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 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 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 * (M Ω ) 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 * (N Ω ) alg → CH * (M Ω ) alg also induces a surjection f * : CH * (N Ω ) hom → CH * (M Ω ) hom on homologically trivial cycles.

Algebraic cycles on a very special EPW sextic

“…Remark 2.5. The following varieties have finite-dimensional motive: abelian varieties, varieties dominated by products of curves [32], K3 surfaces with Picard number 19 or 20 [38], surfaces not of general type with p g = 0 [23, Theorem 2.11], certain surfaces of general type with p g = 0 [23], [40], [55], Hilbert schemes of surfaces known to have finite-dimensional motive [13], generalized Kummer varieties [57, Remark 2.9(ii)], [21], threefolds with nef tangent bundle [27], [47,Example 3.16], fourfolds with nef tangent bundle [28], log-homogeneous varieties in the sense of [12] (this follows from [28,Theorem 4.4]), certain threefolds of general type [49,Section 8], varieties of dimension ≤ 3 rationally dominated by products of curves [47,Example 3.15], varieties X with A Clearly, if Y has finite-dimensional motive then also X = Y /G has finite-dimensional motive. The nilpotence theorem extends to this set-up: Proposition 2.8.…”

Voisin’s conjecture for zero-cycles on Calabi–Yau varieties and their mirrors

“…The following varieties have finite-dimensional motive: varieties dominated by products of curves (which is the case of the Fermat hypersurfaces) and abelian varieties [Kim05] Remark 2.5. It is a (somewhat embarrassing) fact that all examples known so far of finitedimensional motives happen to be in the tensor subcategory generated by Chow motives of curves (i.e., they are "motives of abelian type" in the sense of [Via11]). That is, the finite-dimensionality conjecture is still unknown for any motive not generated by curves (on the other hand, there exist many motives not generated by curves, cf.…”

Algebraic cycles and triple $K3$ burgers

“…Then S has finite-dimensional motive (in the sense of Kimura [27]). What's more, S has motive of abelian type (in the sense of [49]).…”