Abstract: We study obstructions to weak approximation for connected linear groups and homogeneous spaces with connected or abelian stabilizers over finite extensions of C((X, Y)) or function fields of curves over C((X)). We show that for connected linear groups, the usual Brauer–Manin obstruction works as in the case of tori, using the same dévissage argument. However, this Brauer–Manin obstruction is not enough for homogeneous spaces, so we should somehow combine the Brauer–Manin obstruction with the descent obstruction using torsors under quasi-trivial tori, another natural tool used in the study of such questions, as done by Izquierdo and Lucchini Arteche for the study of obstruction to rational points.

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Ref: H. Zhang, Weak approximation for homogeneous spaces over some two-dimensional geometric global fields