Remarks on absolute de Rham and absolute Hodge cycles

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 HDRj(X/k) = HDRj(X /) and by the GAGA principle one has an isomorphism I from HDRj(X/k) to the Betti cohomology HBj(X ,) ([5], p. 96).

An element of the Chow group CHi(X) has a de Rham class
such that for all embeddings : k
So 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 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, X/>i). In 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 iM) ([9], théorème 5), d log induces the absolute de Rham cycle class map
One composes this map with
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 FiH DR2i(X/k) = FiH DR2i(X /), so it is enough to handle k = , 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 - - X, with * : FiH DR2i(X/)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 [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 HDR2i(X/) in DR2i(X/k).

We denote by : HDRj(X/k) k/1 kHDRj(X/k) the Gauss-Manin connection for the smooth morphism X Spec k of schemes over Spec .

Proposition 4 The sequence

is exact.

Proof. The sequence is obviously a complex.
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.
As HDRj(X 0/k0) is a finite dimensional k0 vector space, any

lies in HDRj(X 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
and 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
such that OS(S) = , and one wants to show that  Im  HDRj(X/).

On X/ one considers the filtration by the subcomplexes f* S/>a X/•-a. It defines a spectral sequence
converging to HDRa+b(X/), whose d 1 differential is the Gauss-Manin connection . As S is affine, one has
We now consider the analytic varieties San = (S )an, Xan = (X)an. The corresponding spectral sequence
which abuts to a+b(X an, Xan) = Ha+b(X an,). 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 E2.
On the other hand, by the regularity of the Gauss-Manin connection, one has
([1], (6.2) and (7.9)).
This implies that (E1ab,d 1) degenerates at E2, and so does (E1ab,d 1). In particular
This 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 [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() HB2i(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 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),) H2i((X an),). The image () of in

lies 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:

First of all must be in

where k0alg is the algebraic closure of k 0 in k.
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, X/>i).

Let f : X S, FiH DRj(X/S) = H0(S,Rjf *X/S>i), such that (S)k = FiH DRj(X/k) 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 smooth.

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 exists

with 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 diagram
with the same properties as above. One has now
[6], 4.13.

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

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

Theorem 12 The sequence

is exact.

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

This group is the E20j term of a spectral sequence converging to j(X, X /S>i(log D)) and defined as in [7] (3.3) on the complex X/S>i(log D). One has
By [6] (0.4) and its analogue in characteristic zero [4] (2.7), E2ab injects into
which is just Ha(S an,Rbf *) by [1] 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, X/>i).

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

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 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 following questions.

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 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

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