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Zbl 0621.35097
Shiota, T.
Characterization of Jacobian varieties in terms of soliton equations. (English)
[J] Invent. Math. 83, 333-382 (1986). ISSN 0020-9910; ISSN 1432-1297/e

An equivalence theorem is stated concerning the two properties of a principally polarized abelian variety X: \par (B) X is isomorphic to the Jacobian variety of a complete smooth curve of genus g over complex numbers; \par (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. \par The implication (B)$\to (A)$ has been proven by {\it 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 {\it D. Mumford}, "Curves and their Jacobians" (1975; Zbl 0316.14010)]. A complete proof of the Novikov conjecture is given.
[A.Bocharov]

MSC 2000:: *35Q99 PDE of mathematical physics and other areas
14H40 Jacobians
14K25 Theta-functions
35G20 General theory of nonlinear higher-order PDE

Keywords: equivalence theorem; abelian variety; Jacobian variety; theta divisor; Riemannian theta function; Kadomtsev-Petviashvili equation; Novikov conjecture

Citations: Zbl 0372.35002; Zbl 0316.14010

Cited in: Zbl 1217.14022 Zbl 1164.37021 Zbl 1132.14032 Zbl 1098.14020 Zbl 0945.14017 Zbl 0922.14031 Zbl 0766.14020 Zbl 0736.14010 Zbl 0699.14055 Zbl 0676.14008 Zbl 0696.14019 Zbl 0685.14024 Zbl 0645.14014 Zbl 0637.14021

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