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Integrable systems approach to the Schottky problem and related questions

Samuel Grushevsky, Yuancheng Xie

TL;DR

The work surveys how integrable systems provide an algebro-geometric path to the Schottky problem, by relating Jacobians and their theta functions to algebro-geometric solutions of the KP hierarchy. It develops the machinery of commuting differential operators, Jacobians and theta functions, and Baker–Akhiezer functions, to recast the Schottky problem in terms of the Kummer variety and its trisecants. A central contribution is Krichever’s proof that a flex line on the Kummer variety characterizes Jacobians, connecting analytic KP-type constraints with geometric criteria. This approach yields a powerful bridge between the differential-algebraic world of integrable hierarchies and the geometric theory of curves and abelian varieties, with potential implications for the broader Schottky problem and its variants.

Abstract

We give a somewhat informal introduction to the integrable systems approach to the Schottky problem, explaining how the theta functions of Jacobians can be used to provide solutions of the KP equation, and culminating with the exposition of Krichever's proof of Welters' trisecant conjecture in the most degenerate (flex line) case.

Integrable systems approach to the Schottky problem and related questions

TL;DR

The work surveys how integrable systems provide an algebro-geometric path to the Schottky problem, by relating Jacobians and their theta functions to algebro-geometric solutions of the KP hierarchy. It develops the machinery of commuting differential operators, Jacobians and theta functions, and Baker–Akhiezer functions, to recast the Schottky problem in terms of the Kummer variety and its trisecants. A central contribution is Krichever’s proof that a flex line on the Kummer variety characterizes Jacobians, connecting analytic KP-type constraints with geometric criteria. This approach yields a powerful bridge between the differential-algebraic world of integrable hierarchies and the geometric theory of curves and abelian varieties, with potential implications for the broader Schottky problem and its variants.

Abstract

We give a somewhat informal introduction to the integrable systems approach to the Schottky problem, explaining how the theta functions of Jacobians can be used to provide solutions of the KP equation, and culminating with the exposition of Krichever's proof of Welters' trisecant conjecture in the most degenerate (flex line) case.
Paper Structure (21 sections, 38 theorems, 239 equations)

This paper contains 21 sections, 38 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. Commuting differential operators
  3. Curves and their Jacobians
  4. Jacobians of curves --- the analytic definition
  5. Complex tori and abelian varieties
  6. The Schottky problem
  7. Jacobians of curves --- the algebraic definition
  8. Theta functions on Jacobians
  9. Geometry of Theta Divisors
  10. Weil reducibility
  11. Weil reducibility from the analytic viewpoint
  12. Trisecants of Kummer varieties
  13. Baker-Akhiezer Functions and the KP Hierarchy
  14. The Baker-Akhiezer functions
  15. The KP hierarchy
  16. ...and 6 more sections

Key Result

Theorem 2.2

For any fixed $x_0 \in \mathbb C$ (resp. $x_0\in\mathbb R$) there exists a unique formal solution $\psi$ of eq:normalizedeigenvalueproblem of the form where $\xi_s$ are holomorphic (resp. smooth) functions of $x$, and such that the solution is normalized at the point $x_0$ in the sense that $\xi_0(x) \equiv 1$ and $\xi_s(x_0) = 0$ for all $s > 0$.

Theorems & Definitions (113)

  • Theorem 2.2
  • Remark 2.3
  • proof
  • Remark 2.4
  • Corollary 2.5
  • proof
  • Theorem 2.7
  • Remark 2.8
  • proof
  • Corollary 2.9
  • ...and 103 more