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This article is cited in 4 scientific papers (total in 4 papers)
Étale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$
K. Banerjeea, V. Guletskiĭb a Harish-Chandra Research Institute, Allahabad, India
b Department of Mathematical Sciences, University of Liverpool, Liverpool, UK
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 $H^{2p-1}(Y)_\mathrm{van}$. Then we apply these general results to sections of a smooth cubic fourfold in $\mathbb P^5$.
Bibliography: 33 titles.
Keywords:
algebraic cycles, Chow schemes, $l$-adic étale monodromy, Picard-Lefschetz formulae, cubic fourfold hypersurfaces.
Received: 19.02.2019 and 18.11.2019
Citation:
K. Banerjee, V. Guletskiǐ, “Étale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$”, Sb. Math., 211:2 (2020), 161–200
Linking options:
https://www.mathnet.ru/eng/sm9240https://doi.org/10.1070/SM9240 https://www.mathnet.ru/eng/sm/v211/i2/p3
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