Let X be a smooth proper variety over a field k of characteristic zero. For any embedding
of k into the field of complex numbers , the valued points of X form
a complex manifold denoted by X. By base change for the de Rham cohomology
/) and by the GAGA principle one has an isomorphism I
to the Betti cohomology HBj(X
,) (, p. 96).
Remarks on absolute de Rham and absolute Hodge
An element of the Chow group CHi(X) has a de Rham class
that for all embeddings : k
is an absolute Hodge cycle, a notion defined by Deligne , §2, which we slightly
modify, as we are only interested here in de Rham cohomology (see , open question
Definition 1 A class FiH
DR2i(X/k) is said to be an absolute Hodge cycle if for all
embeddings : k , I() lies in HB2i(X
On the other hand, such an algebraic cycle has an absolute de Rham class in 2i(X,
fact, there is an absolute differential
inducing an absolute differential
where KiM is the Zariski sheaf of Milnor K theory. As CHi(X) = Hi(X,K
théorème 5), d log induces the absolute de Rham cycle class map
composes this map with
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 FiH
/), so it is enough to handle
k = , in which case the compatibility is proven in , (22.214.171.124) and (126.96.36.199) 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
- - X, with
* : FiH
DR2i(G/), 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
 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
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 HDR2i(X/) in
We denote by : HDRj(X/k)
kHDRj(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.
0/k0) k0(S), where k0 (S) k and S is a smooth affine variety mapping to
S0. If x Ker, then x lies in the kernel of
to prove exactness, one has to see that
Denote by f : X = X0 ×S0S S the smooth proper morphism obtained by base change
S S0 of f0. Making S smaller, one may assume that there is
that OS(S) = , and one wants to show that Im HDRj(X/).
Let k0 k be the field of definition of X. One has X = X0 k0k, 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 X0 OS
0k0 = X0.
0/k0) is a finite dimensional k0 vector space, any
On X/• one considers the filtration by the subcomplexes f*
X/•-a. It defines a
converging to HDRa+b(X/), whose d
1 differential is the Gauss-Manin connection . As S is
affine, one has
now consider the analytic varieties San = (S )an, Xan = (X)an. The corresponding
which abuts to a+b(X
an, Xan•) = Ha+b(X
an,). This spectral sequence is, according to
Deligne (, (2.77) and (15.6)) the Leray spectral sequence, and by , (4.1.1) (i), it
degenerates at E2.
On the other hand, by the regularity of the Gauss-Manin connection, one has
(6.2) and (7.9)).
This implies that (E1ab,d
1) degenerates at E2, and so does (E1ab,d
1). In particular
proves the required exactness by base change to (S).
Remark 5 In fact, even if S is not affine, there is a Leray spectral sequence for the de Rham
cohomology  (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 .
Corollary 6 If is an absolute Hodge cycle, then it is an absolute de Rham cycle.
Proof. By  (2.5), we know that = 0, where is as in (4) for j = 2i. Then we apply
Corollary 7 If is an absolute de Rham cycle such that I() HB2i(X
,) for some
embedding : k , then is an absolute Hodge cycle.
Proof. In fact, this is  (2.6). More precisely, choose S as in the proof of 4 and
HDR2i(X/S) restricting to . The embeddings (S) k - - define a valued point of
S, which we still denote by , such that () H2i((X
an),). The image
() of in
Therefore |(Xan)s is rational for all s, in particular for those s coming from an embedding
: k .
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:
where k0alg is the algebraic closure of k
0 in k.
First of all must be in
Secondly 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,
Let f : X S, FiH
DRj(X/S) = H0(S,Rjf
*X/S>i), such that
(S)k = FiH
as in the proof of 4. Let f : X S be the smooth proper morphism obtained from f by
base change OS , 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
Definition 9 A class FiH
DRj(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,
fU : XU = X×SU U, gU : YU = Y×T U U, such that there is an isomorphism
: XU YU, with gU o = fU, *(
OTOU) = OSOU, for U small enough. As fulfills
(*) on S, it fulfills (*) on any blow up : U S such that a commutative diagram
the properties: -1, = f
U,-1-1 are normal crossing divisors, X
U, and U
are smooth. Choose U such that extends to : U T, with a commutative
the same properties as above. One has now
This implies in particular that classes of moderate growth build a k subvectorspace of
Notation 11 We denote this subvectorspace by FiH
DRj(X,k)log, and by j(X,
inverse image in j(X,
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,
group is the E20j term of a spectral sequence converging to j(X, X
/S>i(log D)) and
defined as in  (3.3) on the complex X/S>i(log D). One has
 (0.4) and its analogue in characteristic zero  (2.7), E2ab injects into
which is just Ha(S
*) by  II, §6.
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
vanishes. Therefore lies in the image of j(X,
Remark 13 If the transcendence degree of k is < 1, then of course the sequence
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 HDRj(X/k), such that I
(V ) is
a Hodge substructure of HDRj(X
,). In the light of the above results, one can examine the
Question 14 Is V stable under the Gauss-Manin connection?
For this, one would like I-1[I
(V ) HBj(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 HDRj(X
0/k0) k0(S). Then Wan on San is generated by a
local system F.
Question 15 In the above situation, is the monodromy representation associated to F defined
Again, one can split up 14 into two parts as in 8. Moreover, the knowledge of 14 does not imply
the knowledge of 15.
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