Torsion points on cohomology support loci: from D-modules to Simpson's theorem

TL;DR

This paper investigates the structure of cohomology support loci for D-modules on complex abelian varieties, establishing conditions for torsion points in their components and providing a new proof of Simpson's theorem.

Contribution

It introduces new criteria for the presence of torsion points in cohomology support loci and offers a novel proof of Simpson's theorem using D-module techniques.

Findings

Conditions under which irreducible components contain torsion points

New proof of Simpson's theorem on Green-Lazarsfeld sets

Identification of cases where D-modules are defined over number fields or have Z-structures

Abstract

We study cohomology support loci of regular holonomic D-modules on complex abelian varieties, and obtain conditions under which each irreducible component of such a locus contains a torsion point. One case is that both the D-module and the corresponding perverse sheaf are defined over a number field; another case is that the D-module underlies a grade-polarizable mixed Hodge module with a Z-structure. As a consequence, we obtain a new proof for Simpson's result that Green-Lazarsfeld sets are translates of subtori by torsion points.