Integrable linear equations and Riemann-Schottky type problems

Igor Krichever

A connection discovered by Mumford of the celebrated Fay trisecant formula with a theory of soliton equations eventually had led Welters to his remarkable conjecture: an indecomposable principally polarized abelian variety X is the Jacobian of a curve if and only if there exists a trisecant of its Kummer variety K(X). It was motivated by Gunning's theorem and by another famous conjecture: the Jacobians of curves are exactly the indecomposable principally polarized abelian varieties whose theta-functions provide explicit solutions of the so-called KP equation. The latter was proposed earlier by Novikov and was unsettled at the time of the Welter's work. It was proved later by T. Shiota and until recently has remained the most effective solution of the classical Riemann-Schottky problem.

The characterization of the Jacobians proposed by the trisecant conjecture is much stronger. The proof of this conjecture based on a notion of integrable linear equations and a new type of cubic identities for the theta-functions valid for the case of
Jacobians on the theta-divisor will be presented. We will also discuss applications of integrable equations of the soliton theory for the characterization problem of Prym varieties.