Characteristic cycles and the microlocal geometry of the Gauss map, I

TL;DR

This paper introduces new approaches to understanding Tannakian Galois groups of holonomic D-modules on abelian varieties through principal bundles and microlocalization, revealing connections to Weyl group orbits and minuscule representations.

Contribution

It presents two innovative methods: one using Fourier-Mukai transform to interpret Galois groups, and another constructing a microlocalization functor linking characteristic cycles to Weyl group orbits.

Findings

Galois groups are almost connected.

Microlocalization relates characteristic cycles to Weyl group orbits.

Provides bounds for decompositions of subvarieties.

Abstract

We propose two new approaches to the Tannakian Galois groups of holonomic D-modules on abelian varieties. The first is an interpretation in terms of principal bundles given by the Fourier-Mukai transform, which shows that they are almost connected. The second constructs a microlocalization functor relating characteristic cycles to Weyl group orbits of weights. This explains the ubiquity of minuscule representations, and we illustrate it with a Torelli theorem and with a bound for decompositions of a given subvariety as a sum of subvarieties. The appendix sketches a twistor variant that may be useful for D-modules not coming from Hodge theory.