Descent obstruction for a connected linear algebraic group is Brauer-Manin obstruction
The descent theory for tori was first established by Colliot-Thélène and Sansuc and was refined by Skorobogatov for groups of multiplicative type. Harari and Skorobogatov introduced descent obstruction for a general algebraic group and compared such descent obstruction with Brauer-Manin obstruction. By various efforts of Poonen, Demarche, Stoll and Skorobogatov, it has been proved that descent obstruction is equivalent to étale Brauer-Manin obstruction for smooth projective and geometrically integral varieties. In this talk, we show that any descent obstruction for a connected linear group itself is already Brauer-Manin obstruction. As an application, we extend such equivalence to smooth geometrically integral varieties. This is a joint work with Yang Cao.