Documenta Math. 111 On the Chow Groups of Quadratic Grassmannians

In this text we get a description of the Chow-ring (mod- ulo 2) of the Grassmanian of the middle-dimensional planes on arbi- trary 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 decomposi- tion 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).


A. Vishik 1 Introduction
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 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)| 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 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 knownsee 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.

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) 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 , where the identification is given by the rule: , 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): 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 we get the exact sequences: , where X n−1 is the bundle with the fiber C.In particular, where ρ = c 1 (O(1)), and c(E)(t) = dim(E) i=0 c i (E)t dim(E)−i is the total Chern polynomial of the vector bundle E. Consider the open subvariety Mn : ) is an isomorphism, and πn−1 : Mn → G(n−1) is an n-dimensional affine bundle over G(n − 1).Proposition 2.1 There is split exact sequence

Documenta Mathematica 10 (2005) 111-130
Notice that the choice of a point y ∈ Q\T x,Q gives a section s : G(n − 1) → Mn of the affine bundle πn−1 : Mn → G(n − 1).And the composition ϕ • gn • 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 Thus, ker(j ).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 Then the same is true for j n−1 * .And we get the desired split exact sequence.(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 . 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, 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 From Proposition 2.1 we get the commutative diagram with exact rows:

A. Vishik
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: ) is given by the following rule on the additive generators above: Proof: Let us introduce the elementary cycles

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: 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: By Proposition 2.4, the ring homomorphism is a surjection, and it's kernel is generated as an ideal by the elements 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: is proven for all m < n and all 0 < k < m, and (2) m and (3) m are proven for all m < n.
has the section s (given by the rule: 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 )) = 0, as well as the conditions (1) n,k and (2) n , we can rewrite the last expression as: Since g * n is injective (the map g n is birational), we get: The statement (3) n is proven.

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 u the image of u under the map CH * → CH * /2. , Proof: Consider the diagram: Then it easily follows by the induction on n, that z 2 k = z 2k (where we assume z r = 0 if r > n).Thus, we have surjective ring homomorphism 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∈Bγ 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 Λ := { i∈Λ β i} β∈B .We call the specialization Λ good if Λ ⊂ {1, . . ., n}.Suppose I be some finite set of natural numbers.Let us define the element z I ∈ CH * (G(n))/2 by the formula: where we assume z r = 0, if r > n, and the sum is taken over all simple specializations Λ of the multisubset I = i∈I {i}.Actually, 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 z I form basis (Proposition 3.4(1)).Proof: If I contains an element r > n, then 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: , where the sum is taken over all simple specializations of the multisubset I\{i 1 , i 2 }.Since z 2 i = z 2i , this expression is zero.
We immediately get the (modulo 2) version of the Pieri formula proved by H.Hiller and B.Boe: where we omit terms z J with J ⊂ {1, . . ., n} (in particular, if J contains some element with multiplicity > 1).
Proof: z I∪j = Λ l∈Λ 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 z I • z j , and the sum over the specializations of the type (2, i) is equal to 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.(2) i∈I z i = Λ z 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 z i we have lexicographical filtration.Consider the linear map ε CH * (G(n))/2 sending i∈I z i to z I .Then the associated graded map: gr(ε) is the identity.Thus, ε is invertible, and the set {z I } I⊂{1,...,n} form a basis.
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: given by the rule: w 1 (x Λ ) := z Λ , and w 2 (y Λ ) := j∈Λ z j .Then, by the definition of z I , w 1 = w 2 • ϕ.Then w 2 = w 1 • ψ, which implies that i∈I z i = Λ z 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) z i • z j = z i,j + z i+j , where the first term is omitted if i = j and the second if

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 z i .For convenience, let us put z j ∈ CH j (G(m)) to be zero for j > m.
Theorem 4.1 Proof: Use induction on n.The base is trivial.Suppose the statement is true for (n − 1).Since c(V n )(t) = t n , we have: ρ n+1 = 0.Then, by Theorem 2.5 and the assumption above, g * n (z j ) = ρ j + π * n−1 (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: Now, the statement follows from the injectivity of g * n .

Main theorem
Let X be some variety over the field k.We will denote: 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| k = G(n).In this section we will show that, as an algebra, C * (G(n, Q)) is generated by the elementary cycles z i contained in it.
Let F (n, Q) be the variety of complete flags (l 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: . 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 ⊕ n i=1 O. Thus, modulo 2, the Chow ring of We get: Theorem 5.1 There is a ring isomorphism 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: Notice, that the change of scalar map CH . Thus, we have: 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 h(j) (and h(1) as h).For arbitrary set of variables u = {u 1 , . . ., u r } let us define the degree i polynomials σ i (u) and σ −i (u) from the equation: Since σ −i (h(i)) is the ±-monic polynomial in h i with coefficients in the subring, generated by h(i + 1), we get: CH * (F n ) is a free module over the subring the natural projection between full flag varieties.We will denote by the same symbol z I the images of z I in CH * (F (n, Q))/2.The following statement is the key for the Main Theorem.
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 : Proof: Consider variety Y × X Y with the natural projections π 1 and π 2 on the first and second factor, respectively.Then divisor ∆ , we have: , where the projection P Y (U ) → Y is given by π 2 and c(U -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.
Let us compute the action of φ on basis elements 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: 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. A. Vishik . Since φ is a ring homomorphism, we get: (as usually, one should omit 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 is Cartesian, and since π ′ is smooth, we have an equality: * .It follows from Lemmas 5.5 and 5.7 that Thus, modulo the kernel of multiplication by  Let us prove by induction on the size of z, that z belongs to the subring of C * (G(n, Q)) generated by z j 's.The base of induction, s = 1 is trivial.Notice that m( i∈I z i ) = z I .Thus, the size of z ′ = z − a:s(Ia)=s(z) i∈Ia z i is smaller than that of z.But by the Lemma 5.9, all the z i 's appearing in this expression belong to C * (G(n, Q)).By the inductive assumption, z ′ belongs to the subring of C * (G(n, Q)) generated by z j 's.Then so is 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
)), Proof: It immediately follows from the Main Theorem 5.8, Proposition 3.1, and the fact that z 2 s = z 2s (or 0, if 2s > n).

Now we can introduce:
Documenta Mathematica 10 (2005) 111-130 = 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: .
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: On the other hand, from the projection P F (r) (W r ) θr → F r , we have sequences: (see [4,Example 3.2.11]).It remains to notice, that in K ), to get the equality: The statement now easily follows by induction on r.Now, it is easy to compute the characteristic classes of the quadratic Grassmannians.
, where h j are the roots of the tautological vector bundle V r on G(r).

Documenta Mathematica 10 (2005) 111-130
A. Vishik 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: and the statement follows.Now we can prove the following Conjecture of G.Berhuy (proven by him for n 4): 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 2 i = 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 Let A := Z[σ 1 , . . ., σn ].We have ring homomorphism ψ : A → CH * (G(n)) sending σi 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) ( 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 ).
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: 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 ), Documenta Mathematica 10 (2005) 111-130 A. Vishik 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,

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.

Documenta Mathematica 10 (
2005) 111-130Notice, 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 z i contained in it.Proof:Let z be an element of C * (G(n, Q)).It can be expressed as a linear combination of the basis elements z I 's.Let us define the size s(z) of the element z = z Ia as the maximum of sizes of I a involved.Let m(z) be the main term of z, that is, a:s(Ia)=s(z) z Ia .Lemma 5.9 Let z = a z Ia ∈ C * (G(n, Q)).Let s(I a ) = s(z), and i ∈ I a .Then the elementary cycle 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(z) = 1 j n d j (z) • D(z j ), where d j (z) ∈ CH * (G(n, Q)| k )/2, and the elements D(z j ) ∈ CH * (F n )/2 are linearly independent.Since D is defined over the base field, D(z) ∈ C * (F (n, Q)), and, by the Statement 5.2, d j (z) ∈ C * (G(n, Q)).Clearly, m(d j (z)) = d j (m(z)).It is easy to see that d js . . .d j2 (z) = z i , since for arbitrary I b with s(I b ) < s = s(z) we have: d js . . .d j2 (z I b ) = 0, or 1, and for I c = I a with s(I c ) = s, d js . . .d j2 (z Ic ) is either 0, or has degree different from i. Thus, z i ∈ C * (G(n, Q)).