DISTINGUISHED CYCLES ON VARIETIES WITH MOTIVE OF ABELIAN TYPE AND THE SECTION PROPERTY

Fu L, Vial C (2020)
JOURNAL OF ALGEBRAIC GEOMETRY 29(1): 53-107.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
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.
Erscheinungsjahr
2020
Zeitschriftentitel
JOURNAL OF ALGEBRAIC GEOMETRY
Band
29
Ausgabe
1
Seite(n)
53-107
ISSN
1056-3911
eISSN
1534-7486
Page URI
https://pub.uni-bielefeld.de/record/2941901

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Fu L, Vial C. DISTINGUISHED CYCLES ON VARIETIES WITH MOTIVE OF ABELIAN TYPE AND THE SECTION PROPERTY. JOURNAL OF ALGEBRAIC GEOMETRY. 2020;29(1):53-107.
Fu, L., & Vial, C. (2020). DISTINGUISHED CYCLES ON VARIETIES WITH MOTIVE OF ABELIAN TYPE AND THE SECTION PROPERTY. JOURNAL OF ALGEBRAIC GEOMETRY, 29(1), 53-107. doi:10.1090/jag/729
Fu, L., and Vial, C. (2020). DISTINGUISHED CYCLES ON VARIETIES WITH MOTIVE OF ABELIAN TYPE AND THE SECTION PROPERTY. JOURNAL OF ALGEBRAIC GEOMETRY 29, 53-107.
Fu, L., & Vial, C., 2020. DISTINGUISHED CYCLES ON VARIETIES WITH MOTIVE OF ABELIAN TYPE AND THE SECTION PROPERTY. JOURNAL OF ALGEBRAIC GEOMETRY, 29(1), p 53-107.
L. Fu and C. Vial, “DISTINGUISHED CYCLES ON VARIETIES WITH MOTIVE OF ABELIAN TYPE AND THE SECTION PROPERTY”, JOURNAL OF ALGEBRAIC GEOMETRY, vol. 29, 2020, pp. 53-107.
Fu, L., Vial, C.: DISTINGUISHED CYCLES ON VARIETIES WITH MOTIVE OF ABELIAN TYPE AND THE SECTION PROPERTY. JOURNAL OF ALGEBRAIC GEOMETRY. 29, 53-107 (2020).
Fu, Lie, and Vial, Charles. “DISTINGUISHED CYCLES ON VARIETIES WITH MOTIVE OF ABELIAN TYPE AND THE SECTION PROPERTY”. JOURNAL OF ALGEBRAIC GEOMETRY 29.1 (2020): 53-107.