An invariant for varieties in positive characteristic

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by  G. van der Geer

AN INVARIANT FOR VARIETIES IN POSITIVE
CHARACTERISTIC
2
0
0
G. VAN DER GEER AND T. KATSURA
2

n
a Abstract. We introduce an invariant of varieties in positive characteristic
J whichgeneralizesthea-numberofanabelianvariety. Wecalculateitinsome
examples anddiscussitsmeaningformoduli.
5
2

]
G 1. Introduction
A
Varieties in positive characteristic can differ markedly from varieties in charac-
.
h teristic zero. One aspect that makes this clear is Hodge theory. If X is a smooth
t varietyinpositivecharacteristicforwhichtheHodge-to-deRhamspectralsequence
a
m degeneratesthenthedeRhamcohomologygroupsHm(X)possesstwofiltrations,a
dR
decreasingfiltrationcalledtheHodgefiltrationF•andanincreasingfiltrationcalled
[
theconjugatefiltrationG . Theconjugatefiltrationistheanalogueofthe complex
•
1 conjugate of the Hodge filtration in characteristic zero. But while in characteristic
v
zerothe Hodge filtrationand the conjugate filtrationarealwaystransversal,this is
6
no longer the case in positive characteristic and this gives rise to interesting new
4
2 invariantsofalgebraicvarietiesinpositivecharacteristic. Varietiesinpositivechar-
1 acteristicforwhichtheHodgefiltrationandthe conjugatefiltrationaretransversal
0 are called ‘ordinary’ and resemble in some way smooth complex varieties, while
2
varieties with non-transversalfiltrations resemble singular complex varieties.
0
TherelativepositionofthetwofiltrationsonthedeRhamcohomologyisencoded
/
h by a double coset of a Weyl group and is rather complicated. We introduce an
t
a invariantwhich measuresthe positionof the firststep ofthe conjugate filtrationin
m theHodgefiltration. IfX isanabelianvarietythenthisinvariantcoincideswiththe
: a-number as defined by Oort. The non-transversalityis related to the cohomology
v
of the sheaves B Ωi introduced by Illusie.
i 1
X In this paper after some preliminaries we define the a-number and show that
r this definition coincides with the definition of the a-number for abelian varieties
a
andindicatethemeaningofthisnumberforthemoduliofabelianvarieties. Thisis
relatedtoworkofOgusonthe orderofsingularityofthe Hasselocusinthe moduli
of Calabi-Yau varieties. We then show how to calculate the a-number for Fermat
varieties using the Poincar´eresidue map. After that we show an inequality for the
a-numberofCalabi-Yauthreefoldsandfinishwitha discussionofthe a-numberfor
a family of quintic Calabi-Yau varieties.
2. The Two Spectral Sequences
LetX beasmoothcompletealgebraicvarietydefinedoveranalgebraicallyclosed
field k of characteristic p>0. Let X′ be the base-change of X with respect to the
1991 Mathematics Subject Classification. 14K10.
ANUMBER.TEXFebruary 1, 2008
1
2 G. VAN DER GEER AND T. KATSURA
Frobenius homomorphism of k. The absolute Frobenius F : X −→ X factors
abs
through the relative Frobenius F : X−F→X′ −→ X. There are two spectral
abs
sequences converging to the de Rham cohomology: the first is the Hodge spectral
sequence
Eij =Hj(X,Ωi)=⇒Hi+j(X).
1 dR
This spectral sequence arises from the filtration Ω≥i of the de Rham complex Ω•.
The second spectral sequence is the so-called conjugate spectral sequence which
comesfromthetheLerayspectralsequencefortherelativeFrobeniusF :X −→X′,
that yields a spectral sequence
Ei,j =Hi(X′,H(F Ω• ))=⇒Hi+j(X/k).
2 ∗ X/k dR
But the Cartier operator yields an isomorphism of sheaves on X′
C−1: Ωi −∼→Hi(F (Ω• )),
X′/k ∗ X/k
so that we can rewrite this as
Eij =Hi(X′,Hj(F (Ω• )))=Hi(X′,Ωj )=⇒H∗ (X).
2 ∗ X/k X′/k dR
We assumethat the Hodge-to-deRhamspectralsequence forX degeneratesatthe
E -level. This happens for example if p>dim(X) andX canbe lifted to the Witt
1
vectors W (k), see [De-Il]. Then the de Rham cohomology carries a filtration, the
2
Hodge filtration
F• : Hm =F0 ⊃F1 ⊃...⊃Fm,
dR
with graded pieces
griHm(X)=Hm−i(X,Ωi).
dR
Moreover, then also the conjugate spectral sequence degenerates at the E -level,
2
leading to the so-called conjugate filtration
G :(0)⊂G ⊂G ⊂...⊂G =Hm(X)
• 0 1 m dR
with graded pieces
gr Hm(X)=Hi(X′,Ωm−i).
i dR X′
Moreover, if m = n := dim(X) then we have a non-degenerate pairing h, i on the
de Rham cohomology
Hn (X)×Hn (X)→H2n(X)∼=k
dR dR dR
Note that for the conjugate spectral sequence we have
E2n0 =Hn(X′,H0(Ω•X′/k))=Hn(X′,OX′)
and since Fa∗bs(Hn(X,OX)) = Hn(X′,OX′) we see that G0 = Fa∗bs(Hn(X,OX)).
In particular, the composition
Hn(X,O )−F→∗ G −→F0/F1 ∼=Hn(X,O ).
X 0 X
is the Hasse-Witt map.
AN INVARIANT FOR VARIETIES IN POSITIVE CHARACTERISTIC 3
3. The relative position of two filtrations
Giventwo filtrationsonthe de Rhamcohomologyit is naturalto comparethem
toobtaininterestinginformationonthevariety. Ingeneral,ifonehastwofiltrations
on a finite-dimensional k-vector space V
F• :V =F0 ⊃F1 ⊃...⊃Fm, G :G ⊂G ⊂...⊂G =V
• 0 1 m
such that rank(G ) = rank(Fm−i) the relative position of two such filtrations is
i
encodedbyanelementofadoublecosetofaWeylgroupofthegenerallineargroup
G = GL(V). Sometimes, the vector space V is provided with a non-degenerate
pairing h, i and then we can consider the group G =GSp(V) instead of GL(V).
We fix a maximal torus T and a Borel subgroup B containing T and let W
0
be the Weyl group of G. The (partial) flags F• and G determine a parabolic
•
subgroups P and P which are conjugate under W. Let W = W be the Weyl
F G F G
group of F•. Then the relative position of F• and G is given by an element of
•
w(F•,G )∈W \W/W .
• F G
To define it, recall that if B denotes the variety of Borel subgroups of G then we
have a bijection φ : W−∼→G\B×B, given by w 7→ the orbit of (B,wBw−1). We
chooseBorelsubgroupsB ⊂P andB ⊂P anddefinew(F•,G )asthedouble
F F G G •
coset containing (B ,B ) and this is independent of the choices of B and B .
F G F G
Note that the double coset W \W/W is in bijection with P \G/P , cf. [9].
F G G F
Definition 3.1. Let X be a smooth variety of dimension n in characteristicp>0
for which the Hodge-to-de Rham spectral sequence degenerates and let m be an
integer with 0 ≤ m ≤ 2n. We define the pointer w (X) of X as the element of
m
W \W/W associated to the two filtrations {F•} and {G } on Hm(X).
F G • dR
Wecanrefinethisconsiderablybychoosingfullfiltrationsrefiningthefiltrations
F• and G
•
V =Φ0 ⊃Φ1 ⊃...⊃Φdim(V),
Γ ⊂Γ ⊂...⊂Γ =V
0 1 dim(V)
that are compatible with the action of Frobenius in the following sense. Using
crystalline cohomology we know that F/pi acts on Fi/Fi+1 and the induced fil-
tration of Φ• should be stable under F/pi. Moreover, the filtration Φ• induces
via Gi/Gi−1 ∼= Fi−1/Fi the filtration Γ•. Comparing the two filtrations gives the
finer invariant. If we carry this out for principally polarized abelian varieties we
retrieve the Ekedahl-Oort type of the abelian variety, cf. [12] where this invariant
was introduced in a different way. We refer to [3] and to [2], where this will be
worked out in detail for abelian varieties, K3-surfaces and Calabi-Yau’s and leads
to stratifications on moduli spaces (cf. also [9], [4]).
4. The a-number of a variety in positive characteristic
We are now interested in the case of the cohomology group Hm(X) with m =
dR
n = dim(X). We shall assume throughout that the Hodge-to-de Rham spectral
sequence degenerates at the E -level. Since the pointer w (X) is a complicated
1 n
invariant we look first at the position of G , the image of Hn(X,O ) under the
0 X
Frobenius operator in the Hodge filtration F• on Hn (X). We distill an invariant
dR
in the following way.
4 G. VAN DER GEER AND T. KATSURA
Definition 4.1. The a-number of X is the maximum step in the Hodge fitration
that contains F∗ (Hn(X,O )):
abs X
a(X):=max{j :FjHn (X)containsF∗ (Hn(X,O ))},
dR abs X
or equivalently,
a(X)=max{j :Fj ⊇G }.
0
Note that 0≤a(X)≤n=dim(X) and that a(X)>0 if and only if the Hasse-
Witt map vanishes. Of course, by going from w (X) to a(X) we loose a lot of
n
information. If we wish to retain more information we can refine the a-number
slightly by taking the a-vector (a ,a ,... ,a )(X) with
0 1 m
a (X):=dimG ∩Fj j =0,1,... ,m.
j 0
Sometimes,ifHn(X,O )vanishes,wecanstilldefineananalogueofthea-number.
X
Consider for example the case of a smooth cubic hypersurface in P8. The Hodge
numbers hij with i+j =7 are
h0,7 =h7,0 =0, h1,6 =h6,1 =0, h2,5 =h5,2 =1, h3,4 =h4,3 =360.
and we can define
a′(X)=max{j :Fj ⊇G }.
2
Question 4.2. i) Can we give geometric interpretations of these numbers? ii) Is a
varietywith a(X)=dim(X)rigid? ; i.e., are there deformations that preservethe
a-number?
5. Closed differential forms
Let X be a smooth projective variety of dimension n in characteristic p > 0
and let F : X −→ X be the absolute Frobenius. We consider the direct image
abs
F O . As a sheaf of abelian groups it is just O , but its O -module structure
abs∗ X X X
is different: f ◦g =fpg. One way to define the sheaf B Ωi is via the short exact
1 X
sequence
0→Ωi−1 −→F Ωi−1−d→B Ωi →0.
X abs∗ X 1 X
Here we view B Ωi as an O -module. It can also be viewed as a locally free
1 X X
subsheaf of F Ωi on X′. Here F denotes the relative Frobenius. Moreover,we set
∗ X
B Ωi =0. Inductively we define
0 X
B Ωi =C−1(B Ωi ),
j+1 X j X
where C :Z Ωi →Ωi is the Cartier operator. Similarly, we define
1 X X
Z Ωi =Ωi , Z Ωi =Ωi
0 X X 1 X X,d-closed
and inductively
Z Ωi =C−1(Z Ωi ) for j ≥1.
j+1 X j X
We canview these aslocally free subsheavesof FjΩi on X(pj), the base change of
∗ X
X under the j-th power of relative Frobenius. The inverse Cartier operator gives
rise to an isomorphism
C−j :Ωi −∼=→Z Ωi /B Ωi .
X(pj) j X j X
We have a perfect pairing
F Ωi ⊗F Ωn−i −→Ωn, (ω ,ω )7→C(ω ∧ω ).
abs∗ X abs∗ X X 1 2 1 2
AN INVARIANT FOR VARIETIES IN POSITIVE CHARACTERISTIC 5
On the other hand we have an exact sequence
0→Ωi −→F Ωi −d→B Ωi+1 →0.
X abs∗ X 1 X
and
0→B Ωn−i −→F Ωn−i−C→Ωn−i →0.
1 X abs∗ X X
This induces a perfect pairing
B Ωi+1⊗B Ωn−i →Ωn.
1 X 1 X X
Definition 5.1. (Illusie, Raynaud)We callthe varietyX ordinaryif the cohomol-
ogy groups Hi(X,B Ωj ) vanish for j ≥1 and all i.
1 X
This implies that all global forms are closed, just as in characteristic 0.
Lemma 5.2. If X is ordinary then a(X)=0.
Proof. Consider the exact sequence
0→O −F→O −d→B Ω1 →0.
X X 1 X
In cohomology this gives
Hn−1(X,B Ω1 )→Hn(X,O )→Hn(X,O )→Hn(X,B Ω1 )
1 X X X 1 X
from which it follows that the Hasse-Witt map has trivial kernel and thus that
G 6⊆F1(Hn (X)).
0 dR
6. Abelian Varieties
Let X be an abelian variety of dimension g over an algebraically closed field k.
We denote by X[p] the kernel of multiplication by p. It is a group scheme of order
p2g. The classical a-number of X (cf. [11]) is defined by
a(X):=dim Hom (α ,X).
k k p
Here α is the group scheme of order p that is the kernel of Frobenius acting
p
on the additive group G . Note that Hom (α ,X), the space of group scheme
a k p
homomorphisms of α to X, is in a natural way a vector space over k. If we let
p
A(X) be the maximal subgroup scheme of X[p] annihilated by the operators F
(Frobenius) and V (Verschiebung), then A(X) is the union of the images of all
group scheme homomorphisms α →X and we have
p
a(X)=log ordA(X).
p
We have 0≤a(X)≤g.
It is well-knownthat the contravariantDieudonn´e module D(X) of X[p] can be
identified with the first de Rham cohomology H1 (X) of X. The k-vector space
dR
H1 (X) then carries two operators F and V and the Dieudonn´e module of A(X)
dR
coincides with the intersection ker(V)∩ker(F) on D(X). If we identify the kernel
of F with H0(X,Ω1 ) ⊂ H1 (X) then the Dieudonn´e module of A(X) may be
X dR
identified with the kernel of V acting on H0(X,Ω1 ). We thus find
X
D(A(X))∼=ker(V :H0(X,Ω1 )→H0(X,Ω1 ))∼=H0(X,B Ω1 ).
X X 1 X
and we have
a(X)=dim H0(X,B Ω1 ).
k 1 X
6 G. VAN DER GEER AND T. KATSURA
We also have the following relation:
F(H1(X,O ))∩H0(X,Ω1 )=H0(X,B Ω1 ). (1)
X X 1 X
This follows from the fact that Im(F) = Ker(V). We now show that for abelian
varieties the classical a-number and our a-number coincide.
Proposition 6.1. IfX isanabelian varietyofdimensiong thenthetwonotionsof
a-numbercoincide: if F• and G aretheHodge andconjugate filtrationon Hg (X)
• dR
we have
dim Hom (α ,X)=max{j :G ⊂Fj}.
k k p 0
Proof. Recall that Hg (X) = ∧gH1 (X) and if we write H1 (X) = V ⊕ V
dR dR dR 1 2
with V = H0(X,Ω1 ) and V a complementary subspace, then the Hodge fil-
1 X 2
tration on Hg (X) is Fr = g ∧jV ⊗ ∧g−jV . We have F(Hg(X,O )) =
dR j=r 1 2 X
F(∧gH1(X,O )) = ∧gF(H1(X,O )). If we write F(H1(X,O )) = A⊕B with
X P X X
A=H0(X,B Ω1 )andB acomplementaryspace,then∧g(A⊕B)=∧aA⊗∧g−aB
1 X
with a=dim(A). From this and (1) it is clear that F(Hg(X,O )) lies in Fa, but
X
not in Fa+1.
We now show that the a-number has some meaning for the geometry of moduli
spaces.
LetT(a)bethelocusinsidethemodulispaceA ofprincipallypolarizedabelian
g
varietieswith a-number≥a. HereA isviewedasanalgebraicstack,orwe should
g
add a sufficient level structure to our abelian varieties. Over A the de Rham
g
bundleH1 possesestwosubbundles ofrankg,the Hodgebundle Eandthe kernel
dR
F ofF. ThenT(a)is definedas the degeneracylocus whereE∩F has rankatleast
a. It is known that dimT(a)=g(g+1)/2−a(a+1)/2, cf.[3], [2].
Proposition 6.2. ThelocusT(a)in A is smoothoutsideT(a+1)andthenormal
g
space to T(a) at [X] with a(X)=a can be identified with Sym2(H0(X,B Ω1 )).
1 X
Proof. An infinitesimal deformation of the principally polarized abelian variety X
is given by a symmetric g × g-matrix T = (t ) which can be interpreted as a
ij
symmetric endomorphism of H0(X,Ω1 ), cf. [8]. This deformation preserves the
X
a-numberofX ifitkeepsthe kernelH0(X,B Ω1 )ofV actingonH0(X,Ω1 ),that
1 X X
is, T is a symmetric endomorphism of this subspace. The principal polarization
identifies this subspace with its dual, and this gives us the result.
Proposition 6.3. Let X be an abelian variety with a(X)=g. Then the multiplic-
ity of the point [X] on T(1) is g.
Proof. If X is a principally polarized abelian variety with a(X) = g we choose a
base ω ,... ,ω of H0(X,Ω1 ) and complete it to a basis of H1 (X) as η ,... ,η
1 g X dR 1 g
such that hω ,η i = δ . Then for a deformation with parameter a symmetric
i j ij
g×g-matrix (t ) we have for i=1,... ,g
ij
Fη =ω + t η
i 1 ij j
ω =V( Xt η ).
i ij i
Then we get X
F(η ∧···∧η )=ω ∧···∧ω +···+det(t )η ∧···∧η .
1 g 1 g ij 1 g
AN INVARIANT FOR VARIETIES IN POSITIVE CHARACTERISTIC 7
But the equation of T(1) is
hF(η ∧···∧η ),η ∧···∧η i=det(t ).
1 g 1 g ij
7. Cohomology of Hypersurfaces
LetX beanirreduciblesmoothprojectivehypersurfaceofdegreedinPn+1given
by an equation f = 0. Then the primitive cohomology Hn (X) in the middle
dR 0
dimension can be described by the Poincar´e residue map as follows. Consider the
rational differential forms on projective space
n+1
Ω
Ω= (−1)ix dx ∧...∧dxˆ ∧...∧dx and Ω∗ = .
i 0 i n+1
x ...x
0 n+1
i=0
X
Note that Ω∗ is a logarithmic form: in affine coordinates we have Ω∗ = du /u ∧
1 1
...∧du /u .
n+1 n+1
We let L be the k-vector space generated by the monomials xw =xw0...xwn+1
0 n+1
with n+1w ≡0(modd) and we let L′ ⊂L be the subspace generatedby the xw
i=0 i
with all w ≥1, i.e., the elements in L′ are divisible by x ...x . For an element
i 0 n+1
xw ∈PL′ we set γ(w)= w /d.
i
On L we have operators D defined by
i
P
∂xw ∂f ∂f
D (xw)=x +x xw =w xw+x xw. (2)
i i i i i
∂x ∂x ∂x
i i i
There is a natural map
xw
φ:L′ −→Hn+1(Pn+1\X), xw 7→(−1)γ−1(γ−1)! Ω∗.
dR fγ
Now we use the Poincar´e residue map
Res:Hn+1(Pn+1\X)−→Hn (X)
dR dR
and observe that the composition ρ:=Res·φ factors through quotient
n+1
W′ =L′/L′∩ D L.
i
i=0
X
andwegetanisomorphismW′ ∼=Hn (X) whichiscompatiblewiththepoleorder
dR 0
filtrationonthesourceandtheHodgefiltrationonthetarget,cf.[7],cf.also[1],[13].
The image of Hn(X,O ) under the action of absolute Frobenius on Hn (X) is up
X dR
to a multiplicative constant given by raising xw/fγ to the p-th power and then
reducing the pole order by making use of the relations (2).
8. Fermat Varieties
We give examples of surfaces with a-number equal to 0,1 or 2.
Theorem 8.1. Let p be a prime different from 5. Let X be the Fermat surface in
characteristic p defined by 3 x5 =0 in P3. Then the a-number of X satisfies
i=0 i
P 0 p≡1(mod5)
a(X)= 1 p≡2, 3(mod5)

2 p≡4(mod5).

8 G. VAN DER GEER AND T. KATSURA
Proof. We use the description of the (primitive) cohomology H2(X,O ) with the
X
Poincar´eresidue map
Ax x x x
P :V −→H2 (X), A7→Res( 0 1 2 3Ω∗)
dR f3
withV thevectorspaceofhomogeneouspolynomialsofdegree11. Abasisofthe4-
dimensionalspaceH2(X,O ) is givenbyα =P((x x x x )3/x )for i=0,... ,3.
X i 0 1 2 3 i
Since Ω∗ is a logarithmic form the image of α under Frobenius is given by the
0
Poincar´eresidue of
(x x x x )3p(x x x )p
0 1 2 3 1 2 3 Ω∗
f3p
and using the relations −5x5xw = w xw for any monomial xw obtained from (2)
i i
this is equivalent to an expression res(gΩ∗) with g given by
(x x x x )3x x x /f3 p≡1(mod5)
0 1 2 3 1 2 3
(x x x x )(x x x )2/f2 p≡2(mod5)
0 1 2 3 1 2 3
x4(x x x )2/f2 p≡3(mod5)
0 1 2 3
x2x x x /f p≡4(mod5)
0 1 2 3
and similarly for the other α . This pole order implies that a(X) is as indicated.
i
Theorem 8.2. LetX betheCalabi-Yau FermatvarietyinPr givenbytheequation
xr+1 + ...+ xr+1 = 0. Then the a-number of X is the natural number a with
0 r
0≤a≤r−1 and a≡p−1(modr+1).
Proof. Under the Poincar´e residue map (x ...x )rΩ∗/fr maps to a generator of
0 r
Hr−1(X,O ). The image under Frobenius is (x ...x )pr/fpr × Ω∗ and using
X 0 r
again the relations (2) we can reduce the pole order pr modulo r+1. But pr ≡
−p(modr+1) and from this the result easily follows.
Thereexistvarietiesofarbitrarydimensionwithmaximala-numbernamelyabelian
varieties which are products of supersingular elliptic curves. Besides these abelian
varieties the following Calabi-Yau varieties give examples of such varieties.
Corollary 8.3. Let X be the (p−1)-dimensional Fermat variety defined by
xp+1+xp+1+···+xp+1 =0
0 1 p
in projective space of dimension p. Then the a-number of the Calabi-Yau variety
X is equal to p−1=dim(X).
Let X be the 7-dimensional cubic in P8 defined by 8 x3 = 0 in P8. Then
i=0 i
H7(X,O ) and H6(X,Ω1 ) vanish, while H5(X,Ω2 ) is 1-dimensional. Let a′(X)
X X X P
be the maximumj suchthatthe Hodge stepFj contains G (which canbe seenas
2
an image of H5(X,Ω2) under Frobenius divided by p2 in H2 (X), cf. [6]).
dR
Theorem 8.4. Let X be the 7-dimensional cubic in P8 defined by 8 x3 = 0.
i=0 i
Then we have:
P
2 p≡1(mod3)
a′(X)=
(5 p≡2(mod3).
AN INVARIANT FOR VARIETIES IN POSITIVE CHARACTERISTIC 9
Proof. Using again the description of primitive cohomology with the Poincar´e
residue map we consider the form
(x ...x )2
0 8 Ω∗
f6
and then see that the image under Frobenius is equivalent to the Poincar´e residue
of gΩ∗ with g given by (x ...x )2/f6 if p ≡ 1(mod3) and (x ...x )/f3 if p ≡
0 8 0 8
2(mod3).
9. Calabi-Yau Varieties
LetX beann-dimensionalCalabi-Yauvariety. Suchavarietyhasaheighth(X).
Thereareseveralpossibledefinitionsofthis, e.g. withformalgroups. Herewetake
as definition
h−1:=max{dimHn−1(X,B Ω1 ):i∈Z }.
i X ≥0
Onecanuse the methods of[4]to seethat this definitionagreeswith the definition
using the formal group Φn associated to X. The natural inclusion B Ω1 ⊆ Ω1
i X X
induces a natural map β :Hn−1(X,B Ω1 )→Hn−1(X,Ω1 ).
i i X X
Lemma 9.1. If the natural map β :Hn−1(X,B Ω1)→Hn−1(X,Ω1) is not injec-
i i
tive, then lim dimHn−1(X,B Ω1 )=∞.
i→∞ i X
Proof. The argument is similar to the cases of K3 surfaces, cf. [4].We give the
argument for i = 1. Suppose that β has a non-trivial kernel represented by a
1
cocycle {df }. Then we have a relation
α1,...,αn
df = (−1)jω .
α1,...,αn α1,...,αˆj,...,αn
Since the Cartier map H0(U,Ω1 X) → H0(U,Ω1) is surjective on affine sets U
U,closed U
we can find closed forms ω˜ and regular functions g on ∩U
α1,...,,αˆj,...,αn α1,...,αn αi
and obtain a new relation
fp−1 df +dg = (−1)jω˜ .
α1,...,αn α1,...,αn α1,...,αn α1,...,,αˆj,...,αn
X
Onenowchecksdirectlythatthecocycleatthelefthandsiderepresentsanelement
of Hn−1(X,B Ω1 ) which does not lie in the image of Hn−1(X,B Ω1 ). One also
2 X 1 X
checks that this element is non-zero. The argument for other i is similar.
Proposition 9.2. If h6=∞ then 1≤h≤h1,n−1+1.
Proof. If h6=∞ all the maps β are injective.
i
Remark 9.3. Wehaveh1(Θ)=h1(Ωn−1),soweaprioricouldexpectastratification
of the moduli with h steps; but the example of K3 surfaces shows that h cannot
assume all values between 1 and h1,1+1. Therefore one should find bounds for h.
Proposition 9.4. Let X be a Calabi-Yau variety such that the Hodge-to-de Rham
spectral sequence degenerates. Then we have either h(X) = 1 and a(X) = 0, or
2≤h(X)<∞ and a(X)=1, or h(X)=∞ and a(X)≥1.
10 G. VAN DER GEER AND T. KATSURA
Proof. We have h(X) = 1 if and only if the map F : Hn(X,O ) → F0/F1 =
abs X
Hn(X,O ) is non-zero and this is equivalent to F (Hn(X,O )) 6⊂ F1. So we
X abs X
have h(X)≥2 if and only if the map F maps Hn(X,O ) to F1 and then we have
X
the projection Hn(X,O ) −→ F1 −→ F1/F2 = Hn−1(X,Ω1 ). Now recall that
X X
we have an isomorphism d:O /FO ∼=B Ω1 and from the exact sequence
X X 1 X
F
0→O −→O −→O /FO →0
X X X X
wethusgetHn−1(X,B Ω1 )∼=Hn(X,O ). WeclaimthattheimageofHn(X,O )
1 X X X
under F in F1/F2 is the image of Hn−1(X,B Ω1 ) in Hn−1(X,Ω1 ). This can be
1 X X
checked by a direct computation: if {f } is a Cech n-cocycle representing a gener-
I
atorg ofHn(X,O )then F(g)is representedby {fp} andthis iscohomologousto
X I
0 in Hn(X,O ). So there exists an (n−1)-cocycle h such that δ(h) = fp. The
X J
image of g in F1/F2 is represented by dh . Now use the proof of the preceding
J
Proposition.
Corollary 9.5. Let X be the Fermat Calabi-Yau variety of degree r + 1 in Pr.
Assumethatthecharacteristicpdoesnotdivider+1andsatisfiesp6≡2(mod r+1).
Then the height h(X) is equal to 1 or ∞. Moreover, h(X) = 1 if and only if
p≡1(modr+1).
Remark 9.6. Thecorollaryholdsalsowithouttheconditionthatp6≡2(mod r+1),
but for the proof we then need to use Jacobi sums.
10. Relation with a Result of Ogus
In [10] Ogus proved a result on the order of vanishing of the Hasse-invariant of
a family of Calabi-Yau varieties that is closely related to the a-number introduced
here. Letf :X →S beafamily ofCalabi-YauvarietiessuchthatforeachfibreX
s
the Hodge-to-de Rham spectral sequence degenerates and such that the Kodaira-
Spencer mapping T → R1f (T ) is surjective. Then Ogus proves that the
S/k ∗ X/S
Hasse-invariant vanishes to order i at s if G (X ) ⊂ Fi(X ). Moreover, under an
0 s s
additional natural assumption on X/S the order of vanishing is exactly equal to
the a-number of X .
s
As an example we consider the family of Calabi-Yau hypersurfaces in P4 given
by
X : x5−5αx ···x =0.
α i 1 5
We have h1,1 = 1, h2,1 = 101 anXd b = 204. The Hasse-Witt invariant can be
3
calculated (cf. [6], §2.3)
[(p−1)/5]
(5m!)
H(α)= αp−1−5m.
(m!)5
m=0
X
This is a polynomial of degree p−1 in α. For α=0 we find by using Theorem 8.2
and the formula for H(α)
Proposition 10.1. Let p be a prime 6=5. Then the a-number of X is determined
0
by
a(X )=ord H(α)≡p−1(mod5).
0 0