Unramified cohomology, integral coniveau filtration and Griffiths group

Abstract

We prove that the k-th unramified cohomology H k nr (X, Q/Z) 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 H k+1 (X, Z) by its coniveau 2 filter, and when k is even, whose next graded piece is controlled by the Griffiths group Griff k/2+1 (X). The first filter is a generalization of the Artin-Mumford invariant (k = 2) and the Colliot-Thélène-Voisin invariant (k = 3). We also give a homological analogue.