Remarks on motives of abelian type
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 MK 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 Ω 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 Ω. In particular, we show that Murre’s conjecture (D) for products of curves over Ω 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∗(MΩ)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 finitedimensional, which induces a surjection f∗ : CH∗(NΩ)alg → CH∗(MΩ)alg also induces a surjection f∗ : CH∗(NΩ)hom → CH∗(MΩ)hom on homologically trivial cycles.