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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)$$\to (A)$$ 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)$$\to (B)$$ 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:
 35Q99 Partial differential equations of mathematical physics and other areas of application 14H40 Jacobians, Prym varieties 14K25 Theta functions and abelian varieties 35G20 Nonlinear higher-order PDEs
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