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Cycles on the generic abelian threefold. (English) Zbl 0725.14006
For a smooth projective variety X over \({\mathbb{C}}\) denote by \(R^ 2(X)\) the group of codimension two algebraic cycles homologically equivalent to zero modulo the subgroup of those cycles algebraically equivalent to zero. - In this paper it is proved that \(R^ 2(X)\otimes {\mathbb{Q}}\) is infinite dimensional when X is the generic abelian variety of dimension three. The proof of the theorem is based on a result of Ceresa producing a nonzero element in \(R^ 2(J(C))\otimes {\mathbb{Q}}\) where C is the generic curve of genus three.

14C25 Algebraic cycles
14J30 \(3\)-folds
32J17 Compact complex \(3\)-folds
14M07 Low codimension problems in algebraic geometry
14K99 Abelian varieties and schemes
Full Text: DOI
[1] Bass H, Milnor J and Serre J P, Solution of the Congruence Subgroup ProblemPubl. I.H.E.S. 33 (1967) 59–137 · Zbl 0174.05203
[2] Ceresa G,C is not algebraically equivalent toC in its Jacobian,Ann. Math.,117 (1983) 285–291 · Zbl 0538.14024
[3] Clemens H, Homological equivalence modulo algebraic equivalence is not finitely generatedPubl. I.H.E.S. 58 (1983) pp. 19–38 · Zbl 0529.14002
[4] Deligne P and Mumford D, The Irreducibility of the space of curves of a given genusPubl. I.H.E.S. 36 75–110 · Zbl 0181.48803
[5] Griffiths P A, On the periods of certain rational integrals,Ann. Math.,90 (1969) 460–541 · Zbl 0215.08103
[6] Igusa J,Theta Functions, Springer-Verlag (1972)
[7] Mumford D,Geometric Invariant Theory, Springer-Verlag (1965)
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