INCOMPRESSIBILITY OF ORTHOGONAL GRASSMANNIANS

NIKITA A. KARPENKO

Abstract. We prove the following conjecture due to Bryant Mathews (2008). Let Q be the orthogonal grassmannian of totally isotropic i-planes of a non-degenerate quadratic form q over an arbitrary field (where i is an integer satisfying 1 ≤ i ≤ (dim q)/2). If the degree of each closed point on Q is divisible by 2^{i} and the Witt index of q over the function field of Q is equal to i, then the variety Q is 2-incompressible.

Theorem 7, proved below, has been conjectured in the Ph.D. thesis [12, page 24] (a preprint with the conjecture appeared one year earlier).

We start with some development of the theory of canonical dimension of general projective homogeneous varieties (which might be of independent interest). We fix a prime p. Let G be a semisimple affine algebraic group over a field F such that G_{E} is of inner type for some finite galois field extension E/F of degree a power of p (E = F is allowed). Let X be a projective G-homogeneous F -variety. We refer to [5] for a definition and discussion of the notion of canonical p-dimension cdim_{p} X of X. Actually, canonical p-dimension is defined in the context of more general algebraic varieties. For any irreducible smooth projective variety X, cdim_{p} X is the minimal dimension of a closed subvariety Y ⊂ X with a 0-cycle of p-coprime degree on Y_{F (X)}. Recall that a smooth projective X is p-incompressible, if it is irreducible and cdim_{p} X = dim X.

Proposition 1. For d := cdim_{p} X, there exist a cycle class α ∈ CH^{d} X_{F (X)} (over F (X)) of codimension d and a cycle class β ∈ CH_{d} X (over F) of dimension d such that the degree of the product β_{F (X)} · α is not divisible by p.

Proof. We use the Chow motives with coefficients in 𝔽_{p} := ℤ/pℤ (as defined in [3, Chapter XII]) and write Ch for the Chow group CH modulo p.

Let U(X) be the upper motive of X. By definition, U(X) is a direct summand of the motive M(X) of X such that Ch^{0} U(X) ≠ 0. By [5, Theorem 5.1 and Proposition 5.2], U(X) is also a direct summand of M(X)(d − m), where m := dim X. The composition

M(X) → U(X) → M(X)(d − m)

is given by a correspondence f ∈ Ch^{d}(X × X); the composition

M(X)(d − m) → U(X) → M(X)

is given by a correspondence g ∈ Ch_{d}(X × X). The composition of correspondences g ∘ f ∈ Ch_{m}(X × X) is a projector on X such that U(X) = (X, g ∘ f). In particular, the multiplicity mult(g ∘ f) of the correspondence g ∘ f is 1 ∈ 𝔽_{p}. Taking for α the pull-back of f with respect to the morphism

Date: June 2011.

Key words and phrases. Algebraic groups, quadratic forms, projective homogeneous varieties, Chow groups and motives. Mathematical Subject Classification (2010): 14L17; 14C25.

Spec F (X) × X → X × X

induced by the generic point of the first factor, and taking for β the push-forward of g with respect to the projection of X × X onto the first factor, we get that

deg(β_{F (X)} · α) = mult(g ∘ f) = 1 ∈ 𝔽_{p}.  □

Corollary 2. The canonical p-dimension cdim_{p} X of X is the minimal integer d such that there exists a cycle class α ∈ Ch^{d} X_{F (X)} and a cycle class β ∈ Ch_{d} X with deg(β_{F (X)} ·α) = 1 ∈ 𝔽_{p}.

Proof. We only need to show that cdim_{p} X ≤ d. The proof is similar to [11, Proof of ≤ in Theorem 5.8]. We may assume that β = [Y] for a closed irreducible d-dimensional subvariety Y ⊂ X. Since the product [Y]_{F (X)} · α can be represented by a cycle on Y_{F (X)}, the variety Y_{F (X)} has a 0-cycle of p-coprime degree. Therefore cdim_{p} X ≤ dim Y = d. □

Corollary 3. In the situation of Proposition 1, for any field extension L/F, the change of field homomorphism Ch_{d} X → Ch_{d} X_{L} is non-zero.

Proof. The image of β ∈ Ch_{d} X in Ch_{d} X_{L} is non-zero because deg(β_{L(X)} · α_{L(X)}) = 1. □

Remark 4. If the variety X is generically split (meaning that the motive of X_{F (X)} is a sum of Tate motives which, in particular, implies that the adjoint algebraic group acting on X is of inner type), then [11, Theorem 5.8] says that cdim_{p} X is the minimal d with non-zero Ch_{d} X → Ch_{d} X_{L} for any L. Corollary 3 can be considered as a generalization of a part of [11, Theorem 5.8] to the case of a projective G-homogeneous variety X which is not necessarily generically split with G not necessarily of inner type. Note that the statement of [11, Theorem 5.8] in whole fails in such generality. Corollary 2 is its correct replacement (giving the original statement in the case of generically split X).

Lemma 5. In the situation of Proposition 1, let α, α′ ∈ Ch^{d} X_{F (X)} and β, β′ ∈ Ch_{d} X be cycle classes with deg(β_{F (X)} · α) = 1 = deg(β′_{F (X)} · α′). Then

deg(β_{F (X)} · α′) ≠ 0 ≠ deg(β′_{F (X)} · α).

Proof. We fix an algebraically closed field containing F (X) and write ¯· when considering a variety or a cycle class over that field. The surjectivity of the pull-back with respect to the flat morphism Spec F (X)×X → X ×X induced by the generic point of the first factor of the product X × X, tells us that the group Ch^{d}(X̄ × X̄) contains a rational (i.e., “coming from F ”) cycle class of the form [X̄]×ᾱ+⋯+γ̄×[X̄] with some γ ∈ Ch^{d} X_{F (X)}, where … is in the sum of products Ch^{i} X̄ ⊗ Ch^{j} X̄ with 0 < i, j < d and i + j = d. Multiplying by [X̄]×β̄, we get a rational cycle class of the form [X̄]×[pt]+⋯+γ̄×β̄, where pt is a rational point on X̄ and … is now in the sum of Ch^{i} X̄ ⊗ Ch_{i} X̄ with 0 < i < d. The composition of the obtained correspondence with itself equals [X̄]×[pt]+⋯+deg(γ ·β)(γ̄ ×β̄). Since an appropriate power of this correspondence is a multiplicity 1 projector (cf. [9, Corollary 3.2] or [7]) and d = cdim_{p} X, it follows by [5, Theorem 5.1] that deg(γ · β) ≠ 0. Now multiplying [X̄] × ᾱ + ⋯ + γ̄ × [X̄] by β̄ × [X̄], transposing, and raising to (p − 1)th power (by means of composition of correspondences), we get a rational cycle of the form [X̄] × [pt] + ⋯ + ᾱ × β̄.

Similarly, there is a rational cycle of the form [X̄] × [pt] + ⋯ + ᾱ′ × β̄′. One of its compositions with the previous one produces [X̄]×[pt]+⋯+deg(β′ ·α)(ᾱ′ ×β̄), therefore deg(β′ ·α) ≠ 0 ∈ 𝔽_{p}. The other composition produces [X̄]×[pt]+⋯+deg(β ·α′)(ᾱ×β̄′), so that deg(β · α′) ≠ 0. □

We specify as follows the settings of Proposition 1. The prime p is now 2. The algebraic group G is now O^{+}(q) for a non-degenerate quadratic form q (one may take E = F if dim q is odd or disc q is trivial, otherwise E can be the quadratic field extension of F given by the discriminant of q). We set n := dim q. For any integer i with 0 ≤ i ≤ n/2, let Q_{i} be the variety of i-dimensional totally isotropic subspaces in q. In particular, Q_{0} = Spec F. For i with 0 < i < n/2, Q_{i} is a projective G-homogeneous variety.

Corollary 6. If cdim_{2} Q_{i} = cdim_{2} Q′_{i−1} = dim Q′_{i−1} for some some i with 0 < i < n/2, where Q′_{i−1} is the orthogonal grassmannian of totally isotropic (i − 1) planes of a (n − 2)- dimensional quadratic form q′ over F (Q_{1}) Witt-equivalent to q_{F (Q_{1})}, then deg CH Q_{i} ∋ 2^{i−1}.

Proof. The statement being trivial for i = 1, we may assume that i ≥ 2.

For d := cdim_{2} Q_{i}, using Proposition 1, we choose some α ∈ CH^{d} Q_{i F (Q_{i})} and β ∈ CH_{d} Q_{i} with odd deg(β_{F (Q_{i})}·α). Note that cdim_{2} Q_{i F (Q_{1})} = cdim_{2} Q′_{i−1} = d. We construct some special α′ ∈ CH^{d} Q_{i F (Q_{1})(Q_{i})} and β′ ∈ CH_{d} Q_{i F (Q_{1})} with deg(β′_{F (Q_{1})(Q_{i})} · α′) = 1 as follows. Let Q_{1⊂i} be the variety of (1, i)-flags of totally isotropic subspaces in q together with the evident projections Q_{1⊂i} → Q_{1}, Q_{i}. We define β′ as the pull-back via Q_{1⊂i F (Q_{1})} → Q_{1 F (Q_{1})} followed by the push-forward via Q_{1⊂i F (Q_{1})} → Q_{i F (Q_{1})} of the rational point class l_{0} on Q_{1 F (Q_{1})}. We define α′ as the product z_{i−1} … z_{1}, where z_{j} is the pull-back via Q_{1⊂i F (Q_{1})(Q_{i})} → Q_{1 F (Q_{1})(Q_{i})} followed by the push-forward via Q_{1⊂i F (Q_{1})(Q_{i})} → Q_{i F (Q_{1})(Q_{i})} of the class l_{j} of a j-dimensional projective subspace on Q_{1 F (Q_{1})(Q_{i})}. The degree condition on α′ and β′ is satisfied by [14, Statement 2.13]. Fixing an algebraically closed field containing F (Q_{1})(Q_{i}), we see by Lemma 5 that the product β̄·ᾱ′ is an odd degree 0-cycle class on Q̄_{i}. Moreover, the class β̄ is rational. Since 2z̄_{j} is rational for every j < (n − 2)/2 (by the reason that 2l_{j} is rational), the class 2^{i−1}β̄ᾱ is also rational and it follows that 2^{i−1} ∈ deg CH Q_{i}. □

We come to the main result of this note. It is known for i = 1 by [4] (the proof is essentially contained already in [13]; the characteristic 2 case has been treated later on in [3]). The case of maximal i, i.e., of i = [n/2], is also known and is discussed in the beginning of the proof. For i = 2 and odd-dimensional q, it has been proved in [12] (the proof for i = 2 given here is different; in particular, it does not make use of the motivic decompositions of [2] for products of projective homogeneous varieties).

Theorem 7. Let q be a non-degenerate quadratic form over a field F. Let i be an integer satisfying 1 ≤ i ≤ (dim q)/2. If the degree of every closed point on Q_{i} is divisible by 2^{i} and the Witt index of the quadratic form q_{F (Q_{i})} equals i, then the variety Q_{i} is 2-incompressible (i.e., cdim_{2} Q_{i} = dim Q_{i}).

Proof. We set where n := dim q. Note that for i = n/2 (and even n) the condition on closed points on Q_{n/2} ensures that disc q is non-trivial. In particular, Q_{n/2} is irreducible.

In general, for even n, the variety Q_{n/2} is isomorphic to the orthogonal grassmannian of totally isotropic (n/2 − 1)-planes of q^{1} considered as a variety over F, where q^{1} is any 1-codimensional non-degenerate subform in q_{K}, and K is the quadratic étale F -algebra given by the discriminant of q. Therefore the statement of Theorem 7 for i = n/2 follows from the statement for i = (n − 1)/2. By this reason, we do not consider the case of i = n/2 below. In particular, Q_{i} below is a projective G-homogeneous variety.

We induct on n. There is nothing to prove for n < 3. Below we are assuming that n ≥ 3.

Over the field F (Q_{1}), the motive of Q_{i F (Q_{1})} decomposes as follows (cf. [6], [8] or [1]):

M(Q′_{i−1}) ⊕ M(Q′_{i})((dim Q_{i} − dim Q′_{i})/2) ⊕ M(Q′_{i−1})(dim Q_{i} − dim Q′_{i−1}),

where, as in Corollary 6, Q′_{j} is the variety of j-dimensional totally isotropic subspaces of a quadratic form q′ over the field F (Q_{1}) such that q_{F (Q_{1})} is isomorphic to the orthogonal sum of q′ and a hyperbolic plane. Since n′ := dim q′ = n − 2 < n, the variety Q′_{i−1} is 2-incompressible by the induction hypothesis (more precisely, the induction hypothesis is applied if i ≥ 2, for i = 1 the statement if trivial). Indeed, since the extension F (Q_{1})/F is a tower of a purely transcendental extension followed by a quadratic one, the degree of any closed point on Q′_{i−1} is divisible by 2^{i−1}; the Witt index of q′_{F (Q_{1})(Q′_{i−1})} is i − 1, that is, the Witt index of q_{F (Q_{1})(Q′_{i−1})} is i because the field extension F (Q_{1})(Q′_{i−1})(Q_{i})/F (Q_{i}) is purely transcendental.

By [10, Theorem 1.1] (cf. [6]), it follows that the motive of Q′_{i−1} decomposes in a direct sum of one copy of U(Q′_{i−1}) and shifts of U(Q′_{j}) with various j ≥ i. The motive of Q′_{i} decomposes in a direct sum of shifts of U(Q′_{j}) with various j ≥ i. Note that the condition on the Witt index of the form q_{F (Q_{i})} ensures that for any j ≥ i the motive U(Q′_{i−1}) is not isomorphic to U(Q′_{j}). Therefore the complete motivic decomposition of Q_{i F (Q_{1})} contains one copy of U(Q′_{i−1}), one copy of U(Q′_{i−1})(dim Q_{i} − dim Q′_{i−1}) and no other shifts of U(Q′_{i−1}).

The complete decomposition of U(Q_{i})_{F (Q_{1})} contains the summand U(Q′_{i−1}). If it also contains the shift of U(Q′_{i−1}), then cdim_{2} Q_{i} = dim Q_{i} and we are done. Otherwise, by [6, Lemma 1.2 and Remark 1.3], cdim_{2} Q_{i} = cdim_{2} Q′_{i−1} = dim Q′_{i−1}, and we get by Corollary 6 that Q_{i} has a closed point of degree not divisible by 2^{i}. □

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UPMC Univ Paris 06, Institut de Mathématiques de Jussieu, F-75252 Paris, FRANCE

Web page: www.math.jussieu.fr/~karpenko
E-mail address: karpenko at math.jussieu.fr