Motives and representability of algebraic cycles on threefolds over a field

Abstract

Abstract. We study algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in Q. We decompose the motive of a non-singular projective threefold X with representable algebraic part of CH 0 (X) into Lefschetz motives and the Picard motive of a certain abelian variety, isogenous to the Griffiths' intermediate Jacobian J 2 (X) when the ground field is C. In particular, it implies motivic finite-dimensionality of Fano threefolds over a field. We also prove representability of zero-cycles on several classes of threefolds fibred by surfaces with algebraic H 2 . This gives another new examples of three-dimensional varieties whose motives are finite-dimensional.

On the finite dimensionality of a K3 surface

“…On the contrary, in the cases described in (i), (ii), (iii), where ρ(X ) = 9, φ i is not an isometry. This follows from [18,2,5] because dim T X,Q is odd.…”

“…The conjecture is known for curves, for abelian varieties and for some surfaces: rational surfaces, Godeaux surfaces, Kummer surfaces, surfaces with p g = 0 which are not of general type, surfaces isomorphic to a quotient (C × D)/G, where C and D are curves and G is a finite group. It is also known for Fano 3-folds (see [5]). In all these known cases the motive h(X ) lies in the tensor subcategory of M rat (k) generated by abelian varieties.…”

Categorical representability and intermediate Jacobians of Fano threefolds

“…In this case, the rational representability of A i Q (X) for i ≥ 2 is known ( [Mur83]). In [GG08] it is proved that A 3 Q (X) is rationally representable if and only if the Chow motive of X has a given Chow-Künneth decomposition.…”

“…Anyway, they are satisfied by a quite big class of smooth projective threefolds with κ X < 0. The Chow-Künneth decomposition for the listed varieties is provided by [NS09] for conic bundles and by [GG08] in any other case. In the following list the references point out the most general results about strong representability and incidence property.…”

“…It is nevertheless clear that rational representability is strongly related to the structure of the motive of X. For example, if X is a threefold, then rational representability of all the A i Q (X) is equivalent to the existence of a specific Chow-Künneth decomposition [GG08]. A first point to note is then that fully faithful functors should hold motivic maps, as stated in the following conjecture by Orlov.…”