Insufficiency of the algebraic Brauer--Manin obstruction for homogeneous spaces

Abstract

Over any number field containing a root of unity of odd prime order, we construct a homogeneous space of $\mathrm{SL}_n$ with finite $2$-nilpotent geometric stabilizers, with a constant unramified algebraic Brauer group, which has no rational point but has local points in every completion of the ground field. This yields the first example of transcendental Brauer--Manin obstruction for homogeneous spaces of connected linear algebraic groups. Our method exploits a previous idea by Borovoi and Kunyavskii.