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Rationality on integral Chow groups – Main version
In this section we continue to use notation introduced in the previous sections and we deal with Question I.0.1 still in the context of smooth projective quadrics but for integral Chow groups CH. In the aftermath of the Main Tool Lemma, A. Vishik adressed similar questions for integral Chow groups CH instead of Chow groups modulo 2. Namely, he proved the following integral version of the Main Tool Lemma (see [51, Theorem 3.1]).
Theorem III.4.1 (Vishik). Let Y be a smooth quasi-projective variety over a eld F of characteristic 0 and let Q be a smooth projective quadric with i1(Q) > 1. Then any F(Q)- rational element y 2 CHm(Y ) , with m < dim(Q)=2, is rational.
In the above statement, the assumption that the rst Witt index i1(Q) of Q is strictly greater than 1 means that Q has a projective line dened over the generic point of Q (such quadrics are quite widespread). Once again, the use of symmetric operations in the algebraic cobordism theory forced
A. Vishik to work with a smooth quasi-projective variety Y over a eld of characteristic 0. However, we proved a similar result using only Chow theory itself, which allows one to get a valid statement in any characteristic dierent from 2 (since Chow theory does not rely on resolution of singularities) and to get rid of the assumption of quasi-projectivity for Y (see [9, Theorem 3.1]).
Theorem III.4.2. Let Y be a smooth variety over a eld F of characteristic dierent from 2 and let Q be a smooth projective quadric with i1(Q) > 1. Then any F(Q)-rational element y 2 CHm(Y ) , with m < dim(Q)=2, is the sum of a rational and an exponent 2 element. Once again, the version of A. Vishik remains stronger in the sense that his use of symmetric operations in the algebraic cobordism theory allowed him to get rid of the exponent 2 element appearing in our conclusion. The main idea of the proof of Theorem III.4.2 (inspired by the proof of Theorem III.4.1) is as follows. First of all, we consider the F(Q)-rational element y 2 CHm(Y ) as the coordinate on h0 of a rational cycles x 2 CH m (Q Y ), and we use x mod 2, the 1-primordial cycle in Ch(QQ) and the Steenrod operations on Chow groups modulo 2 to form \special cycles ».
Then we choose carefully some integral representatives of these special cycles and we obtain y as a specic linear combination of rational cycles (modulo 2-torsion). Most of material needed for the proof can be found Chapter XIII and Chapter XV of the book [7].
Remark III.4.3. Let Q be a smooth projective quadric over F of positive dimension (in that case, Q is geometrically integral) given by a quadratic form ‘. Since for isotropic Q, any F(Q)-rational element (in any codimension) is rational, one can make the assumption that the quadric Q is anisotropic in order to prove Theorem III.4.2. In particular, Q is not completely split and one can consider the rst Witt index i1(‘) of ‘, which we simply denote as i1.
Rationality on integral Chow groups – A stronger version
In this section, we continue to use notation introduced in the previous section. The following result is stronger than Theorem III.4.2 although its statement is less eloquent. Let K=F be an extension and X be an F-variety. In the following proof, an element x 2 CH(XK) is called rational if it is in the image of the change of eld homomorphism CH(X) ! CH(XK).
In the same way as before, the following theorem is a generalization of [51, Proposition 3.7] to any eld of characteristic dierent from 2 (although, putting aside characteristic.
Theorem III.5.1 is still weaker than the original version in the sense that an exponent 2 element appears in the conclusion).
Theorem III.5.1. Assume that m < n=2 and i1 > 1, and let E=F be an extension such that i0(QE) > m. Then, for any y 2 CHm(YF(Q)) there exists 2 CHm(Y ) and an exponent 2 element 2 CHm(YE(Q)) such that yE(Q) = E(Q) + .
Proof. We proceed the same way as in the proof of Theorem III.4.2. Let us x an element x 2 CHm(Q Y ) mapped to y under the surjection CHm(Q Y ) CHm(YF(Q)).
Filtrations on Grothendieck ring of projective homogeneous varieties
In this section, we prove two statements concerning ltrations on Grothendieck ring of certain class of projective homogeneous varieties. Those propositions play a crucial role in the proof of Theorem IV.0.1 (see Section IV.4). We use notation introduced in Section II.4.
We have seen in Section II.4 that for any any smooth variety X over a eld F and for any integer i 0, the term i(X) of codimension i of the -ltration on the Grothendieck ring K(X) is contained in the the term i(X) of codimension i of the topological ltration.
The following proposition provides us a way to get the existence of a variety X for which the two ltrations actually coincide when dealing with a certain class of projective homogeneous varieties. The method of proof is largely inspired by the proof of [24, Theorem 6.4 (2)] by N. Karpenko and A. Merkurjev.
Proposition IV.1.1. Let G0 be a split connected semisimple linear algebraic group over a eld F and let B be a Borel subgroup of G0. There exist an extension E=F and a cocycle 2 H1(E;G0) such that the topological ltration and the -ltration on K((G0=B)) coincide.
Proof. Let n be an integer such that G0 GLn and let us set S := GLn and E := F(S=G0). We denote by T the E-variety S S=G0 Spec(E) given by the generic ber of the projection S ! S=G0. Note that since T is clearly a G0-torsor over E, there exists a cocycle 2 H1(E;G0) such that the smooth projective variety X := T=BE is isomorphic to (G0=B). We claim that the Chow ring CH(X) is generated by Chern classes.
Indeed, the morphism h : X ! S=B induced by the canonical G0-equivariant morphism T ! S being a localization, the associated pull-back h : CH(S=B) ! CH(X).
Generically split projective homogeneous varieties
In this section, we present a motivic decomposition result due to V.Petrov, N. Semenov and K. Zainoulline (see [40, Theorem 5.17]) and we introduce in a more general context the basis of the method we will use in Section IV.4 to prove Theorem IV.0.1.
Let X be a projective homogeneous variety under an algebraic group G over a eld F. The variety X is said to be generically split if the group G splits over the generic point of X (e.g any projective homogeneous variety X under a group G of type F4 or E8 which has no splitting extension of degree coprime to 3 or 5 respectively).
Assume furthermore that G is semisimple, then such a generically split G-variety X presents the interest that for any prime p, its Chow motive M(X; Z=pZ) with coecients in Z=pZ decomposes as a sum of twists of an indecomposable motive Rp(G), called Rost motive, by mean of the following theorem.
Theorem IV.2.1 (Petrov, Semenov, Zainoulline). Let G be a semisimple linear algebraic group over a eld F and let p be a prime. Then for any generically split projective homoge- neous variety X under G one has the motivic decomposition M(X; Z=pZ) ‘ M i0 Rp(G)(i)ai .
Table of contents :
I Introduction
II Basic material
II.1 Denition and basic properties of Chow groups
II.2 Further properties of Chow groups
II.3 Steenrod operations on Chow groups modulo 2
II.4 Grothendieck rings
III Quadrics
III.1 Decomposition on Chow groups of projective quadrics
III.2 Rationality on Chow groups modulo 2 – Main result
III.3 Rationality on Chow groups modulo 2 – Other results
III.4 Rationality on integral Chow groups – Main version
III.5 Rationality on integral Chow groups – A stronger version
IV Exceptional projective homogeneous varieties
IV.1 Filtrations on Grothendieck ring of projective homogeneous varieties
IV.2 Generically split projective homogeneous varieties
IV.3 J-invariant
IV.4 Proof of the result
V Principal homogeneous space for SL1(A)
V.1 Preliminaries
V.2 Proof of the result
V.3 Link with Chapter IV – Exceptional projective homogeneous varieties
VI Special correspondences
VI.1 A conjecture of A. Vishik
VI.2 Rationality of special correspondences on quadrics
VI.3 A partial answer to the conjecture
VI.4 Special correspondences on A-trivial varieties
A Milnor K-Theory
B Chern classes
C Correspondences on Chow groups
D Torsors of algebraic groups
Bibliography
