The Tannakian Schottky Conjecture in Genus Five

TL;DR

This paper uses Tannakian formalism to characterize Jacobians of genus up to 5 and provides evidence for a conjecture relating to abelian varieties, employing Chern-Mather classes as a key tool.

Contribution

It establishes a Tannakian-based criterion that uniquely characterizes Jacobians in genus five, extending to bielliptic Prym loci in all genera, advancing understanding of the Tannakian Schottky problem.

Findings

Characterization of Jacobians up to genus 5 using Tannakian groups

Extension of results to bielliptic Prym loci in all genera

Introduction of a Chern-Mather class based detection criterion

Abstract

Using the Tannakian formalism, one can attach to a principally polarized abelian variety a reductive group, along with a representation. We show that this group and the representation characterize Jacobians in genus up to $5$. More generally, our results hold on the bielliptic Prym locus in all genera. This gives the first evidence towards a recent conjecture by Weissauer and Kr\"amer. The main tool in our proof is a criterion for detecting Jacobians relying on Chern-Mather classes.