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

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Integrable Systems and Quantum Groups

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References

  • [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.

    Article  MathSciNet  MATH  Google Scholar 

  • [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.

    Article  MathSciNet  MATH  Google Scholar 

  • [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 1988

    Google Scholar 

  • [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)

    Article  MathSciNet  MATH  Google Scholar 

  • [AIK] Altman, A. Iarrobino, A., Kleiman, S.: Irreducibility of the Compactified Jacobian. Proc. Nordic Summer School, 1–12 (1976)

    Google Scholar 

  • [AK1] Altman, A. B., Kleiman S. L.: Foundation of the theory of Fano schemes, Comp. Math. Vol. 34, Fasc. 1, 1977, pp. 3–47

    MathSciNet  MATH  Google Scholar 

  • [AK2] Altman, A. B., Kleiman S. L.: Compactifying the Picard Scheme, Advances in Math. 35, 50–112 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  • [At] Atiyah, M.: Vector bundles over an elliptic curve, Proc. Lond. Math. Soc. 7, 414–452 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  • [AvM] Adler, M., van Moerbeke, P.: Completely integrable systems, Euclidean Lie algebras, and curves, Advances in Math. 38, (1980), 267–379.

    Article  MathSciNet  MATH  Google Scholar 

  • [B1] Beauville, A.: Beauville, A.: Jacobiennes des courbes spectrales et systèmes hamiltoniens complètement intégrables. Acta Math. 164, 211–235 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • [B2] Beauville, A.: Varietes Kähleriennes dont la premiere classe de Chern est nulle. J. Diff. Geom. 18, p. 755–782 (1983).

    MathSciNet  MATH  Google Scholar 

  • [Bi] Biswas, I.: A remark on a deformation theory of Green and Lazarsfeld, J. reine angew. Math. 449 (1994) 103–124.

    MathSciNet  MATH  Google Scholar 

  • [Bo] Bogomolov, F.: Hamiltonian Kähler manifolds, Soviet Math Dokl. 19 (1978), 1462–1465.

    MATH  Google Scholar 

  • [Bn] Bottacin, F.: Thesis, Orsay, 1992.

    Google Scholar 

  • [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)

    MathSciNet  MATH  Google Scholar 

  • [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 

  • [BK] Beilinson, A., Kazhdan, D.: Flat Projective Connections, unpublished(1990).

    Google Scholar 

  • [BL] Beauville, A., Laszlo, Y.: Conformal blocks and generalized theta functions, Comm. Math. Phys., to appear.

    Google Scholar 

  • [BNR] Beauville, A., Narasimhan, M. S., Ramanan, S.: Spectral curves and the generalized theta divisor, J. Reine Angew. Math. 398, 169–179 (1989)

    MathSciNet  MATH  Google Scholar 

  • [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.

    Article  MathSciNet  MATH  Google Scholar 

  • [CG] Clemens, H., Griffiths, P.: The intermediate Jacobian of the cubic threefold. Annals of Math. 95 281–356 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  • [De1] Deligne P.: Theoreme de Lefschetz et criteres de degenerescence de suites spectrales, Publ. Math. IHES, 35(1968) 107–126.

    Article  MathSciNet  MATH  Google Scholar 

  • [De2] Deligne, P.: letter to C. Simpson, March 20, 1989

    Google Scholar 

  • [D1] Donagi, R.: The tetragonal construction, Bull. Amer. Math. Soc. (N.S.) 4 (1981), 181–185.

    Article  MathSciNet  MATH  Google Scholar 

  • [D2] Donagi, R.: Decomposition of spectral covers, in: Journees de Geometrie Algebrique D'Orsay, Asterisque 218 (1993), 145–175.

    Google Scholar 

  • [D3] Donagi, R.: Abelianization of Higgs bundles, preprint.

    Google Scholar 

  • [D4] Donagi, R.: Spectral covers, To appear in: Proceedings of special year in algebraic geometry 1992–1993 Mathematical Sciences Research Insitute.

    Google Scholar 

  • [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.

    MathSciNet  MATH  Google Scholar 

  • [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 

  • [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 

  • [DR] Desale, U. V., Ramanan, S.: Classification of Vector Bundles of Rank 2 on Hyperelliptic Curves. Invent. Math. 38, 161–185 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  • [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 Mathematics

    Google Scholar 

  • [F1] Faltings, G.: Stable G-bundles and Projective Connections, Jour. Alg. Geo 2 (1993), 507–568.

    MathSciNet  MATH  Google Scholar 

  • [F2] Faltings, G.: A proof of the Verlinde formula, to appear in Jour. Alg. Geo.

    Google Scholar 

  • [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.

    MathSciNet  MATH  Google Scholar 

  • [GH] Griffiths, P. A., Harris, J.: Principles of algebraic geometry. Pure and Applied Mathematics. Wiley-Interscience 1978

    Google Scholar 

  • [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 221

    Google Scholar 

  • [HS] Hilton P., Stammbach U.: A course in homological algebra, Graduate Texts in Mathematics, Vol. 4, Springer-Verlag, New York-Berlin

    Google Scholar 

  • [H1] Hitchin, N.J.: The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59–126.

    Article  MathSciNet  MATH  Google Scholar 

  • [H2] Hitchin, N.J.: Stable bundles and integrable systems. Duke Math. J. 54, No 1 91–114 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  • [H3] Hitchin, N.J.: Flat Connections and Geometric Quantization. Comm. Math. Phys. 131, 347–380 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  • [Hur] Hurtubise J.: Integrable systems and Algebraic Surfaces, Preprint 1994

    Google Scholar 

  • [IVHS] Carlson J., Green, M., Griffiths, P., and Harris, J.: Infinitesimal variations of Hodge structure (I). Compositio Math. 50 (1983), 109–205.

    MathSciNet  MATH  Google Scholar 

  • [K] Kanev, V.: Spectral curves, simple Lie algebras and Prym-Tjurin varieties, Proc. Symp. Pure Math. 49 (1989), Part I, 627–645.

    Article  MathSciNet  MATH  Google Scholar 

  • [Ka] Kawamata, Y.: Unobstructed deformations-a remark on a paper of Z. Ran J. Alg. Geom. 1 (1992) 183–190

    MathSciNet  MATH  Google Scholar 

  • [Kn] Knörrer, H.: Geodesics on the Ellipsoid. Invent. Math. 59, 119–143 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  • [Ko] Kobayashi, S.: Simple vector bundles over symplectic Kähler manifolds. Proc. Japan Acad. Ser A Math. Sci. 62, 21–24 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  • [Ko-P] Kouvidakis, A., Pantev, T.: automorphisms of the moduli spaces of stable bundles on a curve., Math. Ann., to appear.

    Google Scholar 

  • [KP1] Katzarkov, L., Pantev, T.: Stable G 2 bundles and algebrically completely integrable systems, Comp. Math. 92 (1994), 43–60.

    MathSciNet  MATH  Google Scholar 

  • [KP2] Katzarkov, L., Pantev, T.: Representations of Fundamental groups whose Higgs bundles are pullbacks, J. Diff. Geo. 39 (1994), 103–121.

    MathSciNet  MATH  Google Scholar 

  • Laumon G.: Un analogue global du cône nilpotent, Duke Math. J. 57 (1988), 647–671.

    Article  MathSciNet  MATH  Google Scholar 

  • [LM1] Li, Y., Mulase, M.: Category of morphisms of algebraic curves and a characterization of Prym varieties. Preprint

    Google Scholar 

  • [LM2] Li, Y., Mulase, M.: Hitchin systems and KP equations. Preprint

    Google Scholar 

  • [Ma1] Markman, E.: Spectral curves and integrable systems, Compositio Math. 93, 255–290, (1994).

    MathSciNet  MATH  Google Scholar 

  • [Ma2] Markman, E.: In preparation.

    Google Scholar 

  • [Me] Merindol, J. Y.: Varietes de Prym d'un Revetement Galoisien, prep. (1993).

    Google Scholar 

  • [Mor] Morrison, D.: Mirror Symmetry and Rational Curves on Quintic Threefolds: A Guide for Mathematicians, J. Am. Math. Soc. 6 (1993), 223.

    Article  MathSciNet  MATH  Google Scholar 

  • [Mo] Moser, J.: Various aspects of integrable Hamiltonian systems, in: Dynamical Systems 1978, Prog. Math. Vol 8, 233–290.

    Google Scholar 

  • [MS] McDaniel, A., Smolinsky, L.: A Lie theoretic Galois Theory for the spectral curves of an integrable system II, prep. (1994).

    Google Scholar 

  • [Mu1] Mukai S.: Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. math. 77, 101–116 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  • [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 

  • [Mul] Mulase, M.: Cohomological structure in soliton equations and Jacobian varieties, J. Diff. Geom. 19 (1984), 403–430.

    MathSciNet  MATH  Google Scholar 

  • [Mum1] Mumford, D.: Lectures on curves on an algebraic surface, Princeton University Press, 1966.

    Google Scholar 

  • [Mum2] Mumford, D.: Tata Lectures on Theta II, Birkhaeuser-Verlag, Basel, Switzerland, and Cambridge, MA 1984

    MATH  Google Scholar 

  • [Mum3] Mumford, D.: Prym varieties I, in: Contribution to analysis, Acad. Press (1974), 325–350.

    Google Scholar 

  • [Nit] Nitsure, N.: Moduli space of semistable pairs on a curve., Proc. London Math. Soc. (3) 62 (1991) 275–300

    Article  MathSciNet  MATH  Google Scholar 

  • [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 

  • [Ra] Ran, Ziv.: Lifting of cohomology and unobstructedness of certain holomorphic maps. Bull. Amer. Math. Soc. (N.S.), 26 (1992), no. 1, 113–117.

    Article  MathSciNet  MATH  Google Scholar 

  • [Re] Reid, M.: The complete intersection of two or more quadrics, Thesis, Cambridge (GB) 1972.

    Google Scholar 

  • [Rec] Recillas, S.: Jacobians of curves with g 14 's are Pryms of trigonal curves, Bol. Soc. Mat. Mexicana 19 (1974) no. 1.

    Google Scholar 

  • [Se] Seshadri, C. S.: Fibrés vectoriels sur les courbes algébriques. asterisque 96 1982

    Google Scholar 

  • [Sc] Scognamillo, R.: Prym-Tjurin Varieties and the Hitchin Map, preprint (1993).

    Google Scholar 

  • [Sch] Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22 (1973), 211–319.

    Article  MathSciNet  MATH  Google Scholar 

  • [Sh] Shiota, T.: Characterization of Jacobian varieties in terms of soliton equations, Inv. Math. 83 (1986) 333–382.

    Article  MathSciNet  MATH  Google Scholar 

  • [Sim1] Simpson, C.: Moduli of representations of the fundamental group of a smooth projective variety I, II. Preprint

    Google Scholar 

  • [Sim2] Simpson, C.: Higgs bundles and local systems. Inst. Hautes Etudes Sci. Publ. Math. (1992), No. 75 5–95.

    Article  MathSciNet  MATH  Google Scholar 

  • [Sim3] Simpson, C.: Nonabelian Hodge Theory, Proceedings of the International Congress of Mathematics, Kyoto, Japan, 1990.

    Google Scholar 

  • [SW] Segal, G., Wilson, G.: Loop groups and equations of KdV type. Inst. Hautes Etudes Sci. Publ. Math. (1985), No. 61 5–65.

    Article  MathSciNet  MATH  Google Scholar 

  • [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.

    Chapter  Google Scholar 

  • [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.

    Article  MathSciNet  MATH  Google Scholar 

  • [TV1] Treibich, A., J.-L. Verdier, Solitons elliptiques. The Grothendieck Festschrift, Vol. III, 437–480, Birkhauser, Boston, 1990.

    Google Scholar 

  • [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 

  • [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. 1

    Google Scholar 

  • [Ty2] Tyurin, A. N.: The geometry of moduli of vector bundles. Russian Math. Surveys 29:6 57–88 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  • [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.

    MATH  Google Scholar 

  • [We] Weinstein, A.: The Local Structure of Poisson Manifolds. J. Diff. Geom. 18, 523–557 (1983)

    MathSciNet  MATH  Google Scholar 

  • [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)

    Article  MathSciNet  MATH  Google Scholar 

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  1. University of Pennsylvania, Pennsylvania, USA

    Ron Donagi

  2. University of Michigan, Michigan, USA

    Eyal Markman

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  1. Ron Donagi
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Mauro Francaviglia Silvio Greco

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Donagi, R., Markman, E. (1996). Spectral covers, algebraically completely integrable, hamiltonian systems, and moduli of bundles. In: Francaviglia, M., Greco, S. (eds) Integrable Systems and Quantum Groups. Lecture Notes in Mathematics, vol 1620. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094792

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