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Description
The topological Brauer group of a smooth projective complex variety is by definition the torsion part of its third degree Betti cohomology. It is an important stale birational invariant, for example it has been used by Artin and Mumford to prove the existence of non-stably rational, unirational varieties. One can see it as the "purely topological" part of the Brauer group: for this reason, understanding the former is a natural problem arising when is working with the latter, for example in hyper-Kähler geometry.
The topological Brauer group of hyper-Kähler varieties has been computed only in the cases of K3^n-type and Kum_2-type manifolds, and it turns out to be zero. One is then led to speculate whether it always vanishes for hyper-Kähler manifolds. In this talk, we confirm this for all Kum_2n-type manifolds, extending the previously known case of n=1, and give a low bound on its torsion for all other Kum-type manifolds: this is joint work with Moritz Hartlieb.