Igor Krichever

Integrable linear equations of the soliton theory and Riemann-Schottky type problems

Abstract: The remarkable Welter's trisecant 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 $K(X)$, was motivated by the celebrated 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 Jacobains proposed by the trisecant conjecture is much stronger. The proof of this conjecture based on an notion of integrable linear equations and new type 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.