Unramified cohomology, integral coniveau filtration and Griffiths groups

We prove that the degree k unramified cohomology H-nr(k)(X, Q/Z) of a smooth complex projective variety X with small CH0(X) has a filtration of length [k/2], whose first piece is the torsion part of the quotient of Hk+1(X, Z) by its coniveau 2 subgroup, and whose next graded piece is controlled by the Griffiths group Griff(k/2+1)(X) when k is even and is related to the higher Chow group CH(k+3)/2(X, 1) when k is odd. The first piece is a generalization of the ArtinMumford invariant (k = 2) and the Colliot-Thelene-Voisin invariant (k = 3). We also give an analogous result for the H-cohomology Hd-k(X, H-d (Q/Z)), d = dim X.