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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.

MSC:
14C25 Algebraic cycles
14J30 \(3\)-folds
32J17 Compact complex \(3\)-folds
14M07 Low codimension problems in algebraic geometry
14K99 Abelian varieties and schemes
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