Holonomic D-modules on abelian varieties

TL;DR

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

Contribution

It establishes the geometric structure of cohomology support loci and connects the Fourier-Mukai transform of holonomic D-modules to perverse coherent sheaves, supporting a conjecture about hyperk"ahler perverse sheaves.

Findings

Cohomology support loci are finite unions of linear subvarieties.

Fourier-Mukai transform maps the standard t-structure to a perverse coherent t-structure.

Transform of simple holonomic D-modules are intersection complexes.

Abstract

We study the Fourier-Mukai transform for holonomic D-modules on complex abelian varieties. Among other things, we show that the cohomology support loci of a holonomic D-module are finite unions of linear subvarieties, which go through points of finite order for objects of geometric origin; 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; and that Fourier-Mukai transforms of simple holonomic D-modules are intersection complexes in this t-structure. This supports the conjecture that Fourier-Mukai transforms of holonomic D-modules are "hyperk\"ahler perverse sheaves".