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

Abstract

We show that any Weyl group orbit of weights for the Tannakian group of semisimple holonomic {{\mathscr{D}}}-modules on an abelian variety is realized by a Lagrangian cycle on the cotangent bundle. As applications we discuss 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.

Semicontinuity of Gauss maps and the Schottky problem

“…cit., this is only a weak solution to the Schottky problem: In general the Gauss loci in the above corollary have more than one irreducible component and the Jacobian locus is only one of them. The theory of D-modules allows to refine the degree of the Gauss map to representation theoretic invariants that might distinguish the Jacobian locus [26].…”

“…The above is still only a weak solution to the Schottky problem, though χ IC ( ) also appears as the dimension of an irreducible representation of a certain reductive group which gives more information [26,Sect. 4].…”