On generically split generic flag varieties

Abstract

Let G be a split semisimple algebraic group over an arbitrary field F , let E be a G-torsor over F , and let P be a parabolic subgroup of G. The quotient variety X := E/P , known as a flag variety, is generically split, if the parabolic subgroup P is special. It is generic, provided that the G-torsor E over F is a standard generic G k -torsor for a subfield k ⊂ F and a split semisimple algebraic group G k over k with (G k )F = G.For any generically split generic flag variety X, we show that the Chow ring CH X is generated by Chern classes (of vector bundles over X). This implies that the topological filtration on the Grothendieck ring of X coincides with the computable gamma filtration. The results were already known in some cases including the case where P is a Borel subgroup.We also provide a complete classification of generically split generic flag varieties and, equivalently, of special parabolic subgroups for split simple groups.Contents 497 any P -torsor over any extension field of the base field is trivial. This is equivalent to the fact that the generic flag variety X := E/P is generically split, see Lemma 7.1.In the present paper, we show that a generically split generic flag variety X = E/P has the same property as in the above particular case with P = B: the topological filtration on the Grothendieck ring K(X) coincides with the gamma filtration. This confirms that the introduced by Grothendieck gamma filtration is a good approximation of the topological filtration. And not only in the well-known sense that they both coincide after tensoring by Q.The reason for this coincidence (with Z-coefficients) is our main result (Theorem 7.3) saying that the ring CH X coincides with its Chern subring. In its turn, Theorem 7.3 is a consequence of the similar statement for the Chow ring of the classifying space of P , see Proposition 5.5.The assumption that P is special is essential. For instance, taking for G a spinor group Spin(n), we may take a parabolic subgroup P ⊂ G such that the variety X = E/P is a projective quadric. Then the gamma filtration on K(X) is different from the topological filtration provided that n 9. Indeed, as shown in [9], the component of codimension 2 of the graded ring associated with the gamma filtration contains an element of order 2; on the other hand, by [8, Theorem 6.1 and § 3.1], the component of codimension 2 of the graded ring associated with the topological filtration is torsion-free for such n.For an overview of sections of the paper we refer to the table of contents: the titles of sections are self-explaining. Note that the paper ends with a classification of special parabolic subgroups (or, equivalently, of generically split generic flag varieties).

A remark on connective K-theory

“…To get the most from Theorem 3.1, let us put more restrictions on X: assume that X is a generic flag variety (as defined in the introduction) given by a split semisimple group G and a special parabolic subgroup P ⊂ G. By [7,Corollary 7.4], the Chow filtration on K(X) coincides in this case with the Chern filtration. Therefore CK(X) is given by the terms of the Chern filtration as long as Conjecture 1.1 holds for G.…”