The Bogomolov multiplier of a finite group G is defined as the subgroup of the Schur multiplier consisting of the cohomology classes vanishing after restriction to all abelian subgroups of G. This invariant of G plays an important role in birational geometry of quotient spaces V/G. We show that in many cases the vanishing of the Bogomolov multiplier is guaranteed by the rigidity of G in the sense that it has no outer class-preserving automorphisms.
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Acknowledgements. The first author was supported in part by the National Center for Theoretic Sciences (Taipei Office). The second author was supported in part by the Minerva Foundation through the Emmy Noether Research Institute of Mathematics and by the Israel Science Foundation, grant 1207/12; this paper was mainly written during his visit to the NCTS (Taipei) in 2012. Support of these institutions is gratefully acknowledged. We thank Y. Ginosar for valuable discussions. Our special thanks go to the referee who discovered the example in the last item of the last remark and generously suggested to include it in the paper.
- Bogomolov multiplier
- Class-preserving automorphisms
- Shafarevich-Tate set
- Unramified Brauer group