etale monodromy and rational equivalence for -cycles on cubic hypersurfaces in

Banerjee, K and Guletskii, V (2020) etale monodromy and rational equivalence for -cycles on cubic hypersurfaces in. SBORNIK MATHEMATICS, 211 (2). pp. 161-200.

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Official URL: http://dx.doi.org/10.1070/sm9240

Abstract

<jats:title>Abstract</jats:title><jats:p>Let<jats:inline-formula><jats:tex-math><?CDATA $k$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn3.gif" xlink:type="simple" /></jats:inline-formula>be an uncountable algebraically closed field of characteristic<jats:inline-formula><jats:tex-math><?CDATA $0$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn4.gif" xlink:type="simple" /></jats:inline-formula>, and let<jats:inline-formula><jats:tex-math><?CDATA $X$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn5.gif" xlink:type="simple" /></jats:inline-formula>be a smooth projective connected variety of dimension<jats:inline-formula><jats:tex-math><?CDATA $2p$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn6.gif" xlink:type="simple" /></jats:inline-formula>, embedded into<jats:inline-formula><jats:tex-math><?CDATA $\mathbb{P}^m$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn7.gif" xlink:type="simple" /></jats:inline-formula>over<jats:inline-formula><jats:tex-math><?CDATA $k$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn3.gif" xlink:type="simple" /></jats:inline-formula>. Let<jats:inline-formula><jats:tex-math><?CDATA $Y$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn8.gif" xlink:type="simple" /></jats:inline-formula>be a hyperplane section of<jats:inline-formula><jats:tex-math><?CDATA $X$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn5.gif" xlink:type="simple" /></jats:inline-formula>, and let<jats:inline-formula><jats:tex-math><?CDATA $A^p(Y)$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn9.gif" xlink:type="simple" /></jats:inline-formula>and<jats:inline-formula><jats:tex-math><?CDATA $A^{p+1}(X)$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn10.gif" xlink:type="simple" /></jats:inline-formula>be the groups of algebraically trivial algebraic cycles of codimension<jats:inline-formula><jats:tex-math><?CDATA $p$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn11.gif" xlink:type="simple" /></jats:inline-formula>and<jats:inline-formula><jats:tex-math><?CDATA $p+1$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn12.gif" xlink:type="simple" /></jats:inline-formula>modulo rational equivalence on<jats:inline-formula><jats:tex-math><?CDATA $Y$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn8.gif" xlink:type="simple" /></jats:inline-formula>and<jats:inline-formula><jats:tex-math><?CDATA $X$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn5.gif" xlink:type="simple" /></jats:inline-formula>, respectively. Assume that, whenever<jats:inline-formula><jats:tex-math><?CDATA $Y$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn8.gif" xlink:type="simple" /></jats:inline-formula>is smooth, the group<jats:inline-formula><jats:tex-math><?CDATA $A^p(Y)$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn9.gif" xlink:type="simple" /></jats:inline-formula>is regularly parametrized by an abelian variety<jats:inline-formula><jats:tex-math><?CDATA $A$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn13.gif" xlink:type="simple" /></jats:inline-formula>and coincides with the subgroup of degree<jats:inline-formula><jats:tex-math><?CDATA $0$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn4.gif" xlink:type="simple" /></jats:inline-formula>classes in the Chow group<jats:inline-formula><jats:tex-math><?CDATA $\operatorname{CH}^p(Y)$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn14.gif" xlink:type="simple" /></jats:inline-formula>. We prove that the kernel of the push-forward homomorphism from<jats:inline-formula><jats:tex-math><?CDATA $A^p(Y)$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn9.gif" xlink:type="simple" /></jats:inline-formula>to<jats:inline-formula><jats:tex-math><?CDATA $A^{p+1}(X)$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn10.gif" xlink:type="simple" /></jats:inline-formula>is the union of a countable collection of shifts of a certain abelian subvariety<jats:inline-formula><jats:tex-math><?CDATA $A_0$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn15.gif" xlink:type="simple" /></jats:inline-formula>inside<jats:inline-formula><jats:tex-math><?CDATA $A$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn13.gif" xlink:type="simple" /></jats:inline-formula>. For a very general hyperplane section<jats:inline-formula><jats:tex-math><?CDATA $Y$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn8.gif" xlink:type="simple" /></jats:inline-formula>either<jats:inline-formula><jats:tex-math><?CDATA $A_0=0$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn16.gif" xlink:type="simple" /></jats:inline-formula>or<jats:inline-formula><jats:tex-math><?CDATA $A_0$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn15.gif" xlink:type="simple" /></jats:inline-formula>coincides with an abelian subvariety<jats:inline-formula><jats:tex-math><?CDATA $A_1$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn17.gif" xlink:type="simple" /></jats:inline-formula>in<jats:inline-formula><jats:tex-math><?CDATA $A$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn13.gif" xlink:type="simple" /></jats:inline-formula>whose tangent space is the group of vanishing cycles<jats:inline-formula><jats:tex-math><?CDATA $H^{2p-1}(Y)_{\mathrm{van}}$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn18.gif" xlink:type="simple" /></jats:inline-formula>. Then we apply these general results to sections of a smooth cubic fourfold in<jats:inline-formula><jats:tex-math><?CDATA $\mathbb{P}^5$?></jats:tex-math><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="MSB_211_2_161ieqn19.gif" xlink:type="simple" /></jats:inline-formula>.</jats:p><jats:p>Bibliography: 33 titles.</jats:p>

Item Type:	Article
Additional Information:	38 pages
Uncontrolled Keywords:	algebraic cycles, Chow schemes, l-adic etale monodromy, Picard-Lefschetz formulae, cubic fourfold hypersurfaces
Depositing User:	Symplectic Admin
Date Deposited:	05 Dec 2018 09:43
Last Modified:	03 Oct 2023 00:38
DOI:	10.1070/SM9240
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3029507