Generic Flag Varieties

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Generic Flag Varieties It is a very common situation in Algebra that a type of objects is considered over field extensions of some fixed base field. Among the objects of the given type, there is usually a particularly simple one -- the split object. Its properties are well-understood and the problems of interest are easily solved for it. The nature of a general object of the given type is usually too complicated to be addressed directly. An important step towards its understanding is the study of the so-called generic object. This can be considered as the farthest opposite to the split one. It is characterized by the property that every object is its specialization. The use of generic objects is an important ingredient in many proofs. Surprisingly, generic objects are more accessible than the general (i.e., arbitrary) ones. At the same time, information on generic objects provides us with some important information on the general objects. Also generic objects provide a testing ground for general conjectures. This project suggests the study of generic objects arising from algebraic groups as described below. It is again a common situation in Algebra that objects of the given type are classified by G-torsors (or principle G-homogeneous spaces), for certain split semisimple algebraic group G. In such situations, the study of generic objects becomes the study of generic G-torsors. The main objective of the present project is to achieve a complete understanding of the Chow ring CH(E/B), where B is a Borel subgroup of G. The quotient variety E/B is a smooth projective variety known as a generic flag variety. The Grothendieck ring K(E/B) of the variety E/B is endowed with the filtration by codimension of support. Let GK(E/B) be the associated graded ring. Main Question on the Chow ring CH(E/B) asks whether the canonical surjective homomorphism of graded rings CH(E/B) onto GK(E/B) is an isomorphism. Equivalently, it asks if the connective K-theory of E/B is free of torsion. In this project, I want to study this question and some related problems. A breakthrough in this direction will be a major achievement in the field of algebraic groups. The question has already been answered by positive for several types of G. As a consequence, the Chow ring as well as the connective K-theory are completely understood for such G. On the other hand, it has been recently shown that the answer is negative for the spinor group Spin(17). One of the related problems is the problem of comparison of the Chow ring of the classifying space BG of G with the representation ring R(G) viewed as the K-theory of BG. In a recent joint work with A. Merkurjev, this problem has been approached by introducing an analogue of the filtration by codimension of support on R(G). For further progress, we plan to construct a G-equivariant version of the connective K-theory and, in particular, the connective K-theory of BG.