Algebraic cycles and refined unramified cohomology

Abstract:
We introduce refined unramified cohomology groups. This notion allows us to give in arbitrary degree a cohomological interpretation of the failure of integral Hodge- or Tate-type conjectures, of l-adic Griffiths groups, and of the subgroup of the Griffiths group that consists of torsion classes with trivial transcendental Abel-Jacobi invariant. Our approach simplifies and generalizes to cycles of arbitrary codimension previous results of Bloch-Ogus, Colliot-Thélène-Voisin, Voisin, and Ma that concerned cycles of codimension two or three. As an application, we give for any i>2 the first example of a uniruled smooth complex projective variety for which the integral Hodge conjecture fails for codimension i-cycles in a way that cannot be explained by the failure on any lower-dimensional variety.