# 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.