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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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##### References:
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