On the Height of Calabi-Yau Varieties

in Positive Characteristic

G. van der Geer, T. Katsura

Received: April 1, 2003 Revised: September 5, 2003

Communicated by Peter Schneider

Abstract. We study invariants of Calabi-Yau varieties in positive characteristic, especially the height of the Artin-Mazur formal group. We illustrate these results by Calabi-Yau varieties of Fermat and Kummer type.

2000 Mathematics Subject Classification: 14J32, 14L05, 14F

1. Introduction

The large measure of attention that complex Calabi-Yau varieties drew in recent years stands in marked contrast to the limited attention for their counterparts in positive characteristic. Nevertheless, we think these varieties deserve a greater interest, especially since the special nature of these varieties lends itself well for excursions into the largely unexplored territory of varieties in positive characteristic. In this paper we mean by a Calabi-Yau variety a smooth complete variety of dimension n over a field with dim H^{i}(X, O_{X}) = 0 for i = 1, …, n−1 and with trivial canonical bundle. We study some invariants of Calabi-Yau varieties in characteristic p > 0, especially the height h of the Artin-Mazur formal group for which we prove the estimate h ≤ h^{1,n−1} + 1 if h ≠ ∞. We show how this invariant is related to the cohomology of sheaves of closed forms. It is well-known that K3 surfaces do not possess non-zero global 1-forms. The analogous statement about the existence of global i-forms with i = 1 and i = n − 1 on a n-dimensional Calabi-Yau variety is not known and might well be false in positive characteristic. We show that for a Calabi-Yau variety of dimension ≥ 3 over an algebraically closed field k of characteristic p > 0 with no non-zero global 1-forms there is no p-torsion in the Picard variety and Pic/pPic is isomorphic to NS/pNS with NS the Néron-Severi group of X. If in addition X does not have a non-zero global 2-form then NS/pNS ⊗_{F_{p}} k maps injectively into H^{1}(X, Ω_{X}^{1}). This yields the estimate ρ ≤ h^{1,1} for the Picard number. We also study Calabi-Yau varieties of Fermat type and of Kummer type to illustrate the results.

2. The Height of a Calabi-Yau Variety

The most conspicuous invariant of a Calabi-Yau variety X of dimension n in characteristic p > 0 is its height. There are several ways to define it, using crystalline cohomology or formal groups. In the latter setting one considers the functor F_{X}^{r} : Art → Ab defined on the category of local Artinian k-algebras with residue field k by

F_{X}^{r}(S) := Ker{H_{et}^{r}(X × S, 𝔾_{m}) ⟶ H_{et}^{r}(X, 𝔾_{m})}.

According to a theorem of Artin and Mazur [2], for a Calabi-Yau variety X and r = n this functor is representable by a smooth formal group Φ_{X} of dimension 1 with tangent space H^{n}(X, O_{X}). Formal groups of dimension 1 in characteristic p > 0 are classified up to isomorphism by their height h which is a natural number ≥ 1 or ∞. In the former case (h ≠ ∞) the formal group is p-divisible, while in the latter case the formal group is isomorphic to the additive formal group 𝔾̂_{a}. For a non-singular complete variety X over an algebraically closed field k of characteristic p > 0 we let W_{m}O_{X} be the sheaf of Witt rings of length m, which is coherent as a sheaf of rings. It has three operators F, V and R given by F (a_{0}, …, a_{m}) = (a_{0}^{p}, …, a_{m}^{p}), V (a_{0}, …, a_{m}) = (0, a_{0}, …, a_{m}) and R(a_{0}, …, a_{m}) = (a_{0}, …, a_{m−1}) satisfying the relations RV F = F RV = RF V = p. The cohomology groups H^{i}(X, W_{m}O_{X}) with the maps induced by R form a projective system of finitely generated W_{m}(k)-modules. The projective limit is the cohomology group H^{i}(X, W O_{X}). Note that this need not be a finitely generated W (k)-module. It has semi-linear operators F and V. Let X be a Calabi-Yau manifold of dimension n. The vanishing of the groups H^{i}(X, O_{X}) for i ≠ 0, n and the exact sequence

0 → W_{m−1}O_{X} → W_{m}O_{X} → O_{X} → 0

imply that H^{i}(X, W_{m}O_{X}) vanishes for i = 1, …, n − 1 and all m > 0, hence H^{i}(X, W O_{X}) = 0. We also see that restriction R : W_{m}O_{X} → W_{m−1}O_{X} induces a surjective map H^{n}(X, W_{m}O_{X}) → H^{n}(X, W_{m−1}O_{X}) with kernel H^{n}(X, O_{X}). The fact that F and R commute implies that if the induced map F : H^{n}(X, W_{m}O_{X}) → H^{n}(X, W_{m}O_{X}) vanishes then F : H^{n}(X, W_{i}O_{X}) → H^{n}(X, W_{i}O_{X}) vanishes for i < m too. It also follows that H^{n}(X, W_{i}O_{X}) is a k-vector space for i < m. It is known by Artin-Mazur [2] that the Dieudonné module of the formal group Φ_{X} is H^{n}(X, W O_{X}) with W O_{X} the sheaf of Witt vectors of O_{X}. This implies the following result, cf. [3] where we proved this for K3-surfaces. We omit the proof which is similar to that for K3 surfaces.

Theorem 2.1. For a Calabi-Yau manifold X of dimension n we have the following characterization of the height:

h(Φ_{X}) = min{i ≥ 1: [F : H^{n}(W_{i}O_{X}) → H^{n}(W_{i}O_{X})] ≠ 0}.

We now connect this with de Rham cohomology. Serre introduced in [14] a map D_{i} : W_{i}(O_{X}) → Ω_{X}^{1} of sheaves in the following way:

D_{i}(a_{0}, a_{1}, …, a_{i−1}) = a_{0}^{p^{i−1}−1}da_{0} + … + a_{i−2}^{p−1}da_{i−2} + da_{i−1}.

It satisfies D_{i+1}V = D_{i}, and Serre showed that this induces an injective map of sheaves of additive groups

D_{i} : W_{i}O_{X}/F W_{i}O_{X} → Ω_{X}^{1}.  (1)

The exact sequence 0 → W_{i}O_{X} ⟶^{F} W_{i}O_{X} ⟶ W_{i}O_{X}/F W_{i}O_{X} → 0 gives rise to an isomorphism

H^{n−1}(W_{i}O_{X}/F W_{i}O_{X}) ≅ Ker[F : H^{n}(W_{i}O_{X}) → H^{n}(W_{i}O_{X})].  (2)

Proposition 2.2. If h ≠ ∞ then the induced map

D_{i} : H^{n−1}(X, W_{i}O_{X}/F W_{i}O_{X}) → H^{n−1}(X, Ω_{X}^{1})

is injective, and dim Im D_{i} = min{i, h − 1}.

Proof. We give a proof for the reader’s convenience. Take an affine open covering {U_{i}} of X. Assuming some D_{ℓ} is not injective, we let ℓ be the smallest natural number such that D_{ℓ} is not injective on H^{n−1}(W_{ℓ}O_{X}/F W_{ℓ}O_{X}). Let α = {f_{I}} with f_{I} = (f_{I}^{(0)}, …, f_{I}^{(ℓ−1)}) ∈ Γ(U_{i_{0},...,i_{n−1}}, W_{ℓ}O_{X}) represent a nonzero element of H^{n−1}(W_{ℓ}O_{X}/F W_{ℓ}O_{X}) such that D_{ℓ}(α) is zero in H^{n−1}(Ω_{X}^{1}). Then there exists elements ω_{J} = ω_{j_{0}j_{1}...j_{n−2}} in Γ(U_{j_{0}} ∩ … ∩ U_{j_{n−2}}, Ω_{X}^{1}) such that

∑_{j=0}^{ℓ−1}(f_{I}^{(j)})^{p^{ℓ−j−1}}d log f_{I}^{(j)} = ∑_{j} ω_{I_{j}},

where the multi-index I_{j} = {i_{0}, …, i_{n−1}} is obtained from I by omitting i_{j}. By applying the inverse Cartier operator we get an equation

∑_{j=0}^{ℓ−1}(f_{I}^{(j)})^{p^{ℓ−j}}d log f_{I}^{(j)} + df_{I}^{(ℓ)} = ∑_{j} ω̃_{I_{j}}

for certain functions f_{I}^{(ℓ)} and differential forms ω̃_{I_{j}} with C(ω̃_{I_{j}}) = ω_{I_{j}}. Since α is an non-zero element of H^{n−1}(W_{ℓ}O_{X}/F W_{ℓ}O_{X}), the element β = (f_{I}^{(0)}, …, f_{I}^{(ℓ−1)}, f_{I}^{(ℓ)}) gives a non-zero element of H^{n−1}(W_{ℓ+1}O_{X}/F W_{ℓ+1}O_{X}). In view of (2) for i = ℓ + 1 the element β gives a non-zero element β̃ of H^{n}(W_{ℓ+1}O_{X}) such that F (β̃) = 0 in H^{n}(W_{ℓ+1}O_{X}). Take the element α̃ in H^{n}(W_{ℓ}O_{X}) which corresponds to the element α under the isomorphism (2) for i = ℓ. Then we have F (α̃) = 0 in H^{n}(W_{ℓ}O_{X}), and R^{ℓ}(α̃) ≠ 0 in H^{n}(X, O_{X}) by the assumption on ℓ. Therefore, we have R^{ℓ}(β̃) ≠ 0 in H^{n}(X, O_{X}), and the elements V^{j}R^{j}(β̃) for j = 0, …, ℓ generate H^{n}(W_{ℓ+1}O_{X}). Hence the Frobenius map is zero on H^{n}(W_{ℓ+1}O_{X}). Repeating this argument, we conclude that the Frobenius map is zero on H^{n}(W_{i}O_{X}) for any i > 0 and this contradicts the assumption h ≠ ∞.

Corollary 2.3. If the height h of an n-dimensional Calabi-Yau variety X is not ∞ then h ≤ dim H^{n−1}(Ω_{X}^{1}) + 1.

Definition 2.4. A Calabi-Yau manifold X is called rigid if dim H^{n−1}(X, Ω_{X}^{1}) = 0.

Please note that the tangent sheaf Θ_{X} is the dual of Ω_{X}^{1}, hence by the triviality of the canonical bundle it is isomorphic to Ω_{X}^{n−1}. Therefore, by Serre duality the space of infinitesimal deformations H^{1}(X, Θ_{X}) is isomorphic to the dual of H^{n−1}(X, Ω_{X}^{1}).

Corollary 2.5. The height of a rigid Calabi-Yau manifold X is either 1 or ∞.

3. Cohomology Groups of Calabi-Yau Varieties

Let X be a Calabi-Yau variety of dimension n over k. The existence of Frobenius provides the de Rham cohomology with a very rich structure from which we can read off characteristic p properties. If F : X → X^{(p)} is the relative Frobenius operator then the Cartier operator C gives an isomorphism

H^{j}(F_{∗}Ω_{X/k}^{•}) = Ω_{X,d-closed}^{j}/dΩ_{X}^{j−1} ⟶ Ω_{X^{(p)}}^{j}

of sheaves on X^{(p)}. We generalize the sheaves dΩ_{X}^{j−1} and Ω_{X,d-closed}^{j} by setting (cf. [6])

B_{0}Ω_{X}^{j} = (0), B_{1}Ω_{X}^{j} = dΩ_{X}^{j−1}, B_{m+1}Ω_{X}^{j} = C^{−1}(B_{m}Ω_{X}^{j}).

and

Z_{0}Ω_{X}^{j} = Ω_{X}^{j}, Z_{1}Ω_{X}^{j} = Ω_{X,d-closed}^{j}, Z_{m+1}Ω_{X}^{j} = Ker(dC^{m}).

Note that we have the inclusions

\begin{matrix}0 = B_{0}Ω_{X}^{j} ⊂ B_{1}Ω_{X}^{j} ⊂ … ⊂ B_{m}Ω_{X}^{j} ⊂ … \\ … ⊂ Z_{m}Ω_{X}^{j} ⊂ … ⊂ Z_{1}Ω_{X}^{j} ⊂ Z_{0}Ω_{X}^{j} = Ω_{X}^{j} \end{matrix}

and that we have an exact sequence

0 → Z_{m+1}Ω_{X}^{j} ⟶ Z_{m}Ω_{X}^{j} ⟶^{dC^{m}} dΩ_{X}^{j} → 0.

Alternatively, the sheaves B_{m}Ω_{X}^{j} and Z_{m}Ω_{X}^{j} can be viewed as locally free subsheaves of (F^{m})_{∗}Ω_{X}^{j} on X^{(p^{m})}. Duality for the finite morphism F^{m} implies that for every j ≥ 0 there is a perfect pairing of O_{X^{(p^{m})}}-modules F_{∗}^{m}Ω_{X}^{j} ⊗ F_{∗}^{m}Ω_{X}^{n−j} ⟶ Ω_{X^{(p^{m})}}^{n} given by (α, β) ↦ C^{m}(α ∧ β). This induces perfect pairings of O_{X^{(p^{m})}}-modules

B_{m}Ω_{X}^{j} ⊗ F_{∗}^{m}Ω_{X}^{n−j}/Z_{m}Ω_{X}^{n−j} → Ω_{X^{(p^{m})}} and

Z_{m}Ω_{X}^{j} ⊗ F_{∗}^{m}Ω_{X}^{n−j}/B_{m}Ω_{X}^{n−j} → Ω_{X^{(p^{m})}}

Now we have an isomorphism F_{∗}^{m}Ω_{X}^{j}/Z_{m}Ω_{X}^{j} ≅ B_{m}Ω_{X}^{j+1} induced by the map d. Going back to the interpretation of the B_{m}Ω_{X}^{j} as sheaves on X we find in this way for 1 ≤ j ≤ n and m > 0 perfect pairings

B_{m}Ω_{X}^{j} ⊗ B_{m}Ω_{X}^{n+1−j} → Ω_{X}^{n} (ω_{1} ⊗ ω_{2}) ↦ C^{m}(ω_{1} ∧ ω_{2}).

We first note another interpretation for B_{m}Ω_{X}^{1}: the injective map of sheaves of additive groups D_{m} : W_{m}(O_{X})/F W_{m}(O_{X}) → Ω_{X}^{1} induces an isomorphism

D_{m} : W_{m}(O_{X})/F W_{m}(O_{X}) ⟶ B_{m}Ω_{X}^{1}.  (3)

We write h^{i}(X, −) for dim_{k} H^{i}(X, −). Note that duality implies h^{i}(B_{m}Ω_{X}^{n}) = h^{n−i}(B_{m}Ω_{X}^{1}).

Proposition 3.1. We have h^{i}(X, B_{m}Ω_{X}^{1}) = 0 unless i = n or i = n − 1. If i = n − 1 or i = n we have

h^{i}(B_{m}Ω_{X}^{1}) = \begin{cases}min\{m, h − 1\} & if h ≠ ∞ \\ m & if h = ∞. \end{cases}

Proof. The statement about h^{n−1}(B_{m}Ω_{X}^{1}) follows from (3) and the characterization of the height given in Section 2. The other statements follow from the long exact sequence associated with the short exact sequence

0 ⟶ O_{X} ⟶^{F} O_{X} ⟶^{d} dO_{X} ⟶ 0

and the exact sequence

0 → B_{m} → B_{m+1}⟶^{C^{m}}B_{1} → 0.  (4)

The details can safely be left to the reader. This concludes the proof.

The natural inclusions B_{i}Ω_{X}^{j} ↪ Ω_{X}^{j} and Z_{i}Ω_{X}^{j} ↪ Ω_{X}^{j} of sheaves of groups on X induce homomorphisms

H^{1}(B_{i}Ω_{X}^{j}) → H^{1}(Ω_{X}^{j}) and H^{1}(Z_{i}Ω_{X}^{j}) → H^{1}(Ω_{X}^{j})

whose images are denoted by Im H^{1}(B_{i}Ω_{X}^{j}) and Im H^{1}(Z_{i}Ω_{X}^{j}). Note that we have a non-degenerate cup product pairing

⟨ , ⟩ : H^{n−1}(X, Ω_{X}^{1}) ⊗ H^{1}(X, Ω_{X}^{n−1}) → H^{n}(X, Ω_{X}^{n}) ≅ k.

Lemma 3.2. The images Im H^{n−1}(B_{i}Ω_{X}^{1}) and Im H^{1}(Z_{i}Ω_{X}^{n−1}) are orthogonal to each other for the pairing ⟨ , ⟩.

Proof. From the definitions it follows that for elements α ∈ H^{n−1}(B_{i}Ω_{X}^{1}) and β ∈ H^{1}(Z_{i}Ω_{X}^{n−1}) we have C^{i}(α ∧ β) = 0. The long exact sequence associated to

0 → B_{1}Ω_{X}^{n} → Z_{1}Ω_{X}^{n} → Ω_{X}^{n} → 0  (5)

together with the fact that H^{n}(Z_{i}Ω_{X}^{n}) = H^{n}(Ω_{X}^{n}) for i ≥ 0 implies that C acts without kernel on H^{n}(Ω_{X}^{n}). This proves the required orthogonality.

Lemma 3.3. If h ≠ ∞ we have dim ImH^{1}(X, Z_{i}Ω_{X}^{n−1}) = dim H^{1}(Ω_{X}^{n−1}) − i for 0 ≤ i ≤ h − 1.

Proof. If the height h = 1 then we have H^{n−1}(B_{i}Ω_{X}^{1}) = 0 by (4) and moreover the vanishing of H^{i}(X, dΩ_{X}^{n−1}) and the exact sequence

0 → Z_{i+1}Ω_{X}^{n−1} ⟶ Z_{i}Ω_{X}^{n−1} ⟶^{dC^{i}} dΩ_{X}^{n−1} → 0  (6)

imply that Im H^{1}(X, Z_{i}Ω_{X}^{n−1}) = H^{1}(X, Ω_{X}^{n−1}) for i ≥ 1. For 2 ≤ h < ∞, we know by Proposition 2.2 that Im H^{n−1}(X, B_{i}Ω_{X}^{1}) ⊂ H^{n−1}(X, Ω_{X}^{1}) is of dimension min{i, h − 1}. The exact sequence (6) gives an exact sequence

k ⟶ H^{1}(Z_{i+1}Ω_{X}^{n−1}) ⟶^{ψ_{i+1}} H^{1}(Z_{i}Ω_{X}^{n−1}) ⟶ k

from which we deduce that either dim ψ_{i+1}(H^{1}(Z_{i+1}Ω_{X}^{n−1})) = dim H^{1}(Z_{i}Ω_{X}^{n−1}) + 1 or dim ψ_{i+1}(H^{1}(Z_{i+1}Ω_{X}^{n−1})) = dim H^{1}(Z_{i}Ω_{X}^{n−1}). By induction dim ImH^{1}(Z_{i}Ω_{X}^{n−1}) is at least dim H^{1}(Ω_{X}^{n−1}) − i. On the other hand, by Proposition 3.1 we have dim ImH^{1}(Z_{i}Ω_{X}^{n−1}) ≤ dim H^{1}(Ω_{X}^{n−1}) − i for i ≤ h − 1.

Lemma 3.4. If X is a Calabi-Yau manifold of dimension n with h = ∞ then

(Im H^{n−1}(X, B_{i}Ω_{X}^{1}))^{⊥} = Im H^{1}(Z_{i}Ω_{X}^{n−1}).

Proof. We prove this by induction on i. By the exact sequence (5) we have dim H^{i}(X, dΩ_{X}^{n−1}) = 1 for i = 0, 1. Thus, by the exact sequence (6) we see that the difference dim ImH^{1}(Z_{i}Ω^{n−1}) − dim ImH^{1}(Z_{i+1}Ω^{n−1}) is equal to 0 or 1, and we have an exact sequence

H^{1}(Z_{i+1}Ω^{n−1}) ⟶^{ϕ} H^{1}(Z_{i}Ω^{n−1}) ⟶^{dC^{i}} H^{1}(dΩ_{X}^{n−1}).

Assume that Im H^{n−1}(B_{j−1}Ω^{1}) ≠ Im H^{n−1}(B_{j}Ω^{1}) for j ≤ i and Im H^{n−1}(B_{i}Ω^{1}) = Im H^{n−1}(B_{i+1}Ω^{1}). By Lemma 3.2,

ImH^{1}(Z_{i−1}Ω^{n−1}) ⊃ ImH^{1}(Z_{i}Ω^{n−1})

and ImH^{1}(Z_{i−1}Ω^{n−1}) ≠ ImH^{1}(Z_{i}Ω^{n−1}) for j ≤ i. Suppose ImH^{1}(Z_{i}Ω^{n−1}) ≠ ImH^{1}(Z_{i+1}Ω^{n−1}). The natural homomorphism ϕ : H^{1}(Z_{i+1}Ω^{n−1}) → H^{1}(Z_{i}Ω^{n−1}) is not surjective. Since H^{1}(dΩ_{X}^{n−1}) ≅ k, we see that dC^{i} : H^{1}(Z_{i}Ω^{n−1}) → H^{1}(dΩ_{X}^{n−1}) is surjective and we factor it as

H^{1}(Z_{i}Ω^{n−1}) ⟶^{C^{i}} H^{1}(Ω^{n−1}) ⟶^{d} H^{1}(dΩ_{X}^{n−1}).

Since dC^{i} is surjective, d is not the zero map on C^{i}(H^{1}(Z_{i}Ω^{n−1})). Therefore, we have

C^{i}(H^{1}(Z_{i}Ω^{n−1})) ⊄ Im H^{1}(Z_{1}Ω^{n−1}).

Take an affine open covering of X, and take any Čech cocycle C^{i}(η) = {C^{i}(η_{jk})} of C^{i}(H^{1}(Z_{i}Ω^{n−1})) with respect to this affine open covering. Take any element ζ ∈ H^{n−1}(B_{i}Ω^{1}). Then there exists an element ζ̃ such that C^{i}(ζ̃) = ζ. We consider the image of the element C^{i}(η) ∧ ζ in H^{n}(X, Ω_{X}^{n}). Then, we have

C^{i}(η̃) ∧ ζ = C^{i}(η̃) ∧ C^{i}(ζ̃) = C^{i}(η ∧ ζ̃)

Since ImH^{n−1}(B_{2i}Ω^{1}) = ImH^{n−1}(B_{i}Ω^{1}), the image of ζ̃ in H^{n−1}(Ω_{X}^{1}) is contained in ImH^{n−1}(B_{i}Ω^{1}). As ImH^{1}(Z_{i}Ω^{n−1})) is orthogonal to ImH^{n−1}(B_{i}Ω^{1}), we see that η∧ζ̃ is zero in H^{n}(X, Ω_{X}^{n}), and we have C^{i}(η∧ζ̃) = 0 in H^{n}(X, Ω_{X}^{n}). Therefore, we see that the image of C^{i}(H^{1}(Z_{i}Ω^{n−1})) in H^{1}(Ω_{X}^{n−1}) is orthogonal to ImH^{n−1}(B_{i}Ω^{1}) and we have

C^{i}(H^{1}(Z_{i}Ω^{n−1})) ⊂ ImH^{n−1}(B_{i}Ω^{1})^{⊥} ⊂ ImH^{1}(Z_{i}Ω^{n−1}) ⊂ ImH^{1}(Z_{1}Ω^{n−1}),

a contradiction. Hence, we have ImH^{1}(Z_{i}Ω^{n−1}) = ImH^{1}(Z_{i+1}Ω^{n−1}). Collecting results we get the following theorem.

Theorem 3.5. If X is a Calabi-Yau variety of dimension n and height h then for i ≤ h − 1 we have

ImH^{n−1}(X, B_{i}Ω_{X}^{1})^{⊥} = ImH^{1}(X, Z_{i}Ω_{X}^{n−1}).

One reason for our interest in the spaces ImH^{1}(X, Z_{i}Ω_{X}^{n−1}) comes from the fact that they play a role as tangent spaces to strata in the moduli space as in the analogous case of K3 surfaces, cf. [3]. We intend to come back to this in a later paper.

4. Picard groups

We suppose that X is a Calabi-Yau variety of dimension n ≥ 3. We have the following result for the space of regular 1-forms.

Proposition 4.1. All global 1-forms are indefinitely closed: for i ≥ 0 we have H^{0}(X, Z_{i}Ω_{X}^{1}) = H^{0}(X, Ω_{X}^{1}). The action of the Cartier operator on this space is semi-simple.

Proof. Since the sheaves B_{i}Ω_{X}^{1} have non non-zero cohomology in degree 0 and 1 the exact sequence

0 → B_{i}Ω_{X}^{1} ⟶ Z_{i}Ω_{X}^{1} ⟶^{C^{i}} Ω_{X}^{1} → 0.

implies dim H^{0}(Z_{i}Ω_{X}^{1}) = dim H^{0}(Ω_{X}^{1}). Since the natural map H^{0}(Z_{i}Ω_{X}^{1}) → H^{0}(Ω_{X}^{1}) is injective, we have H^{0}(Z_{i}Ω_{X}^{1}) = H^{0}(Ω_{X}^{1}). The second assertion follows from H^{0}(B_{i}Ω_{X}^{1}) = 0. It is well known that for a p^{−1}-linear semi-simple homomorphism λ on a finitedimensional vector space V the map λ − id_{V} is surjective. This means that we have a basis of logarithmic differential forms Cω = ω.

Corollary 4.2. If id denotes the identity homomorphism on H^{0}(X, Ω_{X}^{1}) the map C − id : H^{0}(X, Ω_{X}^{1}) → H^{0}(X, Ω_{X}^{1}) is surjective.

Proposition 4.3. Suppose that X is a smooth complete variety for which all global 1-forms are closed and such that C gives a bijection H^{0}(X, Z_{1}Ω_{X}^{1}) ⟶ H^{0}(X, Ω_{X}^{1}). Then we have an isomorphism

H^{0}(X, Ω_{X}^{1}) ≅ Pic(X)[p] ⊗_{Z} k.

Proof. Let L be a line bundle representing an element [L] of order p in Pic(X). Then there exists a rational function g ∈ k(X)^{∗} such that (g) = pD, where D is a divisor corresponding to L. One observes now by a local calculation that dg/g is a regular 1-form and thus defines an element of H^{0}(X, Ω_{X}^{1}). Conversely, if ω is a global regular 1-form with Cω = ω then ω can be represented locally as df_{i}/f_{i} with respect to some open cover {U_{i}}. From the relation df_{i}/f_{i} = df_{j}/f_{j} we see d log(f_{i}/f_{j}) = 0 and this implies d(f_{i}/f_{j}) = 0. Hence we see that f_{i}/f_{j} = ϕ_{ij}^{p} form some 1-cocycle {ϕ_{ij}}. This cocycle defines a torsion element of order p of Pic(X). These two maps are each others inverse and the result follows. We are using the notation Pic(X) (resp. NS(X)) for the Picard group (resp. Néron-Severi group) of X. If L is a line bundle with transition functions {f_{ij}} then d log f_{ij} represents the first Chern class of L. In this way we can define a homomorphism

φ_{1} : Pic(X) ⟶ H^{1}(Z_{1}Ω_{X}^{1}), [L] ↦ c_{1}(L) = {df_{ij}/f_{ij}}

which obviously factors through Pic(X)/pPic(X).

Proposition 4.4. The homomorphism φ_{1} : Pic(X)/pPic(X) ⟶ H^{1}(X, Z_{1}Ω_{X}^{1}) is injective.

Proof. We take an affine open covering {U_{i}}. Suppose that there exists an element [L] such that φ_{1}([L]) = 0. Then there exists a d-closed regular 1-form ω_{i} on an affine open set U_{i} such that df_{ij}/f_{ij} = ω_{j} − ω_{i} on U_{i} ∩ U_{j} and we have df_{ij}/f_{ij} = C(ω_{j}) − C(ω_{i}). Therefore, we have ω_{j} − C(ω_{j}) = ω_{i} − C(ω_{i}) on U_{i} ∩ U_{j}. This shows that there exists an regular 1-form ω on X such that ω = ω_{i} − C(ω_{i}) on U_{i}. By Corollary 4.2, there exists an element ω′ ∈ H^{0}(Ω_{X}^{1}) such that (C − id)ω′ = ω. Replacing ω_{i} + ω′ by ω_{i}, we have

df_{ij}/f_{ij} = ω_{j} − ω_{i}

with C(ω_{i}) = ω_{i}. Then, there exists an regular function f_{i} on U_{i} such that ω_{i} = df_{i}/f_{i}. So we have d log f_{ij} = d log(f_{j}/f_{i}). Therefore, there exists a regular function φ_{ij} on U_{i} ∩ U_{j} such that f_{ij} = (f_{j}/f_{i})φ_{ij}^{p}. Thus [L] is a p-th power. We conclude that φ_{1} : Pic(X)/pPic(X) → H^{1}(Z_{1}Ω_{X}^{1}) is injective.

Proposition 4.5. The natural homomorphism H^{1}(Z_{1}Ω_{X}^{1}) → H_{DR}^{2}(X) is injective.

Proof. Let {U_{i}} be an affine open covering of X. A Čech cocycle {ω_{ij}} in H^{1}(Z_{1}Ω_{X}^{1}) is mapped to {(0, ω_{ij}, 0)} in H_{DR}^{2}(X). Suppose this element is zero in H_{DR}^{2}(X). Then there exist elements ({f_{ij}}, {ω_{i}}) with f_{ij} ∈ Γ(U_{i} ∩ U_{j}, O_{X}) and ω_{i} ∈ Γ(U_{i}, Ω_{X}^{1}) such that

f_{jk} − f_{ik} + f_{ij} = 0 ω_{ij} = df_{ij} + ω_{j} − ω_{i} dω_{i} = 0.

Since {f_{ij}} gives an element of H^{1}(O_{X}) and H^{1}(O_{X}) = 0, there exists an element {f_{i}} such that f_{ij} = f_{j} − f_{i} on U_{i} ∩ U_{j}. Therefore, we have ω_{ij} = (df_{j} + ω_{j}) − (df_{i} + ω_{i}). Since d(df_{i} + ω_{i}) = 0, we conclude that {ω_{ij}} is zero in H^{1}(Z_{1}Ω_{X}^{1}).

The results above imply the following theorem.

Theorem 4.6. The natural homomorphism Pic(X)/pPic(X) ⟶ H_{DR}^{2}(X) is injective.

Let us point out at this point that for a Calabi-Yau manifold X the Picard group Pic(X) is reduced and coincides with the Néron-Severi group NS(X) because NS(X) = Pic(X)/Pic^{0}(X) and Pic^{0}(X) vanishes because of H^{1}(X, O_{X}) = 0.

Lemma 4.7. For a Calabi-Yau manifold X of dimension n ≥ 3 with non nonzero global 1-forms Pic(X) has no p-torsion.

Proof. Take an affine open covering {U_{i}} of X. Assume {f_{ij}} represents an element [L] ∈ Pic(X) which is p-torsion. Then, there exist regular functions f_{i} ∈ H^{0}(U_{i}, O_{X}^{∗}) such that f_{ij}^{p} = f_{i}/f_{j}. The df_{i}/f_{i} on U_{i} glue together to yield a regular 1-form ω on X. Since H^{0}(X, Ω_{X}^{1}) = 0, we see ω = 0, i.e., df_{i} = 0. Therefore, there exist regular functions g_{i} ∈ H^{0}(U_{i}, O_{X}^{∗}) such that f_{i} = g_{i}^{p}. Hence, we have {f_{ij}} ∼ 0 and we see that Pic(X) has no p-torsion.

Lemma 4.8. Let X be a Calabi-Yau manifold X of dimension n ≥ 3 with no non-zero global 2-forms. Then, the homomorphism

Pic(X)/pPic(X) ⟶ H^{1}(Ω_{X}^{1})

defined by {f_{ij}} ↦ {df_{ij}/f_{ij}} is injective.

Proof. By the assumption H^{0}(X, Ω_{X}^{2}) = 0 we have H^{0}(X, dΩ_{X}^{1}) = 0. Therefore, from the exact sequence

0 → Z_{1}Ω_{X}^{1} ⟶ Ω_{X}^{1} ⟶^{d} dΩ_{X}^{1} → 0,

we deduce a natural injection H^{1}(Z_{1}Ω_{X}^{1}) ⟶ H^{1}(Ω_{X}^{1}). So the result follows from Lemma 4.4.

Theorem 4.9. Let X be a Calabi-Yau manifold X of dimension n ≥ 3 with H^{0}(X, Ω_{X}^{i}) = 0 for i = 1, 2. Then the natural homomorphism

NS(X)/pNS(X) ⊗_{F_{p}} k ⟶ H^{1}(Ω_{X}^{1}) = H_{dR}^{2}(X)

is injective and the Picard number satisfies ρ ≤ dim_{k} H^{1}(Ω_{X}^{1}).

Proof. Suppose that this homomorphism is not injective. Then with respect to a suitable affine open covering {U_{i}} there exist elements {f_{ij}^{(ν)}} representing non-zero elements in NS(X)/pNS(X), such that

∑_{ν=1}^{ℓ} a_{ν}df_{ij}^{(ν)}/f_{ij}^{(ν)} = 0 in H^{1}(Ω_{X}^{1})

for suitable a_{ν} ∈ k. We take such elements with the minimal ℓ. We may assume a_{1} = 1 and we have a_{i}/a_{j} ∉ F_{p} for i ≠ j. By Lemma 4.8, we have ℓ ≥ 2. There exists ω_{i} ∈ H^{1}(U_{i}, Ω_{X}^{1}) such that

∑_{ν=1}^{ℓ} a_{ν}df_{ij}^{(ν)}/f_{ij}^{(ν)} = ω_{j} − ω_{i}  (1)

on U_{i} ∩ U_{j}. There exists an element ω̃_{i} ∈ H^{1}(U_{i}, Ω_{X}^{1}) such that C(ω̃_{i}) = ω_{i}. Therefore, taking the Cartier inverse, we have

∑_{ν=1}^{ℓ} ã_{ν}df_{ij}^{(ν)}/f_{ij}^{(ν)} + dg_{ij} = ω̃_{j} − ω̃_{i}

with ã_{ν} ∈ k, ã_{ν}^{p} = a_{ν}, and suitable dg_{ij} ∈ H^{0}(U_{i} ∩U_{j}, dO_{X}). Since {df_{ij}^{(ν)}/f_{ij}^{(ν)}} is a cocycle, we see that {dg_{ij}} ∈ H^{1}(X, dO_{X}). Since H^{1}(X, dO_{X}) = 0, there exists an element dg_{i} ∈ H^{0}(U_{i}, dO_{X}) such that dg_{ij} = dg_{j} − dg_{i}. Therefore, we have

∑_{ν=1}^{ℓ} ã_{ν}df_{ij}^{(ν)}/f_{ij}^{(ν)} = (ω̃_{j} − dg_{j}) − (ω̃_{i} − dg_{i}).  (2)

Subtracting (2) from (1) we get a non-trivial linear relation with a smaller ℓ in H^{1}(Ω_{X}^{1}), a contradiction. Remark. In the case of a K3 surface X the natural homomorphism

NS(X)/pNS(X) ⊗_{F_{p}} k ⟶ H_{DR}^{2}(X)

is not injective if X is supersingular in the sense of Shioda. Ogus showed that the kernel can be used for describing the moduli of supersingular K3 surfaces, cf. Ogus[10]. So the situation is completely different in dimension ≥ 3.

5. Fermat Calabi-Yau manifolds

Again p is a prime number and m a positive integer which is prime to p. Let f be a smallest power of p such that p^{f} ≡ 1 mod m and put q = p^{f}. We denote by F_{q} a finite field of cardinality q. We consider the Fermat variety X_{m}^{r}(p) over F_{q} defined by

X_{0}^{m} + X_{1}^{m} + … + X_{r+1}^{m} = 0

in projective space P^{r+1} of dimension r + 1. The zeta function of X_{m}^{r} over F_{q} was calculated by A. Weil (cf. [18]). The result is:

Z(X_{m}^{r}/F_{q}, T) = \frac{P (T)^{(−1)^{r−1}}}{(1 − T)(1 − qT) … (1 − q^{r}T)},

where P (T) = ∏_{α}(1 − j(α)T) with the product taken over a set of vectors α and j(α) is a Jacobi sum defined as follows. Consider the set

A_{m,r} = {(a_{0}, a_{1}, …, a_{r+1}) ∈ Z^{r+2} | 0 < a_{i} < m, ∑_{j=0}^{r+1}a_{j} ≡ 0(mod m)},

and choose a character χ : F_{q}^{∗} → C^{∗} of order m. For α = (a_{0}, a_{1}, …, a_{r+1}) ∈ A_{m,r} we define

j(α) = (−1)^{r} ∑ χ(v_{1}^{a_{1}}) … χ(v_{r+1}^{a_{r+1}}),

where the summation runs over v_{i} ∈ F_{q}^{∗} with 1 + v_{1} + … + v_{r+1} = 0. Thus the j(α)’s are eigenvalues of the Frobenius map over F_{q} on the ℓ-adic étale cohomology group H_{et}^{r}(X_{m}^{r}, Q_{ℓ}). Now, let ζ = exp(2πi/m) be a primitive m-th root of unity, and K = Q(ζ) the corresponding cyclotomic field with Galois group G = Gal(K/Q). For an element t ∈ (Z/mZ)^{∗} we let σ_{t} be the automorphism of K defined by ζ ↦ ζ^{t}. The correspondence t ↔ σ_{t} defines an isomorphism (Z/mZ)^{∗} ≅ G and we shall identify G with (Z/mZ)^{∗} by this isomorphism. We define a subgroup H of order f of G by H = {p^{j} mod m | 0 ≤ j < f} Let {t_{1}, …, t_{g}} with t_{i} ∈ Z/mZ^{∗}, be a complete system of representatives of G/H with g = |G/H|, and put

A_{H}(α) = ∑_{t∈H}[∑_{j=1}^{r+1}⟨ta_{j}/m⟩],

where [a] (resp. ⟨a⟩) means the integral part (resp. the fractional part) of a rational number a. Choose a prime ideal P in K lying over p; it has norm N(P) = p^{f} = q. If P_{i} denotes the prime ideal P^{σ_{−t_{i}}^{−1}} we have the prime decomposition (p) = P_{1} ⋯ P_{g} in K and Stickelberger’s theorem tells us that

(j(α)) = ∏_{i=1}^{g} P_{i}^{A_{H}(t_{i}α)},

where t_{i}α = (t_{i}a_{0}, …, t_{i}a_{r+1}). For the details we refer to Lang[7] or Shioda-Katsura[16]. Now we restrict our attention to Fermat Calabi-Yau manifolds X_{m}^{r}(p) with m = r + 2.

Theorem 5.1. Assume r ≥ 2. Let Φ^{r} be the Artin-Mazur formal group of the r-dimensional Calabi-Yau variety X = X_{r+2}^{r}(p). The height h of Φ^{r} is equal to either 1 or ∞. Moreover, h = 1 if and only if p ≡ 1 (mod r + 2).

Before we give the proof of this theorem we state a technical lemma.

Lemma 5.2. Under the notation above, assume [∑_{j=1}^{r+1}⟨ta_{j}/(r + 2)⟩] = 0 with t ∈ (Z/(r + 2)Z)^{∗}. Then a_{j} = t^{−1} in (Z/(r + 2)Z)^{∗} for all j = 0, 1, …, r + 1.

Proof. Since t ∈ (Z/(r+2)Z)^{∗}, we have ⟨ta_{j}/(r+2)⟩ ≥ 1/(r+2). Suppose there exists an index i such that ta_{i} ≢ 1 (mod r + 2). Then we have the inequality ⟨ta_{i}/(r+2)⟩ ≥ 2/(r+2) and thus ∑_{j=1}^{r+1}⟨ta_{ρ}/(r+2)⟩ ≥ 1, which contradicts the assumption. So we have ta_{i} ≡ 1 (mod r + 2) and a_{j} ≡ t^{−1} for j = 1, …, r + 1. Since a_{0} + a_{1} + … + a_{r+1} ≡ 0 (mod r + 2), we conclude a_{0} ≡ t^{−1}. Proof of the theorem. The Dieudonné module D(Φ^{r}) of Φ^{r} is isomorphic to H^{r}(X, W O_{X}). We denote by Q(W) the quotient field of the Witt ring W (k) of k. Then, if h < ∞, we have

h = dim_{Q(W)} H^{r}(X, W O_{X}) ⊗_{W (k)} Q(W).

and by Illusie [6] we know we have

H^{r}(X, W O_{X}) ⊗_{W (k)} Q(W) ≅ H_{cris}^{r}(X) ⊗ Q(W)_{[0,1[}.

According to Artin-Mazur [2], the slopes of H_{cris}^{r}(X) ⊗ Q(W) are given by (ord_{P}q)/f and the (ord_{P}j(α))/f. Hence, the height h is equal to the number of j(α) such that A_{H}(α) < f.

First, assume p ≡ 1 (mod r + 2), i.e. f = 1. Then H = ⟨1⟩ and A_{H}(α) < f = 1 implies A_{H}(α) = 0. Therefore, by Lemma 5.2, we have a_{j} = 1 for all j = 0, 1, …, r + 1 and there is only one α, namely α = (1, 1, …, 1), such that ord_{P}j(α) = 0. So we conclude h = 1 in this case. Secondly, assume p ≢ 1 (modr + 2). By definition, we have f ≥ 2. We now prove that there exists no α such that A_{H}(α) < f. Suppose A_{H}(α) = ∑_{t∈H}[∑_{j=1}^{r+1}⟨ta_{j}/(r + 2)⟩] < f. Then there exists an element t ∈ H such that [∑_{j=1}^{r+1}⟨ta_{j}/(r + 2)⟩] = 0. By Lemma 5.2 we have α = (t^{−1}, t^{−1}, …, t^{−1}). For t′ ∈ H with t′ ≠ t we have [∑_{ρ=1}^{r+1}⟨t′t^{−1}/(r + 2)⟩] ≠ 0. Therefore the inequality yields [∑_{j=1}^{r+1}⟨t′t^{−1}/(r + 2)⟩] = 1 for t′ ∈ H, t′ ≠ t. Since A_{H}(tα) = A_{H}(α) for any t ∈ H, by a translation by t, we may assume α = (1, 1, …, 1), i.e., t = 1. Moreover, we can take a representative of t′ ∈ H such that 0 < t′ < r + 2. Then,

1 = [∑_{j=1}^{r+1}⟨t′t^{−1}/(r + 2)⟩] = [∑_{j=1}^{r+1}⟨t′/(r + 2)⟩] = [∑_{j=1}^{r+1} t′/(r + 2)]
= [(r + 1)t′/(r + 2)]

and we get 1 ≤ (r + 1)t′/(r + 2) < 2. By this inequality, we see t′ = 2. Therefore, we have H = {1, 2}. Since H is a subgroup of (Z/(r + 2)Z)^{∗}, we see that 2^{2} ≡ 1 mod r + 2. Therefore, we have r = 1, which contradicts our assumption. Hence there exists no α such that ord_{P}j(α) < 1 and we conclude h = ∞ in this case. This completes the proof of the theorem. For K3 surfaces we have two notions of supersingularity. We generalize these to higher dimensions.

Definition 5.3. A Calabi-Yau manifold X of dimension r is said to be of additive Artin-Mazur type (‘supersingular in the sense of Artin’) if the height of Artin-Mazur formal group associated with H^{r}(X, O_{X}) is equal to ∞.

Definition 5.4. A non-singular complete algebraic variety X of dimension r is said to be fully rigged (‘supersingular in the sense of Shioda’) if all the even degree étale cohomology groups are spanned by algebraic cycles.

By the theorem above, we know that the Fermat Calabi-Yau manifolds are of additive Artin-Mazur type if and only if p ≢ 1 mod m with m = r + 2. As to being fully rigged we have the following theorem.

Theorem 5.5 (Shioda-Katsura [16]). Assume m ≥ 4, (p, m) = 1 and r is even. Then the Fermat variety X_{m}^{r}(p) is fully rigged if and only if there exists a positive integer ν such that p^{ν} ≡ −1 mod m.

M. Artin conjectured that a K3 surface X is supersingular in the sense of Artin if and only if X is supersingular in the sense of Shioda. He also showed that “if part” holds. In the case of the Fermat K3 surface, i.e, X_{4}^{2}(p), by the two theorems above, we see, as is well-known, that the Artin conjecture holds.

However, in the case of even r ≥ 4, the above two theorems imply that this straightforward generalization of the Artin conjecture to higher dimension does not hold.

6. Kummer Calabi-Yau manifolds

Let A be an abelian variety of dimension n ≥ 2 defined over an algebraically closed field of characteristic p > 0, and G be a finite group which acts on A faithfully. Assume that the order of G is prime to p, and that the quotient variety A/G has a resolution which is a Calabi-Yau manifold X. We call X a Kummer Calabi-Yau manifold. We denote by π : A ⟶ A/G the projection, and by ν : X ⟶ A/G the resolution.

Theorem 6.1. Under the assumptions above the Artin-Mazur formal group Φ_{X}^{n} is isomorphic to the Artin-Mazur formal group Φ_{A}^{n}.

Proof. Since the order of G is prime to p, the singularities of A/G are rational, and we have R^{i}ν_{∗}O_{X} = 0 for i ≥ 1. So by the Leray spectral sequence we have H^{n}(A/G, O_{A/G}) ≅ H^{n}(X, O_{X}) ≅ k and H^{n−1}(A/G, O_{A/G}) ≅ H^{n−1}(X, O_{X}) ≅ 0. It follows that the Artin-Mazur formal group Φ_{A/G}^{n} is pro-representable by a formal Lie group of dimension 1 (cf. Artin-Mazur[2]). Since the tangent space H^{n}(A/G, O_{A/G}) of Φ_{A/G}^{n} is naturally isomorphic to the tangent space H^{n}(X, O_{X}) of Φ_{X}^{n} as above, the natural homomorphism from Φ_{A/G}^{n} to Φ_{X}^{n} is non-trivial. One-dimensional formal groups are classified by their height and between formal groups of different height there are no non-trivial homomorphisms . So the height of Φ_{A/G}^{n} is equal to that of Φ_{X}^{n} and we thus see that Φ_{A/G}^{n} and Φ_{X}^{n} are isomorphic. Since the order of G is prime to p, there is a non-trivial trace map from H^{n}(A, O_{A}) to H^{n}(A/G, O_{A/G}). Therefore, π^{∗} : H^{n}(A/G, O_{A/G}) ⟶ H^{n}(A, O_{A}) is an isomorphism. Therefore, as above we see that the height of Φ_{A/G}^{n} is equal to the height of Φ_{A}^{n}, and that Φ_{X}^{n} is isomorphic to Φ_{A}^{n}. Q.e.d. Though the following lemma might be well-known to specialists we give here a proof for the reader’s convenience.

Lemma 6.2. Let A be an abelian variety of dimension n ≥ 2 and p-rank f(A). The height h of the Artin-Mazur formal group Φ_{A} of A is as follows:

(1) h = 1 if A is ordinary, i.e., f(A) = n,

(2) h = 2 if f(A) = n − 1,

(3) h = ∞ if f(A) ≤ n − 2.

Proof. We denote by H_{cris}^{i}(A) the i-th cristalline cohomology of A and as usual by H_{cris}^{i}(A)_{[ℓ,ℓ+1[} the additive group of elements in H_{cris}^{i}(A) whose slopes are in the interval [ℓ, ℓ + 1[. By the general theory in Illusie [6], we have

H^{n}(A, W (O_{A})) ⊗_{W} Q(W) ≅ (H_{cris}^{n}(A) ⊗_{W} Q(W))_{[0,1[}

with Q(W) the quotient field of W. The theory of Dieudonné modules implies

h = dim_{Q(W)} D(Φ_{A}) = dim_{Q(W)} H^{n}(A, W (O_{A})) ⊗_{W} Q(W) if h < ∞,

and dim_{Q(W)} D(Φ_{A}) = 0 if h = ∞. We know the slopes of H_{cris}^{1}(A) for each case. Since we have

H_{cris}^{n}(A) ≅ ∧^{n}H_{cris}^{1}(A),

counting the number of slopes in [0, 1[ of H_{cris}^{n}(A) gives the result.

Corollary 6.3. Let X be a Kummer Calabi-Yau manifold of dimension n obtained from an abelian variety A as above. Then the height of the Artin-Mazur formal group Φ_{X}^{n} is equal to either 1, 2 or ∞.

Example 6.4. Assume p ≥ 3. Let A be an abelian surface and ι the map A → A sending a ∈ A to its inverse −a ∈ A. We denote by Km(A) the Kummer surface of A, i.e., the minimal resolution of A/⟨ι⟩. Then Φ_{Km(A)}^{2} is isomorphic to Φ_{A}^{2}.

Example 6.5. Assume p ≥ 5, and let ω be a primitive third root of unity. Let E be a non-singular complete model of the elliptic curve defined by y^{2} = x^{3} +1, and let σ be an automorphism of E defined by x ↦ ωx, y ↦ y. We set A = E^{3} and put σ̃ = σ × σ × σ. The minimal resolution X of A/⟨σ̃⟩ is a Calabi-Yau manifold, and the Artin-Mazur formal group Φ_{X}^{3} is isomorphic to Φ_{A}^{3}.

Let ω be a complex number with positive imaginary part, and L = Z + Zω be a lattice in the complex numbers C. From here on, we consider an elliptic curve E = C/L, and we assume that E has a model defined over an algebraic number field K. Then A = E × E × E is an abelian threefold, and we let G ⊆ Aut_{K}(E) be a finite group which faithfully acts on A. We assume that G has only isolated fixed points on A and that the quotient variety A/G has a crepant resolution ν : X → A/G defined over K, [12]. We denote by π the projection A → A/G. For a prime p of K, we denote by X̄ the reduction modulo p of X.

Example 6.6. [K. Ueno [17]] Assume that E is an elliptic curve defined over Q having complex multiplication σ : E → E by a primitive third root of unity. Then G = Z/3Z = ⟨σ⟩ acts diagonally on A = E^{3}. A crepant resolution of A/G gives a rigid Calabi-Yau manifold defined over Q. For a prime number p ≥ 5 the reduction modulo p of A is the abelian threefold given in Example 6.5.

Theorem 6.7. Let X be a Calabi-Yau obtained as crepant resolution of A/G as above. Assume moreover that X is rigid. Then the elliptic curve E has complex multiplication and the intermediate Jacobian of X is isogenous to E.

Corollary 6.8. Under the assumptions as in the theorem, we take a prime p of good reduction for Xand let X̄ be the reduction of X modulo p. Then the height of the formal group ΦX̄ is either 1 or ∞. It is ∞ if and only if the reduction of the intermediate Jacobian variety of X at p is a supersingular elliptic curve.

Example 6.9. We consider the reduction X̄ modulo p of the variety X in the Example 6.6. We assume the characteristic of the residue field of p is not equal to 2 and 3. Then the height h(ΦX̄) = ∞ if and only if the reduction modulo p of the intermediate Jacobian of X is a supersingular elliptic curve, and this is the case if and only if p ≡ 2 (mod 3).

Before we prove the theorem we introduce some notation. We have a natural identification H_{1}(E, Z) = Z + Zω. Fixing a non-zero regular differential form η on E determines a regular three form p_{1}^{∗}η ∧ p_{2}^{∗}η ∧ p_{3}^{∗}η = Ω_{A} on A. We have a natural homomorphism

H^{3}(X, Z) → H_{dR}^{3}(X) → H^{3}(X, O_{X}).

If X is rigid, the corresponding quotient H^{3}(X, O_{X})/H^{3}(X, Z) gives the intermediate Jacobian of X. Since dim_{C} H^{3}(X, O_{X}) = 1, the intermediate Jacobian of X is isomorphic to an elliptic curve. We can define the period map with respect to Ω_{A}:

π_{A} : H_{3}(X, Z) ⟶ C γ ↦ ∫_{γ} Ω_{A}.

By Poincaré duality we can identify π_{A} with the natural projection H^{3}(X, Z) → H^{3}(X, C) → H^{3}(X, O_{X}) = C (cf. Shioda [15], for instance.) There exists a regular 3-form Ω_{X} on X such that Ω_{A} = (ν^{−1} ∘ π)^{∗}Ω_{X}. We can define π_{X} with respect to Ω_{X} for the Calabi-Yau manifold X as well. In order to describe the structure of the intermediate Jacobian of X we look at the period map of an abelian threefold, following the method in Shioda [15] (also see Mumford [8]). Choose a basis u_{1}, u_{2} of H_{1}(E, R). This determines a C-basis H_{1}(E, R) ⊗_{R} C = H_{1}(E, C). If e_{i} for i = 1, 2, 3 is the standard basis of C^{3} then u_{2i−1} = e_{i} and u_{2i} = ωe_{i} for i = 1, 2, 3 form a basis of H_{1}(A, Z) and A = C^{3}/M with M the lattice generated by u_{1}, …, u_{6}. The dual basis is denoted by v_{i}. The basis of H^{1}(A, Z) determines a canonical basis v_{i} ∧ v_{j} ∧ v_{k} of H^{3}(A, Z). The natural homomorphism

p_{A} : H^{3}(A, Z) ⟶ H_{dR}^{3}(A) ⟶ H^{3}(A, O_{A}) ≅ C.

is an element of Hom_{C}(H^{3}(A, C), C) and can be considered as an element of H^{3}(A, C) and is given by

p_{A} = ∑_{i<j<k} det(u_{i}, u_{j}, u_{k})v^{i} ∧ v^{j} ∧ v^{k}.

Therefore the image of p_{A} in C is spanned by the complex numbers 1, ω, ω^{2} and ω^{3} over Z. We now give the proof of Theorem 6.7. Let S be the set of non-free points of the action of G on A. Then the restriction of π to A \ S is étale on A/G \ π(S). Since S is of codimension 3 in A, we have the following diagram:

H^{3}(A, Z) ≅  H^{3}(A \ S, Z)  ⟶^{p_{A}} H^{3}(A, O_{A}) ≅ C
↓ π_{∗}  ↓ (π |_{A\S})_{∗}  ↓ π_{∗}
H^{3}(A/G, Z) ≅ H^{3}(A/G \ π(S), Z) ⟶^{p_{A/G}} H^{3}(A/G, O_{A/G}) ≅ C
↓ ≅  ↓ ν^{∗}
H^{3}(X, Z)  ⟶^{p_{X}} H^{3}(X, O_{X}) ≅ C.

The vertical arrows on the right hand side give an identification of H^{3}(A, O_{A}) and H^{3}(X, O_{X}). Because p does not divide the order of G we see that H^{3}(A, ℚ) maps surjectively to H^{3}(A/G, ℚ) = H^{3}(A, ℚ)^{G}, hence the image of H^{3}(A, ℤ) is commensurable with H^{3}(X, ℤ). Now Im p_{X} is a lattice in C, and Im p_{A} is a lattice in C as well. We know that Im p_{A} is generated by 1, ω, ω^{2} and ω^{3} and thus ω is a quadratic number and the intermediate Jacobian has complex multiplication by Q(ω). Hence the intermediate Jacobian C/Im p_{X} of X is isogenous to E.

7. Questions

We close with two natural basic questions that suggest themselves. Is there a function f(n) such that a Calabi-Yau variety in characteristic p > 0 of dimension n lifts to characteristic 0 if p > f(n)? Note that Hirokado constructed a non-liftable Calabi-Yau threefold in characteristic 3, see [5] (see also [13]). Can a Calabi-Yau variety of dimension 3 in positive characteristic have non-zero regular 1-forms or regular 2-forms?

8. Acknowledgement

This research was made possible by a JSPS-NWO grant. The second author would like to thank the University of Amsterdam for hospitality and the first author would like to thank Prof. Ueno for inviting him to Kyoto, where this paper was finished. Finally we thank the referee for useful comments.

References

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G. van der Geer Faculteit Wiskunde en Informatica University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam The Netherlands geer@science.uva.nl T. Katsura Graduate School of Mathematical Sciences The University of Tokyo Komaba, Meguro-ku Tokyo 153-8914 Japan tkatsura@ms.u-tokyo.ac.jp 114

Documenta Mathematica 8 (2003)