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The Tannakian Schottky Conjecture in Genus Five

Constantin Podelski

TL;DR

This work uses the Tannakian framework attached to a ppav $(A,\Theta)$ to attach a reductive group $G_\Theta$ and representation $\omega_\Theta$, proving that up to genus $5$ these invariants determine Jacobians and extending to the bielliptic Prym locus in all genera. A new Jacobian-detection criterion is developed based on Chern-Mather classes of the intersection cohomology complex and the multiplicativity properties in a ring of Lagrangian cycles, controlled by the problematic locus. The authors compute the characteristic cycles and Chern-Mather classes for Prym theta divisors on the bielliptic Prym locus, showing that non-hyperelliptic fake Jacobians in this locus are actually Jacobians, and derive Schottky-type results in low genus. The results provide explicit CM-class formulas in Jacobian and Prym settings and strengthen the role of Tannakian invariants in distinguishing Jacobians, offering evidence toward Weissauer–Krämer conjectures about the Schottky problem.

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 Krämer. The main tool in our proof is a criterion for detecting Jacobians relying on Chern-Mather classes.

The Tannakian Schottky Conjecture in Genus Five

TL;DR

This work uses the Tannakian framework attached to a ppav to attach a reductive group and representation , proving that up to genus these invariants determine Jacobians and extending to the bielliptic Prym locus in all genera. A new Jacobian-detection criterion is developed based on Chern-Mather classes of the intersection cohomology complex and the multiplicativity properties in a ring of Lagrangian cycles, controlled by the problematic locus. The authors compute the characteristic cycles and Chern-Mather classes for Prym theta divisors on the bielliptic Prym locus, showing that non-hyperelliptic fake Jacobians in this locus are actually Jacobians, and derive Schottky-type results in low genus. The results provide explicit CM-class formulas in Jacobian and Prym settings and strengthen the role of Tannakian invariants in distinguishing Jacobians, offering evidence toward Weissauer–Krämer conjectures about the Schottky problem.

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 . 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 Krämer. The main tool in our proof is a criterion for detecting Jacobians relying on Chern-Mather classes.
Paper Structure (14 sections, 22 theorems, 146 equations, 3 figures)

This paper contains 14 sections, 22 theorems, 146 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A Tannakian criterion to detect Jacobians
  3. The Tannakian formalism
  4. The ring of Lagrangian cycles
  5. The problematic locus
  6. Multiplicativity of the Chern-Mather class
  7. A criterion for detecting Jacobians
  8. The bielliptic Prym locus
  9. Notations
  10. The characteristic cycle of the Prym theta divisor
  11. The Gauss map for bielliptic Pryms
  12. The Chern-Mather class of the Prym theta divisor
  13. Proof of Theorem \ref{['maintheorem: Tannakian Schottky on Bielliptic Prym locus']}
  14. Proof of Theorem \ref{['maintheorem: Tannakian Schottky in dim up to 5']}

Key Result

Theorem 1

For $0\leq g \leq 5$, we have

Figures (3)

  • Figure 3.4:
  • Figure 3.12:
  • Figure 3.15:

Theorems & Definitions (47)

  • Example 1
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Example 2
  • Theorem 3
  • Proposition 2.5
  • proof
  • Remark 2.7
  • Lemma 2.8
  • ...and 37 more