Chow ring of generic flag varieties

Karpenko, Nikita A.

doi:10.1002/mana.201600529

Cited by 10 publications

(8 citation statements)

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“…For the variety of Borel subgroups X = E/B, twisted by E (i.e., E/B is a twisted form of G/B, where B ⊂ G is a Borel subgroup), Karpenko conjectured that the morphism φ is an isomorphism in [4], where he confirmed the conjecture for a simple group G of type A or C. Indeed, by [4,Lemma 4.2] this conjecture is equivalent to the same statement, after replacing the Borel subgroup B by a special parabolic subgroup P of G (i.e., any P -torsor over any field extension of k is trivial). Moreover, in [3,Theorem 1.2] he showed that the conjecture holds for a wider class of groups G including special orthogonal groups and the exceptional groups of types G 2 , F 4 , and simply connected E 6 .…”

Section: Introductionmentioning

confidence: 99%

Maximal orthogonal grassmannians of quadratic forms of dimensions up to $22$

Baek¹

2021

Preprint

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Let X be a connected component of the maximal orthogonal grassmannian of a generic n-dimensional quadratic form q with trivial Clifford invariant. Consider the canonical epimorphism φ from the Chow ring of X to the associated graded ring of the coniveau filtration on the Grothendieck ring of X. In [6] Karpenko proved that φ is an isomorphism for all n ≤ 12 (conjecturally for all n). Recently, in [10] Yagita showed that φ is not an isomorphism for n = 17, 18. In the present paper, together with Yagita's results for n = 17, 18, we show that the map φ is not an isomorphism for all 13 ≤ n ≤ 22. In particualr, the case n = 13 gives the smallest dimensional maximal orthogonal grassmannian whose two graded rings are not isomorphic.

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Section: Introductionmentioning

confidence: 99%

Maximal orthogonal grassmannians of quadratic forms of dimensions up to $22$

Baek¹

2021

Preprint

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“…Let

B

be a Borel subgroup of

G

. There is a conjecture (see [, Conjecture 1.1]) that for the variety

X : = E / B

– a generic variety of complete flags — the canonical epimorphism of the Chow ring

CH X

onto the associated graded ring

G K (X)

of the topological filtration on the Grothendieck ring

K (X)

is an isomorphism. Since the ring

K (X)

is computed and the topological filtration on it is known to coincide with the computable gamma filtration [, Example 2.4], this conjecture, proved in several cases already, provides a way to compute the Chow ring.…”

Section: Introductionmentioning

confidence: 99%

On generically split generic flag varieties

Karpenko

2018

Bull. London Math. Soc.

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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).

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“…Moreover we take G a versal G k -torsor i.e., G is isomorphic to G k(S) -torsor over Spec(k(S)) given by the generic fiber of GL N → S where S = GL N /G k for an embedding G ⊂ GL N ( for the properties of a versal G k -torsor, see §3 below or see [Ga-Me-Se], [To2], [Me-Ne-Za], [Ka1]). When G/B k = F is a versal flag variety, it is known ([Me-Ne-Za], [Ka1]) that CH * (F) is generated by Chern classes. Hence we know gr * γ (F) ∼ = gr * geo (F).…”

Section: Introductionmentioning

confidence: 99%

“…Hence we know gr * γ (F) ∼ = gr * geo (F). Karpenko conjectures that the above I(1) = 0 ( [Ka1], [Ka2]). In this paper, we try to compute gr * γ (G/T ) for G = Spin(2ℓ + 1).…”

Section: Introductionmentioning

confidence: 99%