Abstract

We prove that the degree $k$ unramified cohomology $H_{\mathrm{nr}<!--FUNCTION\; APPLICATION-->}^k(X,\mathbb{Q}∕\mathbb{Z})$ of a smooth complex projective variety $X$ with small ${\mathrm{CH}<!--FUNCTION\; APPLICATION-->}_0(X)$ has a filtration of length $[k∕2]$, whose first piece is the torsion part of the quotient of $H^{k+1}(X,\mathbb{Z})$ by its coniveau $2$ subgroup, and whose next graded piece is controlled by the Griffiths group ${\mathrm{Griff}<!--FUNCTION\; APPLICATION-->}^{k∕2+1}(X)$ when $k$ is even and is related to the higher Chow group ${\mathrm{CH}<!--FUNCTION\; APPLICATION-->}^{(k+3)∕2}(X,1)$ when $k$ is odd. The first piece is a generalization of the Artin–Mumford invariant ($k=2$) and the Colliot-Thélène–Voisin invariant ($k=3$). We also give an analogous result for the $\mathcal{\mathscr{H}}$-cohomology $H^{d-k}(X,{\mathcal{\mathscr{H}}}^d(\mathbb{Q}∕\mathbb{Z}))$, $d=\dim <!--FUNCTION\; APPLICATION-->X$.

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