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Algebraically completely integrable systems and Kummer varieties

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References

  1. Adler, M., van Moerbeke, P.: Completely integrable systems — A systematic approach towards solving integrable systems. (Preprint 1988), to appear in Perspectives in mathematics. New York: Academic Press

    Google Scholar 

  2. Adler, M., van Moerbeke, P.: Geodesic flow onSO(4) and the intersection of quadrics. Proc. Natl. Acad. Sci. USA81, 4613–4616 (1984)

    Google Scholar 

  3. Adler, M., van Moerbeke, P.: The Kowalewski and Henon-Heiles motions as Manakov geodesic flows onSO(4). A two dimensional family of Lax pairs. Commun. Math. Phys.113 (sn4) (1988)

  4. Adler, M., van Moerbeke, P.: The algebraic integrability of geodesic flow onso(4). Invent. Math.67, 297–331 (1982)

    Google Scholar 

  5. Adler, M., van Moerbeke, P.: Completely integrable systems, Euclidean Lie algebras, and curves. Adv. Math.38 (3), 267–317 (1980)

    Google Scholar 

  6. Adler, M., van Moerbeke, P.: Linearization of Hamiltonian systems, Jacobi varieties and representation theory. Adv. Math.38 (3), 318–379 (1980)

    Google Scholar 

  7. Adler, M., van Moerbeke, P.: The complex geometry of the Kowalewski-Painlevé analysis. Invent. Math.97, 3–51 (1989)

    Google Scholar 

  8. Andreotti, A., Mayer, A.: On period relations for Abelian integrals on algebraic curves. Ann. Sc. Norm. Super Pisa21, 189–238 (1967)

    Google Scholar 

  9. Arnold, V.: Mathematical methods of classical mechanics. Berlin Heidelberg New York: Springer 1978

    Google Scholar 

  10. Barth, W.: Abelian surfaces with (1,2)-polarization. Algebraic geometry, Sendai (1985). Adv. Stud. Pure Math.10, 41–84 (1987)

    Google Scholar 

  11. Beauville, A.: Complex algebraic surfaces. Lond. Math. Soc. Lect. Note Ser.68 (1983)

  12. Bechlivanidis, C., van Moerbeke, P.: The Goryachev Chaplygin top, the Toda lattice and the intersection of 5 quadrics in ℙ7. Commun. Math. Phys.110, 317–324 (1987)

    Google Scholar 

  13. Dubrovin, B.A.: Theta functions and nonlinear equations. Usp. Mat. Nauk.36(2), 11–80 (1980) [Transl. Russ. Math. Surv.36(2), 11–92 (1981)

    Google Scholar 

  14. Dubrovin, B., Matveev, V. Novikov, S.: Non-linear equations of Korteweg-de Vries type, finite-zone linear operators, and Ableian varieties. Usp. Mat. Nauk31, 55–136 (1976); Russ. Math. Surv.31, 59–146 (1976)

    Google Scholar 

  15. Fay, J.D.: Theta functions on Riemann surfaces. (Lect. Notes Math., vol. 352) Berlin Heidelberg New York: Springer 1973

    Google Scholar 

  16. Flashchka, H.: The Toda lattice, (1) and (2). Phys. Rev.9, 1924–1925 (1974); Prog. Theor. Phys.51, 703–706 (1974)

    Google Scholar 

  17. Golubev, V.V.: Lectures on integration of the equations of motion of a rigid body about a fixed point. Moscow: Nauka 1953

    Google Scholar 

  18. Griffiths, P., Harris, J.: Principles of algebraic geometry. Pure and applied mathematics. Interscience series of texts, monograph and tracts. New York: Wiley 1978

    Google Scholar 

  19. Griffiths, P.A.: Linearizing flows and a cohomological interpretation of Lax equation. Am. J. Math.107, 1445–1483 (1985)

    Google Scholar 

  20. Greenberg, M.J., Harper, J.R.: Algebraic topology. New York: Benjamin 1981

    Google Scholar 

  21. Knörrer, H.: Geodesic on the ellipsoid. Invent. Math.59, 119–143 (1980)

    Google Scholar 

  22. Lang, S.: Abelian varieties. Berlin Heidelberg New York: Springer 1983

    Google Scholar 

  23. Lyapunov, A.M.: Reports of Kharkov. Math. Soc. Ser. 2,4, 1–2, 81–85 (1893)

    Google Scholar 

  24. MacLane, S.: Homology. New York: Academic Press 1963

    Google Scholar 

  25. McKean, H.: Integrable systems and algebraic curves. Global Analysis. (Lect. Notes Math., vol. 755) Berlin Heidelberg New York: Springer 1979

    Google Scholar 

  26. Mumford, D.: On the equations defining Abelian varieties I–III Invent. Math.1, 287–354 (1966);3, 75–135, 215–244 (1967)

    Google Scholar 

  27. Mumford, D.: Abelian varieties. Oxford: Oxford University Press 1970

    Google Scholar 

  28. Mumford, D.: Varieties defined by quadratic equations. Quest. sulle var. alg., Roma: Edizioni Cremonese 1969

    Google Scholar 

  29. Mumford, D.: Tata lectureson theta I and II. Basel: Birkhäuser 1983

    Google Scholar 

  30. Shimura, G.: Modules des variétés abeliannes polarisees et fonctions modulaires I, II, and III. Seminarie M. Cartan. E.N.S. (1957/58)

  31. Siegel, C.L.: Topics in complex function theory, Vol. I, II, IV. New York: Wiley-Interscience, Interscience Trastcs in Pure and Applied Math. 1971

    Google Scholar 

  32. Spanier, E.H.: Algebraic topology, New York: McGraw-Hill 1966

    Google Scholar 

  33. Spanier, E.H.: The homology of Kummer manifolds. Proc. Am. Math. Soc.7, 155–160 (1956)

    Google Scholar 

  34. Steklov, V.: On the motion of a solid in a liquid. Math. Ann.42, 273–294 (1893)

    Google Scholar 

  35. Weil, A.: On the theory of complex multiplication. Proc. Inter. Symp. on Alg. Number Theory, Tokyo-Nikko, 9–22 (1955)

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Research supported in part by a grant of the Sloan Foundation, Grant Number DD-191

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Piovan, L.A. Algebraically completely integrable systems and Kummer varieties. Math. Ann. 290, 349–403 (1991). https://doi.org/10.1007/BF01459250

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