INCOMPRESSIBILITY OF ORTHOGONAL GRASSMANNIANS OF RANK 2

BRYANT G. MATHEWS

Abstract. For a nondegenerate quadratic form φ on a vector space V of dimension 2n + 1, let X_{d} be the variety of d-dimensional totally isotropic subspaces of V. We give a sufficient condition for X_{2} to be 2-incompressible, generalizing in a natural way the known sufficient conditions for X_{1} and X_{n}. Key ingredients in the proof include the Chernousov-Merkurjev method of motivic decomposition as well as Pragacz and Ratajski’s characterization of the Chow ring of (X_{2})_{E}, where E is a field extension splitting φ.

1. Preliminaries

Before stating our results in the next section, we begin by recalling the notions of canonical p-dimension, p-incompressibility, and higher Witt index.

Let X be a scheme over a field F, and let p be a prime or zero. A field extension K of F is called a splitting field of X (or is said to split X) if X(K) ≠ ∅. A splitting field K is called p-generic if, for any splitting field L of X, there is an F -place K ⇀ L′ for some finite extension L′/L of degree prime to p. In particular, K is 0-generic if for any splitting field L there is an F -place K ⇀ L.

The canonical p-dimension of a scheme X over F was originally defined [1, 7] as the minimal transcendence degree of a p-generic splitting field K of X. When X is a smooth complete variety, the original algebraic definition is equivalent to the following geometric one [7, 10].

Definition 1.1. Let X be a smooth complete variety over F. The canonical pdimension cdim_{p}(X) of X is the minimal dimension of the image of a morphism X′ → X, where X′ is a variety over F admitting a dominant morphism X′ → X with F (X′)/F (X) finite of degree prime to p. The canonical 0-dimension of X is thus the minimal dimension of the image of a rational morphism X −−→ X.

In the case p = 0, we will drop the p and speak simply of generic splitting fields and canonical dimension cdim(X).

For a third definition of canonical p-dimension as the essential p-dimension of the detection functor of a scheme X, we refer the reader to Merkurjev’s comprehensive exposition [10] of essential dimension.

For a smooth complete variety X, the inequalities

cdim_{p}(X) ≤ cdim(X) ≤ dim(X)

are clear from Definition 1.1. Note also that if X has a rational point, then cdim(X) = 0 (though the converse is not true).

Definition 1.2. When a smooth complete variety X has canonical p-dimension as large as possible, namely cdim_{p}(X) = dim(X), we say that X is p-incompressible.

It follows immediately that if X is p-incompressible, it is also incompressible (i.e. 0-incompressible).

We next recall the definitions of absolute and relative higher Witt indices, introduced by Knebusch in [8]. Our discussion follows [5, §90]. The Witt index 𝔦_{0}(φ) of a quadratic form φ is the number of copies of the hyperbolic plane ℍ which appear in the Witt decomposition of φ. Now let φ be a nondegenerate quadratic form over a field F and set F_{0} := F and φ_{0} := φ_{an}, the anisotropic part of φ. We proceed to recursively define F_{k} := F_{k−1}(φ_{k−1}), φ_{k} := (φ_{F_{k}})_{an} for k = 1, 2, …, stopping at F_{h} such that dim φ_{h} ≤ 1.

Definition 1.3. For k ∈ {0, 1, …, h}, the k-th absolute higher Witt index 𝔧_{k}(φ) of φ is defined to be 𝔦_{0}(φ_{F_{k}}). For k ∈ {1, 2, …, h}, the k-th relative higher Witt index 𝔦_{k}(φ) of φ is defined to be the difference

𝔦_{k}(φ) := 𝔧_{k}(φ) − 𝔧_{k−1}(φ).

The 0-th relative higher Witt index of φ is the usual Witt index 𝔦_{0}(φ).

It follows from the definition that

0 ≤ 𝔧_{0}(φ) < 𝔧_{1}(φ) < ⋯ < 𝔧_{h}(φ) = [(dim φ)/2] .

Moreover, it can be shown that the set {𝔧_{0}(φ), …, 𝔧_{h}(φ)} of absolute higher Witt indices of φ is equal to the set of all Witt indices 𝔦_{0}(φ_{K}) for K an extension field of F.

2. Introduction

Let φ be a nondegenerate quadratic form on a vector space V of dimension 2n+1 over a field F. Associated to φ there are smooth projective varieties X_{1}, X_{2}, …, X_{n}, where X_{d} is the variety of d-dimensional totally isotropic subspaces of V. The variety X_{1} is simply the projective quadric hypersurface associated to the quadratic form φ.

We recall the following result proved in [6] and also in [5, Ch. XIV and §90].

Theorem 2.1 (Karpenko, Merkurjev). If the quadric X_{1} is anisotropic, then

cdim_{2}(X_{1}) = cdim(X_{1}) = dim(X_{1}) − 𝔦_{1}(φ) + 1.

In particular, X_{1} is 2-incompressible if and only if 𝔦_{1}(φ) = 1.

At the other extreme is the variety X_{n} of maximal totally isotropic subspaces of V. In [5, Ch. XVI], building on a result of Vishik from [12], the canonical 2dimension of X_{n} is computed in terms of the J-invariant of φ. The following result is a corollary.

Theorem 2.2 (Karpenko, Merkurjev). If deg CH(X_{n}) = 2^{n}ℤ, then X_{n} is 2-incom-pressible.

To compute the canonical 2-dimension of a general X_{d} appears to be difficult because of the complexity of the Chow ring when d ∉ {1, n}. In this paper, we complete a small piece of this general program by determining a sufficient condition for the variety X_{2} to be 2-incompressible. We assume everywhere that n ≥ 3, the n = 2 case having already been dealt with.

Theorem 2.3. If deg CH(X_{2}) = 4ℤ and 𝔦_{2}(φ) = 1, then X_{2} is 2-incompressible. In particular,

cdim_{2}(X_{2}) = cdim(X_{2}) = dim(X_{2}) = 4n − 5.

This result concerning X_{2} is a natural generalization of what is already known about X_{1} and X_{n}. To see this, note that X_{1} being anisotropic means that deg CH(X_{1}) = 2ℤ. Furthermore, deg CH(X_{n}) = 2^{n}ℤ implies that 𝔧_{n−1}(φ) = n − 1, from which it immediately follows that 𝔦_{n}(φ) = 1. One might then conjecture, for general d, that

deg CH(X_{d}) = 2^{d}ℤ, 𝔦_{d}(φ) = 1

are sufficient conditions for X_{d} to be 2-incompressible.

3. Higher Witt indices

In this section we collect two results concerning higher Witt indices which will be needed later.

Proposition 3.1. If deg CH(X_{2}) = 4ℤ, then 𝔧_{1}(φ) = 1.

Proof. Let K be a field of degree 2 over F such that the anisotropic part of φ has a rational point over K. Then 𝔦_{0}(φ_{K}) > 𝔦_{0}(φ). By [5, Prop. 25.1], it follows that 𝔦_{0}(φ_{K}) ≥ 𝔧_{1}(φ). If 𝔧_{1}(φ) ≥ 2 then so is 𝔦_{0}(φ_{K}), which implies that the variety X_{2} has a rational point over K. Since K has degree 2 over F, this contradicts the assumption. Thus 𝔧_{1}(φ) = 1 and 𝔧_{0}(φ) = 0. □

From this proposition, we see that the hypothesis of our Theorem 2.3 implies

𝔧_{2}(φ) = 𝔧_{1}(φ) + 𝔦_{2}(φ) = 1 + 1 = 2.

Proposition 3.2. If 𝔧_{2}(φ) = 2, then 𝔦_{0}(φ_{F (X_{2})}) = 2.

In fact, one need only assume that some absolute higher Witt index of φ is equal to 2 (possibly 𝔧_{0}(φ) or 𝔧_{1}(φ)), but we don’t need this generality for our purposes.

Proof. The variety X_{2} has a rational point over F (X_{2}), so 𝔦_{0}(φ_{F (X_{2})}) ≥ 2.

We prove that 𝔦_{0}(φ_{F (X_{2})}) ≤ 2 by contradiction. If φ̃ is the anisotropic part of φ_{F (φ)}, then φ_{F (φ)} ≃ ℍ ⊥ φ̃, since our assumption 𝔧_{2}(φ) = 2 implies that 𝔧_{1}(φ) = 1. We define two varieties over F′ := F (φ). Let Y_{1} be the projective quadric corresponding to φ̃, and let Y_{2} be the variety of totally isotropic subspaces of dimension 2 with respect to φ̃. Since

𝔦_{0}(φ_{F′(Y_{1})}) = 𝔦_{0}(φ_{F (φ)(φ̃)}) = 𝔧_{2}(φ) = 2,

the variety X_{2} has a rational point over F′(Y_{1}). If 𝔦_{0}(φ_{F (X_{2})}) ≥ 3, then 𝔦_{0} (φ̃_{F′((X_{2})_{F′})}) ≥ 2, so Y_{2} has a rational point over F′ ((X_{2})_{F′}). We thus have two rational morphisms between varieties over F′:

Y_{1} −−→ (X_{2})_{F′} −−→ Y_{2}.

By [2, Lem. 6.1], since X_{2} is smooth and Y_{2} is complete, there exists a rational morphism from Y_{1} to Y_{2}, i.e. (Y_{2})_{F′(Y_{1})} has a rational point. But this is a contradiction, since 𝔧_{1}(φ̃) = 𝔧_{2}(φ) − 1 = 1. □

4. Shapes and multiplicities

A quadratic form φ is split if 𝔦_{0}(φ) = [(dim φ)/2], the greatest possible value. Given a quadratic form φ over F, we fix an extension field E/F such that φ_{E} is split and define φ̄ := φ_{E} and X̄_{d} := (X_{d})_{E}. Let \overline{CH}(X_{d}), called the reduced Chow group, denote the image of the change of field homomorphism CH(X_{d}) → CH(X̄_{d}). Elements of \overline{CH}(X_{d}) will be called rational cycles.

In this section we prove a technical lemma, based on a characterization given by Pragacz and Ratajski in [11] of the Chow ring of the variety X̄_{2}.

Lemma 4.1. If γ ∈ CH^{r}(X̄_{2}) with r ∈ {2n − 3, 2n − 2}, then 2γ is a rational cycle.

We begin by fixing notation and recalling some definitions. Let X_{B} := X̄_{2}, the variety of 2-dimensional isotropic subspaces of V with respect to the split nondegenerate quadratic form φ̄. Recall that V has dimension 2n + 1. Let X_{C} denote the variety of 2-dimensional isotropic subspaces of a vector space W of dimension 2n with respect to a nondegenerate alternating form ψ on W.

The next group of definitions are adapted for our purposes from Macdonald’s [9].

Definition 4.2. A partition is a finite, strictly decreasing sequence of positive integers

λ^{∗} = (λ_{1}^{∗}, λ_{2}^{∗}, …, λ_{s}^{∗}).

The ∗ in our notation will take on the value t for “top” or b for “bottom,” depending on the role the partition plays in a shape, defined below. The length l(λ^{∗}) of the partition above is just s, while the weight of the partition is defined to be |λ^{∗}| := ∑_{k=1}^{s} λ_{k}^{∗}. The empty partition, denoted ∅, is the sequence with no terms.

Partitions are visualized as diagrams of boxes. The diagram D_{λ}^{∗} of a partition λ^{∗} has λ_{k}^{∗} boxes in the kth row, beginning with the top row. For example, the partition (4, 3, 1) has diagram:

Note that the length of a partition is just the number of rows in its diagram, while the weight is just the number of boxes.

A skew diagram D_{μ}^{∗}\D_{λ}^{∗} is obtained by removing the boxes in the intersection of D_{μ}^{∗} and D_{λ}^{∗} from D_{μ}^{∗}. For example, the skew diagram (4, 3, 1)\(2, 1) is as follows:

The remaining definitions are adapted from [11].

Definition 4.3. With n fixed as above, a pair λ = (λ^{t}//λ^{b}) of partitions λ^{t} and λ^{b} is called a shape if λ_{1}^{t}, λ_{1}^{b} ≤ n, l(λ^{t}) ≤ n − 2, l(λ^{b}) ≤ 2, and λ_{n−2}^{t} > l(λ^{b}).

The diagram D_{λ} of a shape is drawn by stacking the diagram D_{λ}^{t} of the first partition on top of the diagram D_{λ}^{b} of the second, with a horizontal line in between. For example, the shape ((4, 2)//(3)) has diagram:

The inequalities in the definition of a shape λ amount to imposing three requirements on its diagram D_{λ}:

(1) D_{λ}^{t} must fit into a rectangle with height n − 2 and width n

(2) D_{λ}^{b} must fit into a rectangle with height 2 and width n

(3) D_{λ} must contain the “triangle” of boxes lying on and above the diagonal beginning at the box in the diagram’s lower-left corner and proceeding to the “northeast”.

We display once again the diagram of the shape ((4, 2)//(3)), this time shading in the triangle of boxes required by the third condition above:

The weight of a shape λ = (λ^{t}//λ^{b}) is defined in terms of the weights of the partitions λ^{t}, λ^{b} as

|λ| := |λ^{t}| + |λ^{b}| − \binom{n − 1}{2}.

This definition is chosen so that the shape π_{0} := ((n−2, n−3, …, 1)//∅) of minimal weight will have weight 0.

We refer to [11] for the definitions of extremal and related components, (μ − λ)- boxes, and compatible shapes.

The set of all shapes, denoted P_{2}, can be mapped bijectively onto bases for each of CH(X_{B}) and CH(X_{C}). We name the maps

σ : P_{2} ⟶ CH(X_{B})
τ : P_{2} ⟶ CH(X_{C})

and we call cycles in the images of the maps basic cycles.

For i = 0, 1, 2, consider the shapes π_{i} := ((n−2+i, n−3, n−4, …, 1)//∅). When n = 5, their diagrams are as follows:

If we set σ_{i} := σ(π_{i}), τ_{i} := τ(π_{i}), then σ_{0}, τ_{0} are the respective multiplicative identities of the Chow rings, while σ_{1}, σ_{2} (resp. τ_{1}, τ_{2}), called special cycles, are the nontrivial Chern classes of the tautological bundle over X_{B} (resp. X_{C}). Pragacz and Ratajski prove that τ_{1}, τ_{2} algebraically generate CH(X_{C}) (Cor. 1.8 and Lem. 3.2), while σ_{1}, σ_{2} only generate CH(X_{B}) after tensoring with ℤ[1/2] (Thm. 10.1).

With the weight |λ| of a shape defined as above, we have σ(λ) ∈ CH^{|λ|}(X_{B}) and τ(λ) ∈ CH^{|λ|}(X_{C}). In particular, codim(σ_{i}) = |π_{i}| = i and codim(τ_{i}) = |π_{i}| = i.

The multiplication rules in CH(X_{B}) and CH(X_{C}) are very similar, differing only by some factors of 2 in certain multiplicities. Indeed, for any shape λ ∈ P_{2}, i = 1, 2, we have the Pieri-type formulas

σ(λ) · σ_{i} = ∑ 2^{e_{B}(λ,μ)}σ(μ)
τ(λ) · τ_{i} = ∑ 2^{e_{C}(λ,μ)}τ(μ)

for multiplying a basic cycle by a special cycle [11, Thms. 2.2 and 10.1]. Here, the sums are over all μ compatible with λ satisfying |μ| = |λ| + i, and e_{B}(λ, μ), e_{C}(λ, μ) are the cardinalities of certain sets of components of the skew diagram D_{μ}^{b}\D_{λ}^{b}. For our purposes, what we need is that for compatible λ, μ with |μ| = |λ| + i, the difference e_{B}(λ, μ) − e_{C}(λ, μ) equals the number of extremal components of D_{μ}^{b}\D_{λ}^{b}. (This follows from the fact that an extremal component is not related and has no (μ − λ)-boxes lying over it, by parts 2 and 4 of the definition [11, Def. 2.1] of compatible shapes.) The skew diagram D_{μ}^{b}\D_{λ}^{b} clearly has at most one extremal component. There is exactly one extremal component if and only if l(μ^{b}) > l(λ^{b}), which by part 5 of the definition of compatible shapes is equivalent to l(μ^{b}) = l(λ^{b}) + 1. There are no extremal components if and only if l(μ^{b}) = l(λ^{b}). Putting all of this together, we conclude that

(1)  e_{B}(λ, μ) − e_{C}(λ, μ) = l(μ^{b}) − l(λ^{b}) ∈ {0, 1}.

To describe the product of several special cycles, we need to extend the notion of compatibility to sequences of shapes. For nonnegative integers a_{1}, a_{2}, define a compatible (a_{1}, a_{2})-chain to be a sequence of shapes

Λ = (π_{0} =λ_{0}, λ_{1}, …, λ_{a_{1}+a_{2}})

such that for i = 1, 2, …, (a_{1} +a_{2}), the shapes λ_{i} and λ_{i−1} are compatible, |λ_{i}| − |λ_{i−1}| ∈ {1, 2}, and |λ_{a_{1}+a_{2}}| = a_{1} + 2a_{2}.

We now can write down formulas for an arbitrary product of special cycles in CH(X_{B}) or CH(X_{C}). In CH(X_{B}), the formula is

σ_{1}^{a_{1}} · σ_{2}^{a_{2}} = ∑_{compatible (a_{1}, a_{2})-chains_{Λ=(λ_{0},λ_{1},...,λ_{a_{1}+a_{2}} )}} 2^{b_{Λ}}σ(λ_{a_{1}+a_{2}}),

where

b_{Λ} = e_{B}(λ_{0}, λ_{1}) + e_{B}(λ_{1}, λ_{2}) + ⋯ + e_{B}(λ_{a_{1}+a_{2}−1}, λ_{a_{1}+a_{2}}),

and in CH(X_{C}), the formula is

τ_{1}^{a_{1}} · τ_{2}^{a_{2}} = ∑_{compatible (a_{1}, a_{2})-chains_{Λ=(λ_{0},λ_{1},...,λ_{a_{1}+a_{2}} )}} 2^{c_{Λ}}τ(λ_{a_{1}+a_{2}}),

where

c_{Λ} = e_{C}(λ_{0}, λ_{1}) + e_{C}(λ_{1}, λ_{2}) + ⋯ + e_{C}(λ_{a_{1}+a_{2}−1}, λ_{a_{1}+a_{2}}).

It follows from equation (1) that

b_{Λ} − c_{Λ} = [l(λ_{1}^{b}) − l(λ_{0}^{b})] + [l(λ_{2}^{b}) − l(λ_{1}^{b})] + ⋯ + [l(λ_{a_{1}+a_{2}}^{b}) − l(λ_{a_{1}+a_{2}−1}^{b})]
(2)  = l(λ_{a_{1}+a_{2}}^{b}) − l(∅)
= l(λ_{a_{1}+a_{2}}^{b}).

We now are ready to prove the lemma.

Proof. It is enough to take γ to be a basic cycle, say γ = σ(λ) for some shape λ of weight r. Since τ_{1} and τ_{2} generate the ring CH(X_{C}), there exist integers u_{j} such that

τ(λ) = ∑_{j=0}^{⌊r/2⌋} u_{j} (τ_{1}^{r−2j} · τ_{2}^{j}) ∈ CH^{r}(X_{C}).

Define

γ′ := ∑_{j=0}^{⌊r/2⌋} u_{j} (σ_{1}^{r−2j} · σ_{2}^{j}) ∈ CH^{r}(X_{B}).

This is a rational cycle, since σ_{1}, σ_{2} are Chern classes of the tautological bundle over X_{B} = X̄_{2} and are therefore defined over the base field F.

It remains to show that γ′ = 2γ. The key is that for any shape λ = (λ^{t}//λ^{b}) of weight 2n − 3 or 2n − 2, l(λ^{b}) = 1. Indeed, it follows easily from the conditions imposed in the definition of a shape λ that l(λ^{b}) = 0 implies |λ| ≤ 2n − 4, and l(λ^{b}) = 2 (the greatest value possible) implies |λ| ≥ 2n − 1. The case n = 5 is illustrated below, where we shade the “π_{0} boxes” which don’t contribute to the weight |λ|. Diagram I corresponds to the shape of maximal weight 2n − 4 = 6 among shapes λ with l(λ^{b}) = 0. Diagram II corresponds to the shape of minimal weight 2n − 1 = 9 among shapes λ with l(λ^{b}) = 2.

Expanding the products in the expression for τ(λ), we get

τ(λ) = ∑_{j=0}^{⌊r/2⌋} u_{j} ∑_{compatible (r−2j, j)-chains_{Λ=(λ_{0},λ_{1},...,λ_{r−j})}} 2^{c_{Λ}}τ(λ_{r−j}).

The maps σ and τ induce a group isomorphism “σ ∘ τ^{−1}” : CH(X_{C}) → CH(X_{B}) which when applied to the equation above yields

(3)  σ(λ) = ∑_{j=0}^{⌊r/2⌋} u_{j} ∑_{compatible (r−2j, j)-chains_{Λ=(λ_{0},λ_{1},...,λ_{r−j})}} 2^{c_{Λ}}σ(λ_{r−j}).

On the other hand, expanding the expression for γ′ yields

(4)  γ′ = ∑_{j=0}^{⌊r/2⌋} u_{j} ∑_{compatible (r−2j, j)-chains_{Λ=(λ_{0},λ_{1},...,λ_{r−j})}} 2^{b_{Λ}}σ(λ_{r−j}),

which differs from the expression for σ(λ) only in that the exponent c_{Λ} has been changed to b_{Λ}. By equation (2) and our length computation above, b_{Λ} − c_{Λ} = l(λ_{r−j}^{b}) = 1 for any compatible (r−2j, j)-chain Λ, since

|λ_{r−j}| = (r − 2j) + 2j = r ∈ {2n − 3, 2n − 2}.

Thus each term on the right-hand side of (4) is twice the corresponding term on the right-hand side of (3) and γ′ = 2σ(λ) = 2γ. □

5. Proof of 2-incompressibility

We now have the ingredients necessary for a proof of the main theorem, whose statement we repeat below.

Theorem 5.1. If deg CH(X_{2}) = 4ℤ and 𝔦_{2}(φ) = 1, then X_{2} is 2-incompressible. In particular,

cdim_{2}(X_{2}) = cdim(X_{2}) = dim(X_{2}) = 4n − 5.

We briefly recall some terminology from [5, §62 and §75]. Let X and Y be schemes with dim X = e. A correspondence of degree zero δ : X ⇝ Y from X to Y is just a cycle δ ∈ CH_{e}(X × Y). The multiplicity mult(δ) of such a δ is the integer satisfying mult(δ) · [X] = p_{∗}(δ), where p_{∗} is the push-forward homomorphism

p_{∗} : CH_{e}(X × Y) → CH_{e}(X) = ℤ · [X].

The exchange isomorphism X × Y → Y × X induces an isomorphism

CH_{e}(X × Y) → CH_{e}(Y × X)

sending a cycle δ to its transpose δ^{t}.

Proof. To prove that a variety X is 2-incompressible, it suffices to show that for any correspondence δ : X ⇝ X of degree zero,

(5)  mult(δ) ≡ mult(δ^{t}) (mod 2).

Indeed, suppose we have f : X′ → X and a dominant g : X′ → X with F (X′)/F (X) finite of odd degree. Let δ ∈ CH(X × X) be the pushforward of the class [X′] along the induced morphism (g, f) : X′ → X × X. By assumption, mult(δ) is odd, so by (5) we have that mult(δ^{t}) is odd. It follows that f_{∗}([X′]) is an odd multiple of [X] and in particular is nonzero, so f is dominant.

We will check that the condition (5) holds for the variety X_{2}. A correspondence of degree zero δ : X_{2} ⇝ X_{2} is just an element of CH_{4n−5}(X_{2} × X_{2}). Using the method of Chernousov and Merkurjev described in [3], we can decompose the motive of X_{2} × X_{2} as follows. See also [4] for examples of similar computations.

We first realize X_{2} as a projective homogeneous variety. Let G denote the special orthogonal group corresponding to the quadratic form φ on V. Let Π be a set of simple roots for the root system Σ of G. If e_{1}, …, e_{n} are the standard basis vectors of ℝ^{n}, we may take

Π = {α_{1} :=e_{1}−e_{2}, …, α_{n−1} :=e_{n−1}−e_{n}, α_{n}:=e_{n}}.

Then X_{2} is a projective G-homogeneous variety, namely the variety of all parabolic subgroups of G of type S, for the subset S = Π\{α_{2}} of the set of simple roots.

Let W denote the Weyl group of the root system Σ. When n ≥ 4, there are six double cosets D ∈ W_{P} \W/W_{P} with representatives w listed in the first column below (where we write α_{k} when we mean the reflection w_{α_{k}}). The second column lists the effect of w^{−1} on the first four e_{i} (the rest not being affected). The third column gives the subset of Π associated to w. When n = 3, there are only five double cosets. In this case, the table may be amended by deleting the final row and removing all mention of e_{4} from the remaining rows.

w (α_{k} here means w_{α_{k}})  w^{−1}(e_{1}, e_{2}, e_{3}, e_{4})  R_{D}
1  (e_{1}, e_{2}, e_{3}, e_{4})  Π\{α_{2}}
α_{2} ⋯ α_{n} ⋯ α_{2}  (e_{1}, −e_{2}, e_{3}, e_{4})  Π\{α_{1}, α_{2}}
(α_{2} ⋯ α_{n} ⋯ α_{2})α_{1}(α_{2} ⋯ α_{n} ⋯ α_{2}) (−e_{2}, −e_{1}, e_{3}, e_{4}) Π\{α_{2}}
α_{1}  (e_{1}, e_{3}, e_{2}, e_{4})  Π\{α_{1}, α_{2}, α_{3}}
α_{2}α_{1}(α_{3} ⋯ α_{n} ⋯ α_{2})  (e_{3}, −e_{2}, e_{1}, e_{4})  Π\{α_{1}, α_{2}, α_{3}}
(α_{2}α_{1})(α_{3}α_{2})  (e_{3}, e_{4}, e_{1}, e_{2})  Π\{α_{2}, α_{4}}

From [3, Thm. 6.3], we deduce the following decomposition of the motive of X_{2} × X_{2}, where the last summand is removed for the case n = 3. We denote by X_{d_{1},d_{2},...,d_{s}} the variety of flags of totally isotropic subspaces of V of dimensions d_{1}, d_{2}, …, d_{s}.

\begin{aligned}M(X_{2} × X_{2}) ≃ & M(X_{2}) ⊕ M(X_{1,2})(2n − 3) ⊕ M(X_{2})(4n − 5) \\ & ⊕ M(X_{1,2,3})(1) ⊕ M(X_{1,2,3})(2n − 2) ⊕ [M(X_{2,4})(4)] \end{aligned}

This in turn yields a decomposition of the middle-dimensional component of the Chow group of X_{2} × X_{2}.

CH_{4n−5}(X_{2} × X_{2}) ≃ CH_{4n−5}(X_{2}) ⊕ CH_{2n−2}(X_{1,2}) ⊕ CH_{0}(X_{2})
⊕ CH_{4n−6}(X_{1,2,3}) ⊕ CH_{2n−3}(X_{1,2,3}) ⊕ [ CH_{4n−9}(X_{2,4})]

It now suffices to check the congruence mult(δ) ≡ mult(δ^{t}) (mod 2) for δ in the image of any of these summands. The embedding of the first summand CH_{4n−5}(X_{2}) is induced by the diagonal morphism X_{2} → X_{2} × X_{2}, so the multiplicities are equal by symmetry.

Any element δ of the third summand CH_{0}(X_{2}) has degree divisible by 4 by assumption, hence its image in the Chow group CH_{0}(X̄_{2}) is divisible by 4. (Here we use that CH_{0}(X̄_{2}) is generated by a single element of degree 1.) The image of δ in CH_{4n−5}(X̄_{2} × X̄_{2}) is then also divisible by 4, and since multiplicity does not change under field extension, mult(δ) ≡ 0 ≡ mult(δ^{t}) (mod 4).

The second summand requires our work from the previous section. Since X_{1,2} is a projective bundle over X_{2}, there is a motivic decomposition M(X_{1,2}) ≃ M(X_{2}) ⊕ M(X_{2})(1), so that

CH_{2n−2}(X_{1,2}) ≃ CH_{2n−2}(X_{2}) ⊕ CH_{2n−3}(X_{2}).

It is enough to consider δ equal to the image of some β ∈ CH_{r}(X_{2}), where r ∈ {2n − 3, 2n − 2}. By the same reasoning as in the previous paragraph, it suffices to show that the image of β in \overline{CH}_{r}(X_{2}) ⊂ CH_{r}(X̄_{2}) is divisible by 2 in CH_{r}(X̄_{2}). Suppose it is not. Then the image β̂ of β in the modulo-2 Chow group

Ch_{r}(X̄_{2}) := CH_{r}(X̄_{2})/2 CH_{r}(X̄_{2})

is nonzero. By [7, Rem. 5.6], the “cellular” variety X̄_{2} is “2-balanced,” i.e. the bilinear form (β̂, γ̂) ↦ deg(β̂ · γ̂) on Ch(X̄_{2}) is nondegenerate. Hence there exists γ ∈ CH^{r}(X̄_{2}) such that

deg(β · γ) ≡ 1 (mod 2).

Since 2γ is rational by our Lemma 4.1, we have

deg \overline{CH}_{0}(X_{2}) ∋ deg(β · 2γ) ≡ 2 (mod 4).

Degree does not change under field extension, so this contradicts our assumption that deg CH(X_{2}) = 4ℤ.

The last three summands of the decomposition are dealt with by the following proposition, whose proof uses our results on higher Witt indices. This will complete the proof of the theorem. □

Proposition 5.2. Let F l := X_{d_{1},d_{2},...,d_{s}} be a variety of totally isotropic flags with d_{s} > 2 and let the correspondence α : F l ⇝ X_{2} × X_{2} induce an embedding

α_{∗} : CH_{r}(F l) ↪ CH_{4n−5}(X_{2} × X_{2}).

Then for any δ in the image of α_{∗}, mult(δ) ≡ 0 ≡ mult(δ^{t}) (mod 2).

Proof. Consider the diagram below of fiber products, where we select either of the projections p_{i} and choose the other morphisms accordingly.

\xymatrix{ & (F l)_{F (X_{2})} \ar@{-->}[dr] \\ (F l × X_{2})_{F (X_{2})} \ar[r] \ar[d] \ar@{-->}[ur] & (X_{2})_{F (X_{2})} \ar[r] \ar[d] & Spec F (X_{2}) \ar[d] \\ F l × X_{2} × X_{2} \ar[r] \ar[d] & X_{2} × X_{2} \ar[r]_{p_{2}}^{p_{1}} & X_{2} \\ F l}

Taking push-forwards and pull-backs, we get the following diagram which commutes except for the triangle at the bottom. The push-forward by p_{i} takes a cycle δ ∈ CH_{4n−5}(X_{2} × X_{2}) to mult(δ) if we chose the first projection p_{1} or to mult(δ^{t}) if we chose the second projection p_{2}.

[figure: CH_{0} ((F l)_{F (X_{2})}); deg; _{CH_{0} ((F l × X_{2})_{F (X_{2})})}; _{CH_{0} ((X_{2})_{F (X_{2})})}; deg; ℤ; _{CH_{4n−5} (F l × X_{2} × X_{2})}; _{CH_{4n−5} (X_{2} × X_{2})}; (mult)∘(transpose); mult; ℤ; CH_{r}(F l); α_{∗}]

Any δ ∈ im(α_{∗}) also lies in the image of CH_{4n−5} (F l × X_{2} × X_{2}), by the definition of the push-forward. Chasing through the diagram, one sees that mult(δ) (and similarly mult(δ^{t})) must lie in deg CH_{0} ((F l)_{F (X_{2})}). By Proposition 3.1, we know that 𝔧_{2}(φ) = 2. Hence by Proposition 3.2, φ_{F (X_{2})} ≃ 2ℍ ⊥ ψ for some anisotropic quadratic form ψ over F (X_{2}). In order for the variety F l_{F (X_{2})} to have a rational point over a field extension K of F (X_{2}), ψ must be isotropic over K, due to the assumption d_{s} > 2. By Springer’s Theorem, if the degree of the extension K is finite then it must be divisible by 2, so

deg CH_{0} ((F l)_{F (X_{2})}) ⊂ 2ℤ.
□

This completes the proof of the theorem.

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12 BRYANT G. MATHEWS

Department of Mathematics, University of California, Los Angeles, CA 90095-1555

E-mail address: bmathews@math.ucla.edu