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The Hodge cohomology and cubic equivalences

Published online by Cambridge University Press:  22 January 2016

Hiroshi Saito
Hiroshi Saito*
Affiliation:
Department of Mathematics, Nagoya University, Nagoya, 464, Japan
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In 1969, Mumford [8] proved that, for a complete non-singular algebraic surface F over the complex number field C, the dimension of the Chow group of zero-cycles on F is infinite if the geometric genus of F is positive. To this end, he defined a regular 2-form ηf on a non-singular variety S for a regular 2-form η on F and for a morphism f: S → SnF, where SnF is the 72-th symmetric product of F, and he showed that ηf vanishes if all 0-cycles f(s), s ∈ S, are rationally equivalent. Roitman [9] later generalized this to a higher dimensional smooth projective variety V.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 94 , June 1984 , pp. 1 - 41
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1984

References

[ 1 ] Bloch, S., Some elementary theorems about algebraic cycles on abelian varieties, Invent. Math., 37 (1975), 215–228.CrossRefGoogle Scholar
[ 2 ] Deligne, P. and Mumford, D., The irreducibility of the space of curves of given genus. IHES, Publ. Math. N° 36, 75–109.Google Scholar
[ 3 ] Griffiths, P. A., On the periods of certain rational integrals: I, II, Ann. of Math., 90 (1969), 460–495 and 496–541.Google Scholar
[ 4 ] Grothendieck, A., La théorie des classes de Chern, Bull. Soc. Math. France, 86 (1958), 137–154.Google Scholar
[ 5 ] Grothendieck, A., and Dieudonne, J., Eléments de géométrie algébrique, IV (Troisième Partie). IHES, Publ. Math. N° 28.Google Scholar
[ 6 ] Hartshorne, R., Residue and duality. Lecture Notes in Math. 20, Springer 1966.Google Scholar
[ 7 ] Katz, N., Le Théorème de Griffiths. Lecture Notes in Math. 340, p. 341. Springer 1973.Google Scholar
[ 8 ] Mumford, D., Rational equivalence of 0-cycles on surfaces, J. Math. Kyoto Univ., 9 (1969), 195–204.Google Scholar
[ 9 ] Roitman, A. A., Г-equivalence of zero-dimensional cycles, Mat. Sb., Tom 86 (128) (1971) 557–570 = Math. USSR Sb., 15 (1971), 555–567.Google Scholar
[10] Roitman, A. A., Rational equivalence of zero-cycles, Mat. Sb. Tom 89 (131) (1972), 567–585 = Math. USSR Sb., 18 (1972), 571–588.Google Scholar
[11] Samuel, P., Méthodes d’algèbre abstraite en géométrie algébrique. Ergebniss, No. 4. Berlin-Gõttingen-Heidelberg, Springer 1955.Google Scholar
[12] Samuel, P., Relations d’équivalence en géométrie algébrique. Proc. Int. Congress Math. 1958, New York, Cambridge Univ. Press 1960.Google Scholar
[13] Weil, A., Sur les critères d’équivalence en géométrie algébrique, Math. Ann., 128 (1954), 95–127.Google Scholar