Ramified descent and transcendental Brauer–Manin obstruction

Given a smooth geometrically connected variety X defined over a number field K and an étale torsor V → U over a Zariski-open U of X, we investigate the problem of which adelic points of X can be approximated by adelic points that lift to a (twist of a) V . The question has long been investigated in the literature when U = X, but less so in the general case. We introduce a Brauer–Manin obstruction to the problem, and provide an example where this obstruction is nontrivial and purely transcendental. This answers in the negative a question posed by Harari at a 2019 workshop. Our example is also an explicit example of a nontrivial transcendental Brauer–Manin obstruction on a smooth compactification of a quotient SL_n /G, with G constant metabelian.