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Complex varieties with infinite Chow groups modulo 2. (English) Zbl 1332.14011

The main result of this paper shows that for a very general principally polarized complex abelian threefold \(X\), the Chow group \(\text{CH}^2(X)/\ell\) is infinite for all prime numbers \(\ell\). The infiniteness comes from pulling back Cerasa cycles by infinitely many different isogenies. By using products \(X\times \mathbb P^{n-3}\) for any \(n\geq 3\), one obtain similar examples in higher dimensions. Furthermore by taking the product with a very general elliptic curve, one obtains the following corollary: For each \(n\geq 4\), there is a smooth complex projective \(n\)-fold \(X\) such that \(\text{CH}^i(X)[\ell]\) is infinite for all \(3\leq i\leq n-1\) and all prime numbers \(\ell\).

MSC:

14C15 (Equivariant) Chow groups and rings; motives
14C25 Algebraic cycles
14H40 Jacobians, Prym varieties
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