An element of the Chow group CH

Definition 1 A class F^{i}H_{
DR}^{2i}(X/k) is said to be an absolute Hodge cycle if for all
embeddings : k , I_{}() lies in H_{B}^{2i}(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
H^{2}(X,O_{
X}) = 0.

At any rate, the existence of motivates the following

Definition 3 A class 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/) in _{
DR}^{2i}(X/k).

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

Proof. The sequence is obviously a complex.

Let k_{0} k be the field of definition of X. One has X = X_{0} _{k0}k, where X_{0} is smooth proper
over k_{0}, and k_{0} = (S_{0}) for a smooth affine variety S_{0} over , such that there is a smooth
proper map f_{0} : X_{0} S_{0} with X_{0} _{OS
0}k_{0} = X_{0}.

As H_{DR}^{j}(X_{
0}/k_{0}) is a finite dimensional k_{0} vector space, any

On

On the other hand, by the regularity of the Gauss-Manin connection, one has

This implies that (E

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 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_{B}^{2i}(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_{DR}^{2i}(X/S) restricting to . The embeddings (S) k - - define a valued point of
S, which we still denote by , such that () H^{2i}((X_{
an})_{},) H^{2i}((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

Secondly must be in

On the other hand, we have seen that if 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 ^{2i}(X, _{
X/}^{>i}).

Let f : X S, F^{i}H_{
DR}^{j}(X/S) = H^{0}(S,R^{j}f_{
*}_{X/S}^{>i}), such that _{
(S)}k = F^{i}H_{
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 F^{i}H_{
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×_{S}U 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}, ^{*}( _{
OT}O_{U}) = _{OS}O_{U}, for U small enough. As _{} fulfills
(*) on S_{}, it fulfills (*) on any blow up _{} : U_{} S_{} such that a commutative diagram
exists

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 ^{j}(X, _{
X/}^{>i})^{log} its
inverse image in ^{j}(X, _{
X/}^{>i}).

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

Thus the spectral sequence degenerates at E_{2}, and _{} comes from ^{j}(X_{}, _{X
}^{>i}(log D)). In
particular _{} comes from ^{j}(X, _{
X/}^{>i}) _{} and the image of in

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}/k_{0}) _{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.

[1] Deligne, P.: Equations différentielles à points réguliers, Springer LN 169 (1970)

[2] Deligne, P.: Théorie de Hodge II, Publ. Math. de l’IHES 40 (1971), 5 - 58

[3] Deligne, P.: Hodge cycles on abelian varieties, in Hodge cycles, Motives and Shimura Varieties, Springer LN 900 (1982), 9 - 101

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

[5] Grothendieck, A.: On the de Rham cohomology of algebraic varieties, Publ. Math. de l’IHES 29 (1966), 95 - 103

[6] Illusie, L.: Réduction semi-stable et décomposition de complexes de de Rham à
coefficients, Duke Math. Journal 60, n^{0}1 (1990), 139 - 185

[7] Katz, N.: Nilpotent connections and the monodromy theorem: applications of a result of Turrittin, Publ. Math. de l’IHES 39 (1970), 175 - 232

[8] Paranjape, K.: Leray spectral sequence for de Rham cohomology, preprint

[9] Soulé, C.: Opérations en K-théorie algébrique, Canad. J. of Math. 37 n^{0}3 (1985),
488 - 550

[10] Srinivas, V.: Gysin maps and cycle classes for Hodge cohomology, preprint

[11] Zucker, S.: Hodge theory with degenerating coefficients: L_{2} cohomology in the
Poincaré metric, Annals of Math. 109 (1979), 415 - 476