Remarks on motives of abelian type

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 pa-per 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 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 con-jectures 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 cy-cles 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 finite-dimensional, 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