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The coniveau filtration and non-divisibility for algebraic cycles. (English) Zbl 0868.14004
The authors give an example of a smooth complete intersection $$X$$ of dimension 3 in $$\mathbb{P}^5$$, defined over a number field $$K$$, and a prime $$\ell$$ such that the Chow group $$\text{CH}^2 (X_{\overline{K}}) \{\ell\}$$ of $$\ell$$-power torsion codimension 2 cycles vanishes and the group $$B^2(X_{\overline{K}})$$ of codimension 2 cycles homologically equivalent to zero is not $$\ell$$-divisible. Hence Bloch’s map $$\lambda^2: \text{CH}^2 (X_{\overline{K}})\{\ell\}\to H^3(X_{\overline{K}}, \mathbb{Q}_\ell/\mathbb{Z}_\ell(2))$$ is the zero map, and the cycle class map $$\text{CH}^2 (X_{\overline{K}})/ \ell\text{CH}^2 (X_{\overline{K}})\to H^4(X_{\overline{K}}, \mathbb{Z}/\ell\mathbb{Z}(2))$$ is not injective.
The variety $$X$$ in the example is a sufficiently general intersection of a smooth quadric in $$\mathbb{P}^5$$, defined over $$\mathbb{Q}$$, with a hypersurface of degree $$d\geq 4$$. As an application of the previous results, it is shown that there exists a smooth proper morphism $$f:{\mathcal W}\to S$$ of quasi-projective varieties over $$\mathbb{C}$$ and a relative cycle $${\mathcal Z}\in Z^2({\mathcal W})$$ whose restriction to every fiber is homologically equivalent to zero, such that $${\mathcal Z}$$ does not become divisible in the Chow group after base change by a finite covering, i.e., there exists an $$n\geq 2$$ such that there is no finite covering $$\pi:T\to S$$ with $$\pi^*{\mathcal Z}=n{\mathcal Z}'$$ in $$\text{CH}^2({\mathcal W}\times_S T)$$.
One of the main ingredients of the proof is to give a criterion for $N^1H^3(X_{\overline{K}}, \mathbb{Z}/\ell\mathbb{Z})\neq H^3(X_{\overline{K}}, \mathbb{Z}/\ell\mathbb{Z})$ ($$N^\bullet$$ denotes the coniveau filtration). To this end, the authors use results of S. Bloch and K. Kato [Publ. Math., Inst. Hautes Étud. Sci. 63, 107-152 (1986; Zbl 0613.14017)] on $$p$$-adic étale cohomology.
Reviewer: J.Nagel (Essen)

MSC:
 14C25 Algebraic cycles 14C05 Parametrization (Chow and Hilbert schemes) 14C35 Applications of methods of algebraic $$K$$-theory in algebraic geometry 14F30 $$p$$-adic cohomology, crystalline cohomology 14F20 Étale and other Grothendieck topologies and (co)homologies
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References:
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