In this article, we study the obstructions to the local-global principle for homogeneous spaces with connected or abelian stabilizers over finite extensions of the field of Laurent series in two variables over the complex numbers and over function fields of curves over . We give examples that prove that the Brauer–Manin obstruction with respect to the whole Brauer group is not enough to explain the failure of the local-global principle, and we then construct a variant of this obstruction using torsors under quasi-trivial tori which turns out to work. In the end of the article, we compare this new obstruction to the descent obstruction with respect to torsors under tori. For that purpose, we use a result on towers of torsors, that is of independent interest and therefore is proved in a separate appendix.
Funding statement: The second author’s research was partially supported by ANID via FONDECYT Grant 1210010 and PAI Grant 79170034.
A A result on towers of torsors
The goal of this appendix is to prove the following result, which is needed in the proof of Theorem 6.4 above. We are greatly indebted to Mathieu Florence for his help with the proof.
Let K be a field of characteristic 0. Let G be a connected linear K-group and T an algebraic K-torus. Let be a G-torsor and let be a T-torsor. Assume that (which holds, for instance, if G is a torus) and that X is geometrically integral. Then there exists a canonical extension
such that the composite is an E-torsor. Moreover, if G is a torus, then E is a torus as well.
For a K-scheme , let be the W(-group)-schemes obtained by base change from , respectively. Consider the group of -automorphisms φ of that are compatible with the action of in the sense that the following diagram commutes:
where a denotes the morphism defining the action of T on Z and the corresponding morphism after base change. The functor defines a group presheaf over the big étale site over K. Denote by the corresponding sheaf and consider the subsheaf defined by taking the subgroup of for each . We have
Indeed, it is well known that the functor over the big étale site of Y is represented, as a Y-scheme, by (cf. for instance [13, Chapter III, Section 1.5]). Moreover, a direct application of Rosenlicht’s Lemma gives us, for geometrically irreducible W,
where M is a free and finitely generated abelian constant sheaf. We deduce then that the quotient sheaf M associated to the presheaf
is a locally free, locally constant sheaf that is finitely generated and abelian. In other words, we have an exact sequence of abelian sheaves
In particular, since M is locally constant, it is representable. And since T is affine, we get by [11, Section III.4, Proposition 1.9] that is represented by an abelian K-group scheme A. We abusively denote by A the sheaf as well.
Since every element in induces an -automorphism of , we have an exact sequence of sheaves
where denotes the sheaf of X-automorphisms of Y.
We claim that π is surjective. In order to prove this, we can replace K by a finite extension. We may assume then that both T and G are split over K. Since G is split and connected and , we have that (cf. [20, Lemma 6.9]). By [20, Proposition 6.10], we get then that the map is surjective. Since T is split, we deduce then that the torsor comes by pullback from a T-torsor . In other words, . The surjectivity is then evident.
Note now that G is clearly a subgroup of . Define then to be the group sheaf corresponding to the preimage of G via π. We get an exact sequence
Since A is abelian, the extension induces an action of the sheafG on the sheafA. By Yoneda’s Lemma, this action is actually an action of the K-group G on the K-group A. Note that T corresponds to the neutral connected component of A and thus it is preserved by the G-action since G is connected. In particular, we may quotient by T in order to get an exact sequence
Then, if one forgets its group structure, corresponds to an M-torsor over the scheme G. By [15, Exposé 8, Proposition 5.1], we know that and hence, since M is locally free and finitely generated, up to taking a finite extension of K. Since representability can be checked over a finite extension (once again by Yoneda’s Lemma), this tells us that is represented by a K-scheme F (which is a K-form of ). Thus we have an exact sequence of K-group-schemes
from where we get a new exact sequence of group sheaves
Once again, since T is an affine group-scheme, [11, Section III.4, Proposition 1.9] tells us that is in fact a scheme and hence a K-group. But now, since M is discrete, it suffices to define E as the identity component of , which clearly fits into an exact sequence
Since is a subgroup of , it is immediate then to check that E acts on Z and that is an E-torsor.
Note finally that the whole construction is clearly canonical. Indeed, the extension is obtained by pullback from the canonical extension of sheaves of automorphism groups. Then we get as theK-group scheme representing and finally we take the identity component, which is a characteristic subgroup. Also, the last assertion when G is a torus follows from [11, Section IV.1, Proposition 4.5]. ∎
The authors would like to thank Mathieu Florence for his enormous help with the result in the appendix. They would also like to thank Jean-Louis Colliot-Thélène and an anonymous referee for all their suggestions that allowed them to considerably improve this text. The second author would like to thank Michel Brion for helpful discussions.
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