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

TL;DR

This paper explores the microlocal geometry of the Gauss map for holonomic D-modules on abelian varieties, revealing how Weyl group orbits correspond to conic Lagrangian cycles and applying these insights to problems in algebraic geometry.

Contribution

It establishes a link between Weyl group orbits and conic Lagrangian cycles in the context of Tannaka groups, providing new tools for geometric and algebraic analysis.

Findings

Weyl group orbits are realized by conic Lagrangian cycles on cotangent bundles.

Application to a weak solution of the Schottky problem in genus five.

Provides obstructions for subvariety decompositions and criteria for Lie algebra simplicity.

Abstract

We show that for the reductive Tannaka groups of semisimple holonomic $\mathscr{D}$-modules on abelian varieties, every Weyl group orbit of weights of their universal cover is realized by a conic Lagrangian cycle on the cotangent bundle. Applications include a weak solution to the Schottky problem in genus five, an obstruction for the existence of summands of subvarieties on abelian varieties and a criterion for the simplicity of the arising Lie algebras.