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Characterization of Jacobian varieties in terms of soliton equations. (English) Zbl 0621.35097

An equivalence theorem is stated concerning the two properties of a principally polarized abelian variety X:

(B) X is isomorphic to the Jacobian variety of a complete smooth curve of genus g over complex numbers;

(A) The theta divisor of X is irreducible, and the Riemannian theta function of X gives a certain family of solutions to the Kadomtsev- Petviashvili equation.

The implication (B)(A) has been proven by I. M. Krichever [Russ. Math. Surv. 32, No.6, 185-213 (1977); translation from Ups. Mat. Nauk 32, No.6(198), 183-208 (1977; Zbl 0372.35002)] and (A)(B) had been conjectured by S. P. Novikov as an answer to Schottky’s problem [see D. Mumford, ”Curves and their Jacobians” (1975; Zbl 0316.14010)]. A complete proof of the Novikov conjecture is given.

Reviewer: A.Bocharov

MSC:
35Q99PDE of mathematical physics and other areas
14H40Jacobians, Prym varieties
14K25Theta-functions
35G20General theory of nonlinear higher-order PDE
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