Remarks on motives of abelian type

Vial C (2017)
Tohoku Mathematical Journal 69(2): 195-220.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
A motive. over a field k is of abelian type if it belongs to the thick and rigid subcategory of Chow motives spanned by the motives of abelian varieties over k. This paper contains three sections of independent interest. First, we show that a motive which becomes of abelian type after a base field extension of algebraically closed fields is of abelian type. Given a field extension K/k and a motive M over k, we also show that M is finite dimensional if and only if M-K is finite-dimensional. As a corollary, we obtain Chow-Kunneth decompositions for varieties that become isomorphic to an abelian variety after some field extension. Second, let Omega be a universal domain containing k. We show that Murre's conjectures for motives of abelian type over k reduce to Murre's conjecture (D) for products of curves over Omega. In particular, we show that Murre's conjecture (D) for products of curves over Omega implies Beauville's vanishing conjecture on abelian varieties over k. Finally, we give criteria on Chow groups for a motive to be of abelian type. For instance, we show that M is of abelian type if and only if the total Chow group of algebraically trivial cycles CH*(M Omega)(alg) is spanned, via the action of correspondences, by the Chow groups of products of curves. We also show that a morphism of motives f : N -> M, with N finite-dimensional, which induces a surjection f* : CH*(N Omega)(alg) -> CH*(M Omega)(alg) also induces a surjection f* : CH*(N Omega)(hom) -> CH*(M Omega)(hom) on homologically trivial cycles.
Stichworte
Algebraic cycles; Chow groups; motives; abelian varieties; finite-dimensionality
Erscheinungsjahr
2017
Zeitschriftentitel
Tohoku Mathematical Journal
Band
69
Ausgabe
2
Seite(n)
195-220
ISSN
0040-8735
Page URI
https://pub.uni-bielefeld.de/record/2913996

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Vial C. Remarks on motives of abelian type. Tohoku Mathematical Journal. 2017;69(2):195-220.
Vial, C. (2017). Remarks on motives of abelian type. Tohoku Mathematical Journal, 69(2), 195-220. doi:10.2748/tmj/1498269623
Vial, C. (2017). Remarks on motives of abelian type. Tohoku Mathematical Journal 69, 195-220.
Vial, C., 2017. Remarks on motives of abelian type. Tohoku Mathematical Journal, 69(2), p 195-220.
C. Vial, “Remarks on motives of abelian type”, Tohoku Mathematical Journal, vol. 69, 2017, pp. 195-220.
Vial, C.: Remarks on motives of abelian type. Tohoku Mathematical Journal. 69, 195-220 (2017).
Vial, Charles. “Remarks on motives of abelian type”. Tohoku Mathematical Journal 69.2 (2017): 195-220.