loci of the base and total space are essentially the same, provided the base space is 1-formal and the monodromy action in degree 1 is trivial. We use reduced multinet structures on line arrangements to construct components of the first characteristic variety of the Milnor fiber in degree 1, and to prove that the monodromy action is non-trivial in degree 1. For an arbitrary line arrangement, we prove that the triviality of the monodromy in degree 1 can be detected in a precise way, by resonance varieties.

References in corpus (8)

• Topology and geometry of cohomology jump loci
• Toric complexes and Artin kernels
• Geometric and algebraic aspects of 1-formality
• Quasi-Kähler groups, 3-manifold groups, and formality
• Algebraic monodromy and obstructions to formality
• Universal representations of braid and braid-permutation groups
• Cohomology rings and formality properties of nilpotent groups
• First Milnor cohomology of hyperplane arrangements

Cited by in corpus (4)

• The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy
• Cohomology, Bocksteins, and resonance varieties in characteristic 2
• On the cohomology of the Milnor fibre of a hyperplane arrangement
• First Milnor cohomology of hyperplane arrangements