CANONICAL DIMENSION OF ORTHOGONAL GROUPS

NIKITA A. KARPENKO

Abstract. We prove Berhuy-Reichstein’s conjecture on the canonical dimension of orthogonal groups showing that for any integer n ≥ 1, the canonical dimension of SO_{2n+1} and of SO_{2n+2} is equal to n(n + 1)/2. More precisely, for a given (2n + 1)-dimensional quadratic form ϕ defined over an arbitrary field F of characteristic ≠ 2, we establish certain property of the correspondences on the orthogonal grassmanian X of n-dimensional totally isotropic subspaces of ϕ, provided that the degree over F of any finite splitting field of ϕ is divisible by 2^{n}; this property allows to prove that the function field of X has the minimal transcendence degree among all generic splitting fields of ϕ.

1. Results

Let F be an arbitrary field of characteristic ≠ 2, ϕ a non-degenerate (2n+1)-dimensional quadratic form over F (with n ≥ 1), X the orthogonal grassmanian of n-dimensional totally isotropic subspaces of ϕ. The variety X is projective, smooth, and geometrically connected; dim X = n(n + 1)/2. We write d(X) for the greatest common divisor of the degrees of all closed points on X.

In this paper, a field extension E/F is called a splitting field of ϕ, if the Witt index (see [9] for the definition of the Witt index of a quadratic form) of the form ϕ_{E} is maximal (i.e., equal to n). Note that a field extension E/F is a splitting field of ϕ, if and only if the set X(E) is non-empty. We write d(ϕ) for the greatest common divisor of the degrees of all finite splitting fields of ϕ.

Clearly, d(ϕ) = d(X). Moreover, this integer is a power of 2 not exceeding 2^{n}. The equality d(ϕ) = 2^{n} holds if, for example, the even Clifford algebra C_{0}(ϕ) of the quadratic form ϕ is a division algebra. Of course, it is so for the (2n + 1)-dimensional generic quadratic form ⟨t_{1}, …, t_{2n+1}⟩ (defined over the field F (t_{1}, …, t_{2n+1}) of rational functions in variables t_{1}, …, t_{2n+1}).

A splitting field L/F of ϕ is called generic, if it is finitely generated and for any splitting field E/F and any non-zero element a ∈ L there exists an F -place f : L → E such that f(a) is neither 0 nor ∞. The function field F (X) is a generic splitting field of ϕ. In fact, it is even very generic in the sense of [1] (where it is also explained how the “very generic” property implies the “generic” one): indeed, if E/F is a splitting field, the variety X_{E} is rational (as any projective homogeneous variety with a rational point is) and therefore F (X) is contained in a purely transcendental extension of E (in E(X) namely).

Following [1], we define the canonical dimension cd(ϕ) of ϕ as the minimum of the transcendence degrees of all generic splitting fields of ϕ (the canonical dimension of SO_{2n+1} is then the maximum of cd(ϕ) when ϕ runs over all (2n + 1)-dimensional quadratic forms over (finitely generated) extensions of F; the canonical dimension of SO_{2n+2} coincides with the canonical dimension of SO_{2n+1}, see [1]).

Date: July 9, 2004.

Supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00287, KTAGS.

Our main result here reads as follows:

Theorem 1.1. If d(ϕ) = 2^{n}, then cd(ϕ) = n(n + 1)/2. In particular,

cd(SO_{2n+1}) = cd(SO_{2n+2}) = n(n + 1)/2 .

The proof is given in section 2. It immediately follows from Theorem 1.2 (proved too in section 2), dealing with correspondences on X. A similar situation occurs in the proof of [1, th. 11.3] based on [4, th. 2.1] dealing with correspondences on a Severi-Brauer variety. An alternative proof of [4, th. 2.1], making use of a degree formula, is given in [7, §7.2]. However, for the similar statement [4, th. 6.4], concerning correspondences on quadrics (producing a similar to Theorem 1.1 result [5, th. 4.3] on the minimum of transcendence degree of generic isotropy fields of a quadratic form), there is no proof making use of a degree formula. In the present article as well, we use neither degree formulas nor Steenrod operations.

Theorem 1.2. If d(X) = 2^{n}, then the multiplicity of any correspondence α: X ⇝ X is congruent modulo 2 to the multiplicity of the transpose of α. In particular, any rational map X → X is necessarily dominant.

Here by a correspondence X ⇝ X we mean an algebraic cycle on X × X of dimension dim X. The multiplicity mult(α) of such a correspondence α is defined by the formula (pr_{1})_{∗}(α) = mult(α) · [X], where pr_{1} : X × X → X is the projection onto the first factor, while (pr_{1})_{∗} is the push-forward homomorphism of the group of algebraic cycles, see [3] (we do not use any equivalence relation on algebraic cycles yet). For the transpose α^{t} of α we clearly have: mult(α^{t}) · [X] = (pr_{2})_{∗}(α). The statement on rational maps is obtained by consideration of the correspondence given by the closure in X × X of the graph of a given rational map X → X.

Remark 1.3. Assume that d(X) = 2^{n}. Although we have Theorem 1.2, we do not know whether the variety X is 2-incompressible in the sense of [7, §7]. Note that the only known proof of p-incompressibility of Severi-Brauer varieties of p-primary division algebras (p is an arbitrary prime), given in [7, §7.2], makes use of a degree formula while the incompressibility of quadrics with first Witt index 1 [5, cor. 3.4] can not be proved by a degree formula.

On its turn, Theorem 1.2 follows (in a way very similar to the way [4, th. 2.1] follows from [4, cor. 2.3]) from the following computation of the reduced modulo 2 Chow group C̄h(X), defined as the image of the restriction homomorphism Ch(X) → Ch(X̄) of the usual modulo 2 Chow groups, where X̄ is X over an algebraic closure F̄ of F (a general reference for Chow groups is [3]):

Proposition 1.4. If d(X) = 2^{n}, then C̄h^{>0}(X) = 0.

The next section starts with the proof of Proposition 1.4.

2. Proofs

In the proof of Proposition 1.4, we are going to use the description of the integral Chow ring CH(X̄) given in [8] (we borrowed this reference from beautiful Totaro’s paper [10]). The graded ring CH^{∗}(X̄) is isomorphic to the quotient of the polynomial ring ℤ[e_{1}, …, e_{n}] by the ideal generated by the polynomials

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}

with i = 1, …, n (e_{i} should be understood to mean 0 for i > n in this formula), where the degree of e_{i} is i. The element of CH(X̄) corresponding to the class of e_{i} is a special Schubert class; we still write e_{i} for it. For any i, the element 2e_{i} is the i-th Chern class of the tautological vector bundle on the grassmanian, therefore is rational, that is, lies in the integral reduced Chow group C̄H(X) ⊂ CH(X̄).

For any subset I of the set {1, 2, …, n}, let us define an element e_{I} ∈ CH(X̄) as the product ∏_{i∈I} e_{i}. Defining |I| as ∑_{i∈I} i, we have codim e_{I} = |I|. The element e_{{1,2,...,n}} of the maximal codimension 1 + 2 + ⋯ + n = dim X is equal to the class of a rational point.

A basis of the modulo 2 Chow group Ch(X̄) is given by the classes of the elements e_{I}, where I runs over all subsets of the set {1, 2, …, n} (in particular, the dimension of Ch(X̄) (as a vector space over ℤ/2ℤ) is equal to 2^{n}).

Proof of Proposition 1.4. Assume the contrary: there exists a homogeneous element α ∈ C̄H(X) of a positive codimension such that α (mod 2) is a non-zero element of C̄h(X). Decomposing α in a sum of some e_{I} (without repetitions) plus 2β with some β ∈ CH(X̄), let us fix a set I such that the element e_{I} occurs in the decomposition. Let J be the complement of I. Let m be the number of elements in J (note that m < n). Then the product 2^{m}e_{J} is rational. We claim that the degree of the rational 0-cycle α · (2^{m}e_{J}) is an odd multiple of 2^{m}: indeed, the product e_{I} ·(2^{m}e_{J}) = 2^{m}e_{{1,2,...,n}} has the degree 2^{m}, while the product e_{I′} · (2^{m}e_{J}) for any I′ ≠ I with |I′| = |I| as well as the product (2β) · (2^{m}e_{J}) are 0 modulo 2^{m+1}. We have got a contradiction with the assumption on d(X). □

In the proof of Theorem 1.2, which follows, we use a motivic decomposition of X ×X (in the category of the integral Chow motives), produced in [2]. This motivic decomposition arises from the relative cellular structure on X × X, where the cells are the orbits of the diagonal G-action for G = SO(ϕ). Every summand of this decomposition is a Tate twist of the motive of X. More precisely, there is one copy of the motive of X (without twist, that is, with the zero twist), while the remaining summands have some positive twists (although we do not need the completely precise information, here it is: for any i, the number of summands twisted i times is equal to the rank of the group CH_{i}(X̄)).

To be absolutely precise, we have to say that the motivic decomposition of X given in [2] is not yet the decomposition described above: it also contains motives of certain flag varieties of the tautological vector bundle on X. However the motive of each such flag variety decomposes in the sum of some twists of the motive of X by [6].

Proof of Theorem 1.2. First of all, since X is projective, the multiplicity homomorphism factors through the Chow group, so that we have mult : CH_{N}(X × X) → ℤ, where N = dim X = n(n+1)/2. Since the multiplicity of a cycle is not changed under extensions of the base field, the multiplicity homomorphism factors even through the reduced Chow group, so that we may replace CH(X × X) by C̄H(X × X). Since we are interested in multiplicities modulo 2, we consider the induced homomorphism of the modulo 2 Chow group (still denoted by mult): mult: C̄h_{N}(X × X) → ℤ/2ℤ.

Theorem under proof claims that the image of the homomorphism

f: C̄h(X × X) → ℤ/2ℤ ⊕ ℤ/2ℤ , f: α ↦ ( mult(α), mult(α^{t}))

is contained in the diagonal subgroup of ℤ/2ℤ⊕ℤ/2ℤ. Using the described above motivic decomposition of X ×X, we get a decomposition of C̄h_{N}(X ×X) in the direct sum, where the summands are: one copy of C̄h_{N}(X) and several copies of C̄h_{i}(X) with various i < N. Since C̄h_{i}(X) = 0 for any i < N by Proposition 1.4, the image of the homomorphism f is cyclic. Since on the other hand, f([Δ_{X}]) = (1, 1), the image of f is generated by (1, 1), that is, coincides with the diagonal subgroup of ℤ/2ℤ ⊕ ℤ/2ℤ. □

Proof of Theorem 1.1. We repeat the proof of [1, th. 11.3] using Theorem 1.2 instead of [7, §7.2] (and meaning by X our orthogonal grassmanian instead of a Severi-Brauer variety).

Since the field F (X) is a generic splitting field of ϕ and has the transcendence degree n(n+1)/2, the inequality cd(ϕ) ≤ n(n+1)/2 holds (the assumption on d(ϕ) is not needed for this bound).

If now L is another generic splitting field of ϕ, then we show that tr. deg(L/F) ≥ n(n + 1)/2 as follows. Let Y be a projective model of L/F. Since both F (X) and F (Y) are generic splitting fields of ϕ, there exist rational morphisms f : X → Y and g : Y → X. Moreover, for any non-empty open subset U ⊂ Y, there exists a rational morphism X → Y with an image meeting U, so that we may assume that f and g are composable. Since the rational map X → X given by the composition g ∘ f is dominant by Theorem 1.2, the dimension of Y is at least equal to dim X = n(n + 1)/2. □

References

[1] G. Berhuy and Z. Reichstein. On the notion of canonical dimension for algebraic groups. Linear Algebraic Groups and Related Structures (Preprint Server) 140 (2004, Jun 18), 36 p.

[2] V. Chernousov, A. Merkurjev. Motivic decomposition of projective homogeneous varieties and the Krull-Schmidt theorem. Preprint (2004), 16 p. Available on: www.math.ucla.edu/~merkurev

[3] W. Fulton. Intersection Theory. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984.

[4] N. A. Karpenko. On anisotropy of orthogonal involutions. J. Ramanujan Math. Soc. 15 (2000), no. 1, 1–22.

[5] N. A. Karpenko, A. S. Merkurjev. Essential dimension of quadrics. Invent. Math. 153 (2003), 361– 372.

[6] B. Köck. Chow motif and higher Chow theory of G/P. Manuscripta Math. 70 (1991), no. 4, 363–372.

[7] A. Merkurjev. Steenrod operations and degree formulas. J. Reine Angew. Math. 565 (2003), 13–26.

[8] M. Mimura and H. Toda. Topology of Lie Groups. Providence: ASM (1991).

[9] W. Scharlau. Quadratic and Hermitian Forms. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985.

[10] B. Totaro. The torsion index of the spin group. Preprint (2004), 35 p.

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