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We prove that the degree k unramified cohomology with torsion coefficients ofa smooth complex projective variety X with small CH_0(X) has a filtration oflength [k/2], whose first piece is the torsion part of the quotient of thedegree k+1 integral singular cohomology by its coniveau 2 subgroup, and whosenext graded piece is controlled by the Griffiths group Griff^{k/2+1}(X) when kis even and is related to the higher Chow group CH^{(k+3)/2}(X, 1) when k isodd. The first piece is a generalization of the Artin-Mumford invariant (k=2)and the Colliot-Thelene-Voisin invariant (k=3). We also give an analogousresult for certain H-cohomology groups.

Unramified cohomology, integral coniveau filtration and Griffiths group