Bernstein-Sato ideals and local systems

TL;DR

This paper explores the relationship between Bernstein-Sato ideals and local systems, providing new insights into the structure of cohomology support loci and extending classical results to collections of polynomials and hyperplane arrangements.

Contribution

It generalizes the theorem relating Bernstein-Sato polynomials to monodromy eigenvalues to multiple polynomials and addresses a multi-variable Monodromy Conjecture, proving it for hyperplane arrangements.

Findings

Partial confirmation of the relation between Bernstein-Sato ideals and local systems.

Generalization of the theorem to collections of polynomials.

Proof of the multi-variable Monodromy Conjecture for hyperplane arrangements.

Abstract

The topology of smooth quasi-projective complex varieties is very restrictive. One aspect of this statement is the fact that natural strata of local systems, called cohomology support loci, have a rigid structure: they consist of torsion-translated subtori in a complex torus. We propose and partially confirm a relation between Bernstein-Sato ideals and local systems. This relation gives yet a different point of view on the nature of the structure of cohomology support loci of local systems. The main result is a partial generalization to the case of a collection of polynomials of the theorem of Malgrange and Kashiwara which states that the Bernstein-Sato polynomial of a hypersurface recovers the monodromy eigenvalues of the Milnor fibers of the hypersurface. We also address a multi-variable version of the Monodromy Conjecture, prove that it follows from the usual single-variable Monodromy Conjecture, and prove it in the case of hyperplane arrangements.