Abstract
The structure of the group of 0-cycles modulo rational equivalence on ann-dimensional abelian varietyA over an algebraically closed fieldk is studied. This group forms an augmented ℤ-algebra under Pontryagin product, with augmentation given by the degree map. The (n+1)-st power of the augmentation idealI(=0-cycles of degree 0) is shown to be zero, while for suitablek (e.g.k=complex numbers) then-th power ofI is non-zero. As corollary, every 0-cycle of degree 0 is shown to be rationally equivalent to a sum of intersections of divisors. Partial results, analogous to the isogeny betweenA and Pic0 A, are proved relating quotientsI *r/I *r+1 to cycles of codimensionr onA.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bloch, S., Lieberman, D., Kas,A.: 0-cycles on algebraic surfaces withP g =0. To appear
Lang, S.: Abelian varieties. New York: Interscience-Wiley 1959
Mumford, D.: Rational equivalence of 0-cycles on surfaces. J. Math. Kyoto Univ.9, 195–204 (1968)
Roitman, A.A.: Γ-equivalence of zero-dimensional cycles. Mat. Sb.86 (128), 557–570 (1971)= Math. USSR Sb.15, 555–567 (1971)
Roitman, A.A.: Rational equivalence of zero-cycles. Math. USSR Sb.18 no. 4, 571–588 (1972)
Weil, A.: Variétés abéliennes et courbes algébriques. Paris Herman 1948
Author information
Authors and Affiliations
Additional information
Partially supported by the C.N.R.S. and by a Nato Fellowship
Rights and permissions
About this article
Cite this article
Bloch, S. Some elementary theorems about algebraic cycles on abelian varieties. Invent Math 37, 215–228 (1976). https://doi.org/10.1007/BF01390320
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01390320