Chow Motives Without Projectivity, II

Bloch’s Conjecture and Chow Motives

Let X be a connected smooth projective complex surface. J. Murre [7] constructed a decomposition of Chow motives for X , i.e. there exist mutually orthogonal idempotents πi ∈ CH (X × X)Q as correspondences for 0 ≤ i ≤ 4 such that ∑ i πi is equal to the diagonal and the action of πi on H (X,Q) is the identity for i = j, and vanishes otherwise. The decomposition is not uniquely characterized by t...

Grothendieck Chow-motives of Severi-Brauer varieties

For any central simple algebra, the Grothendieck Chow-motive of the corresponding Severi-Brauer variety is decomposed in a direct sum where each summand is a twisted motive of the Severi-Brauer variety corresponding to the underlying division algebra. It leads to decompositions in other theories (for instance, of K-cohomologies) because of the universal property of the Chow-motives. In the seco...

Chow motives of twisted flag varieties

Let G be an adjoint simple algebraic group of inner type. We express the Chow motive (with integral coefficients) of some anisotropic projective G-homogeneous varieties in terms of motives of simpler G-homogeneous varieties, namely, those that correspond to maximal parabolic subgroups of G. We decompose the motive of a generalized Severi-Brauer variety SB2(A), where A is a division algebra of d...

Extensions of motives and higher Chow groups

Chow Motives of 3-folds and Fiber Spaces

Let k be a field of characteristic zero. For every smooth, projective k-variety Y of dimension n which admits a connected, proper morphism f : Y → S of relative dimension one, we construct idempotent correspondences (projectors) πij(Y ) ∈ CH (Y × Y,Q) generalizing a construction of Murre. If n = 3 and the transcendental cohomology group H tr(Y ) has the property that H tr(Y,C) = f H tr(S,C) + I...