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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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