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HOME > Table of Contents and Abstracts > Vol. 69, No. 2
Tohoku Mathematical Journal
2017
June
SECOND SERIES VOL. 69, NO. 2
Tohoku Math. J.
69 (2017), 195220

Title
REMARKS ON MOTIVES OF ABELIAN TYPE
Author
Charles Vial
(Received October 24, 2014, revised July 21, 2015) 
Abstract.
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 finitedimensional if and only if $M_K$ is finitedimensional. As a corollary, we obtain ChowKünneth 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 $\mathop{CH}_*(M_\Omega)_\mathop{alg}$ is spanned, via the action of correspondences, by the Chow groups of products of curves. We also show that a morphism of motives $f: N \rightarrow M$, with $N$ finitedimensional, which induces a surjection $f_* : \mathop{CH}_*(N_\Omega)_\mathop{alg} \rightarrow \mathop{CH}_*(M_\Omega)_\mathop{alg}$ also induces a surjection $f_* : \mathop{CH}_*(N_\Omega)_\mathop{hom} \rightarrow \mathop{CH}_*(M_\Omega)_\mathop{hom}$ on homologically trivial cycles.
Mathematics Subject Classification.
Primary 14C15; Secondary 14C25, 14K15.
Key words and phrases.
Algebraic cycles, Chow groups, motives, abelian varieties, finitedimensionality.


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