Unramified cohomology, integral coniveau filtration and Griffiths group

TL;DR

This paper establishes a new filtration structure for unramified cohomology of smooth complex projective varieties, linking it to classical invariants and higher Chow groups, with implications for algebraic cycles and cohomological invariants.

Contribution

It introduces a novel filtration for unramified cohomology that generalizes known invariants and connects to Griffiths groups and higher Chow groups.

Findings

Filtration length is [k/2] for degree k unramified cohomology.

First piece generalizes Artin-Mumford and Colliot-Thelene-Voisin invariants.

Results apply to certain H-cohomology groups.

Abstract

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 piece is the torsion part of the quotient of the degree k+1 integral singular cohomology by its coniveau 2 subgroup, and whose next graded piece is controlled by the Griffiths group Griff^{k/2+1}(X) when k is even and is related to the higher Chow group CH^{(k+3)/2}(X, 1) when k is odd. 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 analogous result for certain H-cohomology groups.