Fano varieties with torsion in the third cohomology group

Artin and Mumford showed that a non-trivial torsion subgroup in the third cohomology group of a smooth complex variety is an obstruction to rationality. Using this they presented one of the first examples of a unirational non- rational threefolds. The example is a resolution of singularities of a nodal Fano threefold; however, it is not a smooth Fano threefold itself. Moreover, by the classification one can check that the torsion subgroup in the third cohomology group of any smooth Fano threefold is trivial. In my talk based on the work by Ottem and Rennemo I will discuss the construction of a 4-dimensional Fano variety with a non-trivial torsion subgroup in the third cohomology group.