Vanishing of Degree 3 Cohomological Invariants

TL;DR

This paper investigates conditions under which degree three cohomological invariants vanish for complex algebraic varieties and finite p-groups, linking sheaf cohomology, Chow groups, and group invariants.

Contribution

It establishes a criterion connecting sheaf cohomology and Chow groups that implies the vanishing of degree three invariants for certain algebraic structures.

Findings

Triviality of $H^0(X,\

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Abstract

For a complex algebraic variety $X$, we show that triviality of the sheaf cohomology group $H^0(X,\mathcal{H}^3)$ occurring on the second page of the Bloch-Ogus spectral sequence follows from a condition on the integral Chow group $CH^2X$ and the integral cohomology group $H^3(X, Z)$. In the case that $X$ is an appropriate approximation to the classifying stack $BG$ of a finite $p$-group $G$, this result states that the group $G$ has no degree three cohomological invariants. As a corollary we show that the nonabelian groups of order $p^3$ for odd prime $p$ have no degree three cohomological invariants.