Zero cycles on twisted Cayley plane

V. Petrov, N. Semenov, K. Zainoulline

August 8, 2005

Abstract

In the present paper we compute the group of zero-cycles modulo rational equivalence of a twisted form X of a Cayley plane. More precisely, we show that the degree map deg : CH_{0}(X) → ℤ is injective.

1 Introduction

Let G be a split exceptional group of type E_{6} and G/P a projective Ghomogeneous variety with a stabilizer given by a maximal parabolic subgroup P corresponding to the first (or last) root of the respective Dynkin diagram. This variety is called Cayley plane and is denoted by 𝕆ℙ^{2}. Its geometry is a subject of intensive investigations (see [IM05]).

Let X be a twisted form of the Cayley plane, i.e., a smooth projective variety over a field k which becomes isomorphic to 𝕆ℙ^{2} over the separable closure of k. In this case the variety X can have no rational points and its geometry turns to be very complicated.

In the present paper we compute the group of zero-cycles CH_{0}(X) which is an important geometric invariant of a variety. Namely, we prove:

Theorem. Let k be a field of characteristic different from 2 and 3. Let X be a twisted form of a Cayley plane corresponding to a group of inner type E_{6} over k with trivial Tits algebras. Then the degree map CH_{0}(X) → ℤ is injective.

The history of the question starts with the work of I. Panin [Pa84] where he proved the injectivity of the degree map for Severi-Brauer varieties. For quadrics this was proved by R. Swan in [Sw89]. The case of involutive varieties was considered by A. Merkurjev in [Me95]. For varieties of type F_{4} it was announced by M. Rost.

Our work was mostly motivated by the paper of D. Krashen [Kr05] where he reformulated the question above in terms of R-triviality of certain symmetric powers and proved the injectivity for a wide class of generalized Severi-Brauer varieties and some involutive varieties, hence, generalizing the previously known results by Panin and Merkurjev. Another motivating point was the result of V. Popov [Po05] where he gave a full classification of generically n-transitive actions of a split linear algebraic group G on a projective homogeneous variety G/P. The case of 𝕆ℙ^{2} = E_{6}/P_{6} which is the main point of our investigations provides an example of such an action for n = 3. As a consequence, one can identify the open orbit X^{o} of this action with a homogeneous variety E_{6}/T · Spin_{8}, where T is the torus which is complementary to Spin_{8}. Then the result of Krashen reduces the question of injectivity to the question of R-triviality of the twisted form of X^{o}/S_{3}.

Apart from the main result concerning exceptional varieties we give shortened proofs for injectivity of the degree map for quadrics and Severi-Brauer varieties as well.

As it was mentioned to us by V. Chernousov and M. Rost, it is possible to prove the same results using Rost invariant and Chain Lemma. Our proof doesn’t use these tools but only the geometry and several classical results concerning exceptional groups.

2 Zero cycles and symmetric powers

In this section we collect some facts from [Kr05] on the relations between zero cycles on a projective variety X and R-equivalence classes on symmetric powers of X.

2.1. We systematically use the Galois descent language, i.e., identify a (quasiprojective) variety X over k with the variety X_{k_{sep}} over the separable closure k_{sep} equipped with the action (by semiautomorphisms of X_{k_{sep}}) of the absolute Galois group Γ = Gal(k_{sep}/k). It means that for any σ ∈ Γ we are given by an automorphism φ_{σ} of X_{k_{sep}} over k such that the diagram

\xymatrix{X_{k_{sep}} \ar[d] \ar[r]^{φ_{σ}} & X_{k_{sep}} \ar[d] \\ Spec k_{sep} \ar[r]^{σ^{♯}} & Spec k_{sep}}

commutes, and φ_{στ} = φ_{σ}φ_{τ}. The set of k-rational points of X is precisely the set of k_{sep}-rational points of X_{k_{sep}} stable under the action of Γ.

2.2. Let X be a variety over k. Two rational points p, q ∈ X(k) are called elementary linked if there exists a rational morphism φ: ℙ_{k}^{1} −−→ X such that p, q ∈ Im(φ(k)). The R-equivalence is the equivalence relation generated by this relation. A variety X is called R-trivial if there exists exactly one class of R-equivalence on X, and algebraically R-trivial if X_{K} = X ×_{Spec k} Spec K is R-trivial for any finite field extension K/k.

The n-th symmetric power of X is by definition the variety S^{n}X = X^{n}/S_{n} where S_{n} is the symmetric group acting on X^{n} via permutations.

Let p be a prime number. A field k is called prime to p closed if there is no finite field extension K/k of degree prime to p. For any field k we will denote by k_{p} its prime to p closure that is the algebraic extension of k which is prime to p closed.

Let X be a projective variety over k. By C̃H_{0}(X) we denote the kernel of the degree map:

C̃H_{0}(X) = Ker(deg: CH_{0}(X) → ℤ).

The crucial role in the sequel will play the following results:

2.3 Proposition. ([Kr05, Lemma 1.3]) If C̃H_{0}(X_{k_{p}}) = 0 for each prime p then C̃H_{0}(X) = 0.

2.4 Proposition. ([Kr05, Theorem 1.4]) Suppose that k is prime to p closed and the following conditions are satisfied:

1. S^{p^{n}}X is algebraically R-trivial for some n ≥ 0,

2. For any field K/k such that X(K) ≠ ∅ X_{K} is R-trivial.

Then C̃H_{0}(X) = 0.

2.5. As an easy application we give a sketch of a proof of the well-known result (first appeared in [Pa84, Theorem 2.3.7]) that C̃H_{0}(SB(A)) = 0 where A is a central simple algebra (a more general case of flag varieties is settled in [Kr05]). For simplicity we assume that deg A = p is prime.

We may assume that k is prime to p closed by Proposition 2.3 (for primes q different from p A splits over k_{q}). By Proposition 2.4 it is enough to show that S^{p} SB(A_{K}) is R-trivial for every finite extension K/k (the second hypothesis holds for any twisted flag variety). Making base change we may assume K = k. If A splits, the assertion is trivial; so we assume that A is not split.

According to our conventions (2.1) the variety SB(A) is the variety of all parabolic subgroups P of type P_{1} in PGL_{1}(A ⊗_{k} k_{sep}) with the action of Γ coming from the action on k_{sep}. Therefore S^{p}X is the variety of all unordered p-tuples [P^{(1)}, …, P^{(p)}] of parabolic subgroups of type P_{1} in PGL_{1}(A⊗_{k} k_{sep}). Let U be the open subvariety of S^{p}X defined by the condition that the intersection P^{(1)} ∩ … ∩ P^{(p)} is a maximal torus in PGL_{1}(A ⊗_{k} k_{sep}). Every maximal torus T in PGL_{1}(A ⊗_{k} k_{sep}) is contained in precisely p parabolic subgroups of type P_{1}, whose intersection is T. Therefore U is isomorphic to the variety of all maximal tori in PGL_{1}(A). This variety is known to be rational (and therefore R-trivial since it is homogeneous). Moreover, one can check that if A is not split then the embedding U → S^{p}X is surjective on k-points. So S^{p}X is R-trivial, and we are done.

2.6. The same method can be applied to prove that C̃H_{0}(Q) = 0 for a nonsingular projective quadric Q over a field of characteristic not 2 (the result of Swan [Sw89], see [Me95] and [Kr05] for generalizations).

As above, we may assume that p = 2, Q is anisotropic, and we have to prove that S^{2}Q is R-trivial. Let q be a corresponding quadratic form on a vector space V. Q is the variety of all lines ⟨v⟩ where v ∈ V ⊗_{k} k_{sep} satisfies q(v) = 0, with the obvious action of Γ. So S^{2}Q can be identified with the variety of pairs [⟨v_{1}⟩, ⟨v_{2}⟩] of lines of this kind, with induced action of Γ. Consider the open subvariety U defined by the condition b_{q}(v_{1}, v_{2}) ≠ 0 (b_{q} stands for the polarization of q). Clearly the embedding U → S^{2}Q is surjective on k-points (otherwise the subspace ⟨v_{1}, v_{2}⟩ defines a totally isotropic subspace of V defined over k). So it is enough to check that U is R-trivial.

Consider the open subvariety W of Gr(2, V) consisting of planes H ≤ V ⊗_{k} k_{sep} such that q|_{H} is nonsingular. For every such a plane there exists up to scalar factors exactly one hyperbolic base {v_{1}, v_{2}} over k_{sep}. Therefore the map from U to W sending [⟨v_{1}⟩, ⟨v_{2}⟩] to ⟨v_{1}, v_{2}⟩ is an isomorphism. But any open subvariety of Gr(2, V) is R-trivial, and we are done.

We will use in the sequel the following observation.

2.7 Lemma. Let H ≤ K ≤ G be algebraic groups over k. Suppose that the map H^{1}(k, H) → H^{1}(k, K) is surjective. Then the morphism G/H → G/K is surjective on k-points.

Proof. An element x of G/K(k) is presented by an element g ∈ G(k_{sep}) satisfying the condition γ(σ) = g^{−1} ·^{σ}g lies in K(k_{sep}) for all σ ∈ Γ. But γ is clearly a 1-cocycle with coefficients in K. Therefore by the assumption there exists some h ∈ K such that h^{−1}γ(σ) ·^{σ}h = (gh)^{−1} ·^{σ}(gh) is a 1-cocycle with coefficients in H. But then gh presents an element of G/H(k) which goes to x under the morphism G/H → G/K, and the lemma is proved.

3 The twisted Cayley Plane

From now on we assume that char k ≠ 2, 3. In the present section we prove the theorem of the introduction.

3.1. Let J be a simple exceptional 27-dimensional Jordan algebra over k, N_{J} be its norm (which is a cubic form on J). An invertible linear map f: J → J is called a similitude if there exists some α ∈ k^{∗} (called the multiplier of f) such that N_{J}(f(v)) = αN_{J}(v) for all v ∈ J. The group G = Sim(J) of all similitudes is a reductive group of inner type E_{6}, and every group of inner type E_{6} with trivial Tits algebras can be obtained in this way up to isogeny (see [Ga01, Theorem 1.4]).

The (twisted) Cayley plane 𝕆ℙ^{2}(J) is the variety obtained by the Galois descent from the variety of all parabolic subgroups of type P_{1} (enumeration of roots follows Bourbaki) in Sim(J⊗_{k}k_{sep}) (the action of Γ = Gal(k_{sep}/k) comes from the action on k_{sep}). This variety can be identified with the variety of all lines ⟨e⟩ spanned by elements e ∈ J ⊗ k_{sep} satisfying the condition e × e = 0 (see [Ga01, Theorem 7.2]).

Theorem. C̃H_{0}(𝕆ℙ^{2}(J)) = 0.

Proof.

3.2. Note that if 𝕆ℙ^{2}(J)(K) ≠ ∅ for some extension K/k then 𝕆ℙ^{2}(J ⊗_{k} K) is rational and, since it is homogeneous, is R-trivial.

Let p be a prime number. If p ≠ 3 then 𝕆ℙ^{2}(J)(k_{p}) ≠ ∅. Indeed, choose any cubic étale subalgebra L of J ([Inv, Proposition 39.20]). It splits over k_{p}, and therefore L ⊗_{k} k_{p} contains a primitive idempotent e. As an element of J ⊗_{k} k_{p} it satisfies the condition e × e = 0 (see [SV, Lemma 5.2.1(i)]).

So we assume p = 3. We have to prove that S^{3}𝕆ℙ^{2}(J ⊗_{k} K) is R-trivial for all finite field extensions K/k. Making the base change, it suffices to prove it for K = k. Moreover, we assume that J is not reduced (otherwise 𝕆ℙ^{2}(J) is rational).

3.3. S^{3}(𝕆ℙ^{2}(J)) is the variety of all unordered triples [⟨e_{1}⟩, ⟨e_{2}⟩, ⟨e_{3}⟩] where e_{i} are elements of J ⊗_{k} k_{sep} satisfying the conditions e_{i} × e_{i} = 0, with the natural action of Γ. Denote by U the open subvariety of 𝕆ℙ^{2}(J) defined by the condition

N_{J⊗_{k}k_{sep}}(e_{1}, e_{2}, e_{3}) ≠ 0

(we denote by N the polarization of the norm as well).

The embedding U → S^{3}(𝕆ℙ^{2}(J)) is surjective on k-points. Indeed, if [⟨e_{1}⟩, ⟨e_{2}⟩, ⟨e_{3}⟩] is stable under the action of Γ and N_{J⊗_{k}k_{sep}}(e_{1}, e_{2}, e_{3}) = 0, then ⟨e_{1}, e_{2}, e_{3}⟩ gives by descent a k-defined subspace V of J such that N|_{V} = 0. But then J is reduced ([SV, Theorem 5.5.2]), which leads to a contradiction. So it is enough to show that U is R-trivial.

3.4. Choose a cubic étale subalgebra L in J. Over the separable closure we have

L ⊗_{k} k_{sep} = k_{sep}e_{1} ⊕ k_{sep}e_{2} ⊕ k_{sep}e_{3}

where e_{1}, e_{2}, e_{3} are primitive idempotents. Then e_{i} × e_{i} = 0 in J ⊗_{k} k_{sep},

N_{J⊗_{k}k_{sep}}(e_{1}, e_{2}, e_{3}) = N_{L⊗_{k}k_{sep}}(e_{1}, e_{2}, e_{3}) = 1,

and the triple [e_{1}, e_{2}, e_{3}] is stable under Γ since L is so. Therefore [⟨e_{1}⟩, ⟨e_{2}⟩, ⟨e_{3}⟩] is a k-rational point of U.

By [SV68, Proposition 3.12] G acts transitively on U (in geometric sense, that is over a separable closure). Therefore

U ≃ G/ Stab_{G}([⟨e_{1}⟩, ⟨e_{2}⟩, ⟨e_{3}⟩]).

This stabilizer is defined over k (since it is stable under Γ) and coincides with Stab_{G}(L) (one inclusion is obvious, and the other one follows from the fact that e_{1}, e_{2}, e_{3} are the only elements e of L ⊗_{k} k_{sep} satisfying the condition e × e = 0, up to scalar factors (see [SV, Theorem 5.5.1]).

3.5. Consider the Springer decomposition of J respect to L: J = L ⊕ V. Then the pair (L, V) has a natural structure of a twisted composition, and there is a monomorphism Aut(L, V) → Aut(J) sending a pair (φ, t) (where φ: L → L, t: V → V) to φ ⊕ t: J → J (see [Inv, § 38.A]). Note that Aut(L, V) coincides with the stabilizer of L in Aut(J).

Show that there exists an exact sequence of algebraic groups:

1 ⟶ Aut(L, V) ⟶ Stab_{G}(L) ⟶ R_{L/k}(𝔾_{m}) ⟶ 1,
f ↦ f(1)

where R_{L/k} stands for the Weil restriction. Exactness at the middle term follows from the considerations above and the fact that the stabilizer of 1 in G coincides with Aut(J) ([SV, Proposition 5.9.4]).

Prove the exactness at the last term. A k_{sep}-point of R_{L/k}(𝔾_{m}) is a triple of scalars (α_{0}, α_{1}, α_{2}) ∈ k_{sep}^{∗}×k_{sep}^{∗}×k_{sep}^{∗}. Multiplying f by the scalar transformation with the coefficient (α_{0}α_{1}α_{2})^{\frac{1}{3}} (which is an element of Stab_{G}(L)(k_{sep})) we may assume that α_{0}α_{1}α_{2} = 1. Choose a related triple (t_{0}, t_{1}, t_{2}) of elements of GO^{+}(𝕆_{d}, N_{𝕆_{d}}) (𝕆_{d} is the split Cayley algebra) such that μ(t_{i}) = α_{i}, i = 0, 1, 2 (see [Inv, Corollary 35.5]). Now the transformation f of J defined by

(\begin{matrix}ε_{0} & c_{2} & · \\ · & ε_{1} & c_{0} \\ c_{1} & · & ε_{2} \end{matrix}) ↦ (\begin{matrix}α_{0}ε_{0} & t_{2}(c_{2}) & · \\ · & α_{1}ε_{1} & t_{0}(c_{0}) \\ t_{1}(c_{1}) & · & α_{2}ε_{2} \end{matrix})

lies in Sim(J) by [Ga01, (7.3)], stabilizes L_{k_{sep}} = diag(k_{sep}, k_{sep}, k_{sep}), and sends 1 ∈ J_{k_{sep}} to diag(α_{0}, α_{1}, α_{2}).

Since H^{1}(k, L^{∗}) = 1 by Hilbert ’90, the map H^{1}(k, Aut(L, V)) → H^{1}(k, Stab_{G}(L)) is surjective. By Lemma 2.7 the morphism

G/ Aut(L, V) → G/ Stab_{G}(L) ≃ U

is surjective on k-points. Therefore it suffices to show that G/ Aut(L, V) is R-trivial.

3.6. Consider the morphism

ψ: G/ Aut(L, V) → G/ Aut(J).

By [Kr05, Corollary 3.14] it suffices to show that 1. ψ is surjective on k-points;

2. G/ Aut(J) is R-trivial;

3. The fibers of ψ (which are isomorphic to Aut(J)/ Aut(L, V)) are unirational and R-trivial.

3.7. In order to prove surjectivity of ψ on k-points it is enough by Lemma 2.7 to prove surjectivity of the map H^{1}(k, Aut(L, V)) → H^{1}(k, Aut(J)). Now H^{1}(k, Aut(L, V)) classifies all twisted compositions (L′, V′) which become isomorphic to (L, V) over k_{sep}, and H^{1}(k, Aut(J)) classifies all (exceptional 27-dimensional) Jordan algebras J′. It is easy to verify that the morphism sends (L′, V′) to the Jordan algebra L′⊕V′, and hence the surjectivity follows from the fact that any Jordan algebra admits a Springer decomposition (cf. [Inv, Proposition 38.7]).

3.8. Let W be the open subvariety of J consisting of elements v with N_{J}(v) ≠ 0. Then G acts transitively (in geometric sense) on W (see [SV, Proposition 5.9.3]), and the stabilizer of the point 1 coincides with Aut(J). So G/ Aut(J) ≃ W is clearly R-trivial.

3.9. Consider the variety Y of all étale cubic subalgebras of J. By [Inv, Proposition 39.20(1)] there is a map from an open subvariety J_{0} of regular elements in J to Y (sending a to k[a]), surjective on k-points. Therefore Y is unirational and R-trivial 24-dimensional irreducible variety.

Aut(J) acts on Y naturally. Let L′ be any k-point of Y. The stabilizer of L′ in Aut(J) obviously equals Aut(L′, V′) (J = L′ ⊕ V′ is the Springer decomposition). So the orbit of L′ is isomorphic to Aut(J)/ Aut(L′, V′) and, in particular, has dimension 24. Therefore it is open, and since L′ is arbitrary, the action is transitive. So we have Aut(J)/ Aut(L, V) ≃ Y is unirational and R-trivial, and the proof is completed.

3.10 Remark. The main result of the paper also implies the injectivity of the degree map for the twisted form Y of F_{4}/P_{4}. To see this one observes that (if the splitting field has degree p = 3) both varieties 𝕆ℙ^{2}(J) and Y split completely over the function fields of each other. This together with [Me, Cor. 3.6] implies that the groups of zero-cycles are isomorphic.

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