Twisted projective homogeneous varieties for algebraic groups are projective varieties which are isomorphic to a given projective homogeneous variety after extension to the separable closure of the field.  Important examples include Severi Brauer varieties, which are twisted forms of projective space; generalised Severi Brauer varieties, which are twisted forms of Grassmannians.
We investigate conditions under which  generalised Severi Brauer varieties have rational subvarieties which are forms of given Schubert subvarieties of the associated Grassmannian. For regular Grassmannians, and more generally for any projective homogeneous variety for an algebraic group, the classes of the Schubert subvarieties of a given codimension give a basis for the Chow groups of that codimension.  The Chow groups, even
in codimension 2, for twisted projective homogeneous varieties, are less well understood.
We use our results as well as an analysis of the Chow motive and the topological filtration of the Grothendieck group of generalised Severi Brauer varieties to show that the codimension 2 Chow groups of  certain generalised Severi Brauer varieties  are torsion free.  This is joint work with Caroline Junkins and Danny Krashen.

Einladung von Prof. Dmitry Feichtner-Kozlov (ALTA)