Motives and representability of algebraic cycles on threefolds over a field

Abstract

We study algebraic cycles on threefolds and finite-dimensionality of their motives with coefficients in Q \mathbb Q . We decompose the motive of a non-singular projective threefold X X with representable algebraic part of C H 0 ( X ) CH_0(X) into Lefschetz motives and the Picard motive of a certain abelian variety, isogenous to the Griffiths’ intermediate Jacobian J 2 ( X ) J^2(X) when the ground field is C \mathbb 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 H^2 . This gives new examples of three-dimensional varieties whose motives are finite-dimensional.