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Abelian solutions of the soliton equations and Riemann-Schottky problems. (English. Russian original) Zbl 1164.37021
Russ. Math. Surv. 63, No. 6, 1011-1022 (2008); translation from Usp. Mat. Nauk 63, No. 6, 19-30 (2008).
This is an exposition of the author’s talk on the conference dedicated to S. P. Novikov’s 70th birthday (Moscow, June 2008). In the middle of 1970s Novikov introduced the conjecture that an irreducible Abelian variety is the Jacobian variety of a Riemann surface if and only if a certain theta-functional formula defines a solution to the Kadomtsev-Petviashvili (KP) equation. Therewith the theta function corresponds to the Abelian variety. This conjecture was inspired by Krichever’s result that this formula corresponding to the Jacobian variety of a Riemann surface gives a solution of KP. Hence the conjecture only consisted in proving the converse statement and that was done in the middle of 1980s by T. Shiota [Invent. Math. 83, 333–382 (1986; Zbl 0621.35097)]. We remark that that was the first complete characterization of Abelian varieties (the Riemann-Schottky problem) and without using soliton theory this problem is solved until recently only for varieties of dimension $$4$$ (in dimensions $$\leq 3$$ any irreducible Abelian variety is the Jacobian).
Recently the author found a new approach to characterization of special classes of Abelian varieties in terms of linear problems corresponding to known soliton equations. Successively he found a new solution to the Riemann-Schottky problem for Jacobian varieties [Progress in Mathematics 253, 497–514 (2006; Zbl 1132.14032)], solved an analogue of this problem for the Prym varieties of branched coverings with two branch points, and proved the famous Welters conjecture on a characterization of Jacobian varieties in terms of the existence of a trisecant. This article gives an account of these results.
##### MSC:
 37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions 14H40 Jacobians, Prym varieties 14H42 Theta functions and curves; Schottky problem 35Q53 KdV equations (Korteweg-de Vries equations) 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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