UNITARY GRASSMANNIANS

NIKITA A. KARPENKO

Abstract. We study projective homogeneous varieties under an action of a projective unitary group (of outer type). We are especially interested in the case of (unitary) grassmannians of totally isotropic subspaces of a hermitian form over a field, the main result saying that these grassmannians are 2-incompressible if the hermitian form is generic. Applications to orthogonal grassmannians are provided.

Contents

1. Introduction 1 2. Invariants under a permutation involution 4 3. Some equivariant Chow groups of the point 5 4. Generic varieties of complete flags 6 5. Weil transfer of projective bundles 10 6. Generic maximal unitary grassmannians 13 7. Essential motives of unitary grassmannians 15 8. Generic unitary grassmannians 16 9. Connection with quadratic forms 17 References 20

1. Introduction

In this paper, we study Chow rings, Chow motives, and incompressibility properties of projective homogeneous varieties under an action of an adjoint absolutely simple affine algebraic group of type A over an arbitrary field F. Such a group can be realized as the F -group of K-automorphism of an Azumaya K-algebra with an F -linear unitary involution, where K is a quadratic étale F -algebra. The case where K is split is pretty much studied in the literature (the varieties arising in that case are the generalized Severi-Brauer varieties and, more generally, varieties of flags of right ideals of a central simple F -algebra). In the present paper we are concentrated on the opposite case: K is a field, but the central simple K-algebra is split, i.e., isomorphic to the algebra of endomorphisms of a finite-dimensional vector space V over K. The involution on the algebra in this case is adjoint to some non-degenerate K/F -hermitian form on V.

Let K/F be a separable quadratic field extension, V a vector space over K endowed with a non-degenerate K/F -hermitian form h. By a Jacobson theorem (cf. Corollary 9.2), this classical object is completely determined by an even more classical one: the quadratic form v ↦ h(v, v) on V considered this time as a vector space over F. Moreover, the quadratic forms arising this way from K/F -hermitian forms, are easily described as the tensor products of a non-degenerate (diagonal) bilinear form by the norm form of K/F (which is a non-degenerate binary quadratic form, the quadratic form associated to the hermitian form ⟨1⟩ on K).

Date: 9 October 2011.

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

Although this well-known and elementary observation shows that the study of hermitian forms is equivalent to the study of the “binary divisible” quadratic forms, it does not show that the hermitian forms are not interesting because they are just a more complicated replacement for a well-understood thing. Quite the contrary, first of all this observation shows that the binary divisible quadratic forms form an important class of quadratic forms. For instance, the subclass of quadratic forms divisible by an n-fold Pfister form (i.e., divisible by a product of n binary forms: n − 1 bilinear and 1 quadratic) for n ≥ 2 is less important even if it is a natural generalization.

On the other hand, this observation provides a hope that the world around hermitian forms might give additional tools to study such quadratic forms. And indeed, it turns out that the projective homogeneous varieties related to hermitian forms (in particular, the unitary grassmannians) are more suitable (in particular, more “economical”) geometric objects related to the binary divisible quadratic forms if compared with the orthogonal grassmannians – the geometric objects commonly attached to quadratic forms. The present paper demonstrates and exploits these phenomena.

The main result of the paper is Theorem 8.1 saying that the unitary grassmannians associated to a generic hermitian form are 2-incompressible ^{1} and that essential parts of their motives are indecomposable. The main application (which was the initial motivation) is Corollary 9.4, computing the J-invariant of the quadratic form attached to the generic hermitian form and therefore the maximal value of J-invariant of a quadratic form arising from a hermitian one (of a fixed dimension). Another striking application (in the opposite direction: from quadratic forms to hermitian ones) is the computation of the modulo 2 reduced Chow ring of a unitary grassmannians in terms of the Chow ring of the orthogonal grassmannians made in Corollary 9.10 (for odd-dimensional hermitian forms see Example 9.12). This computation is particularly interesting in the hyperbolic case because the Chow ring of orthogonal grassmannians of hyperbolic forms is very well studied and described, see e.g. [4].

We start in Section 2 with a completely elementary computation of the subring of the invariant elements under a permutation involution on a polynomial ring. It becomes clear already at this point that one can make the life much more agreeable working modulo the norms. This principle will be constantly present in the sequel.

The reason of making the above computation becomes clear in Section 3 where we are interested in the T -equivariant Chow ring of the point Spec F for certain quasi-split torus T: this Chow ring is identified with the subring of Section 2. Our base field F appearing for the first time (after Introduction) at this point, is (and remains until the end of the paper) of arbitrary characteristic.

1We refer to [12] for the definition and main properties of incompressibility and, more generally, of canonical dimension of (projective homogeneous) varieties.

In Section 4, the main objects of our study begin to appear. The torus T (or 𝔾_{m} × T – depending on parity of n) considered in Section 3, turns out to be a maximal torus in a Borel subgroup B of a non-split quasi-split adjoint absolutely simple affine algebraic group G of type A_{n}. Consequently, as already noticed in [18], the T -equivariant Chow group CH_{T} (Spec F) appears in a computation of the reduced Chow group of the complete flag variety E/B given by a generic (in the sense of [10]) G-torsor E (where B ⊂ G is a Borel subgroup). Having shown that the ring CH_{T} (Spec F) is generated modulo the norms by elements of codimension ≤ 2, we get the same statement for the reduced Chow group \overline{CH}(E/B) (Proposition 4.5). Killing in a generic way a (certain) Tits algebra, we come to the complete flag variety of a (generic) hermitian form while keeping the above property (Corollary 4.6).

Next we would like to get from the complete flag variety to the maximal grassmannian. Clearly, this can be achieved by projecting complete flags to their maximal component. But unlikely the case of the orthogonal group, such a projection here is not a chain of projective bundles. It is a chain of quadratic Weil transfers of projective bundles. So, we make some general study of quadratic Weil transfers of projective bundles in Section 5.^{2} Once again, working modulo norms makes it possible to get a very simple (and very similar to the classical case of a projective bundle) description of the Chow ring.

We apply this description in Section 6 getting a description of the reduced Chow group of a generic maximal unitary grassmannian. We still get the generation by codimension ≤ 2, but in the same time the components of codimension 1 and 2 are 0, showing that the reduced Chow group modulo norms is trivial in positive codimensions (Proposition 6.1). This description actually shows that such a grassmannian is 2-incompressible. More than this, an essential part of its Chow motive (with coefficient in 𝔽_{2}) is indecomposable. But the last two statements belong already to Section 8 and are valid for all (generic) grassmannians, not only for the maximal one. Before we come to them, we need a definition of the essential part and some general observations on isotropic grassmannians, made in Section 7.

The notion of the essential motive of a unitary grassmannian, introduced in Definition 7.5, gives one more justification for working ”modulo norms”: the reduced Chow group of the essential motive is identified with the reduced Chow group of the variety modulo the norms.

The proof of Theorem 8.1, the main result of the paper, closely follows [14], where a similar result is proved for the orthogonal grassmannians. The difference is in the case of the maximal grassmannian (the induction base of the proof), which constitutes the main part of the present paper while in [14] it was served by a reference.

The final Section 9 establishes and exploits the connection with the quadratic forms. It starts by (I suppose) well-known Lemma 9.1 and classically known Corollary 9.2. Corollary 9.3 is just a translation of Lemma 9.1 to the language of upper motives. The first interesting result is Corollary 9.4 (discussed above already) computing the maximal possible value of the so-called J-invariant of a quadratic form split by a given separable quadratic field extension. This result was not accessible by methods of quadratic forms themselves, but transferring the problem to the world of hermitian forms, we come to a problem about a generic hermitian form (without any additional condition) which we can solve because such a hermitian form can be obtained from a generic torsor with a help of generic killing of a Tits algebra.^{3}

2This is already needed in the previous section in the moment we are generically killing the Tits algebra, because this is a central simple algebra over an extension of the base field.

The motivic indecomposability which we have in the generic case by Theorem 8.1 gives an isomorphism of motives of Proposition 9.5 (in generic and therefore in general case) which implies an isomorphism of Chow groups of Corollary 9.10. This isomorphism of Chow groups is interesting even in the quasi-split case: for instance, Examples 9.11 and 9.12 provide a description by generators and relations for the Chow group of a quasi-split maximal unitary grassmannian (in terms of such description known in the orthogonal case, [8, §86]).

Acknowledgements. I am grateful to Vladimir Chernousov, Alexander Merkurjev, Burt Totaro, Alexander Vishik, and Maksim Zhykhovich for useful discussions and advices. Special thanks to Alexander Merkurjev for Remark 4.2 and for noticing that the computations of Section 3 are valid for arbitrary étale algebras (or arbitrary quasi-split tori).

2. Invariants under a permutation involution

Let R be a commutative associative unital ring. For any integer r ≥ 0, let us consider the polynomial R-algebra in 2r variables S := R[a_{1}, b_{1}, …, a_{r}, b_{r}]. Let σ be the involution (i.e., a self-inverse automorphism) of the R-algebra S interchanging for every i = 1, …, r the variables a_{i} and b_{i}. We write S^{σ} ⊂ S for the subalgebra of σ-invariant elements.

The subset (1+σ)S := {s+σ(s) | s ∈ S} ⊂ S^{σ}, the image of the norm homomorphism N : S → S, s ↦ s + σ(s) (this is a homomorphism of R-modules), is contained in S^{σ} and is an ideal of S^{σ}.

Lemma 2.1. The quotient S^{σ}/(1 + σ)S is generated as R-algebra by the classes of the products a_{i}b_{i}, i = 1, …, r.

Proof. Induction on r. The statement is trivial for r = 0.

To pass from r − 1 to r ≥ 1, let S′ be the R-subalgebra R[a_{1}, b_{1}, …, a_{r−1}, b_{r−1}] of S. The S′-module S is free with the basis {a_{r}^{i}b_{r}^{j}}_{i,j≥0} so that an element ∑_{i,j≥0} α_{ij}a_{r}^{i}b_{r}^{j} ∈ S with α_{ij} ∈ S′ is σ-invariant if and only if σ(α_{ij}) = α_{ji} for any i, j. It follows that the S′^{σ}/(1 + σ)S′-algebra S^{σ}/(1 + σ)S is generated by the class of a_{r}b_{r}. □

Remark 2.2. In general, the R-algebra S^{σ} itself is not generated by polynomials of degree ≤ 2. Taking for instance R = ℤ and r = 3, we have a σ-invariant integral polynomial P := a_{1}a_{2}a_{3} + b_{1}b_{2}b_{3} which cannot be written as a polynomial in the σinvariant polynomials of degree ≤ 2 in a_{1}, b_{1}, a_{2}, b_{2}, a_{3}, b_{3}. Indeed, let us refer as elementary polynomial to any σ-invariant monomial (with coefficient 1) and to any polynomial of the form M +σ(M), where M is a non-σ-invariant monomial (with coefficient 1). Any degree 3 product of elementary polynomials of degree 1 or 2 either has no monomial with repeated indices of variables (only has monomials like a_{1}a_{2}b_{3}, but no monomials like a_{1}b_{1}a_{3} or a_{1}^{2}a_{3}) or consists only of monomials with repeated indices. If P is a linear combination of such products with coefficients ±1 (which is the case if P is a polynomial in the σ-invariant polynomials of degree ≤ 2), then the part of the combination given by the products of the second type is 0. Each product in the remaining part of the decomposition, as any product of the first type, has the following property: it is a sum of even number of degree 3 elementary polynomials. In particular, P is a linear combination with coefficients ±1 of an even number of elementary polynomials. Any cancelation keeps the parity of this number. However P is just one elementary polynomial.

3Actually, we use the name generic hermitian form in a more concrete and narrow sense (defined before Corollary 4.6) in the main body of the article. At this (non-formal) point, speaking about “generic” we mean a versality property in the spirit of [25, Definition 5.1(2)].

On the other hand, S^{σ} is generated by its degree 1 and 2 components as far as 2 is invertible in R: this can be shown by induction using the identity

2(fgh + f′g′h′) = (fg + f′g′)(h + h′) − (fh′ + f′h)(g + g′) + (gh + g′h′)(f + f′).

3. Some equivariant Chow groups of the point

Let F be a field. We recall the computation of the 𝔾_{m}-equivariant Chow ring

CH_{𝔾_{m}}(Spec F),

c.f. [6, Lemma 4]. For generalities on equivariant Chow groups and rings, we refer to [7] and [3].

The F -torus 𝔾_{m} acts by multiplication on the affine line 𝔸^{1} = 𝔸_{F}^{1} over F. For any integer l ≥ 1, we consider the diagonal action of 𝔾_{m} on 𝔸^{l}. By the homotopy invariance of the equivariant Chow groups, the pull-back ring homomorphism CH_{𝔾_{m}}(Spec F) → CH_{𝔾_{m}}(𝔸^{l}) is an isomorphism. By the localization sequence, the pull-back ring homomorphism CH_{𝔾_{m}}(𝔸^{l}) → CH_{𝔾_{m}}(U_{l}) to the 𝔾_{m}-invariant open subset U_{l} := 𝔸^{l} \ {0} is bijective in codimensions < l (because the complement of U_{l} in 𝔸^{l} is of codimension l). It follows that the ring homomorphism CH_{𝔾_{m}}(Spec F) → CH_{𝔾_{m}}(U_{l}) is bijective in codimensions < l. Since 𝔾_{m} acts freely on U_{l} and U_{l}/𝔾_{m} = ℙ^{l−1}, the ring CH_{𝔾_{m}}(U_{l}) is identified with the usual Chow ring CH(ℙ^{l−1}). It follows that the ring CH_{𝔾_{m}}(Spec F) is freely generated by the element h ∈ CH_{𝔾_{m}}^{1}(Spec F) corresponding to the class of a rational point in CH^{1}(ℙ^{1}) under the isomorphism CH_{𝔾_{m}}^{1}(Spec F) → CH^{1}(ℙ^{1}).

Now we fix an étale F -algebra L (see [19, §18A] for generalities on étale algebras) and consider the Weil transfer R_{L/F} 𝔾_{m} (see [19, 20.5] where it is called corestriction) with respect to L/F of the L-torus 𝔾_{m}. The F -torus R_{L/F} 𝔾_{m} is quasi-split, and any quasisplit F -torus is isomorphic to R_{L/F} 𝔾_{m} for an appropriate choice of L. We would like to describe the graded (by codimension) ring CH_{R_{L/F} 𝔾_{m}}(Spec F).

We fix a separable closure F_{sep} of F, write Γ for the Galois group of F_{sep}/F, and X for the Γ-set of the F -algebra homomorphisms L → F_{sep}. We consider the following graded rings: the polynomial ring ℤ[X] in variables indexed by the elements of X and its subring ℤ[X]^{Γ} of Γ-invariant elements.

Let us do a computation for CH_{R_{L/F} 𝔾_{m}}(Spec F) similar to the above computation of CH_{𝔾_{m}}(Spec F). The torus R_{L/F} 𝔾_{m} acts by multiplication on the affine F -space R_{L/F} 𝔸^{1} (of dimension #X = dim_{F} L). For any integer l ≥ 1, we consider the diagonal action of R_{L/F} 𝔾_{m} on the affine F -space R_{L/F} (𝔸^{l}) = (R_{L/F} 𝔸^{1})^{l}. The action on the open R_{L/F} 𝔾_{m}- invariant subset U_{l} := R_{L/F} (𝔸^{l} \ {0}) is free, and U_{l}/R_{L/F} 𝔾_{m} = R_{L/F} ℙ^{l−1}. We get a ring homomorphism CH_{R_{L/F} 𝔾_{m}}(Spec F) → CH(R_{L/F} ℙ^{l−1}) which is bijective in codimensions < l (because the complement of U_{l} in R_{L/F} 𝔸^{l} is of codimension l).

Since the integral Chow motive (for generalities on motives wee [8, Chapter XII]) of the variety R_{L/F} ℙ^{l−1} is a sum of shifts of the motives of Spec K for certain subfields K ⊂ F_{sep} finite over F (see [16] or [17]), the ring homomorphism CH(R_{L/F} ℙ^{l−1}) → CH(R_{L/F} ℙ^{l−1})_{F_{sep}} identifies the ring CH(R_{L/F} ℙ^{l−1}) with the subring of Γ-invariant elements in CH(R_{L/F} ℙ^{l−1})_{F_{sep}}. It remains to identify the F_{sep}-variety (R_{L/F} ℙ^{l−1})_{F_{sep}} with the product of copies of ℙ^{l−1} indexed by the elements of the set X and therefore to identify CH(R_{L/F} ℙ^{l−1})_{F_{sep}} with the tensor product of the indexed by the elements of X copies of CH(ℙ^{l−1}). Therefore we obtain an isomorphism of graded rings

(3.1)  CH_{R_{L/F} 𝔾_{m}}(Spec F) = ℤ[X]^{Γ}.

Remark 3.2. Any element L → F_{sep} of X gives a homomorphism of F_{sep}-algebras L_{F_{sep}} → F_{sep} and therefore a character of the F_{sep}-torus (R_{L/F} 𝔾_{m})_{F_{sep}}. The obtained map X → (R_{L/F} 𝔾_{m})_{F_{sep}}^{∗} (to the group of characters) induces an isomorphism of graded Γ-rings ℤ[X] → S((R_{L/F} 𝔾_{m})_{F_{sep}}^{∗}) to the symmetric ring of the abelian group (R_{L/F} 𝔾_{m})_{F_{sep}}^{∗}. Therefore the isomorphism 3.1 gives rise to an (actually independent of the choice of L) isomorphism

CH_{T} (Spec F) = S(T_{F_{sep}}^{∗})^{Γ}

(for any quasi-split F -torus T).

Let now K be a separable quadratic field extension of F. Applying the isomorphism 3.1 to the étale F -algebra L = K^{×r}, since R_{L/F} 𝔾_{m} = (R_{K/F} 𝔾_{m})^{×r}, we get

Lemma 3.3. Let T := (R_{K/F} 𝔾_{m})^{×r} for some r ≥ 1. There is an isomorphism of the graded ring CH_{T} (Spec F) with the subring of σ-invariant elements of the ring

ℤ[a_{1}, b_{1}, …, a_{r}, b_{r}]

(where σ is the ring involution exchanging a_{i} and b_{i} for each i = 1, …, r) such that the image of the norm map CH_{T} (Spec K) → CH_{T} (Spec F) corresponds to the image of the map 1 + σ : ℤ[a_{1}, b_{1}, …, a_{r}, b_{r}] → ℤ[a_{1}, b_{1}, …, a_{r}, b_{r}]^{σ}. □

Note that the image of the norm map CH_{T} (Spec K) → CH_{T} (Spec F) is an ideal of the destination ring. Applying Lemma 2.1, we get

Corollary 3.4. The ring CH_{T} (Spec F) modulo the ideal of norms from Spec K is generated by elements of codimension 2. □

Similarly, applying first Lemma 3.1 to the étale F -algebra L = F × K^{×r} and then Lemma 2.1 with R being a ring integral polynomials in one variable, we get

Corollary 3.5. The ring CH_{𝔾_{m}×T} (Spec F) (for T as in Lemma 3.3) modulo the ideal of norms from Spec K is generated by elements of codimension 1 and 2. □

4. Generic varieties of complete flags

Let G be a non-split quasi-split adjoint absolutely simple affine algebraic group over a field F of type A_{n−1}, n ≥ 2, becoming split over a separable quadratic field extension K/F. We set r := ⌊n/2⌋ (the floor of n/2).

Lemma 4.1. The group G contains a Borel subgroup B and a maximal torus T ⊂ B such that

T ≃ \begin{cases}𝔾_{m} × (R𝔾_{m})^{×(r−1)}, if n is even (i.e., n = 2r); \\ (R𝔾_{m})^{×r}, if n is odd (i.e., n = 2r + 1), \end{cases}

where R = R_{K/F} is the Weil transfer with respect to the field extension K/F.

Proof. In the case of n = 2r, let h be the orthogonal sum of r copies of the K/F -hermitian

hyperbolic plane (the K/F -hermitian form on the vector space K^{2} of the matrix (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})).

In the case of n = 2r + 1, we let h be the orthogonal sum of r copies of the hyperbolic plane and of the K/F -hermitian space ⟨1⟩. In both cases, h is a hermitian form on the K-vector space K^{n}. Up to an isomorphism, G is the F -group of automorphisms of the K-algebra End_{K}(K^{n}) with the (unitary) adjoint involution. We may assume that G is this group of automorphisms.

For n = 2r, we have a homomorphism α : (R𝔾_{m})^{×r} → G, mapping an F -point (λ_{1}, …, λ_{r}) ∈ (R𝔾_{m})^{×r}(F) = K^{×} × ⋯ × K^{×} to the automorphism of End_{K} K^{n} which is the conjugation by the diagonal automorphism (λ_{1}, σ(λ_{1})^{−1}, …, λ_{r}, σ(λ_{r})^{−1}) of K^{n} preserving h, where σ is the non-trivial automorphism of K/F. An F -point (λ_{1}, …, λ_{r}) is in the kernel of α if and only if λ_{1} = σ(λ_{1})^{−1} = ⋯ = λ_{r} = σ(λ_{r})^{−1}. It follows that Ker α is the kernel R^{(1)}𝔾_{m} of the norm map R𝔾_{m} → 𝔾_{m} sitting in (R𝔾_{m})^{×r} diagonally. The quotient (R𝔾_{m})^{×r}/ Ker α is therefore a torus isomorphic to 𝔾_{m} × (R𝔾_{m})^{×(r−1)} and to a closed subgroup T ⊂ G. Since the dimension of the torus T is 2r − 1 = n − 1 and n − 1 is the rank of G, the torus T is maximal in G.

For n = 2r + 1, any F -point (λ, λ_{1}, …, λ_{r}) ∈ R^{(1)}𝔾_{m}(F) × (R𝔾_{m})^{×r}(F) produces a preserving h diagonal automorphism (λ, λ_{1}, σ(λ_{1})^{−1}, …, λ_{r}, σ(λ_{r})^{−1}) of K^{n}. The kernel of the resulting homomorphism R^{(1)}𝔾_{m} × (R𝔾_{m})^{×r} → G is R^{(1)}𝔾_{m}, and the quotient by the kernel is the required torus (R𝔾_{m})^{×r} (of dimension 2r = n − 1).

The torus T constructed in both cases is contained in the Borel subgroup B of G which is the stabilizer of the flag of length r of subspaces in K^{n} where the ith subspace is the sum of the first i odd summands of K^{n}. □

Remark 4.2. Actually, for any quasi-split semisimple adjoint affine algebraic group G and for any Borel subgroup B of G (which exists because G is quasi-split), any maximal torus T of B (is maximal in G and) is isomorphic to R_{L/F} 𝔾_{m}, where L is an étale F - algebra with the Γ-set Hom_{F -alg}(L, F_{sep}) isomorphic to the set of vertices of the Dynkin diagram of G. Indeed, this set of vertices is the set X ⊂ (T_{F_{sep}})^{∗} of simple roots given by B, and we get that T ≃ R_{L/F} 𝔾_{m}, because X is a permutation basis of (T_{F_{sep}})^{∗} (X generates (T_{F_{sep}})^{∗} because G is adjoint).

In particular, for G as in Lemma 4.1, T has the required isomorphism type because in this case the action of Γ on X factors through the Galois group of the quadratic separable field extension K/F whose the non-trivial element exchanges the vertices which are symmetric with respect to the middle of the diagram. It follows that L ≃ K^{×r} for n = 2r + 1 and L ≃ F × K^{×(r−1)} for n = 2r (note that #X = n − 1).

For the rest of this section, we fix a torus T ⊂ G and a Borel subgroup B ⊂ G containing T as in Lemma 4.1. We are going to describe the image of CH(E/B) in CH(G/B), where E is a generic (in the sense of [20, §3]) principle homogenous space of G (the homomorphism CH(E/B) → CH(G/B) is defined below).

We need some preliminary observations which are valid for an arbitrary parabolic subgroup P ⊂ G in place of B. Note that by [5], the Chow motive (with integer coefficients) of the F -variety G/P decomposes in a finite direct sum of shifts of the motive of the F -varieties Spec F and Spec K. Consequently, CH(G/P) injects into CH(Ḡ/P̄) = CH(G_{K}/P_{K}) (where Ḡ and P̄ are G and P over an algebraic closure of F) and is identified with the subring of σ-invariant elements of CH(G_{K}/P_{K}), where σ is the involution of the Chow ring CH(G_{K}/P_{K}) induced by the non-trivial automorphism of K/F. More precisely, we are going to get some information on the image of CH(E/P) in the quotient ring CH(G/P)/ Im ( CH(G_{K}/P_{K}) → CH(G/P)).

For any G-torsor E over F, we construct a homomorphism

CH(E/P) → CH(G/P)

as the composition of three ones: the change of field homomorphism

CH(E/P) → CH(E_{F (E)}/P_{F (E)}),

the pull-back

CH(E_{F (E)}/P_{F (E)}) → CH(G_{F (E)}/P_{F (E)})

with respect to the morphism G_{F (E)}/P_{F (E)} → E_{F (E)}/P_{F (E)} induced by the G-equivariant morphism G_{F (E)} → E_{F (E)} taking the identity of G to the generic point of E, and the inverse

CH(G_{F (E)}/P_{F (E)}) → CH(G/P)

of the change of field homomorphism CH(G/P) → CH(G_{F (E)}/P_{F (E)}) (which is an isomorphism because F is algebraically closed in F (E)).

More generally, the homomorphism CH(E/P) → CH(G/P) is defined for any G-torsor E over a regular field extension F̃/F: since K ⊗_{F} F̃ is a field, the change of fields homomorphism CH(G/P) → CH(GF̃/PF̃) is an isomorphism.

Lemma 4.3. Let S be a smooth irreducible F -variety and let E be a G-torsor over S. For a point s ∈ S(L) of S in a regular field extension L/F, let E_{s} be the G-torsor over L given by the fiber of E → S over s. Then the image of CH(E_{s}/P) → CH(G/P) contains the image of CH(E_{θ}/P) → CH(G/P), where θ ∈ S(F (S)) is the generic point of S.

Proof. Let x ∈ S be the image of the point Spec L under the morphism s : Spec L → S. Since x is regular, there exists a system of local parameters on S around x. Therefore the fields F (S) and F (x) are connected by a finite chain of discrete valuation fields, where each next field is the residue field of the previous one. Using the specialization homomorphisms on Chow groups as in [9, Example 20.3.2], we get a homomorphism CH(E_{θ}/P) → CH(E_{x}/P). Composing it with the change of fields homomorphism CH(E_{x}/P) → CH(E_{s}/P), we get a homomorphism CH(E_{θ}/P) → CH(E_{s}/P) which forms a commutative triangle with the homomorphisms to CH(G/P). □

We are turning back to the case of P = B. The following statement is well-known in the case of split G. In our quasi-split case, the proof is almost the same:

Lemma 4.4. For any G-torsor E, the pull-back CH(E/B) → CH(E/T) with respect to the projection E/T → E/B is an isomorphism. In particular, the image of CH(E/B) → CH(G/B) is identified with the image of CH(E/T) → CH(G/T).

Proof. Let U be the unipotent part of B. By [1, Theorem 10.6 of Chapter III], U is a normal subgroup of G possessing a finite increasing chain of normal subgroups U_{i} with each successive quotients isomorphic to 𝔾_{a}, and B is a semidirect product of U by T. Since H^{1}(F, T) is trivial, it follows that H^{1}(F, U_{i} ⋊ T) is trivial for any i (in particular, H^{1}(F, B) is trivial). Therefore the fiber over any point of the projection

E/(U_{i−1} ⋊ T) → E/(U_{i} ⋊ T)

is isomorphic to the affine line U_{i−1}/U_{i} so that the pull-back of Chow groups is an isomorphism by the homotopy invariance of Chow groups, [8, Theorem 57.13]. □

We take as E a generic principle homogeneous space of G as defined in [20, §3]. Thus E is the generic fiber of certain G-torsor over certain smooth absolutely irreducible F -variety having the versal property of [25, Definition 5.1(2)] (we only need the weak version [25, Remark 5.8] of the versality).

In particular, E is a G-torsor over a regular field extension of F, not over F itself. For the sake of simplicity of notation, we let now F be the field of definition of E (and K be the quadratic extension of the new F given by the tensor product of the old K and the new F over the old F).

Proposition 4.5. For E as right above, the image of CH(E/B) in the quotient

CH(G/B)/ Im ( CH(G_{K}/B_{K}) → CH(G/B))

is generated (as a ring) by elements of codimension 1 and 2.

Proof. By Lemma 4.4, the image of

CH(E/B) → CH(G/B)/ Im ( CH(G_{K}/B_{K}) → CH(G/B))

is identified with the image of

CH(E/T) → CH(G/T)/ Im ( CH(G_{K}/T_{K}) → CH(G/T)).

By [18, Theorem 6.4], the image of CH(E/T) in CH(G/T) is identified with the image of CH_{T} (Spec F) in CH_{T} (G) = CH(G/T). The image of

CH_{T} (Spec F) → CH(G/T)/ Im ( CH(G_{K}/T_{K}) → CH(G/T))

coincides with the image of the quotient

CH_{T} (Spec F)/ Im ( CH_{T} (Spec K) → CH_{T} (Spec F)).

This quotient is generated by its elements of codimension 1 and 2 by Corollaries 3.4 and 3.5. □

Note that for any G-torsor E over F, the kernel of the homomorphism CH(E/B) → CH(G/B) is the torsion subgroup so that its image is identified with the Chow group CH(E/B) modulo torsion which we denote by \overline{CH}(E/B) (and call the reduced Chow group). Furthermore, if E is given by a hermitian form h, then it splits over K; therefore the image of the homomorphism of CH(E/B) to CH(G/B) modulo the norms is identified with the ring \overline{CH}(E/B)/N, where N := Im ( CH(E_{K}/B_{K}) → \overline{CH}(E/B)) is the norm ideal.

We are going to consider a generic hermitian form h of dimension n (for a fixed separable quadratic field extension K/F) by which we mean the diagonal K(t_{1}, …, t_{n})/F (t_{1}, …, t_{n})- hermitian form ⟨t_{1}, …, t_{n}⟩, where t_{1}, …, t_{n} are variables. (For a justification of this definition, note that any hermitian form can be diagonalized, [22, Theorem 6.3 of Chapter 7].) Changing notation, we write now F for F (t_{1}, …, t_{n}) and K for K(t_{1}, …, t_{n}).

Corollary 4.6. Let Y be the variety of complete flags of totally isotropic subspaces of the defined above generic hermitian form h. Then the image of

CH(Y) → CH(G/B)/ Im ( CH(G_{K}/B_{K}) → CH(G/B))

is generated (as a ring) by the elements of codimension 1 and 2. Equivalently, the ring \overline{CH}(X)/N is generated by codimension 1 and 2.

Proof. By the specialization argument as in Lemma 4.3, for any regular field extension F′/F and any n-dimensional K′/F′-hermitian form h′, where K′ := K ⊗_{F} F′, the image of CH(Y) → CH(G/B) is contained in the image of CH(Y′) → CH(G/B), where Y′ is the variety of complete flags of h′. Similarly, the image of CH(E/B) → CH(G/B) is contained in the image of CH(Y′) → CH(G/B) for the G-torsor E constructed below. It follows that \overline{CH}(Y)/N coincides with \overline{CH}(E/B)/N so that it suffices to show that the latter ring is generated by elements of codimension ≤ 2.

To construct the G-torsor E, let us take a generic principle G-homogeneous space E′ of [20, §3] and climb over the function field of the Weil transfer of the Severi-Brauer variety of the corresponding central simple algebra. In more details, the torsor E′ (changing notation, we may assume that it is defined over F) corresponds to a degree n central simple K-algebra A endowed with an F -linear unitary involution τ, [19, §29D]. Over the function field F (X) of the Weil transfer X with respect to K/F of the Severi-Brauer K-variety of A, the algebra A splits and the involution τ becomes adjoint with respect to some K(X)/F (X)-hermitian form. We set E := E′_{F (X)}.

Writing now Y for the F -variety of complete flags of right τ-isotropic ideals in A, our aim is to show that CH(Y_{F (X)}) modulo torsion and norms is generated by codimensions ≤ 2. Since the variety Y is isomorphic to E′/B, we know by Proposition 4.5 that CH(Y) is so. The projection Y × X → Y is a Weil transfer of a projective bundle like the one considered in §5. The CH(Y)-algebra CH(Y × X) is generated by codimension 2 by Proposition 5.2. Since the pull-back CH(Y × X) → CH(Y_{F (X)}) is surjective, the CH(Y)- algebra CH(Y_{F (X)}) is generated by codimension 2. □

5. Weil transfer of projective bundles

Let K/F be a separable quadratic field extension. Let X be a smooth geometrically irreducible F -variety. Let E be a vector bundle over X_{K}. We are going to consider the Weil transfer Y → X of the projective bundle P → X_{K} of E with respect to the base change morphism X_{K} → X. We refer to [2, §7.6] and [23, §4] for generalities on Weil transfers of schemes (also called Weil restriction and corestriction in the literature).

We will need the following statement, where P can be any variety with a morphism to X_{K}:

Lemma 5.1. The Weil transfer of P with respect to X_{K} → X can be obtained as the Weil transfer with respect to K/F (which produces an R_{K/F} (X_{K})-scheme) followed by the base change with respect to the “diagonal” morphism X → R_{K/F} (X_{K}) corresponding to the identity under the identification Mor_{F} (X, R_{K/F} (X_{K})) = Mor_{K}(X_{K}, X_{K}).

Proof. We write R for R_{K/F}. Let Y be the fibred product RP ×_{RX_{K}} X. To show that Y is the Weil transfer of P with respect to X_{K} → X, it suffices to check that for any X-scheme S, the set Mor_{X}(S, Y) is naturally identified with the set Mor_{X_{K}}(S_{K}, P) (note that S_{K} := S ×_{F} Spec K = S ×_{X} X_{K}). By properties of the fibred product, we have a natural identification Mor_{X}(S, Y) = Mor_{RX_{K}}(S, RP). The set on the right hand side of this equality is a subset of the set Mor_{F} (S, RP) which is naturally identified with the set Mor_{K}(S_{K}, P). The subset Mor_{RX_{K}}(S, RP) ⊂ Mor_{F} (S, RP) corresponds under this identification to the subset Mor_{X_{K}}(S_{K}, P) ⊂ Mor_{K}(S_{K}, P). Composing two identifications obtained above, we get the identification required. □

Note that applying the base change X_{K} → X to the morphism Y → X, we get the product of two projective bundles over X_{K}: one associated to E, the other to E′, where E′ → X_{K} is the vector bundle obtained from E → X_{K} by the base change X_{K} → X_{K} given by the non-trivial automorphism of K/F.

For any integer i, we have a map CH^{i}(P) → CH^{2i}(Y) (just a map, not a homomorphism) defined as the composition of the map CH^{i}(P) → CH^{2i}(R_{K/F} P) of [16, §3] followed by the pull-back homomorphism CH^{2i}(R_{K/F} P) → CH^{2i}(Y) (we have in mind the construction of Y given in Lemma 5.1 here). The first Chern class of the tautological line vector bundle on P gives us therefore an element c ∈ CH^{2}(Y).

Proposition 5.2. The CH(X)-algebra CH(Y)/ Im ( CH(Y_{K}) → CH(Y)) is generated by the class of the element c.

Proof. We start by the case where X = Spec F. In this case, the ring CH(Y) is identified with the subring R^{σ} of the σ-invariant elements in the ring R := CH(ℙ^{r−1} × ℙ^{r−1}), where r is the rank of the vector bundle E and σ is the factor exchange involution on R. The image of the norm map is the image of the homomorphism (1+σ) : R → R^{σ}. The ring R^{σ} is generated by two elements: h×h and 1×h+h×1. Therefore the quotient R^{σ}/(1+σ)R is generated the class of h × h which is the class corresponding to the element c of the statement.

In order to do the general case, we apply a variant of [26, Statement 2.13]. Let for a moment X and Y be arbitrary smooth F -varieties with irreducible X. For a morphism π : Y → X, consider the (finite) filtration

CH(Y) = F^{0} CH(Y) ⊃ F^{1} CH(Y) ⊃ …

on CH(Y) with F^{i} CH(Y) being the subgroup generated by the classes of cycles on Y whose image in X has codimension ≥ i. For any point x ∈ X, let Y_{x} be the fiber of π over x. For any i, let X^{(i)} be the set of points of X of codimension i. There is a surjection

(5.3)  ⨁_{x∈X^{(i)}} CH(Y_{x}) → F^{i} CH(Y)/F^{i+1} CH(Y),

mapping an element α ∈ CH(Y_{x}) to the class modulo F^{i+1} CH(Y) of the image under the push-forward CH(Y_{T}) → F^{i} CH(Y) of an arbitrary preimage of α under the pull-back epimorphism CH(Y_{T}) →→ CH(Y_{x}), where T ⊂ X is the closure of x and Y_{T} := Y ×_{X} T ↪ Y is the preimage of T under Y → X.

Lemma 5.4 (cf. [26, Statement 2.13]). Let π : Y → X be as above and let ζ be the generic point of X. Let B ⊂ CH(Y) be a CH(X)-submodule such that

(a) the composition B ↪ CH(Y) → CH(Y_{ζ}) is surjective and

(b) for any x ∈ X either the specialization homomorphism s_{x} : CH(Y_{ζ}) → CH(Y_{x}) is surjective or the image of CH(Y_{x}) → F^{i} CH(Y)/F^{i+1} CH(Y) is contained in the image of B ∩ F^{i} CH(Y) → F^{i} CH(Y)/F^{i+1} CH(Y).

Then B = CH(Y).

Proof. We are repeating the proof of [26, Statement 2.13] making necessary modifications. If the specialization homomorphism s_{x} is surjective for a point x ∈ X^{(i)}, then the image of CH(Y_{x}) under (5.3) is contained in the image of [T] · B ⊂ F^{i} CH(Y) ∩ B in the quotient F^{i} CH(Y)/F^{i+1} CH(Y). Otherwise, we know already by the hypothesis that it is contained in the image of B ∩ F^{i} CH(Y). □

We are turning back to the proof of Proposition 5.2 and our particular Y → X. We apply Lemma 5.4, taking for B the CH(X)-submodule of CH(Y) generated by the powers of c and the image of CH(Y_{K}) → CH(Y). For any x ∈ X, the fiber Y_{x} is isomorphic to the Weil transfer of ℙ^{r−1} with respect to the quadratic étale F (x)-algebra F (x) ⊗_{F} K. If ζ is the generic point, F (ζ) ⊗_{F} K is a field. Therefore condition (a) of Lemma 5.4, requiring that the homomorphism B → CH(Y_{ζ}) is surjective, is satisfied.

Condition (b) of Lemma 5.4 is satisfied as well. Indeed, the specialization homomorphism CH(Y_{ζ}) → CH(Y_{x}), x ∈ X, is surjective if the residue field of the point x does not contain a subfield isomorphic to K. We finish the proof by showing that in the opposite case the image of CH(Y_{x}) → F^{i} CH(Y)/F^{i+1} CH(Y) (for i such that x ∈ X^{(i)}) is in the image of F^{i} CH(Y_{K}) → F^{i} CH(Y)F^{i+1} CH(Y).

Let T be the closure of x in X. Let Y_{T} = Y ×_{X} T ↪ Y be the preimage of T under Y → X. The image of the homomorphism CH(Y_{x}) → F^{i} CH(Y)/F^{i+1} CH(Y) coincides with the image of the homomorphism CH(Y_{T}) → F^{i} CH(Y)/F^{i+1} CH(Y) induced by the push-forward. Since x is the generic point of T and F (x) = F (T) ⊃ K, a non-empty open subset U ⊂ T possesses a morphism to Spec K. Its preimage Y_{U} ⊂ Y_{T} is open and also possesses a morphism to Spec K. Therefore (Y_{U})_{K} ≃ Y_{U} ∐ Y_{U} (as F -varieties) and, in particular, the push-forward CH(Y_{U})_{K} → CH(Y_{U}) is surjective.

The varieties and morphisms in play fit in the following commutative diagram:

[figure: Y_{K}; Y; X; (Y_{T})_{K}; Y_{T}; T; (Y_{U})_{K}; Y_{U}; U]

It follows that the image of the push-forward CH(Y_{T})_{K} → CH(Y_{T}) generates CH(Y_{T}) modulo the image of CH(Y_{T} \ Y_{U}). Since the image of CH(Y_{T} \ Y_{U}) → CH(Y) is in F^{i+1} CH(Y), it follows that the image of CH(Y_{T}) → F^{i} CH(Y)/F^{i+1} CH(Y) is contained in the image of F^{i} CH(Y_{K}) → F^{i} CH(Y)/F^{i+1}/ CH(Y). □

We write \overline{CH}(Y)/N for the reduced Chow ring \overline{CH}(Y) modulo the norm ideal N := Im ( \overline{CH}(Y_{K}) → \overline{CH}(Y)). In particular, \overline{CH}(X)/N is defined as \overline{CH}(X)/ Im ( \overline{CH}(X_{K}) → \overline{CH}(X)).

As a particular case of the map CH^{i}(P) → CH^{2i}(Y) considered right before Proposition 5.2, we have a map CH^{i}(X_{K}) → CH^{2i}(X) and we write c_{i} ∈ \overline{CH}^{2i}(X)/N for the class of the image of the Chern class c_{i}(E) ∈ CH^{i}(X_{K}). The element c_{i} can be also defined as the class of the Chern class c_{2i}(E → X), where E → X is the vector bundle obtained by composition of the vector bundle E → X_{K} and the trivial vector bundle X_{K} → X (although this c_{i} is different from the previous one, their classes in \overline{CH}(X)/N coincide).

Corollary 5.5. The \overline{CH}(X)/N-algebra \overline{CH}(Y)/N is generated by the class of c subject to the only one relation ∑_{i=0}^{r} c_{i}c^{r−i} = 0, where r is the rank of the vector bundle E.

Proof. The relation between the powers of c holds in \overline{CH}(Y)/N because it holds in \overline{CH}(Y_{K})/N ⊃ \overline{CH}(Y)/N. Here, abusing notation, we write CH(Y_{K})/N for the quotient of \overline{CH}(Y_{K}) by its subgroup N := Im ( \overline{CH}(Y_{K}) → \overline{CH}(Y)). This subgroup is actually a \overline{CH}(X)/N-submodule so that \overline{CH}(Y_{K})/N is a \overline{CH}(X)/N-module (but not a \overline{CH}(X)/Nalgebra).

In particular, the \overline{CH}(X)/N-module \overline{CH}(Y)/N is generated by 1, c, …, c^{r−1}. The module generators are free, because they are free in \overline{CH}(Y_{K})/N. It follows that the relation is the only one. □

6. Generic maximal unitary grassmannians

Let now h be a generic K/F -hermitian form of dimension n ≥ 2 (defined right before Corollary 4.6). Let r := ⌊n/2⌋. We consider the variety X of r-dimensional totally isotropic subspaces in h. We keep the notation G of Section 4 for the quasi-split algebraic group and let P be a parabolic subgroup in G representing the conjugacy class corresponding to X.

The aim of this section is to describe the image of the homomorphism

CH(X) → CH(G/P)/ Im ( CH(G_{K}/P_{K}) → CH(G/P))

(constructed (in a more general situation) right before Lemma 4.3). We recall that the G-torsor given by h splits over K; therefore the image of the above homomorphism is identified with the ring \overline{CH}(X)/N, where N := Im ( \overline{CH}(X_{K}) → \overline{CH}(X)). (So, \overline{CH}(X)/N is the ring CH(X) modulo the sum of the ideal of the torsion elements and the ideal of the norms.)

Proposition 6.1. The components of positive codimension of the ring \overline{CH}(X)/N are trivial.

Proof. Let I = [1, r] = {1, 2, …, r}. For every J ⊂ I we consider the variety X_{J} of flags of totally isotropic subspaces in h of dimensions given by J.

By induction on l ∈ I, we prove the following statement: the ring \overline{CH}(X_{[l, r]})/N (where N here is the image of norm homomorphism for X_{[l, r]}) is generated by its elements of codimensions 1, its elements of codimension 2, and the Chern classes of the tautological rank 2l vector bundle on X_{[l, r]}. Note that this statement for l = r gives the statement of Proposition 6.1 because X_{{r}} = X and we have the following triviality statements for the generators: according to [11, Proposition 3.9] the elements of codimension 2 as well as the Chern classes of positive codimension of the tautological vector bundle on G/P are trivial in CH(G/P)/N; for odd n, again by [11, Proposition 3.9], the elements of codimension 1 are also trivial in CH(G/P)/N; finally, if n is even, the elements of codimension 1 in CH(X) become trivial in CH(G/P)/N by Lemma 6.2 below (because the discriminant of our hermitian form h is non-trivial).

The induction base l = 1 is given by Corollary 4.6. Now, assuming that l ≥ 2, let us do the passage from l − 1 to l.

The projection X_{[l−1, r]} → X_{[l, r]} is the Weil transfer of a projective bundle considered in section 5 (namely, of the projective bundle over (X_{[l, r]})_{K} given by the dual of (any)one of two rank l tautological vector bundles on (X_{[l, r]})_{K}). Therefore, by Corollary 5.5, the \overline{CH}(X_{[l, r]})/N-algebra \overline{CH}(X_{[l−1, r]})/N is generated by certain codimension 2 element c subject to one relation ∑_{i=0}^{l} c_{i}c^{l−i} = 0, where the coefficients c_{i} are the even Chern classes of the rank 2l tautological vector bundle on X_{[l, r]}. In particular, the \overline{CH}(X_{[l, r]})/N-module \overline{CH}(X_{[l−1, r]})/N is free of rank l.

Now let C ⊂ \overline{CH}(X_{[l, r]})/N be the subring generated by all c_{i} together with the elements of codimensions 1, 2. The coefficients of the above relation are then in C. Therefore the subring of \overline{CH}(X_{[l−1, r]})/N generated by C and c is also free (now as a C-module) of rank l. On the other hand, this subring coincides with the total ring by the induction hypothesis. Indeed, it contains all the elements of codimension 1, 2 in \overline{CH}(X_{[l−1, r]})/N because any such element is a polynomial in c with coefficients of codimension ≤ 2 in \overline{CH}(X_{[l, r]})/N. It also contains the Chern classes of the tautological rank 2(l − 1) vector bundle on X_{[l−1, r]} because they are expressible in terms of c_{i} and c.

It follows that C = \overline{CH}(X_{[l, r]})/N. □

We terminate this section by a study of the case of n = 2r. We do not assume anymore that our n-dimensional hermitian form h is generic, it is arbitrary. We still write X for the variety of r-dimensional totally isotropic subspaces in h. The variety X_{K} is identified with the K-grassmannian of r-planes in V (the vector space of definition of h). Let E be the tautological bundle on the grassmannian. For any i ≥ 0, we set c_{i} := c_{i}(E) ∈ CH^{i}(X_{K}). The ring CH(X_{K}) is generated by the elements c_{i}. In particular, CH^{1}(X_{K}) is (an infinite cyclic group) generated by c_{1}. We recall that \overline{CH}(X) stands for the reduced Chow group and coincides with the image of CH(X) → CH(X_{K}).

The discriminant disc h of the hermitian form h is the class in F^{×}/N(K^{×}) of the signed determinant

(−1)^{n(n−1)/2} · det(h(e_{i}, e_{j}))_{i,j∈[1, n]}

of the Gram matrix of h in a basis e_{1}, …, e_{n} of V.

Lemma 6.2. The group \overline{CH}^{1}(X) is generated by c_{1}, if disc h is trivial; 2c_{1}, if disc h is non-trivial.

Proof. The cokernel of CH^{1}(X) → CH^{1}(X_{K}) is isomorphic to the kernel of Br(F) → Br(F (X)). This kernel is generated by the class of the discriminant algebra of h, [21].

The discriminant algebra of h is the quaternion algebra given by the quadratic extension K/F, the non-trivial automorphism of K/F and disc h. Its Brauer class is always killed by 2 and is non-trivial if and only if disc h is non-trivial. □

7. Essential motives of unitary grassmannians

We fix the following notation. Let K/F be a separable quadratic field extension. Let n be an integer ≥ 0. Let V be a vector space over K of dimension n. Let h be a K/F - hermitian form on V. For any integer r, let X_{r} be the F -variety of r-dimensional totally isotropic subspaces in V (X_{r} is a closed subvariety of the Weil transfer with respect to K/F of the K-grassmannian of r-planes in V; X_{0} = Spec F; X_{r} = ∅ for r outside of the interval [0, n/2]).

We are working with the Grothendieck Chow motives with coefficients in 𝔽_{2}, [8, Chapter XII]. We write M(X) for the motive of a smooth projective F -variety X. We are constantly (and without any further reference) using the Krull-Schmidt principle for the motives of quasi-homogeneous varieties, [17, Corollary 2.2].

Lemma 7.1 ([15, Theorem 15.8]). Assume that the hermitian form h is isotropic: n is ≥ 2 and h ≃ ℍ⊥h′, where ℍ is the hyperbolic plane, h′ a hermitian form of dimension n − 2. For any integer r one has

M(X_{r}) ≃ M(X′_{r−1}) ⊕ M(X′_{r})(i) ⊕ M(X′_{r−1})(j) ⊕ M,

where X′_{r−1} and X′_{r} are the varieties of h′, i = (dim X_{r} − dim X′_{r})/2, j = dim X_{r} − dim X′_{r−1}, and M is a sum of shifts of the F -motive of Spec K.

The following Corollary is also a consequence of a general result of [17] or of [5]:

Corollary 7.2. If h is split (meaning hyperbolic for even n or “almost hyperbolic” for odd n), then M(X_{r}) is a sum of shifts of the motives of Spec F and of Spec K. □

Corollary 7.3. There is a decomposition M(X_{r}) ≃ M_{r} ⊕ M such that the motive M is a sum of shifts of M(Spec K) and for any field extension L/F with split h_{L} the motive M_{r} is split (meaning is a sum of Tate motives).

Proof. Apply [13, Proposition 4.1] inductively to E := F (X_{⌊n/2⌋}) and S := Spec K. Note that the variety S_{E} is still irreducible and has indecomposable motive because K ⊗_{F} E is a field. The hermitian form h_{E} is split. Since X_{⌊n/2⌋}(K) ≠ ∅, the sum M of all copies of shifts of M(Spec K) present in the complete decomposition of M(X_{r}) over E, can be extracted from M(X_{r}) over F. The remaining part M_{r} of the motive of X_{r} has the desired property. □

Remark 7.4. The reduced Chow group (homological or cohomological one) of the motive M (as a subgroup of \overline{Ch}(X_{r})) is the image N of the norm map \overline{Ch}((X_{r})_{K}) → \overline{Ch}(X_{r}) (it is evidently contained in N and coincides in fact with N because N intersects the reduced Chow group of M_{r} trivially). Therefore the reduced Chow group of M_{r} is identified with the quotient \overline{Ch}(X_{r})/N. Here Ch(X_{r}) stands for the Chow group with coefficients in 𝔽_{2} of the variety X_{r} and the reduced Chow group \overline{Ch}(X_{r}) is \overline{CH}(X_{r}) modulo 2 or, equivalently, the image of Ch(X_{r}) → Ch(X̄_{r}). In the quasi-split case (the case where h is hyperbolic or almost hyperbolic), the reduced Chow group in the above statements can be replaced by the usual Chow group (still modulo 2). We refer to [8, §64] for the definition of homological and cohomological Chow group of a motive. The coincidence of descriptions of homological and cohomological Chow groups for the motives M and M_{r} is explained by their symmetry: M ≃ M^{∗}(dim X_{r}) (and the same for M_{r}), where M^{∗} is the dual motive, [8, §65].

Definition 7.5. The motive M_{r} (defined by X_{r} uniquely up to an isomorphism) will be called the essential motive of X_{r} (or the essential part of the motive of X_{r}).

It follows that the decomposition of the essential motive in the isotropic case has precisely the same shape as the decomposition of the motive of an isotropic orthogonal grassmannian [14]:

Corollary 7.6. Under the hypotheses of Lemma 7.1, one has

M_{r} ≃ M′_{r−1} ⊕ M′_{r}(i) ⊕ M′_{r−1}(j),

where M′_{r−1} and M′_{r} are the essential motives of X′_{r−1} and X′_{r}, i = (dim X_{r} − dim X′_{r})/2, j = dim X_{r} − dim X′_{r−1}. □

According to the general result of [17], any summand of the complete motivic decomposition of the variety X_{r} is a shift of the upper motive U(X_{s}) for some s ≥ r or a shift of the motive of the F -variety Spec K. Therefore we get

Corollary 7.7. Any summand of the complete decomposition of the essential motive M_{r} is a shift of the upper motive U(X_{s}) for some s ≥ r. □

8. Generic unitary grassmannians

In the statement below we use the notion of the essential motive M_{r} of the variety X_{r}, introduced in the previous section. It turns out that in the generic case, this motive is indecomposable:

Theorem 8.1. Let h be a generic K/F -hermitian form of an arbitrary dimension n ≥ 0. For r = 0, 1, …, ⌊n/2⌋, the essential motive M_{r} of the variety X_{r} is indecomposable, the variety X_{r} is 2-incompressible.

Proof. We induct on n in the proof of the first statement. The induction base is the trivial case of n < 2. Now we assume that n ≥ 2.

We do a descending induction on r. The case of the maximal r = ⌊n/2⌊ is an immediate consequence of Proposition 6.1. Indeed, one summand of the complete decomposition of the motive M_{r} for such r is the upper motive U(X_{r}) of X_{r}. The remaining summands (if any) are positive shifts U(X_{r})(i) (i > 0) of the upper motive, see Corollary 7.7. But if we have a summand U(X_{r})(i), then the reduced Chow group \overline{Ch}^{i}(M_{r}) is non-zero. However by Remark 7.4, \overline{Ch}(M_{r}) is isomorphic to \overline{CH}(X_{r})/N which is 0 in positive codimensions by Proposition 6.1.

Now we assume that r < ⌊n/2⌋. Since the case of r = 0 is trivial, we may assume that r ≥ 1 and n ≥ 4.

Let L := F (X_{1}). We have h_{L} ≃ ℍ⊥h′, where h′ is a K(X_{1})/F (X_{1})-hermitian form of dimension n − 2 and ℍ is the hyperbolic plane. Note that the hermitian form h′ is still generic.

For any integer s, we write X′_{s} for the variety X_{s} of the hermitian form h′, and we write M′_{s} for the essential motive of the variety X′_{s}. By Corollary 7.6, the motive (M_{r})_{L} decomposes in a sum of three summands:

(M_{r})_{L} ≃ M′_{r−1} ⊕ M′_{r}(i) ⊕ M′_{r−1}(j),

where i := (dim X_{r}−dim X′_{r})/2 and j := dim X_{r}−dim X′_{r−1}. By the induction hypothesis, each of these three summands is indecomposable. It follows (taking into account the duality like in [14, Remark 1.3]) that if the motive M_{r} is decomposable (over F), then it has a summand M with M_{L} ≃ M′_{r}(i) = U(X′_{r})(i). Note that U(X′_{r}) ≃ U((X_{r+1})_{L}). By an analogue of [14, Lemma 1.2], M ≃ U(X_{r+1})(i), showing that U(X_{r+1})_{L} ≃ M′_{r}. By the induction hypothesis, the motive M_{r+1} is indecomposable, that is, U(X_{r+1}) = M_{r+1}. Therefore we have an isomorphism (M_{r+1})_{L} ≃ M′_{r} and, in particular, dim X_{r+1} = dim X′_{r}. However dim X_{r+1} = (r + 1)(2n − 3(r + 1)), dim X′_{r} = r(2(n − 2) − 3r), and the difference is 2n − 2r − 3 ≥ n − 3 > 0 (recall that n ≥ 4 now).

To show that X_{r} is 2-incompressible, we show that its canonical 2-dimension cdim_{2} X_{r} equals dim X_{r}. By [12, Theorem 5.1], cdim_{2} X_{r} = dim U(X_{r}). By the first part of Theorem 8.1, U(X_{r}) = M_{r}. Finally, dim M_{r} = dim X_{r} for an arbitrary (not only for a generic) h. □

Remark 8.2. For arbitrary h, the motive M_{⌊n/2⌋} is indecomposable if and only if the variety X_{⌊n/2⌋} is 2-incompressible. If this is the case, then, as actually showed in the proof of Theorem 8.1, the motives M_{r} are indecomposable and the varieties X_{r} are 2incompressible for all r = 0, 1, …, ⌊n/2⌋.

9. Connection with quadratic forms

Let K/F be a separable quadratic field extension extension, V a finite-dimensional vector space over K, h a non-degenerate K/F -hermitian form on V. For any v ∈ V the value h(v, v) is in F and the map q : V → F, v ↦ h(v, v) is a non-degenerate quadratic form on V considered as a vector over F. Note that the dimension of q is even. The Witt indices of h and q are related as follows:

Lemma 9.1. i(q) = 2i(h).

Proof. For any integer r ≥ 0, the inequality i(h) ≥ r implies i(q) ≥ 2r. Indeed, if i(h) ≥ r, V contains a totally h-isotropic L-subspace W of dimension r. This W is also totally q-isotropic and has dimension 2r over F. Therefore i(q) ≥ 2r.

To finish we prove by induction on r ≥ 0 that i(q) ≥ 2r − 1 implies i(h) ≥ r. This is trivial for r = 0. If r > 0 and i(q) ≥ 2r − 1, then q is isotropic. But any q-isotropic vector is also h isotropic, therefore the K-vector space V decomposes in a direct sum of h-orthogonal subspaces V = U ⊕ V′ such that h|_{U} is a hyperbolic plane. The subspaces U and V′ are also q-orthogonal and q|_{U} is hyperbolic (of dimension 4). For h′ := h|_{V′} and q′ := q|_{V′} it follows that i(h′) = i(h) − 1 and i(q′) = i(q) − 2, and we are done by the induction hypothesis applied to h′ (of course, q′ is the quadratic form given by h′). □

Corollary 9.2 (N. Jacobson, 1940). If the quadratic forms q_{1} and q_{2} corresponding to K/F -hermitian forms h_{1} and h_{2} are isomorphic, then h_{1} and h_{2} are also isomorphic.

Proof. The orthogonal sum q_{1}⊥ − q_{2} is the quadratic form corresponding to h_{1}⊥ − h_{2}. If q_{1} and q_{2} are isomorphic, q_{1}⊥ − q_{2} is hyperbolic, therefore h_{1}⊥ − h_{2} is also hyperbolic implying that h_{1} and h_{2} are isomorphic. □

For any integer r, let X_{r} be the variety of totally h-isotropic r-dimensional K-subspaces in V and let Y_{r} be the variety of totally q-isotropic r-dimensional F -subspaces in V. The variety Y_{n}, where n := dim V, is not connected and has two isomorphic connected components; changing notation, we let Y_{n} be one of its connected component in this case.

Corollary 9.3. For any r, the upper motives of the varieties X_{r}, Y_{2r}, and Y_{2r−1} are isomorphic. In particular, these varieties have the same canonical dimension. □

Theorem 8.1 with Corollary 9.3 makes it possible to compute for any n the greatest (in the sense of inclusion) value of the J-invariant (see [8, §88]) of a non-degenerate quadratic form over a field extension of a fixed base field F given by tensor product an n-dimensional bilinear form by a fixed binary quadratic form (the fixed binary quadratic form should be defined over F, the bilinear form needs not to be defined over F). Only the case of even n is of interest, because the J-invariant is {0} for odd n.

Corollary 9.4. For any even n ≥ 2, the greatest value of J-invariant discussed right above is {1, 3, 5, …, n − 1} (the set of odd integers from 1 till n − 1).

Proof. By [8, Proposition 88.8], the J-invariant of any non-degenerate quadratic form given by tensor product of an n-dimensional bilinear form by a binary quadratic form, is a subset of the given set. Let K/F be the separable quadratic field extension given by the discriminant of the fixed binary quadratic form. We consider the generic n-dimensional K/F -hermitian form h and calculate the J-invariant of the associated quadratic form q. We are using the above notation for the varieties associated to h and to q.

By [8, Theorem 90.3], the canonical dimension of Y_{n} is the sum of the elements of the J-invariant. On the other hand, by Corollary 9.3 and Theorem 8.1, the canonical dimension of Y_{n} is equal to the dimension of X_{n/2} which is

dim X_{n/2} = n^{2}/4 = 1 + 3 + 5 + ⋯ + (n − 1).

Therefore the J-invariant is equal to the given set. □

Turning back to an arbitrary (not necessarily generic) hermitian form, we have

Proposition 9.5. For any r > 0, the motive M_{r} ⊕ M_{r}(dim Y_{2r−1} − dim X_{r}) is isomorphic to a summand of the motive of Y_{2r−1}. If n > 2, then the motive M_{r}⊕M_{r}(dim Y_{2r}−dim X_{r}) is isomorphic to a summand of the motive of Y_{2r}. (For n = 2, the motive M_{r} is isomorphic to a summand of the motive of Y_{2r}.)

Proof. By specialization argument (together with nilpotence principle for projective homogeneous varieties, see [5, Theorem 8.2 and Corollary 8.4]), it suffices to show that the statement holds in the generic case. In this case, by Theorem 8.1, the motive M_{r} is equal to the upper motive U(X_{r}) which, by Corollary 9.3, is isomorphic to U(Y_{2r−1}) as well as to U(Y_{2r}), indecomposable summands of M(Y_{2r−1}) and of M(Y_{2r}). So, M_{r} is an indecomposable summand of M(Y_{2r−1}) and M(Y_{2r}). By duality [8, §65], M_{r}(dim Y_{2r−1} − dim X_{r}) is also an indecomposable summand of M(Y_{2r−1}) and M_{r}(dim Y_{2r} − dim X_{r}) is also an indecomposable summand of M(Y_{2r}). It remains to notice that

dim Y_{2r−1} − dim X_{r} = (2r − 1)(2n − 3r + 1) − r(2n − 3r) > 0

for any n and any r ≤ n/2 and that

dim Y_{2r} − dim X_{r} = r(4n − 6r − 1) − r(2n − 3r) > 0

for any n > 2 and any r ≤ n/2. □

The following statement has been proved in [24]:

Corollary 9.6. M(Y_{1}) = M_{1} ⊕ M_{1}(1).

Proof. Since dim Y_{1} − dim X_{1} = 1, we know by Proposition 9.5 that M_{1} ⊕ M_{1}(1) is a summand of M(Y_{1}). Comparing the ranks (the number of summands in the complete decomposition over an algebraic closure) of the motives , we see that M(Y_{1}) = M_{1} ⊕ M_{1}(1). □

Remark 9.7. By Theorem 8.1, the decomposition of Corollary 9.6 is complete in the generic case (that is to say, the summands of the decomposition are indecomposable).

Let us consider the natural closed imbedding in : X_{r} ↪ Y_{2r} which we have because a totally h-isotropic K-subspace is also a q-isotropic F -subspace. We will assume that the dimension n of the hermitian form h is even. In this case the image N of the norm homomorphism \overline{Ch}(Y_{2r})_{K} → \overline{Ch}(Y_{2r}) is 0 (because the discriminant of q is trivial and therefore Y_{2r} is a projective homogeneous variety of inner type), so that \overline{Ch}(Y_{2r})/N = \overline{Ch}(Y_{2r}).

The following observation is due to Maksim Zhykhovich:

Lemma 9.8. The pull-back in^{∗} : \overline{Ch}(Y_{2r}) → \overline{Ch}(X_{r})/N is surjective, the push-forward in_{∗} : \overline{Ch}(X_{r})/N → \overline{Ch}(Y_{2r}) is injective.

Remark 9.9. In the quasi-split case, another proof of surjectivity of in^{∗} is given in [11, Lemma 4.1].

Proof of Lemma 9.8. We are working with motives in this proof. One may use the Chow motives with coefficients in 𝔽_{2}, but since we are interested in the reduced Chow groups in the end, it is more appropriate to use the \overline{Ch}-motives: they are obtained by replacing Ch by \overline{Ch} in the construction of the Chow motives with coefficients in 𝔽_{2}.

Let us also write in for the graph of the imbedding in. Let π be a projector on X_{r} giving the motive M_{r}. There exists a correspondence α such that π ∘ in^{t} ∘α = π. Indeed, such a correspondence exists in the generic case and can be obtained in the general case by specialization. Since \overline{Ch}(M_{r}) is identified with \overline{Ch}(X_{r})/N, we have π_{∗}(x) ≡ x (mod N) for any x ∈ \overline{Ch}(X_{r}). It follows that x ≡ π_{∗}(x) = π_{∗} in^{∗} α_{∗}(x) ≡ in^{∗} α_{∗}(x) showing that in^{∗} : \overline{Ch}(Y_{2r}) → \overline{Ch}(X_{r})/N is surjective.

Similarly, there exists a correspondence β such that β ∘ in ∘π = π. It follows that x ≡ β_{∗} in_{∗}(x) for any x ∈ \overline{Ch}(X_{r}) showing that in_{∗} : \overline{Ch}(X_{r})/N → \overline{Ch}(Y_{2r}) is injective. □

Since the composition in_{∗} ∘ in^{∗} is the multiplication by the class [X_{r}] ∈ Ch(Y_{2r}) and in^{∗} is a ring homomorphism, we get Corollary 9.10. The quotient ring \overline{Ch}(X_{r})/N is isomorphic to \overline{Ch}(Y_{2r})/I, where the ideal I ⊂ \overline{Ch}(Y_{2r}) is the annihilator of the element [X_{r}] ∈ \overline{Ch}(Y_{2r}). □

Example 9.11. Let us consider the case of n = 2r. The ring Ch(Ȳ_{2r}) is generated by certain elements e_{1}, …, e_{2r−1} of codimensions 1, …, 2r−1 subject to the relations e_{i}^{2} = e_{2i}, where e_{j} := 0 for j > 2r − 1, [8, §86]. If h is generic, then, according to [8, Theorem 87.7] and Corollary 9.4, the subring \overline{Ch}(Y_{2r}) ⊂ Ch(Ȳ_{2r}) is generated by the even-codimensional elements e_{2}, e_{4}, …, e_{2r−2}. The class [X_{r}] ∈ Ch(Ȳ_{2r}) is rational (that means belongs to \overline{Ch}(Y_{2r})) and non-zero (by injectivity of in_{∗}) so that [X_{r}] = e_{2}e_{4} … e_{2r−2} as this product is the only non-zero rational element in codimension

codim_{Y_{2r}} X_{r} = r(2r − 1) − r^{2} = r(r − 1) = 2 + 4 + ⋯ + (2r − 2).

It follows that for n = 2r and hyperbolic h, the ring Ch(X_{r})/N (we put the usual Chow group instead of the reduced one because they both coincide by the reason that h is hyperbolic) is generated by elements e_{1}, e_{3}, e_{5}, …, e_{2r−1} of codimensions 1, 3, 5, …, 2r − 1 subject to relations e_{i}^{2} = 0 for any i.

Example 9.12. We briefly describe the situation with an odd n. In this situation, the norm homomorphism \overline{Ch}(Y_{2r})_{K} → \overline{Ch}(Y_{2r}) can be non-zero. By this reason, \overline{Ch}(Y_{2r}) in the statements of Lemma 9.8 and Corollary 9.10 has to be replaced by the quotient \overline{Ch}(Y_{2r})/N. In particular, the ring \overline{Ch}(X_{r})/N is naturally isomorphic to the quotient of the ring \overline{Ch}(Y_{2r})/N by the annihilator of [X_{r}] ∈ \overline{Ch}(Y_{2r})/N.

Now we assume that n = 2r + 1. The variety Ȳ_{2r} is a rank 2r projective bundle over Ȳ_{2r+1}. The ring Ch(Ȳ_{2r}) is generated by elements e_{i} ∈ Ch^{i}(Ȳ_{2r}), i = 1, 2, …, 2r and e ∈ Ch^{1}(Ȳ_{2r}) subject to the relations e_{i}^{2} = e_{2i} and e^{2r+1} = 0. Let now Y′_{2r} be the variety Y_{2r} over a field extension of F such that h is almost hyperbolic but K is still a field. The subring Ch(Y′_{2r}) ⊂ Ch(Ȳ_{2r}) is generated by e, e_{2}, …, e_{2r} (all the above generators without e_{1}). The image of the norm map is the ideal generated by e. The ring Ch(Y′_{2r})/N is generated by all e_{i} with i ≥ 2 subject to the relations e_{i}^{2} = e_{2i}. The subring \overline{Ch}(Y_{2r})/N ⊂ Ch(Y′_{2r})/N contains the elements e_{2}, e_{4}, …, e_{2r}. In the case of generic h, the subring \overline{Ch}(Y_{2r})/N does not contain any e_{i} with odd i: otherwise the canonical dimension of Y_{2r} (and therefore of X_{r}) would be smaller than

dim Y_{2r} − (2 + 4 + ⋯ + 2r) = r(r + 2) = dim X_{r}.

It follows that the subring \overline{Ch}(Y_{2r})/N is generated by the elements e_{2}, e_{4}, …, e_{2r}. In particular, the only non-zero homogeneous element of dimension dim X_{r} in \overline{Ch}(Y_{2r})/N is the product e_{2}e_{4} … e_{2r} and therefore

[X_{r}] = e_{2}e_{4} … e_{2r} ∈ Ch(Y′_{2r})/N.

The annihilator of [X_{r}] in Ch(Y′_{2r})/N is therefore the ideal generated by e_{2}, …, e_{2r} and it follows that for n = 2r + 1 and almost hyperbolic h the ring Ch(X_{r})/N is generated by elements e_{3}, e_{5}, …, e_{2r−1} subject to relations e_{i}^{2} = 0.

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UPMC Sorbonne Universités, Institut de Mathématiques de Jussieu, Paris, FRANCE

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