Distinguished cycles on varieties with motive of abelian type and the Section Property

Fu, Lie; Vial, Charles

doi:10.1090/jag/729

Authors: Lie Fu and Charles Vial
Journal: J. Algebraic Geom. 29 (2020), 53-107
DOI: https://doi.org/10.1090/jag/729
Published electronically: September 20, 2019
MathSciNet review: 4028066
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Abstract | References | Additional Information

Abstract: A remarkable result of Peter O’Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a (canonical) section. Motivated by Beauville’s splitting principle, we formulate a conjectural Section Property which predicts that for smooth projective holomorphic symplectic varieties there exists such a section of algebra whose image contains all the Chern classes of the variety. In this paper, we investigate this property for (not necessarily symplectic) varieties with a Chow motive of abelian type. We introduce the notion of a symmetrically distinguished abelian motive and use it to provide a sufficient condition for a smooth projective variety to admit such a section. We then give a series of examples of varieties for which our theory works. For instance, we prove the existence of such a section for arbitrary products of varieties with Chow groups of finite rank, abelian varieties, hyperelliptic curves, Fermat cubic hypersurfaces, Hilbert schemes of points on an abelian surface or a Kummer surface or a K3 surface with Picard number at least 19, and generalized Kummer varieties. The latter cases provide evidence for the conjectural Section Property and exemplify the mantra that the motives of holomorphic symplectic varieties should behave as the motives of abelian varieties, as algebra objects.

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Additional Information

Lie Fu
Affiliation: Institut Camille Jordan, Université Claude Bernard Lyon 1, 69622 Villeurbanne cedex, France
MR Author ID: 1016534
Email: fu@math.univ-lyon1.fr

Charles Vial
Affiliation: Faculty of Mathematics, Universität Bielefeld, Universitätsstrasse 25, 33615 Bielefeld, Germany
MR Author ID: 867800
Email: vial@math.uni-bielefeld.de

Received by editor(s): November 27, 2017
Received by editor(s) in revised form: May 2, 2018, May 17, 2018, and July 17, 2018
Published electronically: September 20, 2019
Additional Notes: The first author was supported by the Agence Nationale de la Recherche (ANR) through ECOVA (ANR-15-CE40-0002), HodgeFun (ANR-16-CE40-0011), LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, and Projet Inter-Laboratoire 2017, 2018 by Fédération de Recherche en Mathématiques Rhône-Alpes/Auvergne CNRS 3490.
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