## Tohoku Mathematical Journal 2017 June SECOND SERIES VOL. 69, NO. 2

 Tohoku Math. J. 69 (2017), 195-220

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 finite-dimensional if and only if $M_K$ is finite-dimensional. As a corollary, we obtain Chow--Künneth 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 $N$ finite-dimensional, 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, finite-dimensionality.