题目：Grothendieck--Serre conjecture for reductive torsors
摘要：Torsors under reductive group schemes, as generalizations of vector bundles, are ubiquitous in mathematical physics and arithmetic geometry. Originally conceived by Serre and Grothendieck in 1958, the prototype of the Grothendieck--Serre conjecture predicted that every principal bundle under an algebraic group over a nonsingular algebraic manifold is Zariski-locally trivial if it is generically trivial. With its subsequent generalization, the conjecture predicts that over a regular local ring R, there is no nontrivial torsor under a reductive group G trivializes over the fraction field Frac(R).
This conjecture is long-standing for more than half a century, viewed as a crucial problem in Langlands program, especially for Shimura varieties. The known cases include the toral case settled by Colliot-Thélène and Sansuc, the dim R=1 case addressed by Nisnevich, the case when R contains a field established by Fedorov--Panin and Panin. The remaining mixed characteristic case, except when G is quasi-split and R unramified settled by Cesnavicius, is still widely open. In this talk, I will introduce my several contributions to the Grothendieck--Serre conjecture, including
i)The variant when R is semilocal Dedekind, which is the cornerstone of all higher dimensional results.
ii)The variant when R is semilocal Prufer, predicted by the combination of the original conjecture and the resolution of singularities.
iii)The case when R is a localization of a smooth projective scheme over a DVR. This is a joint work with Panin and Stavrova.
iv)The case when R is a localization of smooth affine algebra over a DVR. This is one of the state of the art of the conjecture, which is a joint work with Fei Liu.