Local systems on analytic germ complements

TL;DR

This paper generalizes the Monodromy Theorem by showing that the cohomology jump loci of rank one local systems on complex analytic germ complements are finite unions of torsion translates of subtori, extending classical monodromy results.

Contribution

It establishes a broader structural description of cohomology jump loci for complex analytic germs, generalizing classical monodromy theorems.

Findings

Cohomology jump loci are finite unions of torsion translates of subtori.

Generalization of the classical Monodromy Theorem.

Eigenvalues of monodromy are roots of unity in this setting.

Abstract

We prove that the cohomology jump loci of rank one local systems on the complement in a small ball of a germ of a complex analytic set are finite unions of torsion translates of subtori. This is a generalization of the classical Monodromy Theorem stating that the eigenvalues of the monodromy on the cohomology of the Milnor fiber of a germ of a holomorphic function are roots of unity.