INCOMPRESSIBILITY OF PRODUCTS BY GRASSMANNIANS OF ISOTROPIC SUBSPACES

NIKITA A. KARPENKO

Abstract. We prove that the product of an arbitrary projective homogeneous variety Y by an orthogonal, symplectic, or unitary Grassmannian X is 2-incompressible if and only if the varieties X_{F (Y)} and Y_{F (X)} are so. Some new properties of incompressible Grassmannians are established on the way.

1. Introduction

Let F be a field and let p be a prime number. We refer to [4] and [14] for definitions and general discussion of canonical p-dimension and p-incompressibility.

We work with projective homogeneous varieties, i.e., twisted flag varieties under semisimple affine algebraic groups. A necessary condition for p-incompressibility of the product X × Y of projective homogeneous F -varieties X and Y is p-incompressibility of the varieties X_{F (Y)} and Y_{F (X)}. This necessary condition is known by [2] to be sufficient in the case where X is a generalized Severi-Brauer variety, that is, the Grassmannian of right ideals of a fixed dimension in a central simple F -algebra. Such a variety is a twisted form of a usual Grassmannian of subspaces of a fixed dimension in a finite-dimensional vector space over F and is projective homogeneous for an algebraic group of the Dynkin type ^{1}A.

In the present paper we show that the above necessary condition is also sufficient in the case where X is a Grassmannian of totally isotropic spaces of a fixed dimension for:

– a non-degenerate quadratic form (orthogonal case, algebraic groups of types B and D), or

– a hermitian form over a separable quadratic extension field of F (unitary case, type ^{2}A), or

– a hermitian form over a quaternion division F -algebra (symplectic case, type C), where in the characteristic 2 case the hermitian form is supposed to be alternating, [12, §4.A].

Note that in the symplectic case, only the Grassmannians of subspaces of integral dimension over the quaternion algebra are considered because the others are not interesting from the viewpoint of the canonical dimension. Also note that p = 2 is the only interesting prime for the varieties treated here.

Date: 23 September, 2014.

Key words and phrases. Quadratic forms; algebraic groups; projective homogeneous varieties; orthogonal, symplectic, and unitary Grassmannians, Chow groups and motives; canonical dimension and incompressibility. Mathematical Subject Classification (2010): 20G15; 14C25.

The author acknowledges a partial support of the French Agence Nationale de la Recherche (ANR)

under reference ANR-12-BL01-0005; his work has been also supported by the start-up grant of the University of Alberta and a Discovery Grant from the National Science and Engineering Board of Canada.

Proving the result, we establish some properties of incompressible Grassmannians which seem to be of interest on their own, even in the most classical orthogonal case. Note however, that we prove the main result – a criterion of incompressibility for X × Y in terms of incompressibility of a Grassmannian X and an arbitrary projective homogeneous variety Y – without possessing a criterion of incompressibility for the factors themselves, even for the more specific X. So, the situation here is different from the situation of [2], dealing with the usual twisted Grassmannians, where a criterion of their incompressibility was already available.

Fortunately, we at least have [6], [7], and [9], telling, roughly, that there are many incompressible orthogonal, unitary, and symplectic Grassmannians.

Acknowledgements. I am grateful to Alexander Merkurjev for asking me the question about possibility of generalization of [2, Theorem 3.1] during my talk at the conference “(A)round forms, cycles and motives” on the occasion of the 80th birthday of Albrecht Pfister in Mainz, Germany. This work has been done during my stay at the Universität Duisburg-Essen and the Max-Planck-Institut für Mathematick in Bonn; I thank them for hospitality.

2. Canonical p-dimension of a fibration

As shown in [2] (see also Corollary 2.6 here), canonical p-dimension of the product X × Y of projective homogeneous F -varieties X and Y has cdim_{p} X + cdim_{p} Y_{F (X)} as an upper bound. We may view X as the base of the projection (a “trivial fibration”) X ×Y → X, and Y_{F (X)} is its generic fiber. In this section, we generalize this upper bound relation to the case of a more general fibration and also we sharpen the upper bound, replacing cdim_{p} X by an, in general, smaller integer cdim′_{p} X defined in terms of X and the function field of the total variety of the fibration.

Here is the type of the fibrations we are interested in. Let G be a quasi-split semisimple affine algebraic group over F becoming split over a finite extension field of a p-power degree, T a G-torsor over F, P a parabolic subgroup of G and P′ a parabolic subgroup of G contained in P. We consider the fibration

π: Z := T/P′ → T/P =: X

of projective homogeneous varieties, and we write Y for its generic fiber. We are using the Chow group Ch with coefficients in 𝔽_{p} := ℤ/pℤ. In particular, the degree homomorphism deg on Ch_{0} of a complete variety takes its values in 𝔽_{p}.

Lemma 2.1. In the above settings, we have

cdim_{p} Z ≤ cdim′_{p} X + cdim_{p} Y,

where cdim′_{p} X is the minimal integer d such that there exist elements α ∈ Ch^{d}(X_{F (Z)}) and β ∈ Ch_{d}(X) with deg(α · β_{F (Z)}) = 1.

Remark 2.2. Replacing F (Z) by F (X) in the definition of cdim′_{p} X, we get cdim_{p}(X), see [6, Corollary 6.2]. Since F (Z) ⊃ F (X), we have cdim′_{p} X ≤ cdim_{p} X.

Remark 2.3. In the case where the parabolic subgroup P′ is special, Lemma 2.1 has been proved in [3, Lemma 5.3]. The proof was more complicated (and the statement – more specific) because [6, Corollary 6.2] was not available at the time.

Proof of Lemma 2.1. By the definition of cdim′_{p} X, for x := cdim′_{p} X we can find

α_{X} ∈ Ch^{x}(X_{F (Z)}) and β_{X} ∈ Ch_{x}(X)

with deg(α · β_{F (Z)}) = 1.

Note that the function field F (X) is the field of definition of the variety Y. By [6, Corollary 6.2], for y := cdim_{p} Y we can find

α_{Y} ∈ Ch^{y}(Y_{F (X)(Y)}) and β_{Y} ∈ Ch_{y}(Y)

with deg(α · β_{F (X)(Y)}) = 1. We are going to produce certain

α′_{Y} ∈ Ch^{y}(Z_{F (Z)}) and β′_{Y} ∈ Ch_{y+dim X}(Z)

out of α_{Y} and β_{Y}.

By [1, Proposition 57.10], the flat pull-back homomorphisms

Ch_{y+dim X}(Z) → Ch_{y}(Y) and Ch^{y}(Z) → Ch^{y}(Y)

are surjective. We define β′_{Y} simply as a preimage of β_{Y}.

In order to define α′_{Y}, let us first replace the base field F by the function field F (Z). This way we get a rational point on Y, therefore the field extension F (X)(Y)/F (X) becomes purely transcendental and the group Ch^{y}(Y_{F (X)(Y)}) containing α_{Y} is identified with Ch^{y}(Y) via the change of field homomorphism. So, we can define α′_{Y} as a preimage of α_{Y} ∈ Ch^{y}(Y) under the surjection Ch^{y}(Z) →→ Ch^{y}(Y). Turning back to our original base field F, we see that α′_{Y} ∈ Ch^{y}(Z_{F (Z)}).

Now we set

α := π^{∗}(α_{X}) · α′_{Y} ∈ Ch^{x+y}(Z_{F (Z)}) and β := π^{∗}(β_{X}) · β′_{Y} ∈ Ch_{x+y}(Z).

We claim that deg(α · β_{F (Z)}) ≠ 0. According to [6, Corollary 6.2], the claim implies that

cdim_{p} Z ≤ x + y = cdim′_{p} X + cdim_{p} Y.

It remains to prove the claim. Checking the claim, we may replace the base field F by any its extension field. Therefore we may assume that the group G and the G-torsor T are split. The projective homogeneous variety Y can be then defined over F. It follows by [15, §3] that the pull-back Ch(X) → Ch(Z) together with the homomorphism Ch(Y) → Ch(Z), defined on an arbitrary basis of Ch(Y) by taking any preimage under Ch(Z) →→ Ch(Y_{F (X)}), induce an isomorphism of additive groups

(2.4)  Ch(X) ⊗_{𝔽_{p}} Ch(Y) ≃ Ch(Z).

The product in Ch(Z), whose degree we are interested in, can be rewritten as the product of π^{∗}([pt]), where pt is a rational point on X, by an element γ ∈ Ch(Z) with the image under Ch(Z) →→ Ch(Y_{F (X)}) given by the class of a rational point on Y. It follows by (2.4) that π^{∗}([pt]) · γ ≠ 0 in Ch_{0}(Z). Since deg : Ch_{0}(Z) → 𝔽_{p} is an isomorphism, we get that deg(π^{∗}([pt]) · γ) ≠ 0. □

Remark 2.5. One may check that the isomorphism (2.4) commutes with the degree homomorphism. This will show that the degree of α · β_{F (Z)} is actually 1, not just ≠ 0.

Corollary 2.6 ([2]). Let X and Y be projective homogeneous varieties. Then

cdim_{p}(X × Y) ≤ cdim_{p} X + cdim_{p} Y_{F (X)}.

In particular, X × Y is p-incompressible only if X_{F (Y)} and Y_{F (X)} are so.

Proof. Since canonical p-dimension is not changed under base field extensions of degree prime to p (see [13, Proposition 1.5(2)]), we may assume that the semisimple affine algebraic groups acting on the projective homogeneous varieties X and Y become of inner type over a finite p-primary extension of F. Therefore we may apply Lemma 2.1 to the projection X × Y → X. Taking into account Remark 2.2, we get Corollary 2.6. □

3. Incompressible orthogonal Grassmannians

In this section, n is a positive integer, φ is a non-degenerate n-dimensional quadratic form over F, and X_{i} with 0 ≤ i < n/2 is the ith orthogonal Grassmannian of φ. Note that we do not consider the case of i = n/2 (with even n), see Remark 4.2.

We recall that dim X_{i} = i(i − 1)/2 + i(n − 2i). (This formula is valid for i = n/2 as well.)

Now assume that n ≥ 3. The variety X_{1} is the projective quadric of φ and the quadratic form φ_{F (X_{1})} is isotropic. Let φ′ be an (n − 2)-dimensional quadratic form Witt-equivalent to φ_{F (X_{1})}. For i with 0 ≤ i < (n − 2)/2, we write X′_{i} for the ith orthogonal Grassmannian of φ′.

Proposition 3.1. For some n ≥ 3 and some i ≥ 1, assume that the variety X_{i} is 2incompressible. Then φ is anisotropic, all the first i higher Witt indexes 𝔦_{1}(φ), …, 𝔦_{i}(φ) of φ are equal to 1, and the variety X′_{i−1} is also 2-incompressible.

Remark 3.2. In [6], the following sufficient condition for 2-incompressibility of X_{i} has been given: 𝔦_{i}(φ) = 1 and the degree of every closed point on X_{i} is divisible by 2^{i}. The degree condition implies that φ is anisotropic and 𝔦_{1}(φ) = ⋯ = 𝔦_{i−1}(φ) = 1. So, Proposition 3.1 shows that a part of the sufficient condition for 2-incompressibility of X_{i} is actually necessary.

Remark 3.3. For i = 1, Proposition 3.1 says that the 2-incompressibility of the projective quadric X_{1} implies that φ is anisotropic and its first Witt index is 1 (the variety X′_{i−1} is just a point and as such is 2-incompressible automatically). This statement is easy to check (as in the proof below), Lemma 2.1 is not used. A known old result due to A. Vishik actually says that X_{1} is 2-incompressible if and only if φ is anisotropic of first Witt index 1. For a proof (based on ideas of [17]) see [1, Theorem 90.2].

Remark 3.4. For odd n, the statement of Proposition 3.1 on maximal i = (n−1)/2 can be easily deduced from the properties of the J-invariant of φ. The original paper introducing J-invariant and establishing its main properties is [18]. The monograph [1, §88], where the J-invariant is replaced by its “opposite”, can also be consulted. In particular, see [1, Theorem 90.3] for the relation between canonical 2-dimension and the J-invariant.

Here is the deduction. For i = (n − 1)/2, let X := X_{i} and let

e_{1} ∈ Ch^{1}(X_{F (X)}), …, e_{i} ∈ Ch^{i}(X_{F (X)})

be the ring generators of Ch(X_{F (X)}) defined as in [1, §86]. We call an element of Ch(X_{F (X)}) rational, if it lies in the image of the change of field homomorphism Ch(X) → Ch(X_{F (X)}). Assuming that X is 2-incompressible, none of e_{1}, …, e_{i} is rational by [1, Theorem 90.3]. It follows by [1, Proposition 88.8] that the quadratic form φ in question is anisotropic and has 𝔦_{1}(φ) = ⋯ = 𝔦_{i−1}(φ) = 1. Therefore 𝔦_{i}(φ) = 1 as well. Finally, replacing the base field F by the function field F (X_{1}) of the quadric X_{1}, the element e_{i} becomes rational, but none of e_{1}, …, e_{i−1}, see [1, Corollary 88.6]. This shows that

cdim_{2} X = 1 + ⋯ + (i − 1) = dim X′_{i−1}

for the now equivalent (in the sense of existence of rational maps in both directions) varieties X = X_{i} and X′_{i−1}, meaning that X′_{i−1} is 2-incompressible.

Proof of Proposition 3.1. If φ is isotropic, the variety X_{i} is equivalent (in the above sense of existence of rational maps in both directions) to the variety X′_{i−1}. Therefore cdim_{2} X_{i} = cdim_{2} X′_{i−1} (see, e.g., [10, Lemma 3.3]). Since dim X_{i} > dim X′_{i−1}, we get a contradiction with the 2-incompressibility of X_{i}.

We have shown that φ is anisotropic. Next, assuming that i ≥ 2, we are going to check the 2-incompressibility of X′_{i−1}. For this, we apply Lemma 2.1 to the fibration π : X_{1⊂i} → X_{1}, where X_{1⊂i} is the 2-flag variety projecting onto X_{1} and X_{i}. Note that the variety X′_{i−1} is the generic fiber of π. The (i − 1)th power h^{i−1} ∈ Ch^{i−1}(X_{1}) of the hyperplane section class h ∈ Ch^{1}(X_{1}) and the class l_{i−1} ∈ Ch_{i−1}(X_{1})_{F (X_{1⊂i})} of an i-dimensional totally isotropic subspace satisfy the relation

deg(l_{i−1} · h_{F (X_{1⊂i})}^{i−1}) = 1.

Therefore cdim′_{2} X_{1} ≤ dim X_{1} − (i − 1) = (n − 2) − (i − 1) = n − i − 1. It follows by Lemma 2.1 that

cdim_{2} X_{1⊂i} ≤ (n − i − 1) + cdim_{2} X′_{i−1}.

Since the flag variety X_{1⊂i} is equivalent to X_{i}, we may replace cdim_{2} X_{1⊂i} by cdim_{2} X_{i} in this inequality. Since dim X_{i} = (n − i − 1) + dim X′_{i−1}, the upper bound on cdim_{2} X_{i}, we obtained, shows that the 2-incompressibility of X_{i} implies that of X′_{i−1}.

We have shown that X′_{i−1} is 2-incompressible. It follows by the preceding part that the quadratic form φ′ is anisotropic. This means that 𝔦_{1}(φ) = 1. Continuing this induction procedure, we get that 𝔦_{1}(φ) = ⋯ = 𝔦_{i−1}(φ) = 1.

To finish the proof of Proposition 3.1, it remains to show that 𝔦_{i}(φ) = 1. Assume that 𝔦_{i}(φ) ≥ 2. In particular, i < [n/2]. Let ψ be a (n−1)-dimensional non-degenerate subform of φ. (Note that the notion of non-degeneracy for quadratic forms in characteristic 2 we are using is that of [1] so that non-degenerate quadratic forms exist in any dimension.) Let Y_{i} be the ith orthogonal Grassmannian of ψ. The condition 𝔦_{i}(φ) ≥ 2 ensures that the varieties X_{i} and Y_{i} are equivalent. Indeed, we always have a rational map Y_{i} −−→ X_{i}; on the other hand, the intersection of an (i + 1)-dimensional totally isotropic subspace of φ_{F (X_{i})} with the hyperplane of definition of ψ_{F (X_{i})} has dimension at least i, showing that Y_{i}(F (X_{i})) ≠ ∅, so that a rational map X_{i} −−→ Y_{i} exists as well. But dim Y_{i} < dim X_{i} contradicting 2-incompressibility of X_{i}. □

4. Products by orthogonal Grassmannians

Theorem 4.1. Let X be an orthogonal Grassmannian of a non-degenerate quadratic form over F and let Y be a projective homogeneous variety under a semisimple affine algebraic group over F. The product X × Y is 2-incompressible if and only if the varieties X_{F (Y)} and Y_{F (X)} are so.

Remark 4.2. One may include in the statement of Theorem 4.1 the case where the quadratic form (let us call it φ) is of even dimension n and X is a component of X_{n/2}. However the proof of this case easily reduces to the case of (n − 2)/2th orthogonal Grassmannian of a (n − 1)-dimensional quadratic form (namely, any (n − 1)dimensional non-degenerate subform in φ_{L}, where L/F is the discriminant field extension of φ).

Proof of Theorem 4.1. By Corollary 2.6 we only need to prove the “if” part.

Since canonical 2-dimension is not changed under base field extensions of odd degree (see [13, Proposition 1.5(2)]), we may assume that there exists a finite 2-primary Galois field extension E/F such that G_{E} is of inner type, where G is the semisimple algebraic group over F acting on Y. This assumption allows us to apply results of [5] and [6, §6].

The variety X is the ith orthogonal Grassmannian X_{i} of a non-degenerate n-dimensional quadratic form φ for some i, n with 0 ≤ i < n/2. We induct on n. For n = 0 the statement we are proving is trivial. We assume that n ≥ 1 below. We also may assume that i ≥ 1 (otherwise, the variety X is just the point Spec F).

Let F′ be the function field of the variety X_{1} and let φ′ be a (n − 2)-dimensional quadratic form over F′ Witt-equivalent to φ_{F′}. We set X′ := X′_{i−1}, the (i−1)st orthogonal Grassmannian of φ′.

We are using Chow motives with coefficients in 𝔽_{2} := ℤ/2ℤ, [1, §64]. The proof below is a modification of [2, Proof of Theorem 3.1].

The motive M(X_{F′}) of the variety X_{F′} decomposes into a direct sum

(4.3)  M(X_{F′}) ≃ M(X′) ⊕ M(X′)(dim X − dim X′) ⊕ N,

where N is a shift of M(X′_{i}), cf. [6, (2.6)]. (The shifting number is (dim X′ + dim X′_{i})/2 but the knowledge of it is not needed in the sequel.) Multiplying by the motive of Y_{F′}, we get that

M(X × Y)_{F′} ≃ M(X′ × Y_{F′}) ⊕ M(X′ × Y_{F′})(dim X − dim X′) ⊕ N,

with N being a shift of M(X′_{i} × Y_{F′}) now. We claim that the variety X′ × Y_{F′} is 2incompressible by the induction hypothesis. To check the claim, we check that the varieties Y_{F′(X′)} and X′_{F′(Y)} are 2-incompressible. To check that Y_{F′(X′)} is 2-incompressible, we check that Y over a larger field F′(X′ × X_{F′}) is so. Since the projective homogeneous varieties X_{1} and X′ possess F (X)-points, the field

F′(X′ × X_{F′}) = F (X)(X_{1})(X′)

is purely transcendental over F (X). And the variety Y_{F (X)} is 2-incompressible (this is the place in the proof of the “if” part of Theorem refmain, where the assumption on Y_{F (X)} is used). Since canonical 2-dimension of projective homogeneous varieties does not change under purely transcendental field extensions, Y over F′(X′ × X_{F′}) is 2-incompressible.

To check that X′_{F′(Y)} is 2-incompressible, we apply Proposition 3.1 to the base field F (Y). The variety X_{F (Y)} over the field F (Y) is assumed to be 2-incompressible. Therefore the variety X′_{F′(Y)} is 2-incompressible by Proposition 3.1.

The claim being proved, writing U(Z) for the upper motive of a projective homogeneous variety Z, we obtain the following decomposition:

(4.4)  M(X × Y)_{F′} ≃ U(X′ × Y_{F′}) ⊕ U(X′ × Y_{F′})(dim X − dim X′) ⊕ N,

with N having the property that no summand of its complete decomposition is isomorphic to a shift of U(X′ × Y_{F′}). Indeed, by 2-incompressibility of X′ × Y_{F′}, no summand of the complete decomposition of the complement of U(X′ × Y_{F′}) in the total motive of the variety if isomorphic to a shift of the upper motive. Besides, since the higher Witt index 𝔦_{i−1}(φ′_{F′(Y)}) = 𝔦_{i}(φ_{F (Y)}) is equal to 1 by Proposition 3.1, there is no correspondence

X′ × Y_{F′} = X′_{i−1} × Y_{F′} ⇝ X′_{i} × Y_{F′}

of odd multiplicity. Therefore, no indecomposable summand of M(X′_{i} ×Y_{F′}) is isomorphic to a shift of U(X′ × Y_{F′}): by [5, Theorem 1.1], such a summand is either a corestriction of a motive over a strict extension field of F or isomorphic to a shift of U(Z) for a projective homogeneous variety Z with an odd multiplicity correspondence to X′_{i} × Y_{F′}. While a summand of the first type is trivially never isomorphic to an upper motive of a projective homogeneous variety, a summand U(Z) of the second type is not isomorphic to U(X′×Y_{F′}) because of absence of an odd multiplicity correspondence X′ × Y_{F′} ⇝ Z. (If such a correspondence would exist, then composing it with the correspondence Z ⇝ X′_{i} × Y_{F′} we would get an odd multiplicity correspondence X′ × Y_{F′} ⇝ X′_{i} × Y_{F′}). In the above arguments, we are constantly using the criterion [8, Corollary 2.15] of isomorphism of the upper motives U(Z) and U(Z′) of two projective homogeneous varieties Z and Z′: U(Z) ≃ U(Z′) if and only if there exist odd multiplicity correspondences Z ⇝ Z′ and Z′ ⇝ Z.

Now, to show that the variety X × Y is 2-incompressible, it suffices by [4, Theorem 5.1] to show that the complete decomposition of U(X × Y) contains the second summand of (4.4). So, we assume the contrary and we look for a contradiction.

By [6, Proposition 2.4], the complete decomposition of U(X × Y)_{F′} contains as a summand the motive U(X′ × Y_{F′}) shifted by the difference

cdim_{2}(X × Y) − cdim_{2}(X′ × Y_{F′}).

Therefore, our assumption implies that the difference is 0, and we come to

cdim_{2}(X × Y) = cdim_{2}(X′ × Y_{F′}) = d,

where d := dim(X′ × Y_{F′}) = dim X′ + dim Y.

By [6, Proposition 6.1], there exist α ∈ Ch^{d}(X × Y)_{F (X×Y)} and β ∈ Ch_{d}(X × Y) with deg(α · β) ≠ 0 ∈ 𝔽_{2}. In the last formula, we consider both cycles over a common field extension of their fields of definition, before we multiply them. We use this convention below in the proof (in similar formulas on degree of products) as well.

Since cdim_{2}(X_{F′}) = dim X′ =: d′, we can find α′ ∈ Ch^{d′}(X_{F′(X)}) and β′ ∈ Ch_{d′}(X_{F′}) with deg(α′ · β′) ≠ 0. Using these α′ and β′ and a rational point pt ∈ Y_{F (Y)}, we get the cycles

α′ × [pt] ∈ Ch^{d}(X × Y)_{F′(X×Y)} and β′ × [Y] ∈ Ch_{d}(X × Y)_{F′},

having the same property as α and β:

deg ((α′ × [pt]) · (β′ × [Z])) ≠ 0.

It follows by [6, Lemma 6.5] that one can ”mix up” the old cycles with the new ones and get the relation

deg ((α′ × [pt]) · β) ≠ 0.

Since α′ × [pt] = (α′ × [Y]) · ([X] × [pt]), the last degree relation can be rewritten as deg(α′ ·β′′) ≠ 0, where β′′ ∈ Ch_{d′}(X_{F (Y)}) is the push-forward of the product ([X]×[pt])·β along the projection (X×Y)_{F (Y)} → X_{F (Y)}. Since the field extension F′(X)/F (X) is purely transcendental, there exists α′′ ∈ Ch^{d′}(X_{F (X)}) mapped to α′ under the change of field homomorphism. Changing notation, we write α′′ for the image of α′′ in Ch^{d′}(X_{F (Y)(X)}).

The cycles α′′ ∈ Ch^{d′}(X_{F (Y)(X)}) and β′′ ∈ Ch_{d′}(X_{F (Y)}) thus constructed have the property deg(α′′ · β′′) ≠ 0. It follows by [6, Corollary 6.2] that cdim_{2}(X_{F (Y)}) ≤ d′. Since

d′ = dim X′ = (i − 1)(i − 2)/2 + (i − 1)(n − 2i) < i(i − 1)/2 + i(n − 2i) = dim X,

the relation cdim_{2}(X_{F (Y)}) ≤ d′ we obtained contradicts the assumption on 2-incompressi-bility of the variety X_{F (Y)}. □

5. Unitary Grassmannians

Let K/F be a separable quadratic field extension, n ≥ 0 an integer, V an n-dimensional vector space over K, and φ a K/F -hermitian form on V. For any integer i with 0 ≤ i ≤ n/2, we write X_{i} for the unitary Grassmannian of totally isotropic i-dimensional K-subspaces in V.

We recall that dim X_{i} = i(2n − 3i).

Assume that n ≥ 2. Then the variety X_{1} is defined and the hermitian form φ_{F (X_{1})} is isotropic. Let φ′ be an (n−2)-dimensional K(X_{1})/F (X_{1})-hermitian form Witt-equivalent to φ. For i with 0 ≤ i ≤ (n − 2)/2, we write X′_{i} for the corresponding unitary Grassmannian of totally isotropic i-dimensional subspaces.

Proposition 5.1. For some integers n and i with 1 ≤ i ≤ n/2, assume that the variety X_{i} is 2-incompressible. Then φ is anisotropic, all the first i higher Witt indexes 𝔦_{1}(φ), …, 𝔦_{i}(φ) of φ are equal to 1, and the variety X′_{i−1} is also 2-incompressible.

Proof. We start by repeating word by word the proof of Proposition 3.1. The first change that occurs in the unitary case compared to the orthogonal one is in the formula for dim X_{i}. But we still have dim X_{i} > dim X′_{i−1} in the setting, so that anisotropy of φ is proved.

As the next step, we apply Lemma 2.1 to the projection π : X_{1⊂i} → X_{1} in order to prove the 2-incompressibility of X′_{i−1}, which is again the generic fiber of π. The base X_{1} of the projection, which was a usual quadric in the orthogonal case, is now a sort of unitary quadric, and we need information on its Chow group analogous to the information we have and have used for usual quadrics. Such an information is provided in [11, §3].

It is shown there that there are elements h ∈ Ch^{2}(X_{1}) and l_{i−1} ∈ Ch_{2(i−1)}(X_{1})_{F (X_{i})} with deg(l_{i−1} · h^{i−1}) = 1. (The notation for the elements we use is similar to the quadric case and differs from [11].) Therefore

cdim′_{2} X_{1} ≤ dim X_{1} − 2(i − 1) = (2n − 3) − 2(i − 1) = 2n − 2i − 1.

It follows by Lemma 2.1 that

cdim_{2} X_{1⊂i} ≤ (2n − 2i − 1) + cdim_{2} X′_{i−1}.

Since the flag variety X_{1⊂i} is equivalent to X_{i}, we may replace cdim_{2} X_{1⊂i} by cdim_{2} X_{i} in this inequality. Since dim X_{i} = (2n − 2i − 1) + dim X′_{i−1}, the upper bound on cdim_{2} X_{i}, we obtained, shows that 2-incompressibility of X_{i} implies that of X′_{i−1}.

We have shown that X′_{i−1} is 2-incompressible. It follows by the preceding part that the quadratic form φ′ is anisotropic. This means that 𝔦_{1}(φ) = 1. Continuing this induction procedure, we get that 𝔦_{1}(φ) = ⋯ = 𝔦_{i−1}(φ) = 1.

To finish the proof of Proposition 3.1, it remains to show that 𝔦_{i}(φ) = 1. Assume that 𝔦_{i}(φ) ≥ 2. In particular, i < [n/2]. Let ψ be a (n−1)-dimensional non-degenerate subform of φ. The form ψ exists because φ can be diagonalized, [16, Theorem 6.3 of Chapter 7]. Let Y_{i} be the ith unitary Grassmannian of ψ. By precisely the same argument as in the oprthogonal case, the condition 𝔦_{i}(φ) ≥ 2 ensures that the varieties X_{i} and Y_{i} are equivalent. But again dim Y_{i} < dim X_{i}, contradicting 2-incompressibility of X_{i}. □

Theorem 5.2. Let X be a unitary Grassmannian of a K/F -hermitian form over F (where K/F is a separable quadratic field extension) and let Y be a projective homogeneous variety under a semisimple affine algebraic group over F. The product X × Y is 2-incompressible if and only if the varieties X_{F (Y)} and Y_{F (X)} are so.

Proof. The proof goes through as that of Theorem 4.1. The motivic decomposition (4.3) is given by [7, Lemma 7.1]. Note that N is now given by a direct sum of a shift of M(X′_{i}) and several shifted copies of Spec K. But this addition of shifted copies of Spec K does not make any difference for the sequel. □

6. Symplectic Grassmannians

Let Q be a quaternion division F -algebra, n ≥ 0 an integer, V a right n-dimensional vector space over Q, φ a hermitian (with respect to the canonical symplectic involution on Q) form on V. In the case of char F = 2, we require φ to be alternating.

For any integer i with 0 ≤ i ≤ n/2, we write X_{i} for the symplectic Grassmannian of totally isotropic i-dimensional Q-subspaces in V. We refer to [9] for a general discussion on these varieties and recall that

dim X_{i} = i(4n − 6i + 1).

Note that besides the symplectic Grassmannians of φ introduced right above, for any odd integer j with 1 ≤ j ≤ n there is the variety of totally isotropic subspaces in V of “dimension” j/2 over Q. Any such variety however is equivalent to the conic of Q and therefore is not interesting regarding the questions we consider in the paper.

Lemma 6.1. Assume that n ≥ 1. There exists an element h ∈ Ch^{4}(X) and for any integer i with 1 ≤ i ≤ n/2 there exists an element l_{i−1} ∈ Ch^{4(i−1)}(X_{1})_{F (X_{i})} such that deg(l_{i−1} · h_{F (X_{i})}^{i−1}) = 1.

Proof. The variety X_{1} is a closed hypersurface (a sort of “symplectic quadric”) in the “projective space” Qℙ(V) – the variety of 1-dimensional Q-subspaces of V, cf. [9]. Note that dim Qℙ(V) = 4(n − 1), where n = dim V. Picking up a hyperplane W in V, we consider the class

H := [Qℙ(W)] ∈ Ch^{4}(Qℙ(V))

and define h ∈ Ch^{4}(X_{1}) as the pull-back of H.

Now we replace the base field F by the function field F (X_{i}), pick up a totally isotropic i-dimensional Q-subspace in V, and let l_{i−1} be its class in Ch_{4(i−1)}(X_{1}). For the closed imbedding in : X_{1} ↪ Qℙ(V) we have in_{∗}(l_{i−1}) = H^{n−i}. Since deg(H^{n−1}) = 1, we get, applying the projection formula for in:

deg(l_{i−1} · h^{i−1}) = deg in_{∗}(l_{i−1} · h^{i−1}) = deg(in_{∗}(l_{i−1}) · H^{i−1}) = deg(H^{n−1}) = 1. □

As in Lemma 6.1, assume that n ≥ 2. Then the variety X_{1} is defined and the hermitian form φ_{F (X_{1})} is isotropic. Let φ′ be an (n − 2)-dimensional hermitian form Witt-equivalent to φ. For i with 0 ≤ i ≤ (n − 2)/2, we write X′_{i} for the corresponding Grassmannian of totally isotropic i-dimensional subspaces.

Proposition 6.2. For some n ≥ 2 and some i ≥ 1, assume that the variety X_{i} is 2incompressible. Then φ is anisotropic, all the first i higher Witt indexes 𝔦_{1}(φ), …, 𝔦_{i}(φ) of φ are equal to 1, and the variety X′_{i−1} is also 2-incompressible.

Proof. Again everything goes through as in the proof of Proposition 3.1 (or also Proposition 5.1), although the symplectic Grassmannians we are working with now satisfy different dimension formulas compared to the varieties we had in the previous sections. Nevertheless we still have dim X_{i} > dim X′_{i−1} in the current setting, so that we obtain the anisotropy of φ.

The next step is, as before, the proof of 2-incompressibility of X′_{i−1}, involving Lemma 2.1 and the projection π : X_{1⊂i} → X_{1}. The information on the Chow group of X_{1} that we need here now is provided by Lemma 6.1. It shows that

cdim′_{2} X_{1} ≤ dim X_{1} − 4(i − 1) = (4n − 5) − 4(i − 1) = 4n − 4i − 1.

It follows by Lemma 2.1 that

cdim_{2} X_{1⊂i} ≤ (4n − 4i − 1) + cdim_{2} X′_{i−1}.

Since again the flag variety X_{1⊂i} is equivalent to X_{i}, we may replace cdim_{2} X_{1⊂i} by cdim_{2} X_{i} in this inequality. Since dim X_{i} = (4n − 4i − 1) + dim X′_{i−1}, the upper bound on cdim_{2} X_{i}, we obtained, shows that 2-incompressibility of X_{i} implies that of X′_{i−1}.

We have shown that X′_{i−1} is 2-incompressible. It follows as in the orthogonal case that 𝔦_{1}(φ) = ⋯ = 𝔦_{i−1}(φ) = 1.

To finish the proof of Proposition 3.1, it remains to show that 𝔦_{i}(φ) = 1. Assume that 𝔦_{i}(φ) ≥ 2. In particular, i < [n/2]. Let ψ be a (n−1)-dimensional non-degenerate subform of φ. Let Y_{i} be the ith unitary Grassmannian of ψ. The condition 𝔦_{i}(φ) ≥ 2 ensures that the varieties X_{i} and Y_{i} are equivalent. But now again dim Y_{i} < dim X_{i} contradicting the 2-incompressibility of X_{i}. □

Theorem 6.3. Let X be a symplectic Grassmannian of a hermitian form over a quaternion F -algebra and let Y be a projective homogeneous variety under a semisimple affine algebraic group over F. The product X × Y is 2-incompressible if and only if the varieties X_{F (Y)} and Y_{F (X)} are so.

Proof. The proof goes through as that of Theorem 4.1. The analogue of the motivic decomposition (4.3) is provided by [9, Lemma 9.1]. Note that N is now given by a direct sum of a shift of M(X′_{i}) and several shifted copies of the motive of the conic of Q. □

Theorems 4.1, 5.2, and 6.3 give all together

Corollary 6.4 (cf. [2, Corollary 3.6]). A product X_{1} × ⋯ × X_{r} × Y, where for each i = 1, …, r the ith factor X_{i} is a generalized Severi-Brauer variety or a (orthogonal, symplectic, or unitary) Grassmannian of isotropic subspaces and Y is an arbitrary projective homogenous variety, is 2-incompressible if and only each factor of the product considered over the function field of the product of the remaining factors is 2-incompressible. □

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

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