Inventiones mathematicae

, Volume 83, Issue 2, pp 333–382 | Cite as

Characterization of Jacobian varieties in terms of soliton equations

  • Takahiro Shiota
Article

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References

  1. 1.
    Arbarello, E., De Concini, C.: On a set of equations characterizing Riemann matrices. Ann. Math.120, 119–140 (1984)Google Scholar
  2. 2.
    Burchnall, J.L., Chaundy, T.W.: Commutative ordinary differential operators. Proc. Lond. Math. Soc.21, 420–440 (1922); Proc. R. Soc. Lond. (A)118, 557–583 (1928); Ibid. Proc. R. Soc. Lond. (A)134, 471–485 (1931)Google Scholar
  3. 3.
    Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations. In: Proceedings RIMS Symp. Nonlinear integrable systems-classical theory and quantum theory (Kyoto 1981) pp. 39–119. Singapore: World Scientific 1983Google Scholar
  4. 4.
    Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. IHES Publ. Math.36, 75–109 (1969)Google Scholar
  5. 5.
    D'Souza, C.: Compactification of generalized Jacobians. Proc. Indian Acad. Sci. (A)88, (5) 419–457 (1979)Google Scholar
  6. 6.
    Dubrovin, B.A.: The Kadomtsev-Petviashvili equation and the relations between the periods of holomorphic differentials on Riemann surfaces. Math. USSR, Izv.19, (2) 285–296 (1982)Google Scholar
  7. 7.
    Fay, J.D.: Theta functions on Riemann surfaces. Lect. Notes Math., vol. 352. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  8. 8.
    Fay, J.D.: On the even-order vanishing of Jacobian theta functions. Duke Math. J.51, 109–132 (1984)Google Scholar
  9. 9.
    Fay, J.D.: Bilinear identities for theta functions. (Preprint)Google Scholar
  10. 10.
    van der Geer, G.: The Schottky problem. In: Proceedings of Arbeitstagung Bonn 1984, Lect. Notes Math., vol. 1111, pp. 385–406. Berlin-Heidelberg-New York-Tokyo: Springer 1985Google Scholar
  11. 11.
    Grothendieck, A.: Sur quelque points d'algèbre homologique. Tôhoku Math. J. (2)9, 119–221 (1957)Google Scholar
  12. 12.
    Gunning, R.C.: Some curves in abelian varieties. Invent. Math.66, 377–389 (1982)Google Scholar
  13. 13.
    Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math.79, 109–326 (1964)Google Scholar
  14. 14.
    Hirota, R.: Direct method of finding exact solutions of nonlinear evolution equations. Lect. Notes Math., vol. 515, pp. 40–68. Berlin-Heidelberg-New York: Springer 1976Google Scholar
  15. 15.
    Krichever, I.M.: Methods of algebraic geometry, in the theory of nonlinear equations. Russ. Math. Surv.32, (6), 185–213 (1977)Google Scholar
  16. 16.
    Lax, P.D.: Periodic solutions of the KdV equation. Commun Pure Appl. Math.28, 141–188 (1975)Google Scholar
  17. 17.
    Manin, Y.I.: Algebraic aspects of nonlinear differential equations. J. Sov. Math.11, 1–122 (1978)Google Scholar
  18. 18.
    McKean, H.P., van Moerbeke, P.: The spectrum of Hill's equation. Invent. Math.30, 217–274 (1975)Google Scholar
  19. 19.
    Mulase, M.: Cohomological structure in soliton equations and Jacobian varieties. J. Differ. Geom.19 403–430 (1984)Google Scholar
  20. 20.
    Mumford, D.: An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg-de Vries equation and related non-linear equations Proceedings Int. Symp. Algebraic Geometry, Kyoto (1977), pp. 115–153. Kinokuniya Book Store, Tokyo, 1978Google Scholar
  21. 21.
    Mumford, D.: Curves and their Jacobians. Ann Arbor: Univ. of Michigan Press 1975Google Scholar
  22. 22.
    Mumford, D.: Tata Lectures on theta. Boston-Basel-Stuttgart: Birkhäuser, vol. 1 (1983), vol. 2 (1984)Google Scholar
  23. 23.
    Novikov, S.P.: The periodic problem for the Korteweg-de Vries equation. Funct. Anal. Appl.8, 236–246 (1974)Google Scholar
  24. 24.
    Sato, M., Sato, Y.: Soliton equations as dynamical systems on infinite dimensional Grassmann manifold. In: Nonlinear partial differential equations in applied science (Proc. U.S.-Japan Seminar, Tokyo 1982), pp. 259–271. Amsterdam-New York: North Holland 1983Google Scholar
  25. 25.
    Sato, M.: Soliton equations and the universal Grassmann manifold (by Noumi in Japanese). Math. Lect. Note Ser No. 18. Sophia University, Tokyo, 1984Google Scholar
  26. 26.
    Segal, G., Wilson, G.: Loop groups and equations of KdV type. IHES Publ. Math.61, 5–65 (1985)Google Scholar
  27. 27.
    Shiota, T.: Soliton equations and the Schottky problem. Xeroxed notes distributed in a seminar at RIMS, Kyoto Univ 1983Google Scholar
  28. 28.
    Welters, G.E.: A criterion for Jacobi varieties. Ann. Math.120, 497–504 (1984)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Takahiro Shiota
    • 1
  1. 1.Department of MathematicsBrandeis UniversityWalthamUSA

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