Definition 1 A class FⁱH _DR²ⁱ(X/k) is said to be an absolute Hodge cycle if for all embeddings : k , I() lies in H_B²ⁱ(X ,).

On the other hand, such an algebraic cycle has an absolute de Rham class in ²ⁱ(X, _X/^>i). In fact, there is an absolute differential

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 = would imply Bloch’s conjecture when H²(X,O _X) = 0.

At any rate, the existence of motivates the following

Definition 3 A class FⁱH _DR²ⁱ(X/k) is said to be an absolute de Rham cycle if it lies in the image of H_DR²ⁱ(X/) in _DR²ⁱ(X/k).

We denote by : H_DR^j(X/k) _k/¹ _kH_DR^j(X/k) the Gauss-Manin connection for the smooth morphism X Spec k of schemes over Spec .

Proposition 4 The sequence

Proof. The sequence is obviously a complex.
Let k₀ k be the field of definition of X. One has X = X_{0 k0}k, where X₀ is smooth proper over k₀, and k₀ = (S₀) for a smooth affine variety S₀ over , such that there is a smooth proper map f₀ : X₀ S₀ with X_{0 OS 0}k₀ = X₀.
As H_DR^j(X ₀/k₀) is a finite dimensional k₀ vector space, any

X/

S/

X/

X

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 E₂ 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 is an absolute Hodge cycle, then it is an absolute de Rham cycle.

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

Corollary 7 If is an absolute de Rham cycle such that I() H_B²ⁱ(X ,) for some embedding : k , then is an absolute Hodge cycle.

Proof. In fact, this is [3] (2.6). More precisely, choose S as in the proof of 4 and H_DR²ⁱ(X/S) restricting to . The embeddings (S) k - - define a valued point of S, which we still denote by , such that () H²ⁱ((X _an),) H²ⁱ((X _an),). The image () of in

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

On the other hand, we have seen that if FⁱH _DR²ⁱ(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 ²ⁱ(X, _X/^>i).
 
Let f : X S, FⁱH _DR^j(X/S) = H⁰(S,R^jf _*X/S^>i), such that _(S)k = FⁱH _DR^j(X/k) as in the proof of 4. Let f : X S be the smooth proper morphism obtained from f by base change O_S , and be . Let f : X S be a compactification of f such that = S - S, D = f^-1() are normal crossing divisors and X is smooth.

Definition 9 A class FⁱH _DR^j(X/k) is said to be of moderate growth if for some (,f) as above, it verifies

Remark 10 The definition 9 does not depend on the couple (,f) choosen. In fact, take (,g) with g : Y T, (T) k, Y_(T)k = X, _(T)k = . Then considering in k a function field (U) containing (S) and (T), one has base changes : U S, : U T, f_U : X_U = X×_SU U, g_U : Y_U = Y×_T U U, such that there is an isomorphism : X_U Y_U, with g_U o = f_U, ^*( _OTO_U) = _OSO_U, for U small enough. As fulfills (*) on S, it fulfills (*) on any blow up : U S such that a commutative diagram exists

f _U,

X _U,

U

U

U

T

This implies in particular that classes of moderate growth build a k subvectorspace of FⁱH _DR^j(X/k).

Notation 11 We denote this subvectorspace by FⁱH _DR^j(X,k)^log, and by ^j(X, _X/^>i)^log its inverse image in ^j(X, _X/^>i).

Theorem 12 The sequence

Proof. We have to prove that if Ker, then it lies in the image of ^j(X, _X/^>i). With the notations as above,

X

X /S

X/S

*

Thus the spectral sequence degenerates at E₂, and comes from ^j(X, _X ^>i(log D)). In particular comes from ^j(X, _X/^>i) and the image of in

X/

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

More generally, one can consider a k subvectorspace V of H_DR^j(X/k), such that I (V ) is a Hodge substructure of H_DR^j(X ,). 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^-1[I (V ) H_B^j(X ,)] to lie in V and to be independent of .

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 _(S)k, W H_DR^j(X ₀/k₀) _k0(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 ?

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