Spectral covers, algebraically completely integrable, hamiltonian systems, and moduli of bundles

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


Modulus Space Vector Bundle Line Bundle Abelian Variety Poisson Structure 
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  1. [AB] Adams, M. R., Bergvelt, M. J.: The Krichever map, vector bundles over algebraic curves, and heisenberg algebras, Comm. Math. Phys. 154 (1993), 265–305.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [AdC] Arbarello, E., De Concini, C.: Another proof of a conjecture of S. P. Novikov on periods and abelian integrals on Riemann surfaces, Duke Math. J. 54 (1987) 163–178.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [AG] Arnol'd, V. I., Givental', A. B.: Symplectic Geometry. In: Arnol'd, V. I., Novikov, S. P. (eds.), Dynamical systems IV, (EMS, vol. 4, pp. 1–136) Berlin Heidelberg New York: Springer 1988Google Scholar
  4. [AHH] Adams, M. R., Harnad, J., Hurtubise, J.: Isospectral Hamiltonian flows in finite and infinite dimensions, II. Integration of flows. Commun. Math. Phys. 134, 555–585 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [AIK] Altman, A. Iarrobino, A., Kleiman, S.: Irreducibility of the Compactified Jacobian. Proc. Nordic Summer School, 1–12 (1976)Google Scholar
  6. [AK1] Altman, A. B., Kleiman S. L.: Foundation of the theory of Fano schemes, Comp. Math. Vol. 34, Fasc. 1, 1977, pp. 3–47MathSciNetzbMATHGoogle Scholar
  7. [AK2] Altman, A. B., Kleiman S. L.: Compactifying the Picard Scheme, Advances in Math. 35, 50–112 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [At] Atiyah, M.: Vector bundles over an elliptic curve, Proc. Lond. Math. Soc. 7, 414–452 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [AvM] Adler, M., van Moerbeke, P.: Completely integrable systems, Euclidean Lie algebras, and curves, Advances in Math. 38, (1980), 267–379.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [B1] Beauville, A.: Beauville, A.: Jacobiennes des courbes spectrales et systèmes hamiltoniens complètement intégrables. Acta Math. 164, 211–235 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [B2] Beauville, A.: Varietes Kähleriennes dont la premiere classe de Chern est nulle. J. Diff. Geom. 18, p. 755–782 (1983).MathSciNetzbMATHGoogle Scholar
  12. [Bi] Biswas, I.: A remark on a deformation theory of Green and Lazarsfeld, J. reine angew. Math. 449 (1994) 103–124.MathSciNetzbMATHGoogle Scholar
  13. [Bo] Bogomolov, F.: Hamiltonian Kähler manifolds, Soviet Math Dokl. 19 (1978), 1462–1465.zbMATHGoogle Scholar
  14. [Bn] Bottacin, F.: Thesis, Orsay, 1992.Google Scholar
  15. [BD] Beauville, A., Donagi, R.: La variété des droites d'une hypersurface cubique de dimension 4. C. R. Acad. Sci. Paris Ser. I t. 301, 703–706 (1985)MathSciNetzbMATHGoogle Scholar
  16. [BG] Bryant, R., Griffiths, P.: Some observations on the infinitestimal period relations for regular threefolds with trivial canonical bundle, in Arithmetic and Geometry, Papers dedicated to I.R. Shafarevich, vol. II, Birkhäuser, Boston (1983), 77–102.Google Scholar
  17. [BK] Beilinson, A., Kazhdan, D.: Flat Projective Connections, unpublished(1990).Google Scholar
  18. [BL] Beauville, A., Laszlo, Y.: Conformal blocks and generalized theta functions, Comm. Math. Phys., to appear.Google Scholar
  19. [BNR] Beauville, A., Narasimhan, M. S., Ramanan, S.: Spectral curves and the generalized theta divisor, J. Reine Angew. Math. 398, 169–179 (1989)MathSciNetzbMATHGoogle Scholar
  20. [CdOGP] Candelas, P., de la Ossa, X. C. Green, P. S., and Parkes L.: A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Phys. Lett. B 258 (1991), 118–126; Nuclear Phys. B 359 (1991), 21–74.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [CG] Clemens, H., Griffiths, P.: The intermediate Jacobian of the cubic threefold. Annals of Math. 95 281–356 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [De1] Deligne P.: Theoreme de Lefschetz et criteres de degenerescence de suites spectrales, Publ. Math. IHES, 35(1968) 107–126.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [De2] Deligne, P.: letter to C. Simpson, March 20, 1989Google Scholar
  24. [D1] Donagi, R.: The tetragonal construction, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 181–185.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [D2] Donagi, R.: Decomposition of spectral covers, in: Journees de Geometrie Algebrique D'Orsay, Asterisque 218 (1993), 145–175.Google Scholar
  26. [D3] Donagi, R.: Abelianization of Higgs bundles, preprint.Google Scholar
  27. [D4] Donagi, R.: Spectral covers, To appear in: Proceedings of special year in algebraic geometry 1992–1993 Mathematical Sciences Research Insitute.Google Scholar
  28. [D5] Donagi, R.: Group law on the intersection of two quadrics. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 2, 217–239.MathSciNetzbMATHGoogle Scholar
  29. [DEL] Donagi, R., Ein, L. and Lazarsfeld, R.: A non-linear deformation of the Hitchin dynamical system, preprint, (Alg. Geom. eprint no. 9504017).Google Scholar
  30. [DM] Donagi, R. and Markman E.: Cubics, integrable systems, and Calabi-Yau three-folds, to appear in the proceedings of the Algebraic Geometry Workshop on the occasion of the 65th birthday of F. Hirzebruch, May 1993. (Alg. Geom. eprint no. 9408004).Google Scholar
  31. [DR] Desale, U. V., Ramanan, S.: Classification of Vector Bundles of Rank 2 on Hyperelliptic Curves. Invent. Math. 38, 161–185 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  32. [EZ] El Zein F. and Zucker S.: Extendability of normal functions associated to algebraic cycles. In Topics in transcendental Algebraic Geometry, ed. by P. Griffiths, Annals of MathematicsGoogle Scholar
  33. [F1] Faltings, G.: Stable G-bundles and Projective Connections, Jour. Alg. Geo 2 (1993), 507–568.MathSciNetzbMATHGoogle Scholar
  34. [F2] Faltings, G.: A proof of the Verlinde formula, to appear in Jour. Alg. Geo.Google Scholar
  35. [Gr] Griffiths, P. A.: Infinitesimal variations of Hodge structure III: determinantal varieties and the infinitesimal invariant of normal functions. Compositio Math. 50 (1983), 267–324.MathSciNetzbMATHGoogle Scholar
  36. [GH] Griffiths, P. A., Harris, J.: Principles of algebraic geometry. Pure and Applied Mathematics. Wiley-Interscience 1978Google Scholar
  37. [Gro] Grothendieck, A.: Techniques de construction et théorèmes d'existence en géométrie algébrique, IV: Les schémas de Hilbert. Séminaire Bourbaki 13e année (1960/61) Exposés 221Google Scholar
  38. [HS] Hilton P., Stammbach U.: A course in homological algebra, Graduate Texts in Mathematics, Vol. 4, Springer-Verlag, New York-BerlinGoogle Scholar
  39. [H1] Hitchin, N.J.: The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59–126.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [H2] Hitchin, N.J.: Stable bundles and integrable systems. Duke Math. J. 54, No 1 91–114 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  41. [H3] Hitchin, N.J.: Flat Connections and Geometric Quantization. Comm. Math. Phys. 131, 347–380 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  42. [Hur] Hurtubise J.: Integrable systems and Algebraic Surfaces, Preprint 1994Google Scholar
  43. [IVHS] Carlson J., Green, M., Griffiths, P., and Harris, J.: Infinitesimal variations of Hodge structure (I). Compositio Math. 50 (1983), 109–205.MathSciNetzbMATHGoogle Scholar
  44. [K] Kanev, V.: Spectral curves, simple Lie algebras and Prym-Tjurin varieties, Proc. Symp. Pure Math. 49 (1989), Part I, 627–645.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [Ka] Kawamata, Y.: Unobstructed deformations-a remark on a paper of Z. Ran J. Alg. Geom. 1 (1992) 183–190MathSciNetzbMATHGoogle Scholar
  46. [Kn] Knörrer, H.: Geodesics on the Ellipsoid. Invent. Math. 59, 119–143 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  47. [Ko] Kobayashi, S.: Simple vector bundles over symplectic Kähler manifolds. Proc. Japan Acad. Ser A Math. Sci. 62, 21–24 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  48. [Ko-P] Kouvidakis, A., Pantev, T.: automorphisms of the moduli spaces of stable bundles on a curve., Math. Ann., to appear.Google Scholar
  49. [KP1] Katzarkov, L., Pantev, T.: Stable G 2 bundles and algebrically completely integrable systems, Comp. Math. 92 (1994), 43–60.MathSciNetzbMATHGoogle Scholar
  50. [KP2] Katzarkov, L., Pantev, T.: Representations of Fundamental groups whose Higgs bundles are pullbacks, J. Diff. Geo. 39 (1994), 103–121.MathSciNetzbMATHGoogle Scholar
  51. Laumon G.: Un analogue global du cône nilpotent, Duke Math. J. 57 (1988), 647–671.MathSciNetCrossRefzbMATHGoogle Scholar
  52. [LM1] Li, Y., Mulase, M.: Category of morphisms of algebraic curves and a characterization of Prym varieties. PreprintGoogle Scholar
  53. [LM2] Li, Y., Mulase, M.: Hitchin systems and KP equations. PreprintGoogle Scholar
  54. [Ma1] Markman, E.: Spectral curves and integrable systems, Compositio Math. 93, 255–290, (1994).MathSciNetzbMATHGoogle Scholar
  55. [Ma2] Markman, E.: In preparation.Google Scholar
  56. [Me] Merindol, J. Y.: Varietes de Prym d'un Revetement Galoisien, prep. (1993).Google Scholar
  57. [Mor] Morrison, D.: Mirror Symmetry and Rational Curves on Quintic Threefolds: A Guide for Mathematicians, J. Am. Math. Soc. 6 (1993), 223.MathSciNetCrossRefzbMATHGoogle Scholar
  58. [Mo] Moser, J.: Various aspects of integrable Hamiltonian systems, in: Dynamical Systems 1978, Prog. Math. Vol 8, 233–290.Google Scholar
  59. [MS] McDaniel, A., Smolinsky, L.: A Lie theoretic Galois Theory for the spectral curves of an integrable system II, prep. (1994).Google Scholar
  60. [Mu1] Mukai S.: Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. math. 77, 101–116 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  61. [Mu2] Mukai S: On the moduli space of bundles on K3 surfaces I, Vector bundles on algebraic varieties, Proc. Bombay Conference, 1984, Tata Institute of Fundamental Research Studies, no. 11, Oxford University Press, 1987, pp. 341–413.Google Scholar
  62. [Mul] Mulase, M.: Cohomological structure in soliton equations and Jacobian varieties, J. Diff. Geom. 19 (1984), 403–430.MathSciNetzbMATHGoogle Scholar
  63. [Mum1] Mumford, D.: Lectures on curves on an algebraic surface, Princeton University Press, 1966.Google Scholar
  64. [Mum2] Mumford, D.: Tata Lectures on Theta II, Birkhaeuser-Verlag, Basel, Switzerland, and Cambridge, MA 1984zbMATHGoogle Scholar
  65. [Mum3] Mumford, D.: Prym varieties I, in: Contribution to analysis, Acad. Press (1974), 325–350.Google Scholar
  66. [Nit] Nitsure, N.: Moduli space of semistable pairs on a curve., Proc. London Math. Soc. (3) 62 (1991) 275–300MathSciNetCrossRefzbMATHGoogle Scholar
  67. [NR] Narasimhan, M. S., Ramanan, M. S.: 20-linear systems on abelian varieties in Vector bundles and algebraic varieties, Oxford University Press, 415–427 (1987)Google Scholar
  68. [Ra] Ran, Ziv.: Lifting of cohomology and unobstructedness of certain holomorphic maps. Bull. Amer. Math. Soc. (N.S.), 26 (1992), no. 1, 113–117.MathSciNetCrossRefzbMATHGoogle Scholar
  69. [Re] Reid, M.: The complete intersection of two or more quadrics, Thesis, Cambridge (GB) 1972.Google Scholar
  70. [Rec] Recillas, S.: Jacobians of curves with g 41's are Pryms of trigonal curves, Bol. Soc. Mat. Mexicana 19 (1974) no. 1.Google Scholar
  71. [Se] Seshadri, C. S.: Fibrés vectoriels sur les courbes algébriques. asterisque 96 1982Google Scholar
  72. [Sc] Scognamillo, R.: Prym-Tjurin Varieties and the Hitchin Map, preprint (1993).Google Scholar
  73. [Sch] Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22 (1973), 211–319.MathSciNetCrossRefzbMATHGoogle Scholar
  74. [Sh] Shiota, T.: Characterization of Jacobian varieties in terms of soliton equations, Inv. Math. 83 (1986) 333–382.MathSciNetCrossRefzbMATHGoogle Scholar
  75. [Sim1] Simpson, C.: Moduli of representations of the fundamental group of a smooth projective variety I, II. PreprintGoogle Scholar
  76. [Sim2] Simpson, C.: Higgs bundles and local systems. Inst. Hautes Etudes Sci. Publ. Math. (1992), No. 75 5–95.MathSciNetCrossRefzbMATHGoogle Scholar
  77. [Sim3] Simpson, C.: Nonabelian Hodge Theory, Proceedings of the International Congress of Mathematics, Kyoto, Japan, 1990.Google Scholar
  78. [SW] Segal, G., Wilson, G.: Loop groups and equations of KdV type. Inst. Hautes Etudes Sci. Publ. Math. (1985), No. 61 5–65.MathSciNetCrossRefzbMATHGoogle Scholar
  79. [Ti] Tian, G.: Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric, in Mathematical Aspects of String Theory (S.T. Yau, ed.), World Scientific, Singapore (1987), 629–646.CrossRefGoogle Scholar
  80. [To] Todorov, A.: The Weil-Petersson geometry of the moduli space of SU(n≥3) (Calabi-Yau) manifolds, I, Comm. Math. Phys. 126 (1989), 325–346.MathSciNetCrossRefzbMATHGoogle Scholar
  81. [TV1] Treibich, A., J.-L. Verdier, Solitons elliptiques. The Grothendieck Festschrift, Vol. III, 437–480, Birkhauser, Boston, 1990.Google Scholar
  82. [TV2] Treibich, A., J.-L. Verdier Varietes de Kritchever des solitons ellptiques de KP. Proceedings of the Indo-French Conference on Geometry (Bombay, 1989), 187–232, 1993.Google Scholar
  83. [Ty1] Tyurin, A. N.: Symplectic structures on the varieties of moduli of vector bundles on algebraic surfaces with p g >0 Math. USSR Izvestiya Vol. 33 (1989), No. 1Google Scholar
  84. [Ty2] Tyurin, A. N.: The geometry of moduli of vector bundles. Russian Math. Surveys 29:6 57–88 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  85. [V] Voisin, C.: Sur la stabilite des sous-varietes lagrangiennes des varietes symplectiques holomorphes. Complex projective geometry (Trieste, 1989/Bergen, 1989), 294–303, Cambridge Univ. Press, Cambridge, 1992.zbMATHGoogle Scholar
  86. [We] Weinstein, A.: The Local Structure of Poisson Manifolds. J. Diff. Geom. 18, 523–557 (1983)MathSciNetzbMATHGoogle Scholar
  87. [Ye] Ye, Yun-Gang: Lagrangian subvarieties of the moduli space of stable vector bundles on a regular algebraic surface with p g >0. Math. Ann. 295, 411–425 (1993)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag 1996

Authors and Affiliations

  1. 1.University of PennsylvaniaPennsylvaniaUSA
  2. 2.University of MichiganUSA

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