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

TL;DR

This paper investigates the relationship between cohomology jump loci, formality, and monodromy in finite Galois covers, using line arrangements and multinets to analyze the triviality of monodromy actions.

Contribution

It establishes conditions under which jump loci of base and total space coincide and uses multinets to analyze monodromy actions in line arrangements.

Findings

Jump loci of base and total space are equivalent under certain conditions.

Multinet structures can be used to construct components of the characteristic variety.

Triviality of monodromy can be detected via resonance varieties.

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.