Holonomic complexes on abelian varieties, Part I

TL;DR

This paper investigates the Fourier-Mukai transform for holonomic D-modules on complex abelian varieties, revealing the structure of cohomology support loci and their relation to perverse coherent t-structures.

Contribution

It establishes the geometric structure of cohomology support loci and connects the standard t-structure to a perverse coherent t-structure via Fourier-Mukai transform.

Findings

Cohomology support loci are finite unions of translates of triple tori.

Translates are by torsion points for objects of geometric origin.

Standard t-structure corresponds to a perverse coherent t-structure under Fourier-Mukai transform.

Abstract

We study the Fourier-Mukai transform for holonomic D-modules on a complex abelian variety. Among other things, we show that the cohomology support loci of a holonomic complex are finite unions of translates of triple tori, the translates being by torsion points for objects of geometric origin; and that the standard t-structure on the derived category of holonomic complexes corresponds, under the Fourier-Mukai transform, to a certain perverse coherent t-structure in the sense of Kashiwara and Arinkin-Bezrukavnikov.