Special correspondences and Chow traces of Landweber-Novikov operations

K. Zainoulline

August 16, 2007

1 Introduction

In paper [Vi07] A. Vishik using the techniques of symmetric operations in algebraic cobordism proved that changing the base field by the function field of a smooth projective quadric doesn’t change the property of being rational for cycles of small codimension. This fact which he calls Main Tool Lemma plays the crucial role in his construction of fields with u-invariant 2^{r} + 1.

In the present paper we prove M.T.L. for the class of varieties introduced by M. Rost in the context of the proof of Bloch-Kato conjecture. Namely, for varieties which possess a special correspondence (see [Ro06, Definition 5.1]). As in the Vishik’s proof the main technical tools are algebraic cobordism of M. Levine and F. Morel [LM07], generalized Rost degree formula and divisibility of Chow traces of certain Landweber-Novikov operations. Therefore, we always assume that our base field k has characteristic 0.

We use the following notation. By \overline{k} we denote the algebraic closure of k and by X\overline{k} the respective base change X ×_{k} \overline{k} of a variety X. By K we denote the function field of X. Given a prime p by \overline{Ch}^{∗}(X) we denote the Chow ring of X modulo p-torsion taken with ℤ/p-coefficients, i.e., \overline{Ch}^{∗}(X) = \overline{CH}^{∗}(X)/p, where \overline{CH}^{∗}(X) = CH^{∗}(X)/(p-tors). We say that a cycle from \overline{Ch}^{∗}(X\overline{k}) is defined over k if it belongs to the image of the restriction map \overline{Ch}^{∗}(X) → \overline{Ch}^{∗}(X\overline{k}).

The following notion will be central in this paper

1.1 Definition. Let X be a smooth proper irreducible variety over a field k of dimension n and p be a prime. Assume that X has no zero-cycles of degree coprime to p. Let d be an integer 0 ≤ d ≤ n. We say X is a d-splitting variety mod p if for any smooth quasi-projective variety Y over k and all integers m satisfying 0 ≤ m < d, the following condition holds

A cycle y ∈ \overline{Ch}^{m}(Y\overline{k}) is defined over k ⟺ y ×\overline{k} \overline{K} is defined over K (1)

1.2 Example. According to the results of A. Vishik

(a) any smooth projective quadric Q of dimension n is a [\frac{n+1}{2}]-splitting variety mod 2 (see [Vi07, Cor. 3.5.(1)]);

(b) a quadric Q is a n-splitting variety (mod 2) if and only if Q possesses a Rost projector (see [Vi07, p. 373]).

The main result of the paper is the following

1.3 Theorem. Let X be a smooth proper irreducible variety over a field of characteristik 0. Assume that X has no zero-cycles of degree coprime to p. If X possesses a special correspondence in the sense of Rost, then X is a \frac{n}{p−1}-splitting variety and the value \frac{n}{p−1} is optimal.

As an application of the techniques used in the proof of 1.3, we provide a complete list of d-splitting projective homogeneous varieties of type F_{4}.

To do this we introduce the following notation. Let G be a simple linear algebraic group over k. We say a projective homogeneous G-variety X is of type D, if the group G\overline{k} has root system of type D. Moreover, if X\overline{k} is the variety of parabolic subgroups of G\overline{k} defined by the subset of simple roots S of D, then we say that X is of type D/P_{S}. In this notation P_{D} defines a Borel subgroup and P_{i}, i ∈ D, defines a maximal parabolic subgroup.

1.4 Corollary. Let X be a projective homogeneous variety of type F_{4} and p be one of its torsion primes (2 or 3). Assume that X has no zero-cycles of degree coprime to p. Then depending on p we have

p = 2: If X is of type F_{4}/P_{4}, then X is a (dim X)-splitting variety. For all other types X is a 3-splitting variety and this value is optimal.

p = 3: X is always a 4-splitting variety and this value is optimal.

2 Mod-p operations

In the present section we introduce certain operations

ϕ_{p}^{q(t)} : Ω^{∗} → \overline{Ch}^{∗} parametrized by q(t) ∈ \overline{Ch}^{∗}[[t]]

from the ring of algebraic cobordism Ω^{∗} of Levine-Morel to the Chow group modulo p-torsion \overline{Ch}^{∗}, where p is a given prime. All the facts used here can be found in [LM07], [Me03] and [Vi06].

Let \overline{CH}^{∗} and \overline{Ch}^{∗} denote the respective Chow groups modulo p-torsion. Consider the commutative diagram

\xymatrix{Ω^{m} \ar[r]^{S_{LN}^{k(p)}} \ar[d]^{pr} & Ω^{m+(p−1)k} \ar[d]^{pr} \\ _{\overline{CH}_{m}} \ar[d] & _{\overline{CH}_{m+(p−1)k}} \ar[d] \\ _{\overline{Ch}_{m}} \ar[r]^{S^{k}} & \overline{Ch}^{m+(p−1)k},}

where S_{LN}^{k(p)} is the Landweber-Novikov operation corresponding to the partition k(p) = _{\underbrace{(p − 1, p − 1, …, p − 1)}_{k−times}}, pr: Ω^{∗} → CH^{∗} → \overline{CH}^{∗} is the natural surjection

and S^{k} is the reduced p-th power operation. By the properties of reduced power operations, S^{k} = 0 if k > m.

Define (cf. [Vi07, Prop. 2.1])

ϕ_{p}^{t^{(p−1)a}} = \frac{pr ∘ S_{LN}^{k(p)}}{p} , where a = k − m.

By commutativity of the diagram ϕ_{p}^{t^{(p−1)a}} is well-defined. If r is not divisible by (p−1), then pr∘S_{LN}^{k(r)} = 0 and we set ϕ_{p}^{t^{r}} = 0. Hence, we have constructed an operation ϕ_{p}^{t^{r}}, r > 0, which maps Ω^{m} to \overline{Ch}^{r+pm}.

Finally, given a power series q(t) ∈ \overline{Ch}^{∗}[[t]] we define

ϕ_{p}^{q(t)} = ∑_{r≥0} q_{r}ϕ_{p}^{t^{r}}, where q(t) = ∑_{r≥0} q_{r}t^{r}.

By definition the operations ϕ_{p}^{q(t)} are additive and respect pull-backs.

Let E be a vector bundle over a smooth variety U, Consider its total Chern class c(E) = ∏_{j}(t + ξ_{j}), where ξ_{j} are the roots. Define c(E)^{(p)}(t) = ∏_{j}(t^{p−1} + ξ_{j}^{p−1}). Let E = T_{U} be the tangent bundle of rank l. Set b_{l}(U) = deg c(−T_{U})^{(p)}. According to [Me03, §6] this number is always divisible by p. Moreover, it is trivial if l is not divisible by (p − 1). We define the Rost number η_{p} as

η_{p}(U) = \frac{b_{l}(U)}{p} ∈ ℤ/p.

We will extensively use the following property of the operation ϕ_{p} which follows from the multiplicativity of Landweber-Novikov operations

2.1 Lemma. (cf. [Vi07, Prop.2.3]) Let U be a smooth projective variety of positive dimension l and [U] be its class in the Lazard ring 𝕃. Let β ∈ Ω^{j}(X). Then

ϕ_{p}^{t^{r}}([U] · β) = η_{p}(U) · S^{k}(pr(β)), where r = (p − 1)(k − j) + lp > 0.

Observe that ϕ_{p}^{t^{r}}([U] · β) = 0 if r is not divisible by (p − 1).

3 Construction of a cycle defined over k

In the present section we prove Theorem 1.3.

I. We proceed following the proof of [Vi07, Thm. 3.1]. Let Y be a smooth projective variety over k. Let y ∈ \overline{Ch}^{m}(Y\overline{k}) be such that y\overline{K} is defined over K. We want to show that y is defined over k for all m < d. Let Ω^{∗} be the algebraic cobordism of Levine-Morel. There is a natural surjection pr: Ω^{∗} → \overline{Ch}^{∗}. Consider the commutative diagram

\xymatrix{ω \ar@{|->}[d] & Ω^{m}(X × Y) \ar[d]_{res} \ar@{->>}[r]^{pr} & \overline{Ch}^{m}(X × Y) \ar@{->>}[r]^{p^{∗}} \ar[d]^{res} & \overline{Ch}^{m}(Y_{K}) \ar[d]^{res} & u \ar@{|->}[d] \\ ω̄ & Ω^{m}(X\overline{k} × Y\overline{k}) \ar@{->>}[r]^{pr} & \overline{Ch}^{m}(X\overline{k} × Y\overline{k}) \ar@{->>}[r]^{p̄^{∗}} & \overline{Ch}^{m}(Y\overline{K}) & y\overline{K}}

where p: Spec(K) × Y → X × Y and the pull-back p^{∗} is surjective because of the localization sequence for Chow groups. By the hypothesis there exists a preimage u of y\overline{K} by means of res. By surjectivity of pr and p^{∗}, there exists a preimage ω of u. Let ω̄ = res(ω).

II. Let X be a variety which possesses a special correspondence. By the results of [Ro06] this implies that

(a) n = p^{s} − 1,

(b) η_{p}(X\overline{k}) ≠ 0 mod p

(c) the Chow motive of X contains an indecomposable summand M which over \overline{k} splits as a direct sum of Tate motives shifted by the multiples of d = \frac{p^{s}−1}{p−1}

M\overline{k} ≃ ⨁_{i=0}^{p−1} ℤ/p\{di\}.

Let π be an idempotent defining M. We can choose ω in such a way that the realization ρ = π_{∗}(ω̄) ∈ Ω^{m}(X\overline{k} × Y\overline{k}) will have the following form (cf. [Vi07, p.368])

ρ = x_{n} × y_{n} + ∑_{i} x_{i} × y_{i} + x_{0} × y_{0},  (2)

where the sum in the middle is taken over all i ∈ {d, 2d, …, (p − 2)d}, x_{i} ∈ Ω_{i}(X\overline{k}), y_{i} ∈ Ω^{m−n+i}(Y\overline{k}) are certain cobordism classes, x_{0} is the class of a rational point, x_{n} = π_{∗}(1) is the class given by a proper birational morphism X̂ → X\overline{k} and pr(y_{n}) = y (cf. [Vi07, Lemma 3.2]).

III. Consider the cycle ϕ_{p}^{t^{r}}(p_{Y ∗}(ρ)) ∈ \overline{Ch}^{∗}(Y\overline{k}) defined over k. It has codimension m, if r = (dp − m)(p − 1) > 0.

3.1 Lemma. ϕ_{p}^{t^{r}}(p_{Y ∗}(ρ)) = η_{p}(X\overline{k}) · y + ϕ_{p}^{t^{r}}(y_{0}).

Proof. Applying Lemma 2.1 to each summand x_{i} × y_{i} in the middle of (2) we obtain

ϕ_{p}^{t^{r}}(p_{Y ∗}(x_{i} × y_{i})) = η_{p}(x_{i}) · S^{(n−i)/(p−1)}(pr(y_{i})).

Since pr(y_{i}) ∈ \overline{Ch}^{m−n+i}(Y\overline{k}) and m−n+i ≤ m−n+(p−2)d = m−d < 0, it must be trivial. Hence, only the very right and left summands of (2) survive.

By Rost Degree Formula [LM07, Cor. 13.8] we have

x_{n} = 1 + ∑_{u_{Z}∈𝕃_{>0}} u_{Z} · [Z → X\overline{k}].  (3)

Since π is an idempotent, π_{∗} ∘ π_{∗}(1) = π_{∗}(1). The latter implies that

π_{∗}(∑ u_{Z} · [Z → X\overline{k}]) = ∑ u_{Z} · π_{∗}([Z → X\overline{k}]) = 0.

Hence, applying η_{p} we get

0 = ∑_{u_{Z}∈𝕃_{>0}} η_{p}(u_{Z} · π_{∗}([Z → X\overline{k}])) = u_{pt} + ∑_{u_{Z}∈𝕃_{<n}} u_{Z} · π_{∗}([Z → X\overline{k}]).

Since η_{p}(u_{Z} · π_{∗}([Z → X\overline{k}])) is divisible by p if u_{Z} ∈ 𝕃_{<n}, we obtain that η_{p}(u_{pt}) = 0 mod p.

Therefore, for the very left summand of (2) we get

ϕ_{p}^{t^{r}}(p_{Y ∗}(x_{n} × y_{n})) = η_{p}([X\overline{k}] + ∑_{u_{Z}∈𝕃_{<n}} u_{Z} · [Z] + u_{pt}) · pr(y_{n}) = η_{p}(X\overline{k}) · y.

Observe that η_{p}(u_{Z} · [Z]) is divisible by p if u_{Z} ∈ 𝕃_{<n}.

Consider the cycle p_{Y ∗}(ϕ_{p}^{t^{r′}}(ρ)) ∈ \overline{Ch}^{∗}(Y\overline{k}) defined over k. It has codimension m, if r′ = (d − m)(p − 1) > 0.

3.2 Lemma. p_{Y ∗}(ϕ_{p}^{t^{r′}}(ρ)) = ϕ^{t^{r}}(y_{0}).

Proof. By definition p_{Y ∗}(ϕ_{p}^{t^{r′}}(x_{i} × y_{i})) = p_{Y ∗}(\frac{1}{p} · pr(S_{LN}^{d(p)}(x_{i} × y_{i}))). By multiplicativity of S_{LN}^{∗} the latter can be written as

p_{Y ∗}(\frac{1}{p} · pr(S_{LN}^{d(p)}(x_{i} × y_{i}))) = p_{Y ∗}(\frac{1}{p} · ∑_{α+β=n} pr(S_{LN}^{α}(x_{i})) × pr(S_{LN}^{β}(y_{i}))) =

= ∑_{α+β=d(p)} \frac{1}{p} · deg (pr(S_{LN}^{α}(x_{i}))) · pr(S_{LN}^{β}(y_{i})) =

Since m < d, y_{i} has negative codimension for all i < n, therefore, pr(S_{LN}^{β}(y_{i})) is divisible by p for all i < n.

On the other hand, we have S^{∗} = S_{∗} · c(−T_{X\overline{k}})^{(p)}, where S_{∗} (resp. S^{∗}) are the (co-)homological operations. Hence,

deg (pr(S_{LN}^{α}(x_{i}))) = deg (S_{∗}(pr(x_{i})) · c(−T_{X\overline{k}})^{(p)}).

Since all Chern classes of the tangent bundle of X\overline{k} are defined over k, and X possesses a special correspondence, according to [Ro06, Lemma 9.3] we obtain that deg (S_{∗}(pr(x_{i}))·c(−T_{X\overline{k}})^{(p)}) is congruent to deg(S_{∗}(pr(x_{i}))) mod p. Since S_{∗} respect push-forwards, deg(S_{∗}(pr(x_{i}))) is trivial mod p for all i > 0. Hence, deg (pr(S_{LN}^{α}(x_{i}))) is divisible by p for all i > 0.

Combining all together we get

p_{Y ∗}(ϕ_{p}^{t^{r′}}(ρ)) = ∑_{|α|=n} \frac{1}{p} deg (pr(S_{LN}^{α}(x_{n})))y + ϕ_{p}^{t^{r}}(y_{0}) = ϕ_{p}^{t^{r}}(y_{0}),

since S_{LN}^{α}(x_{n}) = 0.

By Lemmas 3.1 and 3.2 the following cycle is defined over k

ϕ_{p}^{t^{r}}(p_{Y ∗}(ρ)) − p_{Y_{∗}}(ϕ_{p}^{t^{r′}}(ρ)) = η_{p}(X\overline{k}) · y.

Since η_{p}(X\overline{k}) ≠ 0 mod p, the cycle y is defined over k, therefore, X is a d-splitting variety.

To see that d = \frac{n}{p−1} is an optimal value take Y = X and consider the

cycle y ∈ \overline{Ch}^{d}(X\overline{k}) defining the Tate motive ℤ/p\{n−d\} in the decomposition of the Rost motive M over \overline{k}. Since M splits over K, y\overline{K} is defined over K. By condition (1) it implies that y is defined over k, i.e., M splits over k which contradicts to the indecomposability of M. The theorem is proven.

4 F_{4}-varieties

In the present section we apply our methods to describe all d-splitting varieties of type F_{4}.

I. Let X be a smooth proper irreducible variety over a field k. As in the beginning of the proof of 1.3 given a cycle y ∈ \overline{Ch}^{m}(Y\overline{k}) we construct a cobordism class ω̄ ∈ Ω^{m}(X\overline{k} × Y\overline{k}) defined over k.

Assume that the motive of X contains a motive M = (X, π) such that the idempotent π can be written as

π\overline{k} = γ × γ^{∗} + ∑_{i} x_{i} × x_{i}^{∗},

where γ ∈ Ω_{(p−1)d}(X\overline{k}) is defined over k, γ^{∗} denotes its Poincare dual, i.e., pr(γ · γ^{∗}) = pt and x_{i} ∈ Ω_{i}(X\overline{k}), i ∈ \{0, d, 2d, …, (p − 2)d\}.

Then the realization ρ = π_{∗}(p_{X}^{∗}(γ) · ω̄) ∈ Ω^{m+n−g}(X\overline{k} × Y\overline{k}), g = (p − 1)d, can be written as (cf. (2))

ρ = γ × y_{g} + ∑_{i} x_{i} × y_{i} + x_{0} × y_{0},

where x_{i} ∈ Ω_{i}(X\overline{k}), y_{i} ∈ Ω^{m−g+i}(Y\overline{k}), i ∈ \{d, 2d, …, (p − 2)d\}.

As in the proof of Lemma (3.1) we obtain that for r = (dp−m)(p−1) > 0 and any m < d the following cycle in \overline{Ch}^{m}(Y\overline{k}) is defined over k

ϕ_{p}^{t^{r}}(p_{Y ∗}(ρ)) = η_{p}(γ) · y + ϕ_{p}^{t^{r}}(y_{0}).

The transposed cycle π^{t} defines a direct summand M^{t} = (X, π^{t}) of the motive of X (the one which over \overline{k} contains the generic point of X). The realization ρ′ = π_{∗}^{t}(ω̄) ∈ Ω^{m}(X\overline{k} × Y\overline{k}) can be written as

ρ′ = ∑_{i} x^{(i)} × y^{(i)} + γ^{∗} × y^{(g)},

where x^{(i)} ∈ Ω^{i}(X\overline{k}), i ∈ \{0, d, 2d, …, (p − 2)d\}, y^{(g)} = y_{0}.

Consider the cycle δ = p_{Y ∗}(p_{X}^{∗}(γ) · ϕ_{p}^{t^{r′}}(ρ′)). It is defined over k and for r′ = (d − m)(p − 1) > 0 belongs to \overline{Ch}^{m}(Y\overline{k}). Assume that δ = ϕ_{p}^{t^{r}}(y_{0}), then subtracting it from ϕ_{p}^{t^{r}}(p_{Y ∗}(ρ)) we obtain the cycle η_{p}(γ) · y. Hence, to prove that X is a d-splitting variety it is enough to show that η_{p}(γ) ≠ 0 mod p and δ = ϕ_{p}^{t^{r}}(y_{0}).

II. Observe that for p = 2 we have δ = ϕ_{p}^{t^{r}}(y_{0}) by simple dimension reasons. Indeed, in this case the cycle ρ′ consists only of two terms

ρ′ = x^{(0)} × y^{(0)} + γ^{∗} × y^{(g)},

where the first summand vanishes in δ, since S_{LN}^{α}(x^{(0)}) = 0 if |α| > 0 and the second summand gives the required cycle ϕ_{p}^{t^{r}}(y_{0}).

For p = 3 the cycle ρ′ consists of three terms

ρ′ = x^{(0)} × y^{(0)} + x^{(d)} × y^{(d)} + γ^{∗} × y^{(g)},

where again the first summand vanishes, the last gives ϕ^{t^{r}}(y_{0}) and the middle gives

∑_{α+β=d} deg (pr(γ · S_{LN}^{α}(x^{(d)}))) · \frac{1}{p}pr(S_{LN}^{β}(y^{(d)})), where |α| = \frac{p−2}{p−1}d. (4)

III. Given an F_{4}-variety X we provide a cycle γ satisfying the conditions above as follows

p = 2: If X is generically split, then we may assume that X is of type F_{4}/P_{1}. By the main result of [PSZ] the cycle

γ = H · c_{4}(T_{X\overline{k}}) · c_{7}(T_{X\overline{k}}),

where H is a generator of the Picard group of X\overline{k} defined over k, is the generic point of a Rost motive sitting inside the motive of X. Observe that [γ] = [Q_{3}] is represented by the class of a 3-dimensional quadric. Hence, η_{2}(Q_{3}) = 1 mod 2 and X is a d-splitting variety with d = 3.

If X is not generically split, i.e., X is of the type F_{4}/P_{4}, then M(X) contains a Rost motive with γ = [X], i.e., X is a variety which possesses a special correspondence. Indeed, one can show that X is a norm variety corresponding to the cohomological invariant f_{5}. Hence, X is a n-splitting variety (here n = 15).

p = 3 In this case all F_{4}-varieties are generically split. Hence, by the results of [PSZ] the motive of X contains a generalized Rost motive with the generic point given by the cycle

γ = H^{7}.

The direct computation using the Adjunction formula shows that η_{3}([H^{7}]) ≠ 0 mod 3. Moreover, in this case the expression (4) will have the following form

deg (γS^{2}(pr(x^{(4)}))) · ϕ_{p}^{t^{12−2m}}(y^{(4)}).

which is equal to zero since S^{2}(pr(x^{(4)})) is equal to zero. Hence, X is a d-splitting variety with d = 4.

4.1 Remark. Let X be a d-splitting geometrically cellular variety. As an immediate consequence of the definition of a canonical p-dimension (see [KM06]) we obtain the following inequality

cd_{p}(X) ≥ d.

In the case of a variety of type F_{4}/P_{4} it gives cd_{2}(X) = dim X = 15.

Acknowledgements I am very grateful to Alexander Vishik and Fabien Morel for various discussions on the subject of this paper. This work was partially supported by SFB 701 Bielefeld and INTAS 05-1000008-8118.

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