Seventy years of spectral curves: 1923–1993

Part of the Lecture Notes in Mathematics book series (LNM, volume 1620)


Modulus Space Vector Bundle Line Bundle Elliptic Curve Theta Function 
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  1. [AHH] M.R. Adams, J. Harnad and J. Hurtubise, Dual moment maps into loop algebras, Lett. Math. Phys. 20 (1990), 299–308.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [AHP] M.R. Adams, J. Harnad and E. Previato, Isospectral Hamiltonian flows in finite and infinite dimension, Comm. Math. Phys. 117 (1988), 451–500.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [AvM] M. Adler and P. van Moerbeke, Completely integrable systems, euclidean Lie algebras and curves. Linearization of Hamiltonian systems, Jacobian varieties and representation theory, Adv. in Math. 38 (1980), 267–379.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [AM] M. Adler and J. Moser, On a class of polynomials connected with the Korteweg de Vries equations, Comm. Math. Phys. 61 (1978), 1–30.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [AMcKM] H. Airault, H.P. McKean and J. Moser, Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem, Comm. Pure Appl. Math. 30 (1977), 95–148.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [B1] A. Beauville, Fibrés de rang 2 sur une courbe, fibré déterminant et fonctions thêta, Bull. Soc. Math. France 116 (1988), 431–448.MathSciNetzbMATHGoogle Scholar
  7. [B2] A. Beauville, Fibrés de rang 2 sur une courbe, fibré déterminant et fonctions thêta, II, Bull. Soc. Math. France 119 (1991), 259–291.MathSciNetGoogle Scholar
  8. [B3] A. Beauville, Jacobiennes des courbes spectrales et systèmes hamiltoniens complètement intégrables Acta Math. 164 (1990), 211–235.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [BNR] A. Beauville, M.S. Narasimhan and S. Ramanan, Spectral Curves and the Generalized Theta Divisor, J. Reine Angew. Math. 398 (1989), 169–179.MathSciNetzbMATHGoogle Scholar
  10. [BR] M. Bershadsky and A. Radul, Fermionic fields on Z N curves, Comm. Math. Phys. 116 (1988), 689–700.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [BS] A. Bertram and A. Szenes, Hilbert polynomials of moduli spaces of rank 2 vector bundles II, Topology 32 (1993), 599–609.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Bo] F. Bottacin, Symplectic geometry on moduli spaces of stable pairs, preprint Univ. Paris Sud, 1992.Google Scholar
  13. [BC1] J.L. Burchnall and T.W. Chaundy, Commutative ordinary differential operators, Proc. London Math. Soc. 211 (1923), 420–440.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [BC2] J.L. Burchnall and T.W. Chaundy, Commutative ordinary differential operators, Proc. Roy. Soc. London Ser. A 118 (1928), 557–583.CrossRefzbMATHGoogle Scholar
  15. [C] A.B. Coble, Algebraic Geometry and Theta Functions, AMS Colloquium Publications X, 1961.Google Scholar
  16. [CPP] E. Colombo, G.P. Pirola and E. Previato, Density of elliptic solitons, J. Reine Angew. Math. 451 (1994), 161–169.MathSciNetzbMATHGoogle Scholar
  17. [D] G. Darboux, Théorie Générale des Surfaces II, Chelsea 1972.Google Scholar
  18. [De] P. Dehornoy, Operateurs différentiels et courbes elliptiques, Compositio Math. 43 (1981), 71–99.MathSciNetzbMATHGoogle Scholar
  19. [DLT] P. Deift, L.C. Li and C. Tomei, Matrix factorizations and integrable systems, Comm. Pure Appl. Math. 42 (1989), 443–521.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [DR] J.V. Desale and S. Ramanan, Classification of vector bundles of rank 2 on hyperelliptic curves, Invent. Math. 38 (1976), 161–185.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [Di] J. Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968), 209–242.MathSciNetzbMATHGoogle Scholar
  22. [Do] R. Donagi, Non-Jacobians in the Schottky loci, Ann. of Math. 126 (1987), 193–217.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [DP] R. Donagi and E. Previato, Abelian solitons, in preparation.Google Scholar
  24. [DN] J.-M. Drezet and M.S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), 53–94.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [Du] B.A. Dubrovin, The KP equation and the relations between the periods of holomorphic differentials on Riemann surfaces, Math. USSR-Izv. 19 (1982), 285–296.CrossRefzbMATHGoogle Scholar
  26. [EK] F. Ehlers and H. Knörrer, An algebro-geometric interpretation of the Bäcklund transformation for the Korteweg-de Vries equation, Comment. Math. Helv. 57 (1982), 1–10.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [EF] N. Ercolani and H. Flaschka, The geometry of the Hill equation and of the Neumann system, Philos. Trans. Roy. Soc. London Ser. A 315 (1985), 405–422.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [F] G. Faltings, A proof of the Verlinde formula, preprint 1992.Google Scholar
  29. [F1] J. Fay, Theta Functions, Lecture Notes in Math 352, Springer, Berlin, 1973.zbMATHGoogle Scholar
  30. [F2] J. Fay, On the even-order vanishing of Jacobian theta functions, Duke Math. J. 51 (1984), 109–132.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [Fl] H. Flaschka, Towards and algebro-geometric interpretation of the Neumann system, Tôhoku Math. J. 36 (1984), 407–426.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [vG] B. van Geemen, Schottky-Jung relations and vectorbundles on hyperelliptic curves, Math. Ann. 281 (1988), 431–449.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [vGvdG] B. van Geemen and G. van der Geer, Kummer varieties and the moduli spaces of abelian varieties, Amer. J. Math. 108 (1986), 615–641.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [vGP1] B. van Geemen and E. Previato, Pryn varieties and the Verlinde formula, Math. Annalen 294 (1992), 741–754.CrossRefzbMATHGoogle Scholar
  35. [vGP2] B. van Geemen and E. Previato, On the Hitchin system MSRI Preprint No. 057-94.Google Scholar
  36. [G] T. Gneiting, Kommutierende Differentialoperatoren und Solitonen, Diplomarbeit, Universität Bayreuth 1993.Google Scholar
  37. [GrP] S. Greco and E. Previato, Spectral curves and ruled surfaces: projective models, Vol. 8 of “The curves seminar at Queen's” Queen's Papers in Pure and Appl. Math., No. 88 (1991), pp. F1–F33.MathSciNetGoogle Scholar
  38. [GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York, 1978.zbMATHGoogle Scholar
  39. [G1] F.A. Grünbaum, Commuting pairs of linear ordinary differential operators of orders four and six, Phys. D 31 (1988), 424–433.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [G2] F.A. Grünbaum, The KP equation: an elementary approach to the “rank 2” solutions of Krichever and Novikov, Phys. Lett. A 139 (1989), 146–150.MathSciNetCrossRefGoogle Scholar
  41. [G3] F.A. Grünbaum, Darboux's method and some “rank two” explicit solutions of the KP equation, in Nonlinear evolution equations: integrability and spectral methods, eds. A. Degasperis et al., Manchester Univ. Press, Manchester, UK, 1990, pp. 271–277.Google Scholar
  42. [H] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977.CrossRefzbMATHGoogle Scholar
  43. [Hi1] N. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91–114.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [Hi2] N. Hitchin, Flat connections and geometric quantization, Comm. Math. Phys. 131 (1990), 347–380.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [I] J.-I. Igusa, Theta functions, Springer-Verlag, Berlin, 1972.CrossRefzbMATHGoogle Scholar
  46. [K] A. Kasman, Rank-r KP solutions with singular rational spectral curves, Thesis, Boston Univ. 1995.Google Scholar
  47. [Kn] H. Knörrer, Geodesics on the ellipsoid, Invent. Math. 59 (1980), 119–143.MathSciNetCrossRefzbMATHGoogle Scholar
  48. [Kr1] I.M. Krichever, The integration of non-linear equations by methods of algebraic geometry, Functional Anal. Appl. 12 (1977), 12–26.CrossRefzbMATHGoogle Scholar
  49. [Kr2] I.M. Krichever, Rational solutions of the Zakharov-Shabat equations and completely integrable systems of N particles on a line, J Soviet Math. 21 (1983), 335–345.CrossRefzbMATHGoogle Scholar
  50. [Kr3] I.M. Krichever, Elliptic solutions of the KP equation and integrable systems of particles, Functional Anal. Appl. 14 (1980), 15–31.CrossRefGoogle Scholar
  51. [KN] I.M. Krichever and S.P. Novikov, Holomorphic bundles over algebraic curves and nonlinear equations, Russian Math. Surveys 35 (1980), 53–79.MathSciNetCrossRefzbMATHGoogle Scholar
  52. [L] Y. Laszlo, Un théorème de Riemann pour les diviseurs thêta sur les espaces des modules de fibrés stables sur une courbe, Duke Math. J. 64 (1991), 333–347.MathSciNetCrossRefGoogle Scholar
  53. [LP] G. Latham and E. Previato, Higher rank Darboux transformations, in Singular Limits of Dispersive Waves, eds. N.M. Ercolani et al., Plenum Press, New York, 1994, pp. 117–134.CrossRefGoogle Scholar
  54. [LP2] G.A. Latham and E. Previato, Darboux transformations for higher rank Kadomtsev-Petviashvili and Kridhever-Novikov equations, in KdV '95, eds. M. Hazewinkel et al., Kluwer Acad., Dordrecht, 1995, pp. 405–433.Google Scholar
  55. [LM] Y. Li and M. Mulase, The Hitchin system and the KP equations, preprint 1994.Google Scholar
  56. [M] E. Markman, Spectral curves and integrable systems, Compositio Math. 93 (1994), 255–290.MathSciNetzbMATHGoogle Scholar
  57. [Mo] I.O. Mokhov, Commuting differential operators of rank 3 and nonlinear differential equations, Math. USSR-Izv. 35 (1990), 629–655.MathSciNetCrossRefzbMATHGoogle Scholar
  58. [Mos] J. Moser: I. Three integrable hamiltonian systems connected with isospectral deformations, Adv. Math. 16 (1975), 197–220; II. Geometry of quadrics and spectral theory, The Chern Symposium, Springer-Verlag, 1980, pp. 147–188.MathSciNetCrossRefzbMATHGoogle Scholar
  59. [M1] D. Mumford, Prym varieties I, in Contributions to Analysis, ed. L.V. Ahlfors, AcadOA. Press New York, 1974, pp. 325–350.Google Scholar
  60. [M2] D. Mumford, An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg de Vries equation and related non-linear equations. Intl. Sympos. on Algebraic Geometry, Kyoto (1977), 115–153.Google Scholar
  61. [M3] D. Mumford, Tata lectures on theta II, Progr. Math. 43, Birkhäuser, Boston, 1984.zbMATHGoogle Scholar
  62. [MN] D. Mumford and P. Newstead, Periods of a moduli space of bundles on curves, Amer. J. Math. 90 (1968), 1200–1208.MathSciNetCrossRefzbMATHGoogle Scholar
  63. [NR1] M.S. Narasimhan and S. Ramanan, Generalised Prym varieties and fixed points, J. Indian Math. Soc. 39 (1975), 1–19.MathSciNetzbMATHGoogle Scholar
  64. [NR2] M.S. Narasimhan and S. Ramanan, 2− linear systems on Abelian varieties, Vector bundles on algebraic varieties, Oxford Univ. Press 1987, pp. 415–427.Google Scholar
  65. [N] P. Newstead, Introduction to moduli problems and orbit spaces, TIFR, Bombay, 1978.zbMATHGoogle Scholar
  66. [OP] M.A. Olshanetsky and A.M. Perelomov, Classical Integrable finite-dimensional systems related to Lie algebras, Phys. Rep. 71 (1981), 313–400.MathSciNetCrossRefGoogle Scholar
  67. [O] W.M. Oxbury, Anticanonical Verlinde spaces as theta spaces on Pryms, preprint 1994.Google Scholar
  68. [PR] K. Paranjape and S. Ramanan, On the canonical ring of a curve, Nagata volume, 1987, pp. 503–516.Google Scholar
  69. [PS] A. Pressley and G.B. Segal, Loop Groups, Oxford 1986.Google Scholar
  70. [P1] E. Previato, Generalized Weierstrass ℘-functions and KP flows in affine space, Comment. Math. Helvetici 62 (1987), 292–310.MathSciNetCrossRefzbMATHGoogle Scholar
  71. [P2] E. Previato, Three questions on vector bundles, in Complex Analysis and Geometry, eds. V. Ancona and A. Silva, Plenum Press 1993, pp. 395–398.Google Scholar
  72. [P3] E. Previato, Burchnall-Chaundy bundles, to appear, Proc. Europroj'94, Barcelona.Google Scholar
  73. [PV] E. Previato and J.-L. Verdier, Boussinesq elliptic solitons: the cyclic case, in Proc. Indo-French Conference on Geometry (Bombay, 1989), Hindustan Book Agency, Delhi 1993, pp. 173–185.zbMATHGoogle Scholar
  74. [PW1] E. Previato and G. Wilson, Vector bundles over curves and solutions of the KP equations, Proc. Sympos. Pure Math. 49 (1989), 553–569.MathSciNetCrossRefzbMATHGoogle Scholar
  75. [PW2] E. Previato and G. Wilson, Differential operators and rank two bundles over elliptic curves, Compositio Math. 81 (1992), 107–119.MathSciNetzbMATHGoogle Scholar
  76. [RS] A. Reymann, M. Semenov-Tian-Shansky: Reduction of Hamiltonian systems, affine Lie algebras, and Lax equations I and II, Invent. Math. 54 (1979), 81–100 and 63 (1981), 423–432.MathSciNetCrossRefzbMATHGoogle Scholar
  77. [SM] R. Salvati Manni, Modular varieties with level 2 theta structure, Amer. J. Math. 116 (1994), 1489–1511.MathSciNetCrossRefzbMATHGoogle Scholar
  78. [S] R. Schilling: Generalizations of the Neumann system: a curve theoretical approach I, II and III, Comm. Pure Appl. Math. 40 (1987), 455–522; 42 (1989), 409–422; and 45 (1992), 775–820.MathSciNetCrossRefzbMATHGoogle Scholar
  79. [SW] G.B. Segal and G. Wilson, Loop groups and equations of KdV type, Publ. Math. Inst. Hautes Études Sci. 61 (1985), 5–65.MathSciNetCrossRefzbMATHGoogle Scholar
  80. [SSY] S.I. Svinolupov, V. Sokolov and R. Yamilov, On Bäcklund transformations for integrable evolution equations, Soviet Math. Dokl. 28 (1983), 165–168.zbMATHGoogle Scholar
  81. [Sy] W.W. Symes, Systems of Toda type, inverse spectral problems, and representation theory, Invent. Math. 59 (1980), 13–51.MathSciNetCrossRefzbMATHGoogle Scholar
  82. [Tr] A. Treibich, Tangential polynomials and elliptic solitons, Duke Math. J. 59 (1989), 611–627.MathSciNetCrossRefzbMATHGoogle Scholar
  83. [TV1] A. Treibich and J.-L. Verdier, Solitons Elliptiques, The Grothendieck Festschrift III, Birkhäuser 1990, pp. 437–480.Google Scholar
  84. [TV2] A. Treibich and J.-L. Verdier, Variétés de Kritchever des solitons elliptiques de KP, Proc. Indo-French Conf. on Geometry (Bombay, 1989), Hindustan Book Agency, Delhi, 1993, pp. 187–232.zbMATHGoogle Scholar
  85. [TV3] A. Treibich and J.-L. Verdier, Revêtements tangentiels et sommes de 4 nombres triangulaires, C.R. Acad. Sci. Paris Sér. I Math., 311 (1990), 51–54.MathSciNetzbMATHGoogle Scholar
  86. [Tu] L. Tu, Semistable bundles over an elliptic curve, Adv. Math. 98 (1993), 1–26.MathSciNetCrossRefzbMATHGoogle Scholar
  87. [V] V.S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Springer 1984.Google Scholar
  88. [Ve] J.-L. Verdier, Équations différentielles algebriques, Lecture Notes in Math. 710, Springer 1979, pp. 101–122.MathSciNetCrossRefzbMATHGoogle Scholar
  89. [W] G. Wilson, Algebraic curves and soliton equations, in Geometry Today, eds. E. Arbarello et al., Birkhäuser, Boston, 1985, 303–329.Google Scholar
  90. [Wi] W. Wirtinger, Untersuchungen über Thetafunctionen, Teubner, Leipzig, 1895.zbMATHGoogle Scholar

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© Springer-Verlag 1996

Authors and Affiliations

  1. 1.Boston UniversityBoston

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