On the Chow Groups of Quadratic Grassmannians

A. Vishik^{1}

Received: November 30, 2004 Revised: December 9, 2004

Communicated by Ulf Rehmann

Abstract. In this text we get a description of the Chow-ring (modulo 2) of the Grassmanian of the middle-dimensional planes on arbitrary projective quadric. This is only a first step in the computation of the, so-called, generic discrete invariant of quadrics. This generic invariant contains the “splitting pattern” and “motivic decomposition type” invariants as specializations. Our computation gives an important invariant J(Q) of the quadric Q. We formulate a conjecture describing the canonical dimension of Q in terms of J(Q).

2000 Mathematics Subject Classification: 11E04, 14C15, 14M15 Keywords and Phrases: Quadrics, Chow groups, Grassmannians, Steenrod operations

Contents

1 Introduction 112

2 The Chow ring of the last Grassmannian 113

3 Multiplicative structure 118

4 Action of the Steenrod algebra 120

5 Main theorem 121

6 On the canonical dimension of quadrics 126 1 Introduction

1Support of the Weyl fund is acknowledged. Partially supported by RFBR grant 22005

Documenta Mathematica 10 (2005) 111–130

The current article is devoted to the computation of certain invariants of smooth projective quadrics. Among the invariants of quadrics one can distinguish those which could be called discrete. These are invariants whose values are (roughly speaking) collections of integers. For a quadric of given dimension such an invariant takes only finitely many values. The first example is the usual dimension of anisotropic part of q. More sophisticated example is given by the splitting pattern of Q, or the collection of higher Witt indices - see [7] and [9]. The question of describing the set of possible values of this invariant is still open. Some progress in this direction was achieved by considering the interplay of the splitting pattern invariant with another discrete invariant, called, motivic decomposition type - see [12]. The latter invariant measures in what pieces the Chow-motive of a quadric Q could be decomposed. The splitting pattern invariant can be interpreted in terms of the existence of certain cycles on various flag varieties associated to Q, and the motivic decomposition type can be interpreted in terms of the existence of certain cycles on Q × Q. So, both these invariants are faces of the following invariant GDI(Q), which we will call (quite) generic discrete invariant. Let Q be a quadric of dimension d, and, for any 1 ⩽ m ⩽ [d/2] + 1, let G(m, Q) be the Grassmanian of projective subspaces of dimension (m − 1) on Q. Then GDI(Q) is the collection of the subalgebras

C^{∗}(G(m, Q)) := image(CH^{∗}(G(m, Q))/2 → CH^{∗}(G(m, Q)|\overline{k})/2).

It should be noticed that this invariant has a ”noncompact form”, where one uses powers of quadrics Q^{×r} instead of G(m, Q). The equivalence of both forms follows from the fact that the Chow-motive of Q^{×r} can be decomposed into the direct sum of the Tate-shifts of the Chow-motives of G(m, Q). The varieties G(m, Q)|\overline{k} have natural cellular structure, so Chow-ring for them is a finite-dimensional ℤ-algebra with the fixed basis parametrized by the Young diagrams of some kind. This way, GDI(Q) appears as a rather combinatorial object. The idea is to try to describe the possible values of GDI(Q), rather than that of the certain faces of it. In the present article we will address the computation of GDI(m, Q) for the biggest possible m = [d/2]+1. This case corresponds to the Grassmannian of middle-dimensional planes on Q. It should be noticed, that it is sufficient to consider the case of odd-dimensional quadrics. This follows from the fact that for the quadric P of even dimension 2n and arbitrary codimension 1 subquadric Q in it, G(n + 1, P) = G(n, Q) ×_{Spec(k)} Spec(k\sqrt{det_{±}(P)}). Below we will show that, for m = [d/2] + 1, the GDI(m, Q) can be described in a rather simple terms - see Main Theorem 5.8 and Definition 5.11. The restriction on the possible values here is given by the Steenrod operations - see Proposition 5.12. And at the moment there is no other restrictions known - see Question 5.13 (the author would expect that there is none). Finally, in the last section we show that in the case of a generic quadric, the Grassmannian of middle-dimensional planes is 2-incompressible, which gives a new proof of the conjecture of G.Berhuy and Z.Reichstein (see [1, Conjecture 12.4]). Also, we formulate a conjecture describing the canonical dimension of arbitrary quadric - see Conjecture 6.6. Most of these results were announced at the conference on “Quadratic forms” in Oberwolfach in May 2002. This text was written while I was a member at the Institute for Advanced Study at Princeton, and I would like to express my gratitude to this institution for the support, excellent working conditions and very stimulating atmosphere. The support of the the Weyl Fund is deeply appreciated. I’m very gratefull to G.Berhuy for the numerous discussions concerning canonical dimension, which made it possible for the final section of this article to appear. Finally, I want to thank a referee for suggestions and remarks which helped to improve the exposition and for pointing out a mistake.

2 The Chow ring of the last Grassmannian

Let k be a field of characteristic different from 2, and q be a nondegenerate quadratic form on a (2n+1)-dimensional k-vector space W_{q}. Denote as G(n, Q) the Grassmannian of n-dimensional totally isotropic subspaces in W_{q}. If q is completely split, then the corresponding Grassmanian will be denoted as G(n), and the underlying space of the form q will be denoted as W_{n}. For small n, examples are: G(1) ≅ ℙ^{1}, G(2) ≅ ℙ^{3}, and G(3) ≅ Q_{6} - the 6-dimensional hyperbolic quadric. The Chow ring CH^{∗}(G(n)) has Z-basis, consisting of the elements of the type z_{I}, where I runs over all subsets of {1, …, n} (see [2, Propositions 1,2] and [5, Proposition 4.4]). In particular, rank(CH^{∗}(G(n))) = 2^{n}. The degree (codimension) of z_{I} is |I| = ∑_{i∈I} i, and this cycle can be defined as the collection of such n-dimensional totally isotropic subspaces A ⊂ W_{q}, that

dim(A ∩ π_{n+1−j}) ⩾ #(i ∈ I, i ⩾ j), for all 1 ⩽ j ⩽ n,

where π_{1} ⊂ … ⊂ π_{n} is the fixed flag of totally isotropic subspaces in W_{q}. The element z_{∅} is the ring unit 1 = [G(n)]. Other parts of the landscape are: the tautological n-dimensional bundle V_{n} on G(n), and the embedding G(n − 1) →^{j_{n−1}} G(n) given by the choice of a rational point x ∈ Q. Fixing such a point x, let M_{n} ⊂ G(n) × G(n) be the closed subvariety of pairs (A, B), satisfying the conditions:

x ∈ B, and codim(A ∩ B ⊂ A) ⩽ 1.

The projection on the first factor (A, B) ↦ A defines a birational map g_{n} : M_{n} → G(n). In particular, by the projection formula, the map g_{n}^{∗} : CH^{∗}(G(n)) → CH^{∗}(M_{n}) is injective. On the other hand, the rule (A, B) ↦ (B/x) defines the map π : M_{n} → G(n − 1). Tautological bundle V_{n} is naturally a subbundle in the trivial 2n + 1-dimensional bundle pr^{∗}(W_{n}), which we will denote still by W_{n}. The variety M_{n} can be also described as the variety of pairs B ⊂ C ⊂ W_{n}, where B is totally isotropic, dim(B) = n, dim(C) = n + 1, and x ∈ B. In other words, M_{n} ≅ ℙ_{G(n−1)}(j_{n−1}^{∗}(W_{n}/V_{n})), where the identification is given by the rule:

(A, B) ↦ (A + B)/B (respectively, (B, B) ↦ B^{⊥}/B ).

Clearly, j_{n−1}^{∗}(W_{n}/V_{n}) = (W_{n−1}/V_{n−1}) ⊕ O, and W_{n−1}/V_{n−1}^{⊥} ≅ (V_{n−1}^{∨}). We have a 3-step filtration V_{n−1} ⊂ V_{n−1}^{⊥} ⊂ W_{n−1}, with first and third graded pieces mutually dual. Hence, the top exterior power of W_{n−1} is isomorphic to the middle graded piece (which is a linear bundle): V_{n−1}^{⊥}/V_{n−1} = Λ^{2n−1}W_{n−1} ≅ O. Thus, M_{n} ≅ ℙ_{G(n−1)}(Y_{n−1}), where [Y_{n−1}] = [V_{n−1}^{∨}] + 2[O] ∈ K_{0}(G(n − 1)). We get a diagram

G(n) ←^{g_{n}} ℙ_{G(n−1)}(Y_{n−1}) →^{π_{n−1}} G(n − 1).

Using the exact sequences 0 → A → (A + B) → (A + B)/A → 0 and 0 → B → (A+B) → (A+B)/B → 0, and the fact that q defines a nondegenerate pairing between the spaces (A+B)/B ≅ A/(A∩B) and (A+B)/A ≅ B/(A∩B) for all pairs (A, B) aside from the codimension > 1 subvariety (Δ(G(n))∩M_{n}) ⊂ M_{n}, we get the exact sequences:

0 → g_{n}^{∗}(V_{n}) → X_{n−1} → O(1) → 0, and
0 → π_{n−1}^{∗}(V_{n−1}) ⊕ O → X_{n−1} → O(−1) → 0,

where X_{n−1} is the bundle with the fiber C. In particular,

[g_{n}^{∗}(V_{n})] = [π_{n−1}^{∗}(V_{n−1})] + [O] + [O(−1)] − [O(1)]. Also,
CH^{∗}(ℙ_{G(n−1)}(Y_{n−1})) = CH^{∗}(G(n − 1))[ρ]/(ρ^{2} · c(V_{n−1}^{∨})(ρ)),

where ρ = c_{1}(O(1)), and c(E)(t) = ∑_{i=0}^{dim(E)} c_{i}(E)t^{dim(E)−i} is the total Chern polynomial of the vector bundle E. Consider the open subvariety M̃_{n} := g_{n}^{−1}(G(n)\j_{n−1}(G(n−1))) ⊂ M_{n} The map g̃_{n} : M̃_{n} → G(n)\j_{n−1}(G(n−1)) is an isomorphism, and π̃_{n−1} : M̃_{n} → G(n−1) is an n-dimensional affine bundle over G(n − 1).

Proposition 2.1 There is split exact sequence

0 → CH^{∗−n}(G(n − 1)) →^{j_{n−1}_{∗}} CH^{∗}(G(n)) →^{j_{n−1}^{∗}} CH^{∗}(G(n − 1)) → 0.

Proof: Consider commutative diagram:

G(n)  ⟵^{g_{n}} M_{n} ⟶^{π_{n−1}} G(n − 1)
φ↑  ^{ψ}↑  ∥
G(n)\j_{n−1}(G(n − 1)) ⟵_{g̃_{n}} M̃_{n} ⟶_{π̃_{n−1}} G(n − 1).

Notice that the choice of a point y ∈ Q\T_{x,Q} gives a section s : G(n − 1) → M̃_{n} of the affine bundle π̃_{n−1} : M̃_{n} → G(n − 1). And the composition φ ∘ g̃_{n} ∘ s is equal to j′_{n−1}, where j′_{n−1} is constructed from the point y ∈ Q in the same way as j_{n−1} was constructed from the point x. Thus, the isomorphism CH^{∗}(G(n)\j_{n−1}(G(n − 1))) ⟶^{(π̃_{n−1}^{∗})^{−1}g̃_{n}^{∗}} CH^{∗}(G(n − 1)) together with the lo-

calization at G(n − 1) →^{j_{n−1}} G(n) ←^{φ} (G(n)\G(n − 1)) gives us exact sequence

CH^{∗−n}(G(n − 1)) →^{j_{n−1}_{∗}} CH^{∗}(G(n)) →^{j′_{n−1}^{∗}} CH^{∗}(G(n − 1)) → 0.

Thus, ker(j′_{n−1}^{∗}) = im(j_{n−1}_{∗}). Since it is true for arbitrary pair of points x, y ∈ Q satisfying the condition that the line passing through them does not belong to a quadric, we get: ker(j_{n−1}^{∗}) = im(j_{n−1}_{∗}). On the other hand, the map j′_{n−1}_{∗} : CH^{∗−n}(G(n − 1)) → CH^{∗}(G(n)) is split injective, since (π̃_{n−1})_{∗} ∘ g̃_{n}^{∗} ∘ φ^{∗} ∘ j′_{n−1}_{∗} = id. Then the same is true for j_{n−1}_{∗}. And we get the desired split exact sequence. □

Lemma 2.2 The ring CH^{∗}(G(n)) is generated by the elements of degree ⩽ n.

Proof: It easily follows by induction with the help of Proposition 2.1, and projection formula. □

Proposition 2.3 Let q be 2n + 1-dimensional split quadratic form. Then

(1) The group O(q) acts trivially on CH^{∗}(G(n)).

(2) The maps j_{n−1}^{∗} and j_{n−1}_{∗} do not depend on the choice of a point x ∈ Q.

Proof: Use induction on n. For n = 1 the statement is trivial. Suppose it is true for (n − 1). Let j_{n−1,x} : G(n − 1)_{x} → G(n) be the map corresponding to the point x ∈ Q. For any φ ∈ O(q) such that φ(x) = y, we have the map φ_{x,y} : G(n − 1)_{x} → G(n − 1)_{y} such that j_{n−1,y} ∘ φ_{x,y} = φ ∘ j_{n−1,x}. By the inductive assumption, the maps φ_{x,y}^{∗} and (φ_{x,y})_{∗} = ((φ_{x,y})^{∗})^{−1} define canonical identification of CH^{∗}(G(n − 1)_{x}) and CH^{∗}(G(n − 1)_{y}) which does not depend on the choice of φ. And under this identification,

j_{n−1,x}^{∗} ∘ φ^{∗} = j_{n−1,y}^{∗} and φ_{∗} ∘ (j_{n−1,x})_{∗} = (j_{n−1,y})_{∗}.

Let φ ∈ O(q) be arbitrary element, and x, y ∈ Q be such (rational) points that φ(x) = y. Let z be arbitrary point on Q such that neither of lines l(x, z), l(y, z) lives on Q. Consider reflections τ_{x,z} and τ_{y,z}. They are rationally connected in O(q). Consequently, for ψ := τ_{y,z} ∘ τ_{x,y}, ψ^{∗} = id = ψ_{∗}. Thus, (j_{n−1,x})^{∗} = (j_{n−1,y})^{∗} and (j_{n−1,x})_{∗} = (j_{n−1,y})_{∗}. From Proposition 2.1 we get the commutative diagram with exact rows:

0⟶ CH^{∗−n}(G(n − 1)_{y}) ⟶^{(j_{n−1,y})_{∗}} CH^{∗}(G(n)) ⟶^{j_{n−1,x}^{∗}} CH^{∗}(G(n − 1)_{x}) ⟶ 0
∥  ↑_{φ^{∗}=(φ_{∗})^{−1}}  ∥
0⟶ CH^{∗−n}(G(n − 1)_{x}) ⟶^{(j_{n−1,x})_{∗}} CH^{∗}(G(n)) ⟶^{j_{n−1,y}^{∗}} CH^{∗}(G(n − 1)_{y}) ⟶0.

It implies that φ^{∗} is identity on elements of degree ⩽ n. Since, by the Lemma 2.2, such elements generate CH^{∗}(G(n)) as a ring, φ^{∗} = id = φ_{∗}, and j_{n−1}^{∗}, j_{n−1}_{∗} are well-defined. □

Proposition 2.4 There is unique set of elements z_{i} ∈ CH^{i}(G(n)) defined for all n ⩾ 1 and satisfying the properties:

(0) For G(1) ≅ ℙ^{1}, z_{1} is the class of a point.

(1) As a ℤ-module, CH^{∗}(G(n)) = ⊕_{I⊂{1,...,n}} ℤ · ∏_{i∈I} z_{i}.

(2) j_{n−1}_{∗}(1) = z_{n}.

(3) j_{n−1}^{∗} : CH^{∗}(G(n)) → CH^{∗}(G(n − 1)) is given by the following rule on the additive generators above:

∏_{i∈I} z_{i} ↦ \begin{cases}0, if n ∈ I; \\ ∏_{i∈I} z_{i}, if n ∉ I. \end{cases}

Proof: Let us introduce the elementary cycles z_{i} ∈ CH^{i}(G(n)) inductively as follows: For n = 1, G(1) ≅ ℙ^{1}, and z_{1} is just the class of a point. Let z_{i} ∈ CH^{i}(G(n − 1)), for 1 ⩽ i ⩽ n − 1 are defined and satisfy the condition (1) − (3). Let us define similar cycles on G(n). From the Proposition 2.1 we get: j_{n−1}^{∗} is an isomorphism on CH^{i}, for i < n. Now, for 1 ⩽ i ⩽ n − 1, we define z_{i} ∈ CH^{i}(G(n)) as unique element corresponding under this isomorphism to z_{i} ∈ CH^{i}(G(n − 1)). And put: z_{n} := j_{n−1}_{∗}(1). We automatically get (2) satisfied. Let J ⊂ {1, …, n − 1}. From the projection formula we get:

j_{n−1}_{∗}(∏_{j∈J} z_{j}) = z_{n} · ∏_{j∈J} z_{j}.

Applying once more Proposition 2.1, we get condition (1) and (3). □

Remark: The cycle z_{i} we constructed is given by the set of n-dimensional totally isotropic subspaces A ⊂ W_{n} satisfying the condition: A ∩ π_{n+1−i} ≠ 0 for fixed totally isotropic subspace π_{n+1−i} of dimension (n + 1 − i). Consider the commutative diagram:

G(n) ⟵_{g_{n}} ℙ_{G(n−1)}(Y_{n−1}) ⟶_{π_{n−1}} G(n − 1)
j_{n−1}↑  ^{j}↑  ↑^{j_{n−2}}
G(n − 1) ⟵_{g_{n−1}} ℙ_{G(n−2)}(Y_{n−2}) ⟶_{π_{n−2}} G(n − 2).

By Proposition 2.4, the ring homomorphism

j^{∗} : CH^{∗}(ℙ_{G(n−1)}(Y_{n−1})) → CH^{∗}(ℙ_{G(n−2)}(Y_{n−2})) is a surjection, and it’s kernel is generated as an ideal by the elements π_{n−1}^{∗}(z_{n−1}) and ρ^{2} · c(V_{n−2}^{∨})(ρ). In particular, (j^{∗})^{k} is an isomorphism for all k < n − 1, and the kernel of (j^{∗})^{n−1} is additively generated by π_{n−1}^{∗}(z_{n−1}).

Theorem 2.5 Let V_{n} be tautological bundle on G(n), z_{i} be elements defined in Proposition 2.4, and ρ = c_{1}(O(1)). Then:

(1) g_{n}^{∗}(z_{k}) = ρ^{k} + 2 ∑_{0<i<k} ρ^{k−i}π_{n−1}^{∗}(z_{i}) + π_{n−1}^{∗}(z_{k}), for all 0 < k < n.

(2) g_{n}^{∗}(z_{n}) = ρ^{n} + 2 ∑_{0<i<n} ρ^{n−i}π_{n−1}^{∗}(z_{i}).

(3) c(V_{n})(t) = t^{n} + 2 ∑_{1⩽i⩽n}(−1)^{i}z_{i}t^{n−i}.

Proof: G(1) is a conic and V_{1} ≅ O(−1). Hence, c(V_{1})(t) = t − 2z_{1}. Let now (1)_{m,k} is proven for all m < n and all 0 < k < m, and (2)_{m} and (3)_{m} are proven for all m < n. Since (j^{∗})^{k} : CH^{k}(ℙ_{G(n−1)}(Y_{n−1})) → CH^{k}(ℙ_{G(n−2)}(Y_{n−2})) is an isomorphism for k < n − 1, the condition (1)_{n,k} follows from (1)_{n−1,k} for all such k in view of j^{∗}(ρ) = ρ and j^{∗}(π_{n−1}^{∗}(z_{k})) = π_{n−2}^{∗}(z_{k}) (Proposition 2.4(3)). Analogously, since the kernel (j^{∗})^{n−1} is additively generated by π_{n−1}^{∗}(z_{n−1}), the condition (2)_{n−1} implies that g_{n}^{∗}(z_{n−1}) = ρ^{n−1} +2 ∑_{1⩽i<n−1} ρ^{n−1−i}π_{n−1}^{∗}(z_{i})+ λ · π_{n−1}^{∗}(z_{n−1}), where λ ∈ Z. Since Y_{n−1} = O ⊕ (V_{n−1}^{⊥})^{∨}, the projection π_{n−1} : ℙ_{G(n−1)}(Y_{n−1}) → G(n − 1) has the section s (given by the rule: (B/x) ↦ (B, B)). It satisfies: g_{n} ∘ s = j_{n−1}. Since s^{∗}(π_{n−1}^{∗}(z_{n−1})) = z_{n−1}, s^{∗}(ρ) = 0 and j_{n−1}^{∗}(z_{n−1}) = z_{n−1}, we get λ = 1, which implies (1)_{n,n−1}. Choose some rational point y ∈ Q\T_{x,Q}. By Propositions 2.3(2) and 2.4(2), the cycle z_{n} is defined as the set of such planes A, that y ∈ A. Then the cycle g_{n}^{∗}(z_{n}) is the set of such pairs (A, B), that y ∈ A, x ∈ B and dim(A+B/A) ⩽ 1. Thus A+B = y+B, and g_{n}^{∗}(z_{n}) is given by the section ℙ_{G(n−1)}(O) ⊂ ℙ_{G(n−1)}(Y_{n−1}). Since c(Y_{n−1})(t) = t^{2} · c(π_{n−1}^{∗}(V_{n−1}^{∨}))(t), this class can be expressed as ρ · c(π_{n−1}^{∗}(V_{n−1}^{∨}))(ρ). The last expression is equal to ρ^{n} + 2ρ^{n−1}π_{n−1}^{∗}(z_{1}) + … + 2ρπ_{n−1}^{∗}(z_{n−1}) because of (3)_{n−1}. The statement (2)_{n} is proven. Finally, since [g_{n}^{∗}(V_{n})] = [π_{n−1}^{∗}(V_{n−1})] + [O] + [O(−1)] − [O(1)], g_{n}^{∗}(c(V_{n})(t)) = π_{n−1}^{∗}(c(V_{n−1})(t)) · \frac{t·(t−ρ)}{t+ρ}. In the light of (3)_{n−1}, this is equal to

(t^{n−1} + 2 ∑_{1⩽i⩽n−1}(−1)^{i}π_{n−1}^{∗}(z_{i})t^{n−1−i}) · \frac{t · (t − ρ)}{t + ρ}.

Using the equality ρ^{2}(ρ^{n−1} + 2ρ^{n−2}π_{n−1}^{∗}(z_{1}) + … + 2π_{n−1}^{∗}(z_{n−1})) = 0, as well as the conditions (1)_{n,k} and (2)_{n}, we can rewrite the last expression as:

t^{n} + 2 ∑_{1⩽i⩽n}(−1)^{i}g_{n}^{∗}(z_{i})t^{n−i}.

Since g_{n}^{∗} is injective (the map g_{n} is birational), we get:

c(V_{n})(t) = t^{n} + 2 ∑_{1⩽i⩽n}(−1)^{i}z_{i}t^{n−i}.

The statement (3)_{n} is proven. □

3 Multiplicative structure

The multiplicative structure of CH^{∗}(G(n)) was studied extensively by H.Hiller, B.Boe, J.Stembridge, P.Pragacz and J.Ratajski - see [6], [11]. We can compute this ring structure from Theorem 2.5. Although, we restrict our consideration only to ( mod 2) case, it should be pointed out that the integral case can be obtained in a similar way. Let us denote as \overline{u} the image of u under the map CH^{∗} → CH^{∗} /2.

Proposition 3.1

CH^{∗}(G(n))/2 = ⊗_{1⩽d⩽n; d−odd}(ℤ/2[\overline{z}_{d}]/(\overline{z}_{d}^{2^{m_{d}}})),

where m_{d} = [log_{2}(n/d)] + 1.

Proof: Consider the diagram:

G(n) ⟵^{g_{n}} ℙ_{G(n−1)}(Y_{n−1}) ⟶^{π_{n−1}} G(n − 1).

From Theorem 2.5, g_{n}^{∗}(\overline{z}_{k}) = \overline{ρ}^{k} + π_{n−1}^{∗}(\overline{z}_{k}), for k < n, and g_{n}^{∗}(\overline{z}_{n}) = \overline{ρ}^{n}. Then it easily follows by the induction on n, that \overline{z}_{k}^{2} = \overline{z}_{2k} (where we assume \overline{z}_{r} = 0 if r > n). Thus, we have surjective ring homomorphism

⊗_{1⩽d⩽n; d−odd}(ℤ/2[\overline{z}_{d}]/(\overline{z}_{d}^{2^{m_{d}}})) → CH^{∗}(G(n))/2.

Since the dimensions of both rings are equal to 2^{n}, it is an isomorphism. □

Let J be a set. Let us call a multisubset the collection Λ = ∐_{β∈B} Λ_{β} of disjoint subsets of J. For a subset I of J, we will denote by the same symbol I the multisubset ∐_{i∈I}{i}. Let B = ∐_{γ∈C} B_{γ}, and Λ′_{γ} = ∐_{β∈B_{γ}} Λ_{β}. Then the multisubset Λ′ := ∐_{γ∈C} Λ′_{γ} is called the specialization of Λ. We call the specialization simple if #(B_{γ}) ⩽ 2, for all γ ∈ C. Let J now be some set of natural numbers (it may contain multiple entries). Then to any finite multisubset Λ = ∐_{β∈B} Λ_{β} of J we can assign the set of natural numbers \overline{Λ} := \{∑_{i∈Λ_{β}} i\}_{β∈B}. We call the specialization Λ good if \overline{Λ} ⊂ \{1, …, n\}. Suppose I be some finite set of natural numbers. Let us define the element \overline{z}_{I} ∈ CH^{∗}(G(n))/2 by the formula:

\overline{z}_{I} = ∑_{Λ} ∏_{j∈\overline{Λ}} \overline{z}_{j},

where we assume \overline{z}_{r} = 0, if r > n, and the sum is taken over all simple specializations Λ of the multisubset I = ∐_{i∈I}\{i\}. Actually, \overline{z}_{I} is just reduction modulo 2 of the Schubert cell class z_{I}. This follows from the Pieri formula of H.Hiller and B.Boe (see [6]) and our Proposition 3.3. We do not use this fact, but instead prove directly that \overline{z}_{I} form basis (Proposition 3.4(1)).

Lemma 3.2 If I ⊄ \{1, …, n\}, then \overline{z}_{I} = 0.

Proof: If I contains an element r > n, then \overline{z}_{I} is clearly zero. Suppose now that I contains some element i twice, say as i_{1} and i_{2}. Consider the subgroup Z_{2} ⊂ S_{#(I)} interchanging i_{1} and i_{2} and keeping all other elements in place. We get ℤ/2-action on our specializations. The terms which are not stable under this action will appear with multiplicity 2, so, we can restrict our attention to the stable terms. But such specializations have the property that {i_{1}, i_{2}} is disjoint from the rest of i’s, and the corresponding sum looks as: ∑_{M} ∏_{j∈\overline{M}} \overline{z}_{j} · (\overline{z}_{i}^{2} + \overline{z}_{2i}), where the sum is taken over all simple specializations of the multisubset I\\{i_{1}, i_{2}\}. Since \overline{z}_{i}^{2} = \overline{z}_{2i}, this expression is zero. □

We immediately get the (modulo 2) version of the Pieri formula proved by H.Hiller and B.Boe:

Proposition 3.3 ([6])

\overline{z}_{I} · \overline{z}_{j} = \overline{z}_{I∪j} + ∑_{i∈I} \overline{z}_{(I\i)∪(i+j)},

where we omit terms \overline{z}_{J} with J ⊄ \{1, …, n\} (in particular, if J contains some element with multiplicity > 1).

Proof: \overline{z}_{I∪j} = ∑_{Λ} ∏_{l∈\overline{Λ}} \overline{z}_{l}, where the sum is taken over all simple specializations of the multisubset I ∪ j. We can distinguish two types of specializations:

1) j is separated from I; 2) j is not separated from I, that is, there is β such that Λ_{β} = {i, j}, for some i ∈ I. Let us call the latter specializations to be of type (2, i). Clearly, the sum over specializations of the first kind is equal to \overline{z}_{I} · \overline{z}_{j}, and the sum over the specializations of the type (2, i) is equal to \overline{z}_{(I\i)∪(i+j)}. Finally, the terms with J ⊄ {1, …, n} could be omitted by Lemma 3.2. □

We also get the expression of monomials on z_{i}’s in terms of z_{I}’s.

Proposition 3.4 (1) The set \{\overline{z}_{I}\}_{I⊂\{1,...,n\}} is a basis of CH^{∗}(G(n))/2.

(2) ∏_{i∈I} \overline{z}_{i} = ∑_{Λ} \overline{z}\overline{Λ}, where sum is taken over all good specializations of I.

Proof:

(1) On the ℤ/2-vector space CH^{∗}(G(n))/2 = ⊕_{I⊂\{1,...,n\}}ℤ/2 · ∏_{i∈I} \overline{z}_{i} we have lexicographical filtration. Consider the linear map ε : CH^{∗}(G(n))/2 → CH^{∗}(G(n))/2 sending ∏_{i∈I} \overline{z}_{i} to \overline{z}_{I}. Then the associated graded map: gr(ε) is the identity. Thus, ε is invertible, and the set \{\overline{z}_{I}\}_{I⊂\{1,...,n\}} form a basis.

(2) Consider the ℤ/2-vector spaces W_{1} := ⊕_{Λ}ℤ/2 · x_{Λ}, and W_{2} := ⊕_{Λ}ℤ/2 · y_{Λ}, where Λ runs over all finite multisubsets of ℕ. Consider the linear maps ψ : W_{2} → W_{1} which sends y_{Λ} to the ∑_{Λ′} x_{Λ′}, where the sum is taken over all specializations of Λ, and φ : W_{1} → W_{2} which sends x_{Λ} to the ∑_{Λ′} y_{Λ′}, where the sum is taken over all simple specializations Λ′ of Λ. It is an easy exercise to show that φ and ψ are mutually inverse. Consider the linear surjective maps: w_{1} : W_{1} → CH^{∗}(G(n))/2 and w_{2} : W_{2} → CH^{∗}(G(n))/2 given by the rule: w_{1}(x_{Λ}) := \overline{z}\overline{Λ}, and w_{2}(y_{Λ}) := ∏_{j∈\overline{Λ}} \overline{z}_{j}. Then, by the definition of \overline{z}_{I}, w_{1} = w_{2} ∘ φ. Then w_{2} = w_{1} ∘ ψ, which implies that ∏_{i∈I} \overline{z}_{i} = ∑_{Λ} \overline{z}\overline{Λ}, where the sum is taken over all specializations of I. It remains to notice, that nongood specializations do not contribute to the sum (by Lemma 3.2). □

Examples: 1) \overline{z}_{i} · \overline{z}_{j} = \overline{z}_{i,j} + \overline{z}_{i+j}, where the first term is omitted if i = j and the second if i + j > n. 2) \overline{z}_{i,j,k} = \overline{z}_{i} · \overline{z}_{j} · \overline{z}_{k} + \overline{z}_{i+j} · \overline{z}_{k} + \overline{z}_{j+k} · \overline{z}_{i} + \overline{z}_{i+k} · \overline{z}_{j}.

4 Action of the Steenrod algebra

On the Chow-groups modulo prime l there is the action of the Steenrod algebra. Such action was constructed by V.Voevodsky in the context of arbitrary motivic cohomology - see [13], and then a simpler construction was given by P.Brosnan for the case of usual Chow groups - see [3]. For quadratic Grassmannians we will be interested only in the case l = 2. We can compute the action of the Steenrod squares S^{r} : CH^{∗} /2 → CH^{∗+r} /2 on the cycles \overline{z}_{i}. For convenience, let us put z_{j} ∈ CH^{j}(G(m)) to be zero for j > m.

Theorem 4.1

S^{r}(\overline{z}_{i}) = \binom{i}{r} · \overline{z}_{i+r}

Proof: Use induction on n. The base is trivial. Suppose the statement is true for (n − 1). Since \overline{c}(V_{n})(t) = t^{n}, we have: \overline{ρ}^{n+1} = 0. Then, by Theorem 2.5 and the assumption above, g_{n}^{∗}(\overline{z}_{j}) = \overline{ρ}^{j} + π_{n−1}^{∗}(\overline{z}_{j}), for all j. Using the fact that S^{r} commutes with the pull-back morphisms (see [3]), and the inductive assumption, we get:

\begin{matrix}g_{n}^{∗}(S^{r}(\overline{z}_{i})) = S^{r}(g_{n}^{∗}(\overline{z}_{i})) = S^{r}(\overline{ρ}^{i} + π_{n−1}^{∗}(\overline{z}_{i})) = \\ \binom{i}{r}\overline{ρ}^{i+r} + π_{n−1}^{∗}(\binom{i}{r}\overline{z}_{i+r}) = \binom{i}{r} · g_{n}^{∗}(\overline{z}_{i+r}). \end{matrix}

Now, the statement follows from the injectivity of g_{n}^{∗}. □

5 Main theorem

Let X be some variety over the field k. We will denote:

C^{∗}(X) := image(CH^{∗}(X)/2 → CH^{∗}(X|\overline{k})/2).

Let now Q be a smooth projective quadric of dimension 2n−1, and X = G(n, Q) be the Grassmanian of middle-dimensional projective planes on it. Then X|\overline{k} = G(n). In this section we will show that, as an algebra, C^{∗}(G(n, Q)) is generated by the elementary cycles \overline{z}_{i} contained in it. Let F (n, Q) be the variety of complete flags (l_{0} ⊂ l_{1} ⊂ … ⊂ l_{n−1}) of projective subspaces on Q. Then F (n, Q) is naturally isomorphic to the complete flag variety F_{G(n,Q)}(V_{n}) of the tautological n-dimensional bundle on G(n, Q). On the variety F (n, Q) there are natural (subquotient) line bundles L_{1}, …, L_{n}. The first Chern classes c_{1}(L_{i}), 1 ⩽ i ⩽ n generate the ring CH^{∗}(F (n, Q)) as an algebra over CH^{∗}(G(n, Q)), and the relations among them are: σ_{j}(c_{1}(L_{1}), …, c_{1}(L_{n})) = c_{j}(V_{n}), 1 ⩽ j ⩽ n - see [4, Example 3.3.5]. Let F_{n} be the variety of complete flags of subspaces of the n-dimensional vector space V. It also has natural line bundles L′_{1}, …, L′_{n}. Again, the first Chern classes c_{1}(L′_{i}) generate the ring CH^{∗}(F_{n}). By Theorem 2.5 (3), modulo 2, all Chern classes c_{j}(V_{n}) are the same as the Chern classes of the trivial ndimensional bundle ⊕_{i=1}^{n}O. Thus, modulo 2, the Chow ring of F_{G(n,Q)}(V_{n}) is isomorphic to the Chow ring of F_{G(n,Q)}(⊕_{i=1}^{n}O). We get:

Theorem 5.1 There is a ring isomorphism

CH^{∗}(F (n, Q))/2 ≅ CH^{∗}(G(n, Q))/2 ⊗_{ℤ/2} CH^{∗}(F_{n})/2,

where the map CH^{∗}(G(n, Q)) → CH^{∗}(F (n, Q)) is induced by the natural projection F (n, Q) → G(n, Q), and the map CH^{∗}(F_{n})/2 → CH^{∗}(F (n, Q))/2 is given on the generators by the rule: c_{1}(L′_{i}) ↦ c_{1}(L_{i}).

□

Notice, that the change of scalar map CH^{∗}(F_{n}) → CH^{∗}(F_{n}|\overline{k}) is an isomorphism, and, C^{∗}(F (n, Q)) = C^{∗}(G(n, Q)) ⊗_{ℤ/2} CH^{∗}(F_{n}|\overline{k})/2. Thus, we have:

Statement 5.2 Let v_{1}, …, v_{s} be linearly independent elements of CH^{∗}(F_{n}|\overline{k})/2, and x_{i} ∈ CH^{∗}(G(n, Q)|\overline{k})/2, then x = ∑_{i=1}^{s} x_{i} · v_{i} belongs to C^{∗}(F (n, Q)) if and only if all x_{i} ∈ C^{∗}(G(n, Q)).

□

The ring CH^{∗}(F_{n}) can be described as follows. Let us denote c_{1}(L_{j}) as h_{j}, and the set \{h_{j}, …, h_{n}\} as \underline{h}(j) (and \underline{h}(1) as \underline{h}). For arbitrary set of variables \underline{u} = \{u_{1}, …, u_{r}\} let us define the degree i polynomials σ_{i}(\underline{u}) and σ_{−i}(\underline{u}) from the equation:

∏_{l}(1 + u_{l}) = ∑_{i} σ_{i}(\underline{u}) = (∑_{i} σ_{−i}(\underline{u}))^{−1} .

Statement 5.3 ([4, Example 3.3.5])

CH^{∗}(F_{n}) = ℤ[\underline{h}]/(σ_{i}(\underline{h}), 1 ⩽ i ⩽ n) = ℤ[\underline{h}]/(σ_{−i}(\underline{h}(i)), 1 ⩽ i ⩽ n).

Since σ_{−i}(\underline{h}(i)) is the ±-monic polynomial in h_{i} with coefficients in the subring, generated by \underline{h}(i + 1), we get: CH^{∗}(F_{n}) is a free module over the subring ℤ[\underline{h}(n)]/(σ_{−n}(\underline{h}(n))) = ℤ[h_{n}]/(h_{n}^{n}). Let π : F (n, Q) → F (n − 1, Q) be the natural projection between full flag varieties. We will denote by the same symbol \overline{z}_{I} the images of \overline{z}_{I} in CH^{∗}(F (n, Q))/2. The following statement is the key for the Main Theorem.

Proposition 5.4

π^{∗}π_{∗}(\overline{z}_{I}) = ∑_{i∈I} \overline{z}_{(I\i)} · π^{∗}π_{∗}(\overline{z}_{i}).

Proof: F (n, Q) is a conic bundle over F (n−1, Q) inside the projective bundle ℙ_{F (n−1,Q)}(V), where, in K_{0}(F (n, Q)), π^{∗}[V] = [L_{n}] + [L_{n}^{−1}] + [O]. Sheaf L_{n} is nothing else but the restriction of the sheaf O(−1) from ℙ_{F (n−1,Q)}(V) to F (n, Q).

Lemma 5.5 Let V be a 3-dimensional bundle over some variety X equipped with the nondegenerate quadratic form p. Let π : Y → X be conic bundle of p-isotropic lines in V. Then there is a CH^{∗}(X)-algebra automorphism ϕ : CH^{∗}(Y) → CH^{∗}(Y) of exponent 2 such that

(1) ϕ(c_{1}(O(−1)|_{Y})) = c_{1}(O(1)|_{Y}).

(2) π^{∗}π_{∗}(x) · c_{1}(O(1)) = x − ϕ(x)

Proof: Consider variety Y ×_{X} Y with the natural projections π_{1} and π_{2} on the first and second factor, respectively. Then divisor Δ(Y) ⊂ Y ×_{X} Y defines an invertible sheaf L on Y ×_{X} Y such that L^{2} ≅ π_{1}^{∗}(O(1)) ⊗ π_{2}^{∗}(O(1)) and Δ^{∗}(L) = O(1). Consider the map f := Δ ∘ π_{2} : Y ×_{X} Y → Y ×_{X} Y. Define ϕ : CH^{∗}(Y) → CH^{∗}(Y) as id − Δ^{∗} ∘ f_{∗} ∘ π_{1}^{∗}. The described maps fit into the diagram:

Y ⟵^{π_{1}} Y ×_{X} Y
π↓  ↓^{π_{2}}
X ⟵_{π} Y  ⟶_{Δ} Y ×_{X} Y

with the transversal Cartesian square (π_{1}^{∗}(T_{π}) = T_{π_{2}}). Consequently, π^{∗} ∘ π_{∗} = π_{2}_{∗} ∘ π_{1}^{∗}. Since O(Δ(Y))|_{Y} = O(1), we have:

π^{∗}π_{∗}(x) · c_{1}(O(1)) = Δ^{∗}Δ_{∗}π_{2}_{∗}π_{1}^{∗}(x) = x − ϕ(x).

Consider the map ψ := id−f_{∗} : CH^{∗}(Y ×_{X} Y) → CH^{∗}(Y ×_{X} Y). We claim that ψ is a ring homomorphism. Really, Y ×_{X} Y ≅ ℙ_{Y} (U), where the projection ℙ_{Y} (U) → Y is given by π_{2} and c(U)(t) = t(t−c_{1}(O(1))). Thus CH^{∗}(Y ×_{X} Y) = CH^{∗}(Y)[ρ]/(ρ(ρ − c_{1}(O(1)))), where the map CH^{∗}(Y) → CH^{∗}(Y ×_{X} Y) is π_{2}^{∗}. Notice, that f_{∗}π_{2}^{∗} = Δ_{∗}π_{2}_{∗}π_{2}^{∗} = 0, that is, ψ|_{CH^{∗}(Y)} is the identity. At the same time, ψ(ρ) = ρ − ρ = 0. Since CH^{∗}(Y ×_{X} Y) is free CH^{∗}(Y)-module of rank 2 with the basis 1, ρ, by the projection formula, we get that ψ is an endomorphism of CH^{∗}(Y ×_{X} Y) considered as an CH^{∗}(Y)-algebra.

Since, ϕ = Δ^{∗} ∘ ψ ∘ π_{1}^{∗}, it is a homomorphism of CH^{∗}(X)-algebras. Also, ϕ(c_{1}(O(1))) = −c_{1}(O(1)). Finally, since the composition π_{2}_{∗}π_{1}^{∗}Δ^{∗}Δ_{∗} : CH^{∗}(Y) → CH^{∗}(Y) is equal 2 · id, we get

(Δ^{∗} ∘ f_{∗} ∘ π_{1}^{∗})^{∘2} = 2(Δ^{∗} ∘ f_{∗} ∘ π_{1}^{∗}),

which is equivalent to: ϕ^{∘2} = id. Thus, ϕ is an automorphism of exponent 2. □

Let us compute the action of ϕ on basis elements \overline{z}_{I}. Let σ_{i} be elementary symmetric functions in h_{i}’s. Since h_{i} ∈ CH^{∗}(F (n − 1, Q)), for i < n, we have equality ϕ(h_{i}) = h_{i} for them, and ϕ(h_{n}) = −h_{n}. We know that σ_{i} = (−1)^{i}2z_{i}. We immediately conclude:

Lemma 5.6 ϕ(z_{i}) = z_{i} + ∑_{0<l<i} 2z_{i−l}h_{n}^{l} + h_{n}^{i}.

Lemma 5.7 ϕ(\overline{z}_{I}) = \overline{z}_{I} + ∑_{i∈I} \overline{z}_{(I\i)}\overline{h}_{n}^{i}.

Proof: Let us define the size s(I) of I as the number of it’s elements. Use induction on the size of I. The case of size = 1 is OK by the previous lemma. Suppose the statement is known for sizes < s(I).

Let i be some element of I. We know from Proposition 3.3 that \overline{z}_{I} = \overline{z}_{(I\i)} · \overline{z}_{i} + ∑_{j∈I,j≠i} \overline{z}_{(I\\{i,j\})∪(i+j)}. Since ϕ is a ring homomorphism, we get:

ϕ(\overline{z}_{I}) = ϕ(\overline{z}_{(I\i)}) · ϕ(\overline{z}_{i}) + ∑_{j∈I\i} ϕ(\overline{z}_{(I\\{i,j\})∪(i+j)}) =

(\overline{z}_{(I\i)} + ∑_{l∈I\i} \overline{z}_{(I\\{i,l\})}h_{n}^{l}) · (\overline{z}_{i} + \overline{h}_{n}^{i})+

∑_{j∈I\i} (\overline{z}_{(I\\{i,j\})∪(i+j)} + \overline{z}_{(I\\{i,j\})}\overline{h}_{n}^{i+j} + ∑_{m∈I\\{i,j\}} \overline{z}_{(I\\{i,j,m\})∪(i+j)}\overline{h}_{n}^{m}) =

\overline{z}_{I} + \overline{z}_{(I\i)}\overline{h}_{n}^{i} + ∑_{l∈I\\{i\}}(\overline{z}_{(I\\{i,l\})} · \overline{z}_{i})\overline{h}_{n}^{l} + ∑_{m≠j∈I\\{i\}} \overline{z}_{(I\\{i,j,m\})∪(i+j)}\overline{h}_{n}^{m} =

\overline{z}_{I} + \overline{z}_{(I\i)}\overline{h}_{n}^{i} + ∑_{j∈I\i} \overline{z}_{(I\j)}\overline{h}_{n}^{j} + 2 · ∑_{m≠j∈I\\{i\}} \overline{z}_{(I\\{i,j,m\})∪(i+j)}\overline{h}_{n}^{m} =

\overline{z}_{I} + ∑_{j∈I} \overline{z}_{(I\j)}\overline{h}_{n}^{j}

(as usually, one should omit \overline{z}_{J} with J ⊄ \{1, …, n\}). □

Let p = q ⊥ ℍ. Then Q can be identified with the quadric of projective lines on P passing through fixed rational point y. This identifies the complete flag variety F (r, Q) with the subvariety of F (r + 1, P) consisting of flags containing our point y. We get an embedding i_{r} : F (r, Q) → F (r + 1, P). It is easy to see that the diagram

F (n, Q) ⟶^{i_{n}} F (n + 1, P)
π↓  ↓_{π′}
F (n − 1, Q) ⟶_{i_{n−1}} F (n, P)

is Cartesian, and since π′ is smooth, we have an equality: π^{∗} ∘ π_{∗} ∘ i_{n}^{∗} = i_{n}^{∗} ∘ π′^{∗} ∘ π′_{∗}. It follows from Lemmas 5.5 and 5.7 that

π′^{∗}π′_{∗}(\overline{z}_{I}) · \overline{h}_{n+1} = ∑_{i∈I} \overline{z}_{(I\i)}\overline{h}_{n+1}^{i}.

Thus, modulo the kernel of multiplication by \overline{h}_{n+1}, π′^{∗}π′_{∗}(\overline{z}_{I}) ≡ ∑_{i∈I} \overline{z}_{(I\i)}\overline{h}_{n+1}^{i−1}. But, by the Statement 5.3 and Theorem 5.1, such kernel is generated by \overline{h}_{n+1}^{n}. Since i_{n}^{∗}(\overline{z}_{I}) = \overline{z}_{I}, i_{n}^{∗}(\overline{h}_{n+1}) = \overline{h}_{n} and \overline{h}_{n}^{n} = 0 on F (n, Q), we get:

π^{∗}π_{∗}(\overline{z}_{I}) = i_{n}^{∗}π′^{∗}π′_{∗}(\overline{z}_{I}) = ∑_{i∈I} \overline{z}_{(I\i)}\overline{h}_{n}^{i−1} = ∑_{i∈I} \overline{z}_{(I\i)}π^{∗}π_{∗}(\overline{z}_{i}).

□

Notice, that the elements π^{∗}π_{∗}(z_{i}) belong to CH^{∗}(F_{n})/2, and they are linearly independent (being nonzero and having different degrees). As a corollary, we get:

Main Theorem 5.8 As an algebra, C^{∗}(G(n, Q)) is generated by the elementary classes \overline{z}_{i} contained in it.

Proof: Let \overline{z} be an element of C^{∗}(G(n, Q)). It can be expressed as a linear combination of the basis elements \overline{z}_{I}’s. Let us define the size s(\overline{z}) of the element \overline{z} = ∑ \overline{z}_{I_{a}} as the maximum of sizes of I_{a} involved. Let m(\overline{z}) be the main term of \overline{z}, that is, ∑_{a:s(I_{a})=s(\overline{z})} \overline{z}_{I_{a}}.

Lemma 5.9 Let \overline{z} = ∑_{a} \overline{z}_{I_{a}} ∈ C^{∗}(G(n, Q)). Let s(I_{a}) = s(\overline{z}), and i ∈ I_{a}. Then the elementary cycle \overline{z}_{i} belongs to C^{∗}(G(n, Q)).

Proof: Let i ∈ I_{a}, and I_{a}\i = {j_{2}, …, j_{s}}. Denote the operation π^{∗}π_{∗} as D. Then D(\overline{z}) = ∑_{1⩽j⩽n} d_{j}(\overline{z}) · D(\overline{z}_{j}), where d_{j}(\overline{z}) ∈ CH^{∗}(G(n, Q)|\overline{k})/2, and the elements D(\overline{z}_{j}) ∈ CH^{∗}(F_{n})/2 are linearly independent. Since D is defined over the base field, D(\overline{z}) ∈ C^{∗}(F (n, Q)), and, by the Statement 5.2, d_{j}(\overline{z}) ∈ C^{∗}(G(n, Q)). Clearly, m(d_{j}(\overline{z})) = d_{j}(m(\overline{z})). It is easy to see that d_{j_{s}} … d_{j_{2}}(\overline{z}) = \overline{z}_{i}, since for arbitrary I_{b} with s(I_{b}) < s = s(\overline{z}) we have: d_{j_{s}} … d_{j_{2}}(\overline{z}_{I_{b}}) = 0, or 1, and for I_{c} ≠ I_{a} with s(I_{c}) = s, d_{j_{s}} … d_{j_{2}}(\overline{z}_{I_{c}}) is either 0, or has degree different from i. Thus, \overline{z}_{i} ∈ C^{∗}(G(n, Q)). □

Let us prove by induction on the size of \overline{z}, that \overline{z} belongs to the subring of C^{∗}(G(n, Q)) generated by \overline{z}_{j}’s. The base of induction, s = 1 is trivial. Notice that m(∏_{i∈I} \overline{z}_{i}) = \overline{z}_{I}. Thus, the size of \overline{z}′ = \overline{z} − ∑_{a:s(I_{a})=s(\overline{z})} ∏_{i∈I_{a}} \overline{z}_{i} is smaller than that of \overline{z}. But by the Lemma 5.9, all the \overline{z}_{i}’s appearing in this expression belong to C^{∗}(G(n, Q)). By the inductive assumption, \overline{z}′ belongs to the subring of C^{∗}(G(n, Q)) generated by \overline{z}_{j}’s. Then so is \overline{z}. □

Remark. Actually, for the proof of the Main Theorem one just needs the statement of the Lemma 5.5(1).

Corollary 5.10 For arbitrary smooth projective quadric Q,

C^{∗}(G(n, Q)) = ⊗_{1⩽d⩽n; d−odd}(ℤ/2[\overline{z}_{d2^{l_{d}}}]/(\overline{z}_{d2^{l_{d}}}^{2^{(m_{d}−l_{d})}})),

for certain 0 ⩽ l_{d} ⩽ m_{d} = [log_{2}(n/d)] + 1.

Proof: It immediately follows from the Main Theorem 5.8, Proposition 3.1, and the fact that \overline{z}_{s}^{2} = \overline{z}_{2s} (or 0, if 2s > n). □

Now we can introduce:

Definition 5.11 (1) Let Q be a quadric of dimension 2n − 1. Denote as J(Q) the subset of \{1, …, n\} consisting of those i, for which \overline{z}_{i} ∈ C^{∗}(G(n, Q)).

(2) Let P be a quadric of dimension 2n. Let Q be arbitrary subquadric of codimension 1 in P. Then J(P) is a subset of {0, 1, …, n}, where 0 ∈ J(P) iff det_{±}(P) = 1 and, for i > 0, i ∈ J(P) iff i ∈ J(Q|_{k\sqrt{det_{±}(P)}}).

Remark: The definition (2) above is motivated by the fact that G(n + 1, P) is isomorphic to G(n, Q) ×_{Spec(k)} Spec(k\sqrt{det_{±}(P)}).

It follows from the Main Theorem 5.8 that C^{∗}(G(n, Q)) is exactly the subring of CH^{∗}(G(n, Q)|\overline{k})/2 generated by \overline{z}_{i}, i ∈ J(Q). In particular, J(Q) carries all the information about C^{∗}(G(n, Q)). Notice, that the same information is contained in the sequence {l_{d}}_{d−odd;1⩽d⩽n}. What restrictions do we have on the possible values of J(Q)? Because of the action of the Steenrod operations, we get:

Proposition 5.12 Let i ∈ J(Q), and r ∈ ℕ is such that \binom{i}{r} ≡ 1 ( mod 2), and i + r ⩽ n. Then (i + r) ∈ J(Q).

Proof: j belongs to J(Q) if and only if the cycle \overline{z}_{j} ∈ CH^{j}(G(n, Q)|\overline{k})/2 is defined over the base field. Since \overline{z}_{i} has such a property, and the Steenrod operation S^{r} is defined over the base field too, we get: \overline{z}_{(i+r)} = S^{r}(\overline{z}_{i}) is also defined over the base field. □

The natural question arises:

Question 5.13 Do we have other restrictions on J(Q)? In other words, let J ⊂ {1, …, n} be a subset satisfying the conditions of Proposition 5.12. Does there exist a quadric such that J(Q) = J?

It is not difficult to check that, at least, for n ⩽ 4, there is no other restrictions.

6 On the canonical dimension of quadrics

In this section we will show that in the case of a generic quadric the variety G(n, Q) is 2-incompressible, and also will formulate the conjecture describing the canonical dimension of arbitrary quadric. I would like to point out that the current section would not appear without the numerous discussions with G.Berhuy, who brought this problem to my attention. We start by computing the characteristic classes of the variety G(n, Q). Let W be (2n + 1)-dimensional vector space over k equipped with the nondegenerate quadratic form q. Let F (r) = F (r, Q) be the variety of complete flags (π_{1} ⊂ … ⊂ π_{r}) of totally isotropic subspaces in W. Thus, F (0) = Spec(k), F (1) = Q, etc. ... . We get natural smooth projective maps: ε_{r} : F (r + 1) → F (r) with fibers - quadrics of dimension 2n − 2r − 1. Let L_{i} be the standard subquotient linear bundles L_{i} := π_{i}/π_{i−1}, and h_{i} = c_{1}(L_{i}). The bundle L_{i} is defined on F (r), for r ⩾ i. These divisors h_{i} are the roots of the tautological vector bundle V_{n} studied above.

Proposition 6.1 The Chern polynomial of the tangent bundle T_{F (r)} is equal to:

c(T_{F (r)}) = \frac{∏_{1⩽i⩽r}(1 − h_{i})^{2n+1}}{∏_{1⩽i⩽r}(1 − 2h_{i}) · ∏_{1⩽i<j⩽r} ((1 − h_{j} + h_{i}) · (1 − h_{j} − h_{i}))}.

Proof: Let V_{r} be a tautological vector bundle on F (r). Then V_{r} is an isotropic subbundle of η^{∗}(W) (η : F (r) → Spec(k) is the projection), and on the subquotient W_{r} := V_{r}^{⊥}/V_{r} we have a nondegenerate quadratic form q_{{r}}. Then the variety F (r + 1) is defined as zeroes of this quadratic form. Thus, F (r + 1) is the divisor of the sheaf O(2) on the projective bundle ℙ_{F (r)}(W_{r}), and we have exact sequence:

0 → T_{F (r+1)} → T_{ℙ_{F (r)}(W_{r})}|_{F (r+1)} → O(2)|_{F (r+1)} → 0.

On the other hand, from the projection ℙ_{F (r)}(W_{r}) →^{θ_{r}} F_{r}, we have sequences:

0 → T_{θ_{r}} → T_{ℙ_{F (r)}(W_{r})} → θ_{r}^{∗}(T_{F (r)}) → 0 and
0 → O → W_{r} ⊗ L_{r+1}^{−1} → T_{θ_{r}} → 0.

(see [4, Example 3.2.11]). It remains to notice, that in K_{0}, [W_{r}] = (2n + 1)[O] − ∑_{i=1}^{r}([L_{i}] + [L_{i}^{−1}]), to get the equality:

c(T_{F (r+1)}) = ε_{r}^{∗}(c(T_{F (r)})) · \frac{(1 − h_{r+1})^{2n+1}}{(1 − 2h_{r+1}) · ∏_{i=1}^{r}(1 − h_{r+1} + h_{i})(1 − h_{r+1} − h_{i})}

The statement now easily follows by induction on r. □

Now, it is easy to compute the characteristic classes of the quadratic Grassmannians.

Proposition 6.2

c(T_{G(r)}) = \frac{∏_{1⩽i⩽r}(1 − h_{i})^{2n+1}}{∏_{1⩽i⩽r}(1 − 2h_{i}) · ∏_{1⩽i<j⩽r} ((1 − (h_{j} − h_{i})^{2}) · (1 − h_{j} − h_{i}))},

where h_{j} are the roots of the tautological vector bundle V_{r} on G(r).

Proof: Consider the (forgetting) projection δ_{r} : F (r) → G(r) = G(r, Q). We have natural identification of F (r) with the variety of complete flags corresponding to the tautological bundle V_{r} on G(r) (we will permit ourselves to use the same notation for the tautological bundles on G(r) and F (r) - this is justified by the fact that they are related by the map δ_{r}^{∗}). Using the fact that δ_{r} : F (r) → G(r) can be decomposed into a tower of projective bundles, and [4, Example 3.2.11], we get:

c(T_{δ_{r}}) = ∏_{1⩽i<j⩽r}(1 + h_{j} − h_{i}),

and the statement follows. □

Now we can prove the following Conjecture of G.Berhuy (proven by him for n ⩽ 4):

Theorem 6.3 degree(c_{dim(G(n))}(−T_{G(n)})) ≡ 2^{n} ( mod 2^{n+1}).

Proof: By Proposition 6.2, Chern classes of (−T_{G(n)}) can be expressed as polynomials in the Chern classes of the tautological vector bundle V_{n}. From Theorem 2.5 we know that c_{j}(V_{n}) = σ_{j} = (−1)^{j}2z_{j}, where z_{j} are elementary cycles defined in Proposition 2.4. Since, in K_{0}, [V_{n}] + [V_{n}^{∨}] = 2n[O], we get the relations on σ_{j}:

Lemma 6.4 σ_{i}^{2} = 2(−1)^{i}(σ_{2i} + ∑_{1⩽j<i}(−1)^{j}σ_{j} · σ_{2i−j}).

Proof: It is just the component of degree 2i of the relation

(1 + ∑_{i} σ_{i}) · (1 + ∑_{i}(−1)^{i}σ_{i}) = 1.

□

Let A := ℤ[σ̃_{1}, …, σ̃_{n}]. We have ring homomorphism ψ : A → CH^{∗}(G(n)) sending σ̃_{i} to σ_{i}. It follows from the Lemma 6.4, that for arbitrary f ∈ A there exists some g ∈ A such that g does not contain squares, and ψ(f − g) ∈ 2^{n+1} CH^{∗}(G(n)). If f has degree = dim(G(n)), then g got to be monomial λ · ∏_{1⩽i⩽n} σ_{i}. Moreover, if f was a monomial divisible by 2, or containing square, then λ will be divisible by 2. Consider ideal L ⊂ A generated by 2 and squares of elements of positive degree. Let R be a quotient ring, and φ : A → R be the projection. Since ∏_{1⩽i⩽n} σ_{i} = (−1)^{\binom{n+1}{2}}2^{n} ∏_{1⩽i⩽n} z_{i}, and ∏_{1⩽i⩽n} z_{i} is the class of a rational point (by Proposition 2.4), we get that for arbitrary f ∈ A, the degree(ψ(f)) is divisible by 2^{n}, and for f ∈ L the degree is divisible by 2^{n+1}. Thus, modulo 2^{n+1}, the degree of ψ(f) depends only on φ(f).

In R we have the following equalities:

φ(1 + f^{2}) = φ((1 + f)^{2}) = 1, for any f of positive degree.

Thus,

φ(c(−T_{G(n)})) = φ ( ∏_{1⩽i⩽n}(1 + h_{i}) · ∏_{1⩽i<j⩽n}(1 + h_{i} + h_{j})) .

And in the light of Giambelli’s formula (see [4, Example 14.5.1(b’)]),

φ (c(−T_{G(n)})\binom{n+1}{2}) = φ (σ_{n} · det[σ_{n−2i+j}]_{1⩽i,j⩽n}) .

Since we mod-out the squares, this expression is equal to φ(∏_{i=1}^{n} σ_{i}). Consequently, degree(c_{dim(G(n))}(−T_{G(n)})) ≡ 2^{n} ( mod 2^{n+1}).

□

We recall from [10] that a variety X is p-compressible if there is a rational map X −−→ Y to some variety Y such that dim(Y) < dim(X) and v_{p}(n_{X}) ⩽ v_{p}(n_{Y}), where n_{Z} is the image of the degree map deg : CH_{0}(Z) → ℤ. From the Rost degree formula ([10, Theorem 6.4]) for the characteristic number c_{dim(G(n))} modulo 2 (see [10, Corollary 7.3, Proposition 7.1]), we get:

Proposition 6.5 Let Q be a smooth 2n + 1-dimensional quadric, all splitting fields of which have degree divisible by 2^{n} (we call such Q - generic). Then the variety G(n, Q) is 2-incompressible.

□

Call two smooth varieties X and Y equivalent if there are rational maps X −−→ Y and Y −−→ X. Then let d(X) be the minimal dimension of a variety equivalent to X. Recall from [1] that a canonical dimension cd(q) of a quadratic form q is defined as d(G(n, Q)), where n = [dim(q)/2] + 1. Proposition 6.5 gives another proof of the fact that the canonical dimension of a generic (2n + 1)-dimensional form is n(n + 1)/2, which computes the canonical dimension of the groups SO_{2n+1} and SO_{2n+2} (cf. [8, Theorem 1.1, Remark 1.3]). Our computations of the generic discrete invariant GDI(m, Q) permit to conjecture the answer in the case of arbitrary smooth quadric Q:

Conjecture 6.6 Let Q be smooth projective quadric of dimension d. Then

cd(Q) = ∑_{j∈{1,...,[d+1/2]}\J(Q)} j,

where J(Q) is the invariant from the Definition 5.11.

If Q is generic, then J(Q) is empty, and cd(Q) is indeed equal ∑_{1⩽i⩽n} i = n(n + 1)/2.

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