Characteristic classes and Hilbert-Poincar\'e series for perverse sheaves on abelian varieties

TL;DR

This paper studies the structure of convolution powers of perverse sheaves on abelian varieties, revealing rationality of their generating series and expressing related invariants via characteristic classes, with applications to Prym-Tjurin varieties.

Contribution

It provides new formulas for the Hilbert-Poincaré series of convolution powers and relates these to characteristic classes on the dual abelian variety, advancing understanding of their geometric properties.

Findings

The generating series for the generic rank is rational with a simple form.

Similar rationality results hold for symmetric convolution powers.

Explicit formulas relate characteristic classes to Schur functors on the dual abelian variety.

Abstract

The convolution powers of a perverse sheaf on an abelian variety define an interesting family of branched local systems whose geometry is still poorly understood. We show that the generating series for their generic rank is a rational function of a very simple shape and that a similar result holds for the symmetric convolution powers. We also give formulae for other Schur functors in terms of characteristic classes on the dual abelian variety, and as an example we discuss the case of Prym-Tjurin varieties.