Skip to main content
SpringerLink
Log in

The middle Jacobian of three-dimensional varieties

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

Abstract

The paper gives a survey of the most recent transcendental methods in the theory of three-dimensional algebraic manifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature cited

  1. I. G. Bashmakova, Diophantus and Diophantine Equations [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  2. A. Borel and J. P. Serre, “The Riemann-Roch theorem,” Matematika,5, No. 5 (1961).

  3. A. Weil, Introduction to the Theory of Kähler Manifolds [Russian translation], IL, Moscow (1961).

    Google Scholar 

  4. D. Hilbert and S. Cohn-Vossen, Descriptive Geometry [Russian translation], GITTL, Moscow (1951).

    Google Scholar 

  5. V. A. Iskovskikh and Yu. I. Manin, “Three-dimensional quartics and counterexamples to the problem of Lüroth,” Mat. Sb.,86, 140–166 (1971).

    Google Scholar 

  6. Yu. I. Main, Cubic Forms. Algebra, Geometry, Arithmetic [in Russian], Nauka, Moscow (1972).

    Google Scholar 

  7. I. I. Pyatetskii-Shapiro and I. R. Shafarevich, “On surfaces of type K3,” Izv. Akad. Nauk SSSR, Ser Mat.,34, No. 3, 234–263 (1971).

    Google Scholar 

  8. A. N. Rudakov and I. R. Shafarevich, “Nonseparable morphisms of algebraic surfaces,” Izv. Akad. Nauk SSSR, Ser. Mat.,40, No. 6, 1269–1307 (1976).

    Google Scholar 

  9. A. S. Tikhomirov, “The intermediate Jacobian of a single Fano body of index two,” Izv. Akad. Nauk SSSR, Ser. Mat.,42, No. 3 (1978).

  10. A. N. Tyurin, “On the Fano surface of the nonsingular surface in P4,” Izv. Akad. Nauk SSSR, Ser. Mat.,34, No. 6, 1200–1208 (1970).

    Google Scholar 

  11. A. N. Tyurin, “The geometry of the Fano surface of the nonsingular cubic F⊂P4 and Torelli's theorem,” Izv. Akad. Nauk SSSR, Ser. Mat.,35, 498–529 (1971).

    Google Scholar 

  12. A. N. Tyurin, “The geometry of the Poincaré divisor of a Prym manifold,” Izv. Akad. Nauk SSSR, Ser. Mat.,39, No. 5, 1003–1043 (1975).

    Google Scholar 

  13. A. N. Tyurin, “Five lectures on three-dimensional manifolds,” Usp. Mat. Nauk,27, No. 5, 3–50 (1972).

    Google Scholar 

  14. A. N. Tyurin, “The geometry of moduli of vector bundles,” Usp. Mat. Nauk,29, No. 6, 59–88 (1974).

    Google Scholar 

  15. A. N. Tyurin, “On the intersections of quartics,” Usp. Mat. Nauk,30, No. 6, 51–99 (1975).

    Google Scholar 

  16. F. Hirzebruch, Topological Methods of Algebraic Geometry [Russian translation], Mir, Moscow (1973).

    Google Scholar 

  17. S. Chern, Complex Manifolds [Russian translation], IL, Moscow (1961).

    Google Scholar 

  18. G. B. Shabat, “On the complex structure of regions covering algebraic surfaces,” Funkts. Analiz. Prilozhen.,11, No. 2, 67–76 (1977).

    Google Scholar 

  19. A. Altman and S. Kleiman, “Foundations of the theory of Fano schemes,” Preprint.

  20. A. Andreotti, “On a theorem of Torelli,” Am. J. Math.,80, 801–828 (1958).

    Google Scholar 

  21. A. Andreotti and A. Mayer, “On period relations for abelian integrals,” Ann. Scuola Norm. Sup., Pisa,21, No. 2, 189–238 (1967).

    Google Scholar 

  22. M. Artin and D. Mumford, “Some elementary examples of unirational varieties which are not rational,” Proc. London Math. Soc.,25, No. 1, 75–95 (1972).

    Google Scholar 

  23. A. Beauville, “Variétés de Prym et Jacobiennes intermédiaires,” Preprint.

  24. A. Beauville, “Prym varieties and Schottky problem,” Invent. Math.,41, Fasc 2, 149–196 (1977).

    Google Scholar 

  25. E. Bombieri and D. Mumford, “Enriques' classification of surfaces of char p. III,” Invent. Math.,35, 197–232 (1976).

    Google Scholar 

  26. R. Bott, “Homogeneous vector bundles,” Ann. Math.,66, No. 2, 203–248 (1975).

    Google Scholar 

  27. S. Chern, “On the characteristic classes of complex sphere bundles and algebraic varieties,” Am. J. Math.,75, No. 36, 565–596 (1953).

    Google Scholar 

  28. C. Clemens, “Applications of the theory of Prym varieties,” Proc. Int. Congr. Math. Vancouver, 1974, Vol. I, S. 1 (1975), pp. 415–421.

    Google Scholar 

  29. C. Clemens and Ph. Griffiths, “On the intermediate Jacobian of the cubic threefold,” Ann. Math.,95, No. 2, 281–356 (1972).

    Google Scholar 

  30. P. Delingne and D. Mumford, “The irreducibility of the space of curves of given genus,” Publication IHES, No. 35.

  31. H. Farkas, “On the Schottky relation and its generalization to arbitrary genus,” Ann. Math.,92, No. 1, 57–81 (1970).

    Google Scholar 

  32. Ph. Griffiths, “On periods of integrals on algebraic manifolds. I,” Am. J. Math.,90, 568–626 (1968).

    Google Scholar 

  33. Ph. Griffiths, “On the periods of integrals on algebraic manifolds. II,” Am. J. Math.,90, 805–865 (1968).

    Google Scholar 

  34. Ph. Griffiths, “Complex-analytic properties of certain Zariski open sets on algebraic varieties,” Ann. Math.,94, No. 1, 21–51 (1971).

    Google Scholar 

  35. A. Grothendieck, “Hodge's general conjecture is false for a trivial reason,” Topology,8, No. 3, 299–303 (1969).

    Google Scholar 

  36. A. Grothendieck, EGA, Publ. Math. IHES, 11 (1961).

  37. H. Hironaka, “Resolution of singularities of an algebraic variety over a field of characteristic zero. I–II,” Ann. Math.,79, 109–326 (1964).

    Google Scholar 

  38. W. Hoyt, “On products and algebraic families of Jacobian varieties,” Ann. Math.,77, No. 3, 415–423 (1963).

    Google Scholar 

  39. K. Kodaira, “A theorem on completeness of characteristic systems of complete continuous systems,” Am. J. Math.,81, 477 (1959).

    Google Scholar 

  40. K. Kodaira and D. Spencer, “On deformations of complex analytic structures. I,” Ann. Math.,67, No. 2, 328–401 (1958).

    Google Scholar 

  41. P. Lelong, “Integration sur un ensemble analytique complexe,” Bull. Soc. Math., France,85, 239–262 (1957).

    Google Scholar 

  42. J. Lewittes, “Riemann surfaces and the theta functions,” Acta Math.,111, Nos. 1–2, 37–61 (1964).

    Google Scholar 

  43. D. Lieberman, “Higher Picard varieties,” Ams. J. Math.,90, No. 4, 1165–1199 (1968).

    Google Scholar 

  44. D. Mumford, “Pathologies, I, II, III,” Am. J. Math.,89, No. 1, 94–104 (1967).

    Google Scholar 

  45. D. Mumford, Abelian Varieties. Tata Inst. Studies in Math., Oxford Univ. Press (1970).

  46. D. Mumford, “Prym varieties. I,” Contrib. Anal., New York-London (1974), pp. 325–350.

  47. D. Mumford, “Theta characteristics of an algebraic curve,” Ann. Sci. Ecole Norm. Sup.,4, No. 2, 181–192 (1971).

    Google Scholar 

  48. T. P. Murre, “Reduction of the proof of the nonrationality of the nonsingular cubic threefold to a result of Mumford,” Compos. Math.,27, No. 1, 63–82 (1973).

    Google Scholar 

  49. Problems of Mathematics, Princeton University, Bicentennial Conferences, 1946, Ser. 2, Conf. 2, Princeton, New Jersey.

  50. M. Rosenlicht, “Generalized Jacobian varieties,” Ann. Math.,59, No. 3, 505–530 (1954).

    Google Scholar 

  51. B. Segre, The Nonsingular Cubic Surfaces, Clarendon Press, Oxford (1942).

    Google Scholar 

  52. J. P. Serre, Cohomologie Galoisienne, Lect. Notes Math., No. 5 (1965).

  53. A. Weil, “On Picard varieties,” Am. J. Math.,75, 865–893 (1952).

    Google Scholar 

  54. S. Zucker, “Generalized intermediate Jacobians and the theorem on normal functions,” Invent. Math.,33, No. 3, 185–222 (1976).

    Google Scholar 

Download references

Authors
  1. A. N. Tyurin
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 12, pp. 5–57, 1979.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tyurin, A.N. The middle Jacobian of three-dimensional varieties. J Math Sci 13, 707–745 (1980). https://doi.org/10.1007/BF01084562

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01084562

Keywords

Search

Navigation