Unramified cohomology, integral coniveau filtration and Griffiths groups

We prove that the degree k unramified cohomology with torsion coefficients of a smooth complex projective variety X with small CH_0(X) has a filtration of length [k/2], whose first filter is the torsion part of the quotient of the degree k+1 integral singular cohomology by its coniveau 2 filter, and when k is even, whose next graded piece is controlled by the Griffiths group of codimension k/2+1 cycles. The first filter is a generalization of the Artin-Mumford invariant (k=2) and the Colliot-Thelene-Voisin invariant (k=3). We also give a homological analogue.