eingeben 1

Remarks on absolute de Rham and absolute Hodge cycles

Hélène Esnault^{1)} Kapil H. Paranjape^{2)}

1)Universität - GH - Essen, Fachbereich 6, Mathematik D-45117 Essen , Germany 2)Tata Institute of Fundamental Research Homi Bhabha Road, Bombay 400 005, India

Let X be a smooth proper variety over a field k of characteristic zero. For any embedding σ of k into the field of complex numbers ℂ, the ℂ valued points of X ⊗_{σ} ℂ form a complex manifold denoted by X_{σ}. By base change for the de Rham cohomology H_{DR}^{j}(X/k) ⊗_{σ} ℂ = H_{DR}^{j}(X ⊗_{σ} ℂ/ℂ) and by the GAGA principle one has an isomorphism I_{σ} from H_{DR}^{j}(X/k) ⊗_{σ} ℂ to the Betti cohomology H_{B}^{j}(X_{σ}, ℂ) ([5], p. 96).

An element of the ℚ Chow group CH^{i}(X) ⊗_{ℤ} ℚ has a de Rham class

α ∈ F^{i}H_{DR}^{2i}(X/k) = ℍ^{2i}(X, Ω_{X/k}^{≥i}) ⊂ H_{DR}^{2i}(X/k)

such that for all embeddings σ : k → ℂ

I_{σ}(α) ∈ I_{σ}(F^{i}H_{DR}^{2i}(X/k) ⊗_{σ} ℂ) ∩ H_{B}^{2i}(X_{σ}, ℚ).

So α is an absolute Hodge cycle, a notion defined by Deligne [3], §2, which we slightly modify, as we are only interested here in de Rham cohomology (see [3], open question 2.2).

Definition 1 A class α ∈ F^{i}H_{DR}^{2i}(X/k) is said to be an absolute Hodge cycle if for all embeddings σ : k → ℂ, I_{σ}(α) lies in H_{B}^{2i}(X_{σ}, ℚ).

On the other hand, such an algebraic cycle has an absolute de Rham class in ℍ^{2i}(X, Ω_{X/ℚ}^{≥i}). In fact, there is an absolute differential

d log : O_{X}^{∗} ⟶ Ω_{X/ℚ}^{≥1} [1]

inducing an absolute differential

d log : K_{i}^{M} ⟶ Ω_{X/ℚ}^{≥i} [i]

where K_{i}^{M} is the Zariski sheaf of Milnor K theory. As CH^{i}(X) ⊗_{ℤ} ℚ = H^{i}(X, K_{i}^{M}) ([9], théorème 5), d log induces the absolute de Rham cycle class map

CH^{i}(X) ⊗_{ℤ} ℚ ⟶^{ψ} ℍ^{2i}(X, Ω_{X/ℚ}^{≥i}).

One composes this map with

ℍ^{2i}(X, Ω_{X/ℚ}^{≥i}) ⟶ ℍ^{2i}(X, Ω_{X/k}^{≥i}) = F^{i}H_{DR}^{2i}(X/k)

to obtain the de Rham cycle class map. As we don’t have a reference for this, we indicate how to prove it. By base change F^{i}H_{DR}^{2i}(X/k) ⊗_{σ} ℂ = F^{i}H_{DR}^{2i}(X ⊗_{σ} ℂ/ℂ), so it is enough to handle k = ℂ, in which case the compatibility is proven in [2], (2.2.5.1) and (2.2.5.2) for i = 1. For i > 1, resolving the structure sheaf of an effective cycle by vector bundles, and for a given vector bundle, computing its Chern classes on the Grassmannian bundle G ⟶^{π} X, with π^{∗} : F^{i}H_{DR}^{2i}(X/ℂ) ↪ F^{i}H_{DR}^{2i}(G/ℂ), one reduces the compatibility to the case i = 1.

Remark 2 The existence of the absolute de Rham cycle class is proven in great generality in [10] when X is singular. In fact, this class is convenient to formulate some questions. For example, its injectivity for a surface X over k = ℂ would imply Bloch’s conjecture when H^{2}(X, O_{X}) = 0.

At any rate, the existence of ψ motivates the following

Definition 3 A class α ∈ F^{i}H_{DR}^{2i}(X/k) is said to be an absolute de Rham cycle if it lies in the image of H_{DR}^{2i}(X/ℚ) in ℍ_{DR}^{2i}(X/k).

We denote by ∇ : H_{DR}^{j}(X/k) → Ω_{k/ℚ}^{1} ⊗_{k} H_{DR}^{j}(X/k) the Gauss-Manin connection for the smooth morphism X → Spec k of schemes over Spec ℚ.

Proposition 4 The sequence

H_{DR}^{j}(X/ℚ) ⟶ H_{DR}^{j}(X/k) ⟶^{∇} Ω_{k/ℚ}^{1} ⊗ H_{DR}^{j}(X/k)

is exact.

Proof. The sequence is obviously a complex. Let k_{0} ⊂ k be the field of definition of X. One has X = X_{0} ⊗_{k_{0}} k, where X_{0} is smooth proper over k_{0}, and k_{0} = ℚ(S_{0}) for a smooth affine variety S_{0} over ℚ, such that there is a smooth proper map f_{0} : X_{0} → S_{0} with X_{0} ⊗_{O_{S_{0}}} k_{0} = X_{0}. As H_{DR}^{j}(X_{0}/k_{0}) is a finite dimensional k_{0} vector space, any

α ∈ H_{DR}^{j}(X/k) = H_{DR}^{j}(X_{0}/k_{0}) ⊗_{k_{0}} k

lies in H_{DR}^{j}(X_{0}/k_{0}) ⊗_{k_{0}} ℚ(S), where k_{0} ⊂ ℚ(S) ⊂ k and S is a smooth affine variety mapping to S_{0}. If x ∈ Ker∇, then x lies in the kernel of

H_{DR}^{j}(X_{0} ⊗_{k_{0}} ℚ(S)/ℚ(S)) ⟶ Ω_{ℚ(S)/ℚ}^{1} ⊗ H_{DR}^{j}(X_{0} ⊗_{k_{0}} ℚ(S)/ℚ(S))

and to prove exactness, one has to see that

α ∈ Im (H_{DR}^{j}(X_{0} ⊗_{k_{0}} ℚ(S)/ℚ) ⟶ H_{DR}^{j}(X_{0} ⊗_{k_{0}} ℚ(S)/ℚ(S))).

Denote by f : X = X_{0} ×_{S_{0}} S → S the smooth proper morphism obtained by base change S → S_{0} of f_{0}. Making S smaller, one may assume that there is

β ∈ Ker(H_{DR}^{j}(X /S) ⟶^{∇} Ω_{S/ℚ}^{1} ⊗ H_{DR}^{j}(X /S))

such that β ⊗_{O_{S}} ℚ(S) = α, and one wants to show that β ∈ Im H_{DR}^{j}(X /ℚ).

On Ω_{X /ℚ}^{•} one considers the filtration by the subcomplexes f^{∗}Ω_{S/ℚ}^{≥a} ∧ Ω_{X /ℚ}^{•−a}. It defines a spectral sequence

E_{1}^{ab} = Ω_{S/ℚ}^{a} ⊗ H_{DR}^{b}(X /S)

converging to H_{DR}^{a+b}(X /ℚ), whose d_{1} differential is the Gauss-Manin connection ∇. As S is affine, one has

E_{2}^{ab} = ℍ^{a}(S, Ω_{S/ℚ}^{•} ⊗ H_{DR}^{b}(X /S)).

We now consider the analytic varieties S_{an} = (S ⊗_{ℚ} ℂ)_{an}, X_{an} = (X ⊗_{ℚ} ℂ)_{an}. The corresponding spectral sequence

E_{2,an}^{ab} = ℍ^{a}(S_{an}, Ω_{S_{an}}^{•} ⊗ H_{DR}^{b}(X_{an}/S_{an}))
= ℍ^{a}(S_{an}, Ω_{S_{an}}^{•} ⊗ R^{b}f_{∗}Ω_{X_{an}/S_{an}}^{•})
= H^{a}(S_{an}, R^{b}f_{∗}ℂ)

which abuts to ℍ^{a+b}(X_{an}, Ω_{X_{a}n}^{•}) = H^{a+b}(X_{an}, ℂ). This spectral sequence is, according to Deligne ([11], (2.77) and (15.6)) the Leray spectral sequence, and by [2], (4.1.1) (i), it degenerates at E_{2}. On the other hand, by the regularity of the Gauss-Manin connection, one has

\begin{aligned}E_{2,an}^{ab} & = ℍ^{a}(S ⊗_{ℚ} ℂ, Ω_{S⊗_{ℚ}ℂ/ℂ}^{•} ⊗ H_{DR}^{b}(X ⊗_{ℚ} ℂ/S ⊗_{ℚ} ℂ)) \\ & = E_{2}^{ab} ⊗_{ℚ} ℂ \end{aligned}

([1], (6.2) and (7.9)). This implies that (E_{1}^{ab}, d_{1}) ⊗_{ℚ} ℂ degenerates at E_{2}, and so does (E_{1}^{ab}, d_{1}). In particular

\begin{aligned}H_{DR}^{j}(X /ℚ) & = H^{0}(S, Ω_{S/ℚ}^{•} ⊗ H_{DR}^{j}(X /S)) \\ & = Ker(H^{0}(S, H_{DR}^{j}(X /S)) ⟶ H^{0}(S, Ω_{S/ℚ}^{1} ⊗ H_{DR}^{j}(X/S))). \end{aligned}

This proves the required exactness by base change to ℚ(S).

Remark 5 In fact, even if S is not affine, there is a Leray spectral sequence for the de Rham cohomology [7] (3.3), which again degenerates at E_{2} by the comparison between the Leray spectral sequences for the Betti and the de Rham cohomologiesi, and the regularity of Gauss-Manin. For more on this, see [8].

Corollary 6 If α is an absolute Hodge cycle, then it is an absolute de Rham cycle.

Proof. By [3] (2.5), we know that ∇α = 0, where ∇ is as in (4) for j = 2i. Then we apply (4).

Corollary 7 If α is an absolute de Rham cycle such that I_{σ}(α) ∈ H_{B}^{2i}(X_{σ}, ℚ) for some embedding σ : k → ℂ, then α is an absolute Hodge cycle.

Proof. In fact, this is [3] (2.6). More precisely, choose S as in the proof of 4 and β ∈ H_{DR}^{2i}(X /S) restricting to α. The embeddings ℚ(S) → k ⟶^{σ} ℂ define a ℂ valued point of S, which we still denote by σ, such that β(σ) ∈ H^{2i}((X_{an})_{σ}, ℚ) ⊂ H^{2i}((X_{an})_{σ}, ℂ). The image β(σ) of β in

H^{0}(S_{an}, R^{2i}f_{∗}ℂ) = H^{2i}((X_{an})_{σ}, ℂ)^{π_{1}(S_{an},σ)}

lies in

H^{0}(S_{an}, R^{2i}f_{∗}ℚ) = H^{2i}((X_{an})_{σ}, ℚ)^{π_{1}(S_{an},σ)}.

Therefore β|_{(X_{an})_{s}} is rational for all s, in particular for those s coming from an embedding σ : k → ℂ.

Remark 8 An advantage, if any, to adopt the language of absolute de Rham cycles consists of dividing the question of wether α is absolute Hodge or not into two steps:

First of all α must be in

H_{DR}^{2i}(X_{0}/k_{0}) ⊗_{k_{0}} k_{0}^{alg} = KerH_{DR}^{2i}(X_{0}/k_{0}) ⊗_{k_{0}} k ⟶ Ω_{k/k_{0}}^{1} ⊗_{k_{0}} H_{DR}^{2i}(X_{0}/k_{0}),

where k_{0}^{alg} is the algebraic closure of k_{0} in k. Secondly α must be in

KerH_{DR}^{2i}(X_{0}/k_{0}) ⊗_{k_{0}} k_{0}^{alg} ⟶ Ω_{k_{0}/ℚ}^{1} ⊗_{k_{0}} H_{DR}^{2i}(X_{0}/k_{0}) ⊗_{k_{0}} k_{0}^{alg}.

On the other hand, we have seen that if α ∈ F^{i}H_{DR}^{2i}(X/k) is the class of an algebraic cycle, then not only it is an absolute de Rham cycle, but also it is coming from ℍ^{2i}(X, Ω_{X/ℚ}^{≥i}).

Let f : X → S, β ∈ F^{i}H_{DR}^{j}(X /S) = H^{0}(S, R^{j}f_{∗}Ω_{X /S}^{≥i}), such that β ⊗_{ℚ(S)} k = α ∈ F^{i}H_{DR}^{j}(X/k) as in the proof of 4. Let f_{ℂ} : X_{ℂ} → S_{ℂ} be the smooth proper morphism obtained from f by base change O_{S} ⊗_{ℚ} ℂ, and β_{ℂ} be β ⊗_{ℚ} ℂ. Let \overline{f_{ℂ}} : \overline{X_{ℂ}} → \overline{S_{ℂ}} be a compactification of f_{ℂ} such that Σ = \overline{S_{ℂ}} − S_{ℂ}, D = \overline{f_{ℂ}}^{−1}(Σ) are normal crossing divisors and \overline{X_{ℂ}} is smooth.

Definition 9 A class α ∈ F^{i}H_{DR}^{j}(X/k) is said to be of moderate growth if for some (β, \overline{f_{ℂ}}) as above, it verifies

(∗) β_{ℂ} ∈ H^{0}(\overline{S_{ℂ}}, R^{j}\overline{f_{ℂ∗}}Ω_{\overline{X_{ℂ}}/\overline{S_{ℂ}}}^{≥i}(log D)) ⊂ H^{0}(S_{ℂ}, R^{j}f_{ℂ∗}Ω_{X_{ℂ}/S_{ℂ}}^{≥i})

Remark 10 The definition 9 does not depend on the couple (β, \overline{f_{ℂ}}) choosen. In fact, take (γ, g) with g : Y → T, ℚ(T) ⊂ k, Y ⊗_{ℚ(T)} k = X, γ ⊗_{ℚ(T)} k = α. Then considering in k a function field ℚ(U) containing ℚ(S) and ℚ(T), one has base changes σ : U → S, τ : U → T, f_{U} : X_{U} = X ×_{S} U → U, g_{U} : Y_{U} = Y ×_{T} U → U, such that there is an isomorphism ι : X_{U} → Y_{U}, with g_{U} ∘ ι = f_{U}, ι^{∗}(γ ⊗_{O_{T}} O_{U}) = β ⊗_{O_{S}} O_{U}, for U small enough. As β_{ℂ} fulfills (∗) on \overline{S_{ℂ}}, it fulfills (∗) on any blow up \overline{σ_{ℂ}} : \overline{U_{ℂ}} → \overline{S_{ℂ}} such that a commutative diagram exists

\overline{X_{U,ℂ}} ⟶ \overline{X_{ℂ}}
\overline{f_{U,ℂ}}↓  \sqrt{f_{ℂ}}
\overline{U_{ℂ}} ⟶^{\overline{σ_{ℂ}}} \overline{S_{ℂ}}

with the properties: \overline{σ_{ℂ}}^{−1}Σ, Δ = \overline{f_{U,ℂ}}^{−1}\overline{σ_{ℂ}}^{−1}Σ are normal crossing divisors, \overline{X_{U,ℂ}} and \overline{U_{ℂ}} are smooth. Choose \overline{U_{ℂ}} such that τ extends to \overline{τ_{ℂ}} : \overline{U_{ℂ}} → \overline{T_{ℂ}}, with a commutative diagram

\overline{X_{U,ℂ}} ⟶^{\overline{ι_{ℂ}}} \overline{Y_{ℂ}}
\overline{f_{U,ℂ}}↓  \sqrt{g_{ℂ}}
\overline{U_{ℂ}} ⟶^{\overline{τ_{ℂ}}} \overline{T_{ℂ}}

with the same properties as above. One has now

H^{0}(\overline{U_{ℂ}}, R^{j}\overline{f_{U,ℂ∗}}Ω_{\overline{X_{U,ℂ}}/\overline{U_{ℂ}}}^{≥i}(log Δ)) = H^{0}(\overline{T_{ℂ}}, R^{j}\overline{g_{ℂ∗}}Ω_{\overline{Y_{ℂ}}/\overline{T_{ℂ}}}^{≥i}(log \overline{g_{ℂ}}^{−1}(\overline{T_{ℂ}} − \overline{T_{ℂ}}))

[6], 4.13. This implies in particular that classes of moderate growth build a k subvectorspace of F^{i}H_{DR}^{j}(X/k).

Notation 11 We denote this subvectorspace by F^{i}H_{DR}^{j}(X, k)^{log}, and by ℍ^{j}(X, Ω_{X/ℚ}^{≥i})^{log} its inverse image in ℍ^{j}(X, Ω_{X/ℚ}^{≥i}).

Theorem 12 The sequence

ℍ^{j}(X, Ω_{X/ℚ}^{≥i})^{log} ⟶ F^{i}H_{DR}^{j}(X/k)^{log} ⟶^{∇} Ω_{k/ℚ}^{1} ⊗ F^{i−1}H_{DR}^{j}(X/k)

is exact.

Proof. We have to prove that if α ∈ Ker∇, then it lies in the image of ℍ^{j}(X, Ω_{X/ℚ}^{≥i}). With the notations as above,

β_{ℂ} ∈ H^{0}(\overline{S_{ℂ}}, Ω• ; \overline{S_{ℂ}}(log Σ) ⊗ R^{j}\overline{f_{ℂ∗}}Ω_{\overline{X_{ℂ}}/\overline{S_{ℂ}}}^{≥i−•}(log D)).

This group is the E_{2}^{0j} term of a spectral sequence converging to ℍ^{j}(\overline{X_{ℂ}}, Ω_{\overline{X_{ℂ}}/\overline{S_{ℂ}}}^{≥i}(log D)) and defined as in [7] (3.3) on the complex Ω_{\overline{X_{ℂ}}/\overline{S_{ℂ}}}^{≥i}(log D). One has

E_{2}^{ab} = ℍ^{a}(\overline{S_{ℂ}}, Ω• ; \overline{S_{ℂ}}(log Σ) ⊗ R^{b}\overline{f_{ℂ∗}}Ω_{\overline{X_{ℂ}}/\overline{S_{ℂ}}}^{≥i−•}(log D)).

By [6] (0.4) and its analogue in characteristic zero [4] (2.7), E_{2}^{ab} injects into

ℍ^{a}(\overline{S_{ℂ}}, Ω• ; \overline{S_{ℂ}}(log Σ) ⊗ R^{b}\overline{f_{ℂ∗}}Ω_{\overline{X_{ℂ}}/\overline{S_{ℂ}}}^{•}(log D)),

which is just H^{a}(S_{an}, R^{b}f_{ℂ∗}ℂ) by [1] II, §6. Thus the spectral sequence degenerates at E_{2}, and β_{ℂ} comes from ℍ^{j}(\overline{X_{ℂ}}, Ω≥i ; \overline{X_{ℂ}}(log D)). In particular β_{ℂ} comes from ℍ^{j}(X , Ω_{X /ℚ}^{≥i}) ⊗_{ℚ} ℂ and the image of α in

\frac{F^{i}H_{DR}^{j}(X/k) ⊗_{ℚ} ℂ}{Im ℍ^{j}(X, Ω_{X/ℚ}^{≥i}) ⊗ ℂ} = (\frac{F^{i}H_{DR}^{j}(X/k)}{Im ℍ^{j}(X, Ω_{X/ℚ}^{≥i})}) ⊗ ℂ

vanishes. Therefore α lies in the image of ℍ^{j}(X, Ω_{X/ℚ}^{≥i}).

Remark 13 If the transcendence degree of k is ≤ 1, then of course the sequence

ℍ^{j}(X, Ω_{X/ℚ}^{≥i}) ⟶ F^{i}H_{DR}^{j}(X/k) ⟶^{∇} Ω_{k/ℚ}^{1} ⊗ F^{i−1}H_{DR}^{j}(X/k)

is trivially exact. But if the transcendence degree of k is higher, it is not clear why an absolute Hodge cycle has to be a moderate absolute de Rham cycle.

More generally, one can consider a k subvectorspace V of H_{DR}^{j}(X/k), such that I_{σ}(V ⊗_{σ}ℂ) is a Hodge substructure of H_{DR}^{j}(X_{σ}, ℂ). In the light of the above results, one can examine the following questions.

Question 14 Is V stable under the Gauss-Manin connection?

For this, one would like I_{σ}^{−1}[I_{σ}(V ⊗_{σ} ℂ) ∩ H_{B}^{j}(X_{σ}, ℚ)] to lie in V and to be independent of σ. If so, then V defines a vector bundle W with a flat connection on S, where S is defined as in 4 such that V = W ⊗_{ℚ(S)} k, W ⊂ H_{DR}^{j}(X_{0}/k_{0}) ⊗_{k_{0}} ℚ(S). Then W_{an} on S_{an} is generated by a local system F.

Question 15 In the above situation, is the monodromy representation associated to F defined over ℚ? Again, one can split up 14 into two parts as in 8. Moreover, the knowledge of 14 does not imply the knowledge of 15.

References

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[4] Deligne, P.; Illusie, L.: Relèvements modulo p^{2} et décomposition du complexe de de Rham, Invent. math. 89 (1987), 247 - 270

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