ON GENERIC FLAG VARIETIES OF SPIN(11) AND SPIN(12)

NIKITA A. KARPENKO

Abstract. Let X be the variety of Borel subgroups of a split semisimple algebraic group G over a field, twisted by a generic G-torsor. Conjecturally, the canonical epimorphism of the Chow ring CH X onto the associated graded ring GK(X) of the topological filtration on the Grothendieck ring K(X) is an isomorphism. We prove the new cases G = Spin(11) and G = Spin(12) of this conjecture. On an equivalent note, we compute the Chow ring CH Y of the highest orthogonal grassmannian Y for the generic 11- and 12-dimensional quadratic forms of trivial discriminant and Clifford invariant. In particular, we describe the torsion subgroup of the Chow group CH Y and determine its order which is equal to 16 777 216. On the other hand, we show that the Chow group CH_{0} Y of 0-cycles on Y is torsion-free.

Contents

1. Introduction 1 2. Preliminaries 3 3. Main result 3 4. Torsion 6 5. Some other generic quadratic forms 7 References 8

1. Introduction

Let G be a split semisimple algebraic group over a field k. A (standard) generic Gtorsor E is defined as the generic fiber of the quotient map π : GL(N) → GL(N)/G for an integer N ≥ 1 and an embedding of G into the general linear group GL(N). Thus E is a G-torsor over the function field F := k(GL(N)/G). The above quotient map π is a G-torsor with the (called versal) property that every G-torsor over a field extension of k is isomorphic to a fiber of π, [11, §5.3]. This explains the interest to E as to the “most generic” G-torsor over a field. More specifically, we are interested in the variety X := E/B, where B ⊂ G is a Borel subgroup, which is a twisted (by the generic torsor E) form of the flag variety G/B.

Date: 20 September 2017. Key words and phrases. Quadratic forms over fields; algebraic groups; generic torsors; projective

homogeneous varieties; Chow groups. Mathematical Subject Classification (2010): 20G15; 14C25.

This work has been supported by a Discovery Grant from the National Science and Engineering Board of Canada.

Conjecture 1.1 ([6]). For the above variety X = E/B, the canonical epimorphism

CH X →→ GK(X)

of the Chow ring CH X onto the associated graded ring GK(X) of the topological filtration on the Grothendieck ring K(X) is an isomorphism.

We recall the the above epimorphism is an edge effect of the Brown-Gersten-Quillen spectral sequence; its kernel consists of torsion elements (but is not necessarily the whole torsion subgroup of CH X).

Conjecture 1.1, being already proven for simple G of type A and of type C, for special orthogonal groups as well as for some exceptional groups (see [6] for the references and some proofs), is still widely open in the remaining cases, especially in the case of the spinor groups G = Spin(n). For n ≤ 6 the statement is trivial since the group Spin(n) is special for such n meaning that every Spin(n)-torsor over a field is trivial. Therefore X is isomorphic to G/B for n ≤ 6. In particular, X is cellular and the group CH X is torsion-free forcing the epimorphism in question to be an isomorphism.

Up to now, the only known nontrivial cases were n = 7, 8, 9, 10, where the proof was based on a decomposition of the Chow motive of X in a direct sum of shifts of the Rost motive associated to a 3-fold quadratic Pfister forms. The Rost motive in question is a direct summand of the motive of a 3-dimensional quadric Y, the epimorphism CH Y →→ GK(Y) is easily seen to be an isomorphism, and this gives the similar statement for X. Note that the above argument goes through in the more general case where X is given by an arbitrary Spin(n)-torsor, not necessarily the generic one.

The main result of this paper is Theorem 3.1 which settles the cases of n = 11 and n = 12. Although a generalization of the above motivic decomposition is available for any n (see [9]), the summands are shifts of a so-called generalized Rost motive; for n ≥ 11 it is not related to quadrics anymore and is not a direct summand of any variety of dimension < 8. As a matter of fact, in this paper we do not use motives at all. Our method for treating n = 11, 12 actually works for n = 7, 8 as well, delivering a new and “motivically-free” proof.

Note that in general the destination ring GK(X) of the epimorphism in Conjecture 1.1 is explicit at least in the sense that in each concrete case it can, in principle, be calculated by computer. The reason for that is, first of all, that the Grothendieck ring K(X) is computed for an arbitrary projective homogeneous variety X ([8]). This computation alone is not enough, because the topological filtration on K(X) is still quite mysterious for general X. However, for X as in Conjecture 1.1, it coincides with the explicit gamma filtration (see [3, Corollary 7.4]).

As to the Chow ring CH X in the situation of Conjecture 1.1, it comes with a finite system of generators given by Chern classes of certain (linear) bundles on X. Conjecture 1.1 simply means that all relations between the images of these very explicit generators in the explicit ring GK(X) hold already in CH X.

It is worthy to mention that although the construction of the generic torsor E depends on the choice of the embedding G ↪ GL(N), each of the rings CH X and GK(X) is canonically the same for whatever choice of it (see [6, Lemma 2.1]).

2. Preliminaries

We recall that any Spin(n)-torsor over a field gives rise to an n-dimensional nondegenerate quadratic form over the field with the property that its discriminant and Clifford invariant are trivial. A generic Spin(n)-torsor E gives rise to a generic quadratic form q with the above property. Note that we can avoid quadratic forms over fields of characteristic 2, because by [6, Proposition 3.2] we may work over fields of characteristic 0 when proving Conjecture 1.1.

For any given split semisimple G and any special parabolic P ⊂ G, Conjecture 1.1 for G has an equivalent version, where the Borel subgroup B is replaced by P, [7, Lemma 4.2]. For G = Spin(n), we may choose P the way that the variety X = E/P becomes a connected component of the highest orthogonal grassmannian of q. (For odd n, the highest orthogonal grassmannian is connected, but for even n it consists of two connected components isomorphic to each other.) Namely, P is a maximal parabolic subgroup whose conjugacy class is given by the subset of vertices of the Dynkin diagram of Spin(n) obtained by throwing away the very last vertex. We prove Conjecture 1.1 for G = Spin(11), Spin(12) by showing that the epimorphism CH X →→ GK(X) is an isomorphism for this new choice of X.

The input of the proof consists only of the following two properties of q and X. The first property is specific for n ≤ 12 but does not require q to be generic: by Pfister’s theorem [10], any non-degenerate quadratic form of dimension at most 12 with trivial discriminant and Clifford invariant completely splits over some quadratic field extension of the base field. In contrast, the second property holds for arbitrary n = 2m + 1, 2m + 2 and generic q:

Proposition 2.1. For X as above, the Chow ring CH X is generated by the Chern classes c_{i} ∈ CH^{i} X, i = 1, …, m of the tautological (rank-m) vector bundle on X together with an additional generator e ∈ CH^{1} X satisfying the relation 2e = c_{1}.

Proof. We recall that X = E/P, where E is a generic Spin(n)-torsor and P ⊂ Spin(n) is a maximal parabolic subgroup whose conjugacy class is given by the subset of vertices of the Dynkin diagram of Spin(n) obtained by throwing away the very last vertex. For the variety Y := E/B, the ring CH Y is generated by CH^{1} Y, [5, Example 2.4]. The variety Y, considered over X via the projection Y → X, is the variety of complete flags of the tautological vector bundle on X. It follows by [7, Lemma 4.3] that the ring CH X is generated by CH^{1} X together with c_{1}, …, c_{m}. Finally, the group CH^{1} X is infinite cyclic and its subgroup generated by c_{1} is of index 2. □

3. Main result

Theorem 3.1. Let q be the quadratic form over a field F corresponding to a generic Spin(n)-torsor with n = 11 or n = 12 and let X be a connected component of its highest orthogonal grassmannian. Then the epimorphism CH X →→ GK(X) is an isomorphism.

Remark 3.2. Since the Grothendieck group K(X) is torsion-free, Theorem 3.1 implies that the Chow group of 0-cycles CH_{0} X is torsion-free. In fact, a proof of this statement will appear already on an earlier stage of the proof of Theorem 3.1 (see Corollary 3.4). For every n ≥ 13, the similar statement is open. On the other hand, no example is known to the author, where the highest orthogonal grassmannian of a quadratic form possesses nontrivial torsion in its Chow group of 0-cycles.

By Proposition 2.1, the ring CH X is generated by the Chern classes c_{i} ∈ CH^{i} X, i = 1, …, 5 of the tautological (rank-5) vector bundle on X and an additional element e ∈ CH^{1} X satisfying the relation 2e = c_{1}. The complete list of relations for the elements c_{1}, …, c_{5} is as follows (see [2]):

c_{i}^{2} − 2c_{i−1}c_{i+1} + 2c_{i−2}c_{i+2} − ⋯ + (−1)^{i}2c_{0}c_{2i} = 0 for all i ≥ 1.

Therefore, in order to compute the ring CH X, we only need to find the relations involving the remaining generator e.

The group CH X is (additively) generated by the products of powers of the generators e, c_{2}, …, c_{5}, where the exponent of the power of each of c_{2}, …, c_{5} is at most 1. In order to eliminate higher powers of c_{2}, …, c_{5} one uses the above relations and the argument like in [2, proof of Theorem 2.1].

Let us fix a quadratic field extension L/F such that the quadratic form q_{L} is split. The abelian group CH X_{L} is free and the ring CH X_{L} is generated by the elements e_{i} ∈ CH^{i} X_{L}, i = 1, …, 5 satisfying 2e_{i} = c_{i}, where c_{i} ∈ CH^{i} X_{L} is the ith Chern class of the tautological vector bundle on X_{L} and therefore is the image of c_{i} ∈ CH^{i} X, considered previously, under the change of field homomorphism CH^{i} X → CH^{i} X_{L}. (Our elements e_{i} coincide up to their signs with the elements e_{i} of [1, §86].) In particular, e_{1} ∈ CH^{1} X_{L} is the image of e ∈ CH^{1} X.

The above relations on c_{i} imply (via division by 4) that

e_{i}^{2} − 2e_{i−1}e_{i+1} + 2e_{i−2}e_{i+2} − ⋯ + (−1)^{i−1}2e_{1}e_{2i−1} + (−1)^{i}e_{2i} = 0,

where e_{i} is defined to be 0 for i > 5, and this is a complete list of relations (see [12] – the original proof – or [1, Proposition 86.16]). An additive basis of CH X_{L} consists of the 2^{5} products ∏_{i∈I} e_{i}, where I runs over the subsets of {1, …, 5}, [1, Theorem 86.12]. In particular, the product e_{1} … e_{5} ∈ CH^{15} X_{L} = CH_{0} X_{L} is the class of a rational point (and 15 is the dimension of the variety X).

Let us consider the norm homomorphism N = N_{L/F} : CH X_{L} → CH X. This is a group (and not a ring) homomorphism satisfying the projection formula N(res(x)y) = xN(y) with respect to the change of field (ring) homomorphism res = res_{L/F} : CH X → CH X_{L}, [1, Proposition 56.9]. In particular, the image of N is an ideal of the ring CH X. Since the composition

N ∘ res : CH X → CH X

is the multiplication by 2, the ideal N CH X_{L} ⊂ CH X contains 2 CH X. Besides, the restriction of the inverse order composition

res ∘N : CH X_{L} → CH X_{L}

to the subgroup res CH X ⊂ CH X_{L} is also the multiplication by 2. Since 2^{5} CH X_{L} ⊂ res CH X and the group CH X_{L} is torsion-free, the composition res ∘N itself (on the whole CH X_{L}) is the multiplication by 2. In particular, we have the stronger inclusion 2 CH X_{L} ⊂ res CH X, which can also be checked directly.

Here comes the starting observation which is crucial to the whole computation:

Lemma 3.3. The quotient ring CH X/N CH X_{L} is (1) generated by the classes of e, c_{2}, and c_{4};

(2) additively generated by the products e^{i}, e^{i}c_{2}, e^{i}c_{4}, e^{i}c_{2}c_{4} with i = 0, …, 7;

(3) a vector space over 𝔽_{2} of dimension at most 2^{5}.

We will see later (in Corollary 3.5) that the upper bound in (3) is the precise value of the dimension so that the generators in (2) form a basis.

Proof of Lemma 3.3. (1): We need to exclude the generators c_{3} and c_{5}. In order to exclude c_{3}, let us consider the element x := N(e_{3}) ∈ CH^{3} X. As the group CH^{3} X is generated by e^{3}, ec_{2}, c_{3}, we have x = αe^{3} + βec_{2} + γc_{3} for some integers α, β, γ. It follows that res(x) = (α + 2β)e_{1}e_{2} + 2γe_{3}. On the other hand, res(x) = (res ∘N)(e_{3}) = 2e_{3} so that γ = 1 and α = −2β. We get that

N(e_{3}) = c_{3} + βe(c_{2} − 2e^{2})

showing that the class of c_{3} in the quotient ring CH X/N CH X_{L} is a multiple of ec_{2}. This excludes the generator c_{3}.

To exclude c_{5}, we consider x := N(e_{5}) ∈ CH^{5} X. As the group CH^{5} X is generated by e^{5}, e^{3}c_{2}, e^{2}c_{3}, ec_{4}, c_{2}c_{3}, c_{5}, we have

x = α_{5}e^{5} + α_{3}e^{3}c_{2} + α_{2}e^{2}c_{3} + α_{1}ec_{4} + α_{0}c_{2}c_{3} + αc_{5}

for some integers α_{5}, α_{3}, α_{2}, α_{1}, α_{0}, α. It follows that

res(x) ≡ α_{5}(2e_{2}e_{3} − e_{1}e_{4}) + 2α_{3}(e_{1}e_{4}) + 2α_{2}(e_{2}e_{3}) + 2α_{1}(e_{1}e_{4}) + 2αe_{5} (mod 4).

On the other hand, res(x) = (res ∘N)(e_{5}) = 2e_{5} so that α ≡ 1 (mod 2). It follows that the class of c_{5} in the quotient ring is a polynomial in e, c_{2}, c_{3}, c_{4}. This excludes the generator c_{5} and finishes the proof of (1). (2): Clearly, the quotient ring is generated by the indicated elements if we allow all i ≥ 0. We only need to exclude i ≥ 8. Let x := N(e_{3}e_{5}) ∈ CH^{8} X. The group CH^{8} X is generated by the products

e^{8}, e^{6}c_{2}, e^{5}c_{3}, e^{4}c_{4}, e^{3}c_{2}c_{3}, e^{3}c_{5}, e^{2}c_{2}c_{4}, ec_{2}c_{5}, ec_{3}c_{4}.

Looking at the coordinates of the images of the generators in CH^{8} X_{L} (with respect to the basis {∏_{i∈I} e_{i}}_{I⊂{1,...,5}}), we see that the e_{3}e_{5}-coordinate is nonzero modulo 4 for the generator e^{8} only (the e_{3}e_{5}-coordinate for the image of e^{8} is 2). It follows that the class of e^{8} in the quotient ring CH X/N CH X_{L} is a linear combination of the other generators, finishing the proof of (2). (3): Just count the number of the generators in (2). □

Note that the norm map N : CH X_{L} → CH X is injective, its image N CH X_{L} is a free subgroup of rank 2^{5} in CH X.

Corollary 3.4. The group CH_{0} X is torsion-free.

Proof. It follows by by Lemma 3.3 that CH_{0} X = N CH_{0} X_{L} ≃ ℤ. □

Proof of Theorem 3.1. Besides of the norm homomorphism N : CH X_{L} → CH X, we have the norm homomorphisms N : K(X_{L}) → K(X), inducing the norm homomorphism N : GK(X_{L}) → GK(X). The square

CH X_{L} ⟶ GK(X_{L})
N↓  ^{N}↓
CH X ⟶ GK(X)

commutes, inducing an epimorphism

f : CH X/N CH X_{L} →→ GK(X)/NGK(X_{L}).

Since the group CH X_{L} is torsion-free, the epimorphism CH X_{L} →→ GK(X_{L}) is an isomorphism. Since the group GK(X_{L}) is torsion-free, the norm homomorphism GK(X_{L}) → GK(X) is injective. It follows from the commutative diagram

CH X_{L}  ⟶^{isomorphism}  GK(X_{L})
N↓  ^{N}↓^{monomorphism}
CH X  ⟶  GK(X)
↓  ↓
CH X/N CH X_{L}  ⟶^{f} GK(X)/NGK(X_{L})

that CH X → GK(X) is an isomorphism provided that the lower map f is an isomorphism.

The map f is an epimorphism of finite-dimensional 𝔽_{2}-vector spaces. We prove that it is an isomorphism comparing the dimensions of the spaces. By Lemma 3.3(3), we already have the upper bound 2^{5} on the dimension of the space on the left. The 𝔽_{2}-vector space GK(X)/NGK(X_{L}) on the right is the associated graded space for some filtration on the 𝔽_{2}-vector space K(X)/NK(X_{L}) and therefore has the same dimension as K(X)/NK(X). By [8], the change of field homomorphism K(X) → K(X_{L}) is an isomorphism so that NK(X) = 2K(X). The abelian group K(X) = K(X_{L}) is free of the same rank as CH X_{L}, i.e. of the rank 2^{5}. Thus dim_{𝔽_{2}} GK(X)/NGK(X_{L}) = dim_{𝔽_{2}} K(X)/2K(X) = 2^{5}. □

As a byproduct, we made the statement of Lemma 3.3 more precise:

Corollary 3.5. The 𝔽_{2}-vector space CH X/N CH X_{L} has dimension 2^{5} and the products e^{i}, e^{i}c_{2}, e^{i}c_{4}, e^{i}c_{2}c_{4} with i = 0, …, 7 form its basis. □

This corollary is used in the next section for determination of the torsion subgroup.

4. Torsion

For X as in Theorem 3.1, we can now describe the torsion subgroup of CH X:

Proposition 4.1. The torsion subgroup of CH X is an 𝔽_{2}-vector space of dimension 24; its (homogeneous) basis is given by the elements

e^{i}(c_{2} − 2e^{2}), e^{i}(c_{4} − 2ec_{3} + 2e^{4}), e^{i}(c_{2} − 2e^{2})(c_{4} − 2ec_{3} + 2e^{4}) with i = 0, …, 7.

Proof. The torsion subgroup of CH X vanishes over L and therefore has exponent 2.

The indicated elements, vanishing in CH X_{L}, are of exponent 2 in CH X as well. They are linearly independent over 𝔽_{2} because their images in the quotient CH X/N CH X_{L} are the classes of e^{i}c_{2}, e^{i}c_{4}, e^{i}c_{2}c_{4} (i = 0, …, 7) which are linearly independent by Corollary 3.5. Thus we have constructed a torsion subgroup in CH X of dimension 24. We finish the proof of Proposition 4.1 by showing that 24 is the dimension of the total torsion subgroup of CH X.

The total torsion subgroup of CH X coincides with the kernel of the change of field homomorphism CH X → CH X_{L}. Using the isomorphism CH X → GK(X), we identify it with the kernel of GK(X) → GK(X_{L}). By [4, Proposition 2], the order of the kernel of the latter map coincides with the order of its cokernel divided by the order of the cokernel of K(X) → K(X_{L}). Since K(X) → K(X_{L}) is an isomorphism and replacing GK(X) back to CH X, it remains to compute the order of the cokernel of the change of field homomorphism res : CH X → CH X_{L}.

To simplify the computation, we first recall that the image of res contains 2 CH X_{L}. The quotient CH X_{L}/2 CH X_{L} is an 𝔽_{2}-vector space with the basis {e_{I} := ∏_{i∈I} e_{i}}_{I⊂{1,...,5}}. For any i, the image of c_{i} in CH X_{L}/2 CH X_{L} is trivial and the image of e^{i} is e_{I} for the set I of the 2-powers with the sum i. It follows that the image of res in CH X_{L}/2 CH X_{L} is generated by the part of the basis with I ⊂ {1, 2, 4}. Therefore the dimension we want to compute is 2^{5} − 2^{3} = 24. □

Remark 4.2. Let X be the highest orthogonal grassmannian of an arbitrary non-degene-rate 11-dimensional quadratic form q of trivial Clifford invariant with the property that the ring CH X is generated by CH^{1} X and Chern classes of the tautological vector bundle. Then all results of this and the previous sections hold for X because their proofs do not involve any other condition on X. This observation is exploited in the next (and last) section.

5. Some other generic quadratic forms

We fix an odd integer n ≥ 3. Let k be a field and let F be the rational function field over k in variables t_{ij}, 1 ≤ i ≤ j ≤ n. Let q : F^{n} → F be the quadratic form on the vector space F^{n} given by the formula

q(x_{1}, …, x_{n}) = ∑_{1≤i≤j≤n} t_{ij}x_{i}x_{j}.

Let X be the highest orthogonal grassmannian of q and let Y be the Severi-Brauer variety of the even Clifford algebra of q.

Proposition 5.1. The ring CH X_{F (Y)} is generated by CH^{1} X_{F (Y)} together with the Chern classes of the tautological vector bundle.

Proof. By [2], the ring CH X is generated by the Chern classes of the tautological vector bundle. The projection X × Y → X is a projective bundle so that the CH X-algebra CH(X × Y) is generated by CH^{1}(X × Y). And the pull-back CH(X × Y) → CH X_{F (Y)} with respect to the generic point of Y is an epimorphism of graded CH X-algebras. □

Corollary 5.2. For n ≤ 11, the epimorphism CH X_{F (Y)} →→ GK(X_{F (Y)}) is an isomorphism.

Proof. For n < 11, the statement is known (see also §1). For n = 11 it is new and follows by Remark 4.2. □

Remark 5.3. For k of characteristic ≠ 2, one may replace q by the diagonal quadratic form as in [2, §8]. For k of characteristic 2, one may replace q by the quadratic form as in [2, §9].

Remark 5.4. The quadratic form q is a subform of a unique (up to an isomorphism) non-degenerate (n+1)-dimensional quadratic form q′ of trivial discriminant. The Clifford invariants of q and q′ coincide and the connected component X′ of the highest grassmannian of q′ is isomorphic to X (see [1, Proposition 85.2]). Therefore Proposition 5.1 and Corollary 5.2 also hold for X′ in place of X.

References

[1] Elman, R., Karpenko, N., and Merkurjev, A. The algebraic and geometric theory of quadratic forms, vol. 56 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2008.

[2] Karpenko, N. A. On generic quadratic forms. Linear Algebraic Groups and Related Structures (preprint server) 585 (10 June 2017), 10 pages.

[3] Karpenko, N. A. On generically split generic flag varieties. Linear Algebraic Groups and Related Structures (preprint server) 583 (29 April 2017), 17 pages.

[4] Karpenko, N. A. On topological filtration for Severi-Brauer varieties. In K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), vol. 58 of Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 1995, pp. 275–277.

[5] Karpenko, N. A. Chow groups of some generically twisted flag varieties. Annals of K-Theory 2, 2 (2017).

[6] Karpenko, N. A. Chow ring of generic flag varieties. Math. Nachr. (2017), Early View.

[7] Karpenko, N. A. Chow ring of generically twisted varieties of complete flags. Adv. Math. 306 (2017), 789–806.

[8] Panin, I. A. On the algebraic K-theory of twisted flag varieties. K-Theory 8, 6 (1994), 541–585.

[9] Petrov, V., Semenov, N., and Zainoulline, K. J-invariant of linear algebraic groups. Ann. Sci. École Norm. Sup. (4) 41 (2008), 1023–1053.

[10] Pfister, A. Quadratische Formen in beliebigen Körpern. Invent. Math. 1 (1966), 116–132.

[11] Serre, J.-P. Cohomological invariants, Witt invariants, and trace forms. In Cohomological invariants in Galois cohomology, vol. 28 of Univ. Lecture Ser. Amer. Math. Soc., Providence, RI, 2003, pp. 1–100. Notes by Skip Garibaldi.

[12] Vishik, A. On the Chow groups of quadratic Grassmannians. Doc. Math. 10 (2005), 111–130 (electronic).

Mathematical & Statistical Sciences, University of Alberta, Edmonton, CANADA

E-mail address: karpenko at ualberta.ca, web page:  www.ualberta.ca/~karpenko