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Abelian varieties attached to cycles of intermediate dimension

Published online by Cambridge University Press:  22 January 2016

Hiroshi Saito
Hiroshi Saito*
Affiliation:
Department of Mathematics, Nagoya University, Nagoya, Japan 464
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The group of cycles of codimension one algebraically equivalent to zero of a nonsingular projective variety modulo rational equivalence forms an abelian variety, i.e., the Picard variety. To the group of cycles of dimension zero and of degree zero, there corresponds an abelian variety, the Albanese variety. Similarly, Weil, Lieberman and Griffiths have attached complex tori to the cycles of intermediate dimension in the classical case. The aim of this article is to give a purely algebraic construction of such “intermediate Jacobian varieties.”

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 75 , September 1979 , pp. 95 - 119
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1979

References

[1] Andreotti, A. and Frankel, T., The Lefschetz theorem on hyperplane sections, Ann. of Math., vol. 69 (1959), p. 713.Google Scholar
[2] Beaville, A., Varites de Prym et Jacobiennes intermediaires, Ann. Scient. de ENS, vol. 10 (1977), p. 309.Google Scholar
[3] Bloch, S., An Example in the Theory of Algebraic Cycles, Lecture Note in Math., vol. 551.Google Scholar
[4] Bloch, S., Some elementary theorems about algebraic cycles on an abelian varieties, Inv. Math., vol. 37 (1975), p. 215.Google Scholar
[5] Bloch, S., Kas, A. and Lieberman, D., Zero cycles on surface with pg = 0, Compositio Math., vol. 33 (1976), p. 215.Google Scholar
[6] Chevalley, C., Sur la theorie de la variete de Picard, Amer. J. Math., vol. 82 (1960), p. 433.Google Scholar
[7] Griffiths, P. A., Periods of integrals on algebraic manifolds, I and II, Amer. J. Math., vol. 90 (1968), p. 568 and p. 805.Google Scholar
[8] Griffiths, P. A., Some transcendental methods in the study of algebraic cycles, Lecture Note in Math., vol. 185.Google Scholar
[9] Kleiman, S., Algebraic cycles and the Weil conjectures, in Dix expose sur la cohomologie de schemas, p. 359.Google Scholar
[10] Kleiman, S., Finiteness Theorems for algebraic cycles, Actes Congres Intern. Math. 1970, Tome 1, p. 445.Google Scholar
[11] Lang, S., Abelian Varieties, Interscience Publ. Inc. Google Scholar
[12] Lieberman, D., Higher Picard Varieties, Amer. J. Math., vol. 90 (1968), p. 1165.Google Scholar
[13] Lieberman, D., Intermediate Jacobians, in Algebraic Geometry Oslo 1970, p. 125.Google Scholar
[14] Mattuck, A., Algebraic cycles on abelian varieties, Proceedings of the AMS, vol. 9 (1958), p. 88.Google Scholar
[15] Morikawa, H., Cycles and endomorphisms of Abelian Varieties, Nagoya Math. J., vol. 7 (1954), p. 95.Google Scholar
[16] Mumford, D., Rational equivalence of zero cycles on Surface, J. Math. Kyoto Univ., vol. 9 (1969), p. 195.Google Scholar
[17] Murre, J. P., Algebraic equivalence modulo rational equivalence on a cubic threefold, Compositio Math., vol. 25 (1972), p. 161.Google Scholar
[18] Samuel, P., Relations d’equivalence en geometrie algebrique, Proc. Intern. Cong. Math., Edinbourg (1958), p. 470.Google Scholar
[19] Chevalley, Seminaire C., 2e annee 1958, Anneaux de Chow et applecations, Secret. Math. 1958.Google Scholar
[20] Chevalley, Seminaire C., 3e annne 1958/59, Varietes de Picard, Secret. Math. 1960.Google Scholar
[21] Weil, A., Sur les criteres d’equivalence en geometrie algebrique, Math. Ann., vol. 128 (1954), p. 95.CrossRefGoogle Scholar
[22] Griffiths, P., and Schmid, W., Recent development in Hodge theory: a discussion of techniques and results in Discrete subgroups of Lie groups and applications to moduli, p. 31, Tata Institute of Fundamental Research. Studies in mathematics, 7, Oxford, Oxford University Press, 1975.Google Scholar