Holmstrom Coniveau

See Gillet: K-theory and Intersection theory

There is also something in the K-theory handbook

A very nice paper by Levine on the homotopy coniveau filtration.

Kahn et al: The Bloch-Ogus-Gabber theorem. Gives a general argument for many different CTs at once.

arXiv:0907.3535 Coniveau filtrations and finite dimensionality for pure motives from arXiv Front: math.AG by Charles Vial Let $X$ be a smooth projective variety over a field $k \subset \C$. We define a coniveau filtration $\widetilde{N}$ – a priori different from the usual coniveau filtration $N$ – on the singular cohomology of $X$ with rational coefficients. If $X$ satisfies the Lefschetz conjecture B, we prove that the projectors on the filtered pieces of the cohomology of $X$ are induced by algebraic correspondences. If moreover $X$ is finite dimensional in the sense of Kimura (for example, $X$ could be any abelian variety or any finite quotient thereof) the Chow motive of $X$ admits a refined Chow-Künneth decomposition lifting the coniveau grading on the cohomology of $X$ and we study the behavior of the Chow groups of $X$ under this decomposition. As an application, we give several conjectural descriptions of the rational Chow groups of $X$ depending on the support of the cohomology groups of $X$. We also consider the case where $k$ is an arbitrary field. Results valid in characteristic zero extend to positive characteristic if one is ready to admit that the Lefschetz conjecture holds in general.

nLab page on Coniveau