Abelian pole systems and Riemann-Schottky type systems

TL;DR

This survey explores how soliton theory and pole systems provide new characterizations of Jacobians and Prym varieties among abelian varieties, linking algebraic geometry with integrable systems.

Contribution

It highlights the abelian analogues of pole systems in soliton hierarchies for characterizing Jacobians and Prym varieties, including recent results on curves with involution.

Findings

Characterization of Jacobians via trisecant lines of Kummer varieties

Prym varieties characterized by symmetric quadrisecants of Kummer varieties

Recent results on Jacobians of curves with involution

Abstract

In this survey of works on a characterization of Jacobians and Prym varieties among indecomposable principally polarized abelian varieties via the soliton theory we focus on a certain circle of ideas and methods which show that the characterization of Jacobians as ppav whose Kummer variety admits a trisecant line and the Pryms as ppav whose Kummer variety admits a pair of symmetric quadrisecants can be seen as an abelian version of pole systems arising in the theory of elliptic solutions to the basic soliton hierarchies. We present also recent results in this direction on the characterization of Jacobians of curves with involution, which were motivated by the theory of two-dimensional integrable hierarchies with symmetries.