# On the unramified cohomology of certain quotient varieties

## TL;DR

This paper develops a method to demonstrate non-vanishing of unramified cohomology groups with $\

## Contribution

It introduces a new approach to prove non-vanishing of unramified cohomology for certain quotient varieties, including Kummer varieties, impacting the understanding of the integral Hodge conjecture.

## Key findings

- Non-vanishing of unramified cohomology for specific quotient varieties.
- Construction of new three-dimensional counterexamples to the integral Hodge conjecture.
- Application of the method to Kummer varieties.

## Abstract

In this note, we consider unramified cohomology with $\mathbb{Z}/2$ coefficients for some (degree two) quotient varieties and describe a method that allows one to prove the non-vanishing of these groups under certain conditions. We apply this method to prove a non-vanishing statement in the case of Kummer varieties. Combining this with work of Colliot-Th\'el\`ene and Voisin, we obtain a new type of three-dimensional counterexample to the integral Hodge conjecture.