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Secants of abelian varieties, theta functions, and soliton equations. (English. Russian original) Zbl 0922.14031
Russ. Math. Surv. 52, No. 1, 147-218 (1997); translation from Usp. Mat. Nauk 52, No. 1, 149-224 (1997).
The Riemann Schottky problem is the problem of finding equations for the Jacobian locus in the moduli space of principally polarized abelian varieties $${\mathcal A}_g$$. Novikov was the first to propose to solve the problem by means of soliton equations; to be more precise, Novikov conjectured that a principally polarized abelian variety is a Jacobian if and only if its associated Riemann theta function leads to a solution of the KP-equation in a certain way. The conjecture was finally proven by T. Shiota [Invent. Math. 83, 333-382 (1986; Zbl 0621.35097)]. The present paper gives a survey of similar applications of soliton equations to the Riemann Schottky problem.

##### MSC:
 14K25 Theta functions and abelian varieties 14H42 Theta functions and curves; Schottky problem
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