Table Of ContentGERMS OF DE RHAM COHOMOLOGY CLASSES WHICH VANISH AT
THE GENERIC POINT
HE´LE`NE ESNAULT AND ECKART VIEHWEG
8
9
9
Abstract. We show that hypergeometric differential equations, unitary and Gauß-Manin
1
connections give rise to de Rham cohomology sheaves which do not admit a Bloch-Ogus res-
n
olution [1]. The latter is in contrast to Panin’s theorem [8] asserting that corresponding´etale
a
J cohomology sheaves do fulfill Bloch-Ogus theory.
8 Germes de classes de cohomologie de de Rham qui s’annulent au point
g´en´erique
]
G
R´esum´e. Nous montrons que les syst`emes d’´equations hyperg´eom´etriques, les connexions
A unitaires et de Gauß-Manin donnent lieu `a des faisceaux de cohomologie de de Rham qui
. n’ont pas de r´esolution de Bloch-Ogus [1]. Ce dernier exemple contraste avec le th´eor`eme de
h
Panin [8] affirmant que des faisceaux semblables en cohomologie ´etale v´erifient la th´eorie de
t
a Bloch-Ogus.
m
[
Version abr´eg´ee en franc¸ais. Soit (E, ) une connexion plate sur une vari´et´e lisse S sur un
1 ∇
corps k alg´ebriquement clos en caract´eristique 0. La restriction des faisceaux de cohomologie
v
3 i ((E, )) `a leur valeur au point g´en´erique de S est trivialement injective pour i = 0,1. Afin
HDR ∇
3 de montrer que pour les exemples de (E, ) ´evoqu´es plus haut, cela n’est plus n´ecessairement
0 ∇
le cas pour i = 2, nous forc¸ons l’existence de germes de sections de la fa¸con suivante (voir [5]).
1
0 On remplace S par l’´eclatement d’un point. Cela introduit un diviseur exceptionnel sur lequel
8
la connexion est triviale, et donc acquiert des sections. Par le morphisme de Gysin, ces sections
9
/ fournissent des sections non nulles dans H2 (S,(E, )) qui en particulier s’annulent au point
h DR ∇
g´en´erique de S. Que ces sections ne s’annulent pas dans le germe du faisceau en un point du di-
t
a
viseurexceptionnelprovientd’unehypoth`eseconvenableder´esidusdanslecashyperg´eom´etrique,
m
de g´en´ericit´e dans le cas unitaire, et de la th´eorie de Hodge dans le cas de Gauß-Manin pour une
:
v
famille `a forte variation.
i
X
r
a
Let S be a smooth algebraic variety defined over an algebraically closed field k. Bloch-
Ogus theory [1] provides a acyclic resolution of the Zariski sheaves of ´etale cohomology
i (Z/n(j)) if chark = 0 or if (chark,n) = 1, of de Rham cohomology i if chark = 0,
H´et HDR
and of Betti cohomology i if k = C. Here i denotes the Zariski sheaf associated to the
HB H
presheaf U Hi(U). The first level of the resolution says that the restriction map to the
7→
generic point i : η = Speck(S) S
η
→
i (Z/n(j)) i Hi (η,Z/n(j))
H´et −−→ η∗ ´et
is injective (and similarly for de Rham and Betti cohomology).
Bloch-Ogus theory extends in an obvious way to the sheaves of cohomology i(L) with
H
values in a local system of complex vector spaces L of finite monodromy if k = C, or
equivalently [6] to the de Rham cohomology sheaves i ((E, )) of a locally free sheaf E
HDR ∇
This work has been partly supported by the DFG Forschergruppe ”Arithmetik und Geometrie” and
by the I.H.E.S., Bures-sur-Yvette.
1
2 HE´LE`NE ESNAULT AND ECKART VIEHWEG
with a flat connection , the monodromy of which is finite with respect to one embedding
∇
k C (and hence to all).
⊂
A remarkable generalization of the Bloch-Ogus theory for ´etale cohomology has been
given by I. Panin [8]. Let f : X S be a projective smooth morphism and let L be a
→
local system of free Z/n-modules of finite rank (where n is prime to chark if chark > 0).
Then the Zariski sheaves i (f,L(j)) associated to the presheaves
H´et
U Hi (f−1(U),L(j))
7−→ ´et
have a Bloch-Ogus acyclic resolution on S. In particular the restriction to the generic point
i (f,L(j)) i Hi (f−1(η),L(j))
H´et −−→ η∗ ´et
is injective, as in the classical case “f = identity and L = Z/n”. Panin’s proof strongly
relies on the finiteness of the local systems involved.
This raises the question of a similar theorem for the de Rham cohomology in character-
istic zero. In this note we give negative examples:
0.1. Bundles E with a flat connection for which
∇
(0.1) 2 ((E, )) i H2 (η,(E, ))
HDR ∇ −−→ η∗ DR ∇
isnot injective, or equivalently (over C), localsystems L ofcomplex vector spaces for which
2(L) i H2(η,L) is not injective (see 2.1 and 3.1).
η∗
H →
0.2. Smooth projective morphisms f : X S for which
→
(0.2) 4 (f) i H4 (f−1(η))
HDR −−→ η∗ DR
is not injective (see 1.3 and 1.4). Here i (f) denotes the Zariski sheaf associated to
HDR
U Hi (f−1(U)). Or equivalently, over C,
7→ DR
4(f) i H4(f−1(η)) = i lim H4(f−1(U))
HB −−→ η∗ B η∗ B
−U−⊂→S
is not injective, with a similar notation for Betti cohomology.
Deligne’s theorem [3], saying that R•f C = Rjf C[ j] over k = C, together with [6]
∗ j ∗
⊕ −
imply that the injectivity of
i (f) i Hi (f−1(η))
HDR −−→ η∗ DR
is equivalent to the injectivity
i−j((Rjf Ω• , )) i Hi−j((η,Rjf Ω• , ))
HDR ∗ X/S ∇ −−→ η∗ DR ∗ X/S ∇
for all j, where is the Gauß-Manin connection. Thus in the second example we will
∇
verify that this morphism is not injective, for i j = j = 2.
−
In all examples given the method to construct a germ of a de Rham section is as follows.
0.3. Construction of a section. Let (E′, ′) be defined over a smooth variety S′ of
∇
dimension at least two, and let δ : S S′ be the blow up of a point p S′, with
→ ∈
exceptional divisor F. Let (E, ) = δ∗(E′, ′) be the pullback connection. Then the
∇ ∇
restriction map
H1 (S,(E, )) = H1 (S′,(E′, ′)) H1 (S′ p,(E′, ′)) = H1 (S F,(E, ))
DR ∇ DR ∇ −−→ DR − ∇ DR − ∇
is an isomorphism. Hence the Gysin map
i : H2 (S,(E, )) = H0 (F,(E, ) ) = krankE H2 (S,(E, ))
F DR,F ∇ DR ∇ |F −−→ DR ∇
is injective, and any section i (σ), σ krankE, vanishes at the generic point η of S.
F
∈
GERMS OF DE RHAM COHOMOLOGY CLASSES 3
To show that the maps (0.1) and (0.2) need not be injective, we will show that for
certain σ the image i (σ) is non-zero in the stalk 2 ((E, )) for all q F. The latter
F HDR ∇ q ∈
is equivalent to saying that for any divisor C S, with F C
⊂ 6⊂
(0.3) i (σ) i H2 (S,(E, )).
F 6∈ C DR,C ∇
For any smooth dense open subscheme C of C and for S = S (C C ) one has
0 0 0
− −
(0.4) H2 (S,(E, )) = H2 (S ,(E, )) = H0 (C ,(E, ) ) = H0 (C˜,ν∗(E, ))
DR,C ∇ DR,C0 0 ∇ DR 0 ∇ |C0 DR ∇
where ν : C˜ S is the normalization of C S. One way to think of this is analytically.
→ ⊂
Let L be the kernel of , let λ : S S, j : S C S and j = λ j . Then
0 0 0 0 0 0
∇ → − → ◦
H2(S,L) = H0(S,R1j L) and
C ∗
H2 (S ,L) = H0(S,λ R1j L) = H0(S ,R1j L) = H0(C ,L ).
C0 0 ∗ 0∗ 0 0∗ 0 |C0
As S S has codimension 2 in S, Riλ L = 0 for i = 1,2. Thus by the Leray spectral
0 ∗
− ≥
sequence for j = λ j the restriction map R1j L λ R1j L is an isomorphism. Since
0 ∗ ∗ 0∗
H0(C ,L ) = H0(◦C˜,ν∗L) this concludes the proo→f of (0.4). In other words, we do as if
0 |C0
C was smooth.
This way to force geometrically the existence of sections which have nothing to do with
the connection was used by the first author in [5] for example 2.1. We thank I. Panin for
explaining us the proof of his manuscript [8].
1. Gauß-Manin systems
Let ϕ : Y B be a semi-stable family of curves of genus g 1 with B a smooth
→ ≥
projective curve and Y a smooth projective surface, defined over an algebraically closed
field k of characteristic zero. We assume throughout this section
Assumption 1.1. ϕ ω is ample.
∗ Y/B
Let B be the open subscheme of B with Y = ϕ−1(B ) smooth over B , and as in 0.3
0 0 0 0
let δ : S S′ = B B be the blow up of a point p = (b ,b ) B B with exceptional
0 0 1 2 0 0
→ × ∈ ×
divisor F. Let
f : X = (Y Y ) S S
0 × 0 ×(B0×B0) −−→
be the pullback family. We consider the Gauß-Manin connection (R2f Ω• , ). On the
∗ X/S ∇
de Rham cohomology
H0 (F,(R2f Ω• , ) ) = H2 (Y Y )
DR ∗ X/S ∇ |F DR b1 × b2
one has the F-filtration F0 F1 F2 which defines a pure Hodge-structure after base
⊃ ⊃
extension from k to C.
Claim 1.2. Forσ F0 F1 H2 (Y Y ) and for allq F the image i (σ) is non-zero
∈ { − } DR b1× b2 ∈ F
in the stalk 2 ((R2f Ω• , )) .
HDR ∗ X/S ∇ q
Proof. Let C S be a reduced curve with F C, let ν : C˜ S be the normalization of
C, let δ¯: S¯ ⊂B B be the blow up of p, and6⊂let n : Γ S¯→be the normalization of the
→ ¯× →
closure of C in S. Let us denote by
¯
h : X = (Y Y) Γ Γ,
Γ (B×B)
× × −−→
h : X = (Y Y ) C˜ C˜,
C˜ 0 × 0 ×(B0×B0) −−→
X = X F = P1 Y Y P1
F ×S × b1 × b2 −−→
4 HE´LE`NE ESNAULT AND ECKART VIEHWEG
the induced families of surfaces. The Gysin map i : H2 (X ) H4 (X), followed by
C˜ DR C˜ → DR
the restriction map
ρ : H4 (X) H4 (X ) = H4 (P1 Y Y )
F DR −−→ DR F DR × b1 × b2
equals the restriction map
ρν−1(F) : HD2R(XC˜) −−→ M HD2R(XC˜ ×C˜ {c}) = M HD2R({ν(c)}×Yb1 ×Yb2)
c∈ν−1(F) c∈ν−1(F)
followed by the sum of the Gysin maps
M HD2R({ν(c)}×Yb1 ×Yb2) −−→ HD4R(P1 ×Yb1 ×Yb2).
c∈ν−1(F)
On the other hand, ρν−1(F) factors through the surjective map
ρ : H2 (X ) H0 (C˜,(R2h Ω• , )) = H2 (S,(R2f Ω• , )),
DR C˜ −−→ DR ∗ XC˜/C˜ ∇ DR,C ∗ X/S ∇
and ρ i = ρ′ i ρ where ρ′ is the restriction map
F ◦ C˜ F ◦ C ◦ F
ρ′ : H2 (S,(R2f Ω• , )) H2 (F,(R2f Ω• ), ) ) = H4 (P1 Y Y ),
F DR ∗ X/S ∇ −−→ DR ∗ X/S ∇ |F DR × b1 × b2
and i : H2 (S,(R2f Ω• , )) H2 (S,(R2f Ω• , ))
C DR,C ∗ X/S ∇ −−→ DR ∗ X/S ∇
¯
as in 0.3. By definition, h is a semi-stable family of surfaces, with singular fibres Z =
¯h−1( ) for = n−1(S¯ S) = Γ C˜. Hence R2h¯ Ω• (logZ) is the Deligne extension
∞ ∞ − − ∗ XΓ/Γ
[2] of its restriction to C˜. Thus
H0 (C˜,(R2h Ω• , )) = H0(Γ,Ω•(log ) R2h¯ Ω• (logZ)),
DR ∗ XC˜/C˜ ∇ Γ ∞ ⊗ ∗ XΓ/Γ
and ρ( F0/F1 H2 (X )) = ρ(H2(X , )) H0(Γ,R2h¯ ).
{ } DR C˜ C˜ OXC˜ ⊂ ∗OXΓ
The sheaf
R2h¯ = (δ¯ n)∗(pr∗R1ϕ pr∗R1ϕ )
∗OXΓ ◦ 1 ∗OY ⊗ 2 ∗OY
is dual to (δ¯ n)∗(pr∗ϕ ω pr∗ϕ ω ). Since δ¯ n is finite, the latter is ample and
◦ 1 ∗ Y/B ⊗ 2 ∗ Y/B ◦
H0(Γ,R2h¯ ) = 0. Thus, since the restriction and Gysin maps are morphisms of mixed
∗OXΓ
Hodge structures [4],
(1.1) im(ρ i ) = im(ρ′ i ) F3H4 (P1 Y Y ).
F ◦ C˜ F ◦ C ⊂ DR × b1 × b2
On the other hand, ρ′ i is the multiplication by ( 1) = deg( (F)). Hence for
F ◦ F − OF
σ F0 F1 H2 (Y Y ) one has
∈ { − } DR b1 × b2
(1.2) ρ′ i (σ) F2 F3 H4 (P1 Y Y ).
F ◦ F ∈ { − } DR × b1 × b2
(1.1) and (1.2) imply i (σ) i H2 (S,(E, )), and, as explained in 0.3, this proves the
F 6∈ C DR,C ∇
claim 1.2.
Of course there are lots of families of semi-stable curves ϕ : Y B which satisfy the
→
assumption 1.1. The most elementary one is:
Example 1.3. Choose ϕ : Y B to be a semi-stable non-isotrivial family of elliptic
→
curves.
In fact, some power of the sheaf ϕ ω is the pullback of an ample invertible sheaf on
∗ Y/B
the moduli space M¯ = P1 of stable elliptic curves.
1
GERMS OF DE RHAM COHOMOLOGY CLASSES 5
Example 1.4. There exist smooth families of curves ϕ : Y B of genus g 3 over a
→ ≥
projective curve B with ϕ ω ample, and hence there exist smooth families of surfaces
∗ Y/B
f : X S with S projective, for which the map (0.2) is not injective.
→
Proof. Let M and A be the moduli spaces of curves of genus g with level 3 structure and
g,3 g
of g-dimensional principally polarized abelian varieties, respectively. For g 3 the image
≥
of M in the Baily-Borel compactification of A is a projective manifold whose boundary
g,3 g
has codimension larger than or equal to two. Hence M has a projective compactification
g,3
with the same property. Taking hyperplane intersections one obtains a smooth projective
curve B in M , and thereby a smooth family of curves ϕ : Y B.
g,3
→
In order to show that for B in general position, ϕ ω is ample, we may assume that
∗ Y/B
k = C.
The monodromy representation of the fundamental group of M is irreducible, and for
g,3
B in general position the fundamental group of B maps surjectively to the one of M .
g,3
Hence the monodromy representation for B is irreducible as well.
On the other hand, by [7], 4.10, the sheaf ϕ ω is the direct sum of an ample vec-
∗ Y/B
tor bundle and a vector bundle, flat with respect to the Gauß-Manin connection. The
irreducibility of the monodromy representation implies that the latter is trivial.
2. Hypergeometric equations
As in 0.3 let S′ = P2 D, where D is the union of three lines H , H and H in general
1 2 3
−
position.
Example 2.1. Choose a ,a ,a k such that
1 2 3
∈
i) the elements 1,a ,a k are Q-linearly independent, for 1 i < j 3,
i j
∈ ≤ ≤
ii) a +a +a = 0;
1 2 3
for example, a = √2, a = √3.
1 2
Let ω H0(P2,Ω1 (logD)) be the unique form with res ω = a for i = 1,2,3, and
∈ P2 Hi i
(E′, ′) = ( ,d+ω). As in 0.3 consider the blow up δ : S S′ with exceptional divisor
S′
∇ O →
F and the pullback (E, ). We take a section 0 = σ k = H0(F,(E, ) ) and regard its
F
∇ 6 ∈ ∇ |
image i (σ) under the Gysin map.
F
Claim 2.2. For all reduced curves C S not containing F one has H2 (S,(E, )) = 0.
⊂ DR,C ∇
In particular, 0 = i (σ) 2 ((E, )) for all q F.
6 F ∈ HDR ∇ q ∈
Proof. Let C be the smooth locus of C. As in (0.4)
0
H2 (S,(E, )) = H0 (C ,(E, ) ),
DR,C ∇ DR 0 ∇ |C0
and since (E, ) has rank one, the claim 2.2 is equivalent to (E, ) = ( ,d).
Let δ¯ : S¯ ∇P2 be the blow up of p, and let n : Γ S¯ be t∇he|Cn0o6rmaOliCz0ation of the
closure of C →in S¯. For = n−1δ¯−1(D), and for some m→ N sufficiently large, one has
i
∞ ∈
HD0R(C0,(E,∇)|C0) = H0(Γ,Ω•Γ(log∞)⊗(δ¯◦n)∗(OP2(PmiHi),d+ω)).
The residues of (δ¯ n)∗( P2(PmiHi),d+ω) along x Γ are in
◦ O ∈
(N 0 ) (a m ) for δ¯(n(x)) H H ,
−{ } · i − i ∈ i −Sj6=i j
(N 0 ) (a m )+(N 0 ) (a m ) for δ¯(n(x)) H H .
i i j j i j
−{ } · − −{ } · − ∈ ∩
Indeed, the residue of ( P2(PmiHi),d +ω) along Hi is ai mi, and Γ lies on a surface
O −
X obtained by a sequence of blow ups X S. As well known, if z is a smooth point on
→
a variety Z with local coordinates x ,... ,x , and (E, ) a differential equation defined
1 n
∇
6 HE´LE`NE ESNAULT AND ECKART VIEHWEG
around z by ω = n b dxi + η for a regular form η on Z, then the pullback differential
Pi=1 i xi
equation on the blow up of z has residue n b along the exceptional locus.
Pi=1 i
By the assumption i) in 2.1, the residues of (δ¯ n)∗( P2(PmiHi),d+ω) can not be in
◦ O
Q, and a fortiori (E, ) can not be trivial.
∇ |C0
3. Unitary rank one sheaves
Example 3.1. Let B and B be two non-isogeneous elliptic curves, defined over k = C,
1 2
let L Pic0(B ) be non-torsion, and let be the unique unitary connection on L . Using
i i i i
∈ ∇
the notations introduced in 0.3, we choose S′ = B B and
1 2
×
(E′, ′) = (pr∗L pr∗L ,pr∗ pr∗ ).
∇ 1 1 ⊗ 2 2 1∇1 ⊗ 2∇2
Then for the pullback (E, ) of (E′, ′) on the blow up S of a point p, the map (0.1) will
∇ ∇
not be injective.
In fact, for the exceptional divisor F on S one has H0 (F,(E, ) ) = C whereas for
DR ∇ |F
all reduced curves C S with F C one has:
⊂ 6⊂
Claim 3.2. H2 (S,(E, )) = 0.
DR,C ∇
Proof. Let again C denote the smooth locus of C and let n : Γ S be the normalization.
0
→
By (0.4)
HD2R,C(S,(E,∇)) = HD0R(C0,(E,∇)|C0) = HD0R(Γ,n∗(E,∇)) ⊂ MH0(Γj,p∗j,1L1 ⊗p∗j,2L2)
j
where Γ are the irreducible components of Γ, and where p denotes the restriction of
j j,i
pr δ n : Γ B to Γ . If p is dominant, the image B′ of
i ◦ ◦ → i j j,i i
p∗ : Pic0(B ) Pic0(Γ )
j,i i −−→ j
is isogeneous to B and it is the Zariski closure of the subgroup generated by p∗ L . Hence
i j,i i
if one of the projections, say p , maps Γ to a point p∗ L p∗ L = p∗ L has no global
j,1 j j,1 1⊗ j,2 2 j,2 2
section.
If both, p and p are dominant, the two elliptic curves B′ and B′ are not isogeneous,
j,1 j,2 1 2
hence B′ B′ is finite, and H0(Γ ,p∗ L p∗ L ) = 0.
1 ∩ 2 j j,1 1 ⊗ j,2 2
References
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169 (1970)
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IHES 35 (1968) 107-127
[4] Deligne, P.: Th´eorie de Hodge II, Publ. Math. IHES 40 (1971) 5-58
[5] Esnault, H.: Bloch-Ogus sheaves, handwritten notes, distributed at the ICTP, Sept. 1997
[6] Grothendieck, A.: On the de Rham cohomology of algebraic varieties, Publ. Math. IHES 29 (1966)
95-103
[7] Koll´ar,J.: SubadditivityoftheKodairadimension,in: AlgebraicGeometry,Sendai1985,Adv.Studies
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Universita¨t Essen, FB6 Mathematik, 45 117 Essen, Germany
E-mail address: [email protected]
[email protected]
