Remarks on absolute de Rham and absolute Hodge cycles

 
Hélène   Esnault1)
Kapil  H.   Paranjape2)

 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 s of k into the field of complex numbers C, the C valued points of X  ox sC form a complex manifold denoted by Xs. By base change for the de Rham cohomology HDRj(X/k)  ox sC = HDRj(X  ox sC/C) and by the GAGA principle one has an isomorphism Is from HDRj(X/k)  ox sC to the Betti cohomology HBj(X s,C) ([5], p. 96).
 
An element of the Q Chow group CHi(X)  ox ZQ has a de Rham class
a  (-  F iH2iDR(X/k) = H2i(X, _O_ >Xi/k) < H2iDR(X/k)
such that for all embeddings s : k -->C
Is(a)  (-  Is(F iH2iDR(X/k)  ox s C)  /~\  H2iB(Xs, Q).
So a 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 a  (- FiH DR2i(X/k) is said to be an absolute Hodge cycle if for all embeddings s : k -->C, Is(a) lies in HB2i(X s,Q).

On the other hand, such an algebraic cycle has an absolute de Rham class in H2i(X, _O_ X/Q>i). In fact, there is an absolute differential

dlog : O*X ----> _O_ >X1/Q [1]
inducing an absolute differential
        M       >i
dlog : K i ----> _O_X/Q  [i]
where KiM is the Zariski sheaf of Milnor K theory. As CHi(X)  ox ZQ = Hi(X,K iM) ([9], théorème 5), d log induces the absolute de Rham cycle class map
                y           >i
CHi(X)    ox Z Q ---->  H2i(X, _O_ X/Q).
One composes this map with
H2i(X, _O_ >i ) ---->  H2i(X, _O_ >i ) = FiH2i  (X/k)
         X/Q               X/k        DR
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)  ox sC = FiH DR2i(X  ox sC/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 p - - --> X, with p* : FiH DR2i(X/C)-->FiH DR2i(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 H2(X,O X) = 0.

At any rate, the existence of y motivates the following

Definition 3 A class a  (- FiH DR2i(X/k) is said to be an absolute de Rham cycle if it lies in the image of HDR2i(X/Q) in H DR2i(X/k).

We denote by  \~/ : HDRj(X/k) --> _O_ k/Q1  ox kHDRj(X/k) the Gauss-Manin connection for the smooth morphism X --> Spec k of schemes over Spec Q.

Proposition 4 The sequence

  j               j          \~/            j
HDR(X/Q)   ---->  HDR(X/k)   ---->  _O_1k/Q  ox  H DR(X/k)
is exact.

Proof. The sequence is obviously a complex.
Let k0 < k be the field of definition of X. One has X = X0  ox k0k, where X0 is smooth proper over k0, and k0 = Q(S0) for a smooth affine variety S0 over Q, such that there is a smooth proper map f0 : X0 --> S0 with X0  ox OS 0k0 = X0.
As HDRj(X 0/k0) is a finite dimensional k0 vector space, any

a  (-  HjDR(X/k)   = HjDR(X0/k0)   ox k0 k
lies in HDRj(X 0/k0)  ox k0Q(S), where k0 <Q(S) < k and S is a smooth affine variety mapping to S0. If x  (- Ker \~/ , then x lies in the kernel of
HjDR(X0   ox k0 Q(S)/Q(S)) ---->  _O_1Q(S)/Q  ox  HjDR(X0  ox k0 Q(S)/Q(S))
and to prove exactness, one has to see that
a   (-  Im  (HjDR(X0   ox k0 Q(S)/Q) ---->  HjDR(X0  ox k0 Q(S)/Q(S))).
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
b  (-  Ker(Hj   (X /S)  -- \~/ --> _O_1    ox  Hj  (X /S))
           DR             S/Q     DR
such that b  ox OSQ(S) = a, and one wants to show that b  (-  Im  HDRj(X/Q).
 
On _O_X/Q one considers the filtration by the subcomplexes f*_O_ S/Q>a  /\ _O_ X/Q•-a. It defines a spectral sequence
Ea1b = _O_aS/Q  ox  HbDR(X /S)
converging to HDRa+b(X/Q), whose d 1 differential is the Gauss-Manin connection  \~/ . As S is affine, one has
  ab    a     •       b
E 2 = H  (S,_O_ S/Q  ox  H DR(X /S)).
We now consider the analytic varieties San = (S  ox QC)an, Xan = (X ox QC)an. The corresponding spectral sequence
 ab       a       •       b
E2,an  =  H (San, _O_San  ox  H DR(Xan/San))
      =  Ha(San, _O_•San  ox  Rbf*_O_ •Xan/San)
      =  Ha(San, Rbf*C)
which abuts to Ha+b(X an, _O_Xan) = Ha+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 E2.
On the other hand, by the regularity of the Gauss-Manin connection, one has
 ab       a           •           b
E2,an  =  H (S  ox Q C, _O_S ox QC/C  ox  HDR(X  ox Q C/S   ox Q C))
      =  Ea2b  ox Q C
([1], (6.2) and (7.9)).
This implies that (E1ab,d 1)  ox QC degenerates at E2, and so does (E1ab,d 1). In particular
Hj   (X /Q)  =  H0(S, _O_ •   ox  Hj  (X /S))
  DR                  0 S/Q   j DR            0     1       j
             =  Ker(H  (S,H DR(X  /S)) ---->  H  (S,_O_ S/Q  ox  H DR(X/S))).
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 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 a is an absolute Hodge cycle, then it is an absolute de Rham cycle.

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

Corollary 7 If a is an absolute de Rham cycle such that Is(a)  (- HB2i(X s,Q) for some embedding s : k -->C, then a is an absolute Hodge cycle.

Proof. In fact, this is [3] (2.6). More precisely, choose S as in the proof of 4 and b  (- HDR2i(X/S) restricting to a. The embeddings Q(S) --> k s - - -->C define a C valued point of S, which we still denote by s, such that b(s)  (- H2i((X an)s,Q) < H2i((X an)s,C). The image b(s) of b in

  0       2i         2i          p1(San,s)
H  (San,R  f*C)  = H  ((Xan)s, C)
lies in
H0(San, R2if*Q) =  H2i((Xan)s,Q)p1(San,s).
Therefore b|(Xan)s is rational for all s, in particular for those s coming from an embedding s : k -->C.

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

                  alg
H2iDR(X0/k0)   ox k0 k0 = KerH2iDR(X0/k0)   ox k0 k ----> _O_1k/k0  ox k0 H2iDR(X0/k0),
where k0alg is the algebraic closure of k 0 in k.
Secondly a must be in
      2i              alg       1         2i              alg
KerH  DR(X0/k0)   ox k0 k0 ---->  _O_ k0/Q  ox k0 H DR(X0/k0)  ox k0 k0 .

On the other hand, we have seen that if a  (- 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 H2i(X, _O_ X/Q>i).
 
Let f : X --> S, b  (- FiH DRj(X/S) = H0(S,Rjf *_O_X/S>i), such that b  ox Q (S)k = a  (- FiH DRj(X/k) as in the proof of 4. Let fC : XC --> SC be the smooth proper morphism obtained from f by base change OS  ox QC, and bC be b  ox QC. Let fC : XC --> SC be a compactification of fC such that S = SC - SC, D = fC-1(S) are normal crossing divisors and XC is smooth.

Definition 9 A class a  (- FiH DRj(X/k) is said to be of moderate growth if for some (b,fC) as above, it verifies

           0 ---  j---- >i                0      j     >i
(*) bC  (-  H (SC,R  fC*_O_ XC/SC(log D)) <  H  (SC,R  fC*_O_ XC/SC)

Remark 10 The definition 9 does not depend on the couple (b,fC) choosen. In fact, take (g,g) with g : Y --> T, Q(T) < k, Y ox Q(T)k = X, g  ox Q(T)k = a. Then considering in k a function field Q(U) containing Q(S) and Q(T), one has base changes s : U --> S, t : U --> T, fU : XU = XSU --> U, gU : YU = YT U --> U, such that there is an isomorphism i : XU -->YU, with gU o i = fU, i*(g  ox OTOU) = b  ox OSOU, for U small enough. As bC fulfills (*) on SC, it fulfills (*) on any blow up sC : UC --> SC such that a commutative diagram exists

  XU,C---- -->  XC-

fU,C |,            |, fC
         --
  UC-  --s-C-->  SC-
with the properties: sC-1S, D = f U,C-1sC-1S are normal crossing divisors, X U,C and UC are smooth. Choose UC such that t extends to tC : UC --> TC, with a commutative diagram
  -----  iC   ----
  XU,C ----->   Y C
----            g-
fU,C |,            |,  C
  ---    tC   ---
  UC   ----->   TC
with the same properties as above. One has now
   ---    ----- >i                   ---    ---->i        ---  ---  ---
H0(UC, Rj fU,C*_O_ XU,C/UC(log D)) =  H0(TC, Rj gC*_O_Y-C/TC(log gC-1(TC - TC))
[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 Hj(X, _O_ X/Q>i)log its inverse image in Hj(X, _O_ X/Q>i).

Theorem 12 The sequence

  j     >i   log       i j        log   \~/   1       i-1  j
H  (X, _O_X/Q)   ---->  F H DR(X/k)     ---->  _O_k/Q  ox  F   H DR(X/k)
is exact.

Proof. We have to prove that if a  (- Ker \~/ , then it lies in the image of Hj(X, _O_ X/Q>i). With the notations as above,

         ---                ----
bC  (-  H0( SC,_O_ •-(log S)  ox  RjfC*_O_>i-•-(log D)).
              SC                 XC/SC
This group is the E20j term of a spectral sequence converging to Hj(XC, _O_X C/SC>i(log D)) and defined as in [7] (3.3) on the complex _O_XC/SC>i(log D). One has
          ---                 ----
Eab2 =  Ha(SC, _O_ •-(log S)  ox  Rb fC*_O_ >i--•(log D)).
                SC                  XC/SC
By [6] (0.4) and its analogue in characteristic zero [4] (2.7), E2ab injects into
Ha(S---,_O_•-(log S)  ox  Rbf--_O_•----(log D)),
     C   SC              C* XC/SC
which is just Ha(S an,RbfC *C) by [1] II, 6.

Thus the spectral sequence degenerates at E2, and bC comes from Hj(XC, _O_X C>i(log D)). In particular bC comes from Hj(X, _O_ X/Q>i)  ox QC and the image of a in

                       (                 )
F iHjDR(X/k)   ox Q C       F iHjDR(X/k)
-----j----->i--------=   -----j----->i---   ox  C
Im  H  (X,_O_ X/Q)  ox  C     Im H  (X, _O_X/Q)
vanishes. Therefore a lies in the image of Hj(X, _O_ X/Q>i).

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

        >i            j          \~/                 j
Hj(X,  _O_X/Q) ---->  FiH DR(X/k)  ---->  _O_1k/Q  ox  F i- 1HDR(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 HDRj(X/k), such that I s(V  ox sC) is a Hodge substructure of HDRj(X s,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 Is-1[I s(V  ox sC)  /~\ HBj(X s,Q)] to lie in V and to be independent of s.

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  ox Q(S)k, W < HDRj(X 0/k0)  ox k0Q(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 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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[2]   Deligne, P.: Théorie de Hodge II, Publ. Math. de l’IHES 40 (1971), 5 - 58

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[4]   Deligne, P.; Illusie, L.: Relèvements modulo p2 et décomposition du complexe de de Rham, Invent. math. 89 (1987), 247 - 270

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