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Algebraic cycles on generic abelian varieties. (English) Zbl 0871.14005
In this paper one studies rational Chow groups of generic abelian varieties. First, there are proved results which make the following conjecture plausible:


(1) All codimension \(d\) cycles in the generic abelian variety of dimension \(2g\) are generated by divisors up to torsion, for \(d<g/2\).
(2a) For \(g>2\) and even, there exist codimension \(g/2+1\) cycles which are not generated by divisors.
(2b) For \(g>1\) and odd, there exist codimension \((g+1)/2\) cycles which are homologically trivial but not algebraically trivial.
Proofs are given for (2) in the cases \(g=4,5\). Interesting results about Griffiths groups are given.

MSC:
14C05 Parametrization (Chow and Hilbert schemes)
14K05 Algebraic theory of abelian varieties
14G25 Global ground fields in algebraic geometry
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