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Étale monodromy and rational equivalence for 1-cycles on cubic hypersurfaces in \(\mathbb{P}^5\). (English. Russian original) Zbl 1448.14011

Sb. Math. 211, No. 2, 161-200 (2020); translation from Mat. Sb. 211, No. 2, 3-45 (2020).
Authors’ abstract: Let \(k\) be an uncountable algebraically closed field of characteristic 0, and let \(X\) be a smooth projective connected variety of dimension \(2p\), embedded into \(\mathbb{P}^m\) over \(k\). Let \(Y\) be a hyperplane section of \(X\), and let \(A^p(Y)\) and \(A^{p+1}(X)\) be the groups of algebraically trivial algebraic cycles of codimension \(p\) and \(p+1\) modulo rational equivalence on \(Y\) and \(X\), respectively. Assume that, whenever \(Y\) is smooth, the group \(A^p(Y)\) is regularly parametrized by an abelian variety \(A\) and coincides with the subgroup of degree 0 classes in the Chow group \(\operatorname{CH}^p(Y)\). We prove that the kernel of the push-forward homomorphism from \(A^p(Y)\) to \(A^{p+1}(X)\) is the union of a countable collection of shifts of a certain abelian subvariety \(A_0\) inside \(A\). For a very general hyperplane section \(Y\) either \(A_0=0\) or \(A_0\) coincides with an abelian subvariety \(A_1\) in \(A\) whose tangent space is the group of vanishing cycles \(H^{2p-1}(Y)_{\text{van}} \). Then we apply these general results to sections of a smooth cubic fourfold in \(\mathbb{P}^5\).

MSC:

14C25 Algebraic cycles
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14F30 \(p\)-adic cohomology, crystalline cohomology
14J30 \(3\)-folds
14J35 \(4\)-folds
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References:

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