Fano varieties with torsion in the third cohomology group

TL;DR

This paper constructs the first known examples of Fano varieties exhibiting torsion in their third cohomology group, using double covers of linear sections of symmetric matrix rank loci, and addresses a question on filtrations in rationally connected varieties.

Contribution

It introduces the first examples of Fano varieties with torsion in third cohomology and connects these to questions about coniveau filtrations in rationally connected varieties.

Findings

Constructed Fano varieties with torsion in third cohomology

Linked examples to higher-dimensional analogues of Artin--Mumford threefold

Provided answers to Voisin's question on coniveau filtrations

Abstract

We construct first examples of Fano varieties with torsion in their third cohomology group. The examples are constructed as double covers of linear sections of rank loci of symmetric matrices, and can be seen as higher-dimensional analogues of the Artin--Mumford threefold. As an application, we answer a question of Voisin on the coniveau and strong coniveau filtrations of rationally connected varieties.