A Note on the Formal Groups of Weighted Delsarte Threefolds

One-dimensional formal groups over an algebraically closed field of positive characteristic are classified by their height. In the case of $K3$ surfaces, the height of their formal groups takes integer values between $1$ and $10$, or $\infty$. For Calabi-Yau threefolds, the height is bounded by $h^{1,2}+1$ if it is finite, where $h^{1,2}$ is a Hodge number. At present, there are only a limited number of concrete examples for explicit values or the distribution of the height. In this paper, we consider Calabi-Yau threefolds arising from weighted Delsarte threefolds in positive characteristic. We describe an algorithm for computing the height of their formal groups and carry out calculations with various Calabi-Yau threefolds of Delsarte type.


Introduction
Let k be an algebraically closed field of characteristic p > 0. Let X be a Calabi-Yau threefold over k, by which we mean a smooth projective variety over k of dimension 3 with a trivial canonical sheaf and dim H 1 (X, O X ) = dim H 2 (X, O X ) = 0). In [2], M. Artin and Mazur defined a functor Φ X for X on the category of finite local k-algebras A with residue field k: ) where X A = X × Spec A and G m is the sheaf of multiplicative groups. It is proved in [2] that this functor (for Calabi-Yau varieties in general) is representable by a smooth formal group of dimension 1 = P g . By abuse of notation, we also use Φ X for this formal group and called it the (Artin-Mazur) formal group of X.
A formal group in positive characteristic p is endowed with the multiplicationby-p map. The p-rank of its kernel is called the height of Φ X and we denote it by h := ht Φ X , namely p h = # ker ([p] : Φ X −→ Φ X ).
It is known that one-dimensional formal groups in positive characteristic are determined up to isomorphism by the height. In the case of K3 surfaces, the height takes integer values between 1 and 10 (cf. [1], [9], [15]). In the case of Calabi-Yau threefolds, it is proved in [6] that h is bounded above by h 1,2 + 1, where h 1,2 is a Hodge number of X. Note that it is still a conjecture that h 1,2 is bounded for Calabi-Yau threefolds. visited Noriko Yui several times at the Department of Mathematics and Statistics of Queen's University and at the Fields Institute in Canada. He thanks Professor Yui for many inspiring discussions and is grateful to the two institutions for their hospitality. The author also thanks the Banff International Research Station in Canada for the workshop on Modular Forms in String Theory in 2016 where the main result of this paper was presented.

Mirror symmetry and the height of formal groups
In this section, we consider orbifold Calabi-Yau threefolds (X, Y ) that form a mirror pair (i.e. h 1,1 (X) = h 2,1 (Y ) and h 2,1 (X) = h 1,1 (Y )). Based on a result of [6], we give an uppor bound for the height of their formal groups in terms of Hodge numbers.
Lemma 2.1. Let X be a Calabi-Yau threefold over k and G be a finite symplectic group action on X. Write Y := X/G. Assume that p is coprime to the order of G and that there exists a crepant resolution Y of Y . Then Y is a Calabi-Yau threefold with Φ X ∼ = Φ Y . In particular, the formal groups of X and Y have the same height: Proof. Since p is coprime to the order of G, Y has at most rational singularities and by Theorem 3.1 of [13], we have Φ Y ∼ = Φ Y . Write f : X −→ Y for the quotient map. As X is Calabi-Yau and G is symplectic, there is an isomorphism f * ω X ∼ = ω Y for the canonical sheaves of X and Y . Then for a crepant resolution Y of Y , we Proposition 2.2. Let X be a threefold over k with O X ∼ = ω X and G be a finite symplectic group action on X. Write X and Y for crepant resolutions of X and Y , respectively, constructed as in the following diagram: h ≤ h 1,2 ( X) + 1 and h ≤ h 1,2 ( Y ) + 1.
As ( X, Y ) is a mirror pair of Calabi-Yau threefolds, we find h 1,2 ( Y )+ 1 = h 1,1 ( X)+ 1. Combining these relations, we obtain the asserted inequality. Taking the contrapositive of Proposition 2.2, we find the following. Corollary 2.3. Let X and Y be Calabi-Yau threefolds constructed as in Proposition 2.2. If h > h 1,1 ( X)+1 or h > h 1,2 ( X)+1, then h = ∞ or there exists no symplectic group action G such that ( X, Y ) forms a mirror pair.

Weighted Delsarte Varieties
In order to compute the cohomology groups of weighted Delsarte varieties, we explain some geometric properties of them (see also [8]).
Let Q = (q 0 , · · · , q n ) be an n + 1-tuple of positive integers such that p ∤ q i (0 ≤ i ≤ n) and gcd(q 0 , · · · ,q i , · · · , q n ) = 1 for every 0 ≤ i ≤ n, whereq i means that q i is omitted. The weighted projective n-space over k of type Q, denoted by P n (Q), is the projective variety P n (Q) := Proj k[x 0 , · · · , x n ], where the polynomial algebra is graded by deg(x i ) = q i for 0 ≤ i ≤ n (cf. [5]).
Let m be a positive integer such that p ∤ m. Let A = (a ij ) be an (n + 1) × (n + 1) matrix of integer entries satisfying the conditions We define an (n − 1)-dimensional weighted Delsarte variety in P n (Q) of degree m with matrix A (cf. [3], [11], [8]) to be the weighted projective hypersurface defind by When A is a diagonal matrix, the equation has the form x d0 0 + x d1 1 + · · · + x dn n = 0 and we call it a weighted Fermat variety.
Weighted Delsarte varieties are birational to finite quotients of Fermat varieties and many properties of their cohomology groups can be extracted from those of Fermat varieties (cf. [8], [12], [16]). For instance, write d =| det A | and let F d be the (n − 1)-dimensional Fermat variety of degree d in the usual projective space P n : Then Γ A is a subgroup of Γ and it acts on F d by γ · (y 0 : y 1 : · · · : y n ) = n j=0 λ a0j j y 0 : n j=0 λ a1j j y 1 : · · · : n j=0 λ anj j y n for γ ∈ Γ A and (y 0 : y 1 : · · · : y n ) ∈ F d .
We describe the ℓ-adicétale cohomology of the varieties involved (ℓ is a prime different from p = char k). It is known that the cohomology of Fermat variety F d is decomposed into 1-dimensional pieces parameterized by the characters of Γ. In fact, define A similar property holds for X A since it is birational to the quotient variety F d /Γ A . Here we describe the cohomology of F d /Γ A and the cohomology of X A of dimension 3 will be discussed in the next section.
Lemma 3.2. Let X A be a weighted Delsarte variety in P n (Q) with matrix A. Let Γ A be the group we defined in (3.1) and put Proof. The proof is similar to the case of weighted Delsarte (or diagonal) surfaces (see, for instance, [7] and [8] for details). It follows from the isomorphism

Calabi-Yau threefolds of Delsarte type
In this section, we discuss Calabi-Yau threefolds arising from weighted Delsarte threefolds. Using their cohomology groups, we describe an algorithm of computing the height of their formal groups.
Let X be a weighted projective variety in P n (Q) with Q = (q 0 , · · · , q n ). X is said to be quasi-smooth (cf. [5]) if its affine quasi-cone is smooth outside the origin. For instance, weighted Fermat varieties are quasi-smooth. As a special case of weighted quasi-smooth varities, we observe the following property for quasi-smooth weighted Delsarte varieties. (a) The quotient variety F d /Γ A has at most rational abelian quotient singularities.
(b) If X A is quasi-smooth, then it has at most rational cyclic quotient singularities.
Proof. (a) Since F d is a smooth variety and Γ A is an abelian group, F d /Γ A has at most abelian quotient singularities. In characteristic 0, quotient singularities are known to be rational. Hence we only need to show that F d /Γ A is liftable to characteristic 0 and this follows from conditions (i) and (ii) on matrix A.
(b) A quasi-smooth variety is locally isomorphic to the quotient of a smooth variety by some cyclic group action and this cyclic group is a subgroup of µ qi for some weight q i (cf. [5]). Since p ∤ q i for every i, the group action by a subgroup of µ qi can be lifted to characteristic 0. Hence X A has at most cyclic quotient singularities and they are rational. Now we consider weighted Delsarte threefolds.

Lemma 4.2. Let X A be a weighted Delsarte threefold in P 4 (Q) with matrix A.
Assume that X A is quasi-smooth and m = q 0 + q 1 + q 2 + q 3 + q 4 . Then the dualizing sheaf of X A is trivial and there exists a crepant resolution for X A .
Proof. Since X A is a quasi-smooth hypersurface and of dimension 3, it is known (cf. [4], Proposition 6) to be in general position relative to P 4 (Q) sing (i.e. codim XA (X A ∩ P 4 (Q) sing ) ≥ 2, where P 4 (Q) sing is the singular locus of P 4 (Q)). Hence the dualizing sheaf of X A is computed as [5]). The existence of a crepant resolution for X A is proved in [10].
Definition 4.1. If X A is a quasi-smooth weighted Delsarte threefold with matrix A of degree m with m = q 0 + · · · + q 4 , then a crepant resolution X of X A is called a Calabi-Yau threefold of (weighted) Delsarte type in P 4 (Q) with matrix A. When A is a diagonal matrix, X A is also called a Calabi-Yau threefold of (weighted) Fermat type.
Since the quotient F d /Γ A is birational to X A , it is also birational to X A . Using the cohomological information of F d /Γ A , we write several birational properties of X A . Recall that A = (a ij ), d =| det A | and that F d is the Fermat threefold of degree d in P 4 . We have For each α = (α 0 , α 1 , α 2 , α 3 , α 4 ) ∈ A(X A ), define an integer where < α i /d > denotes the fractional part of α i /d. It takes values α = 0, 1, 2 or 3. If X is a Calabi-Yau threefold of Delsarte type in P 4 (Q) with matrix A, then there exists a unique element with α 0 = 0 (cf. [14], [15]). Note that − α 0 = 3 − 0 = 3. Recall that p is the characteristic of k. Let f be the order of p modulo d. Put To compute the height of the formal group of X, we look for those α with A H (α) < f (that is, the part of the Newton polygon with slope less than 1). As α ≥ 0 and #H = f , the inequality A H (α) < f holds only if tα = 0 for some t ∈ H; in other words, p i α = α 0 for some i. Hence it suffices to consider the H-orbit of α 0 when we calculate the height of the formal group of X. Given α 0 = (α 0 , α 1 , · · · , α 4 ), choose the value of α i as 0 < α i < d and let e = gcd(α 0 , α 1 , · · · , α 4 ). Set Note that α X is now defined modulo d A . Proof. Let H be the orbit of p modulo d A . Write a (resp. b) for the multiplicity of 1 (resp. 2) among the tα X 's for t ∈ H. Depending on whether p i ≡ −1 (mod d A ) or not for some i, there are 2 cases: (i) If H ∋ −1, then A H (α X ) = 0 + 1 + · · · + 1 + 2 + · · · + 2 + 3 = a + 2b + 3 = f + b + 1 > f (note a + b + 2 = f ). Hence the slope A H (α X )/f is greater than 1.
(ii) If H ∋ −1, then A H (α X ) = 0 + 1 + · · · + 1 + 2 + · · · + 2 = a + 2b Therefore the inequality A H (α X )/f < 1 holds if and only if case (ii) occurs with b = 0. This is the case where p i α X = 1 for all i aside from i = 0. (1) Write K for the quotient field of the ring W (k) of Witt vectors over k. Then by [2], The last equivalence follows from Lemma 4.3. (2) Note that as H is a group, A H (α) takes the same value for every α in the H-orbit of α X . Also, note that gcd(α 0 /e, α 1 /e, · · · , α 4 /e) = 1 implies that p i α X = p j α X for i ≡ j (mod f ).
Compared with the case of K3 surfaces, it is less frequent to have p i α X = 1 for all i (0 < i < f ). Hence the infinite height occurs more often than finite values of the height. Proof. Since µ is in 1 ≤ µ < f , we find p µ α X = − α X = 3. Hence Theorem 4.4 shows that h is infinite.
Remark 4.1. The converse to the above corollary is not true.

Calabi-Yau threefolds of weighted Fermat type
In this section, we apply the results of the previous section to weighted Fermat threefolds and compute the height of the formal group of a crepant resolution X. Let X A be a weighted Fermat threefold defined by the equation: When q 0 + q 1 + q 2 + q 3 + q 4 = m, a crepant resolution X of X is Calabi-Yau. Here Yui [16] has observed that there are 147 possibilities for Q = (q 0 , · · · , q 4 ). First we restate Theorem 4.4 for weighted Fermat threefolds.
Proposition 5.1. Let X be a Calabi-Yau threefold of Fermat type in P 4 (Q) of degree m with matrix A. Then α X = (q 0 , q 1 , q 2 , q 3 , q 4 ) and d A = m. Let f be the order of p modulo m. Assume that h := ht Φ X < ∞ is finite. Then h = f .
in this case and from the definition of A(X A ). As d i q i = m and gcd(q 0 , · · · , q 4 ) = 1, we see α X = (q 0 , q 1 , q 2 , q 3 , q 4 ) and d A = m.
Remark 6.1. In Example 6.3, the calculation shows that h > h 1,1 = 11 for some characteristic p. According to Corollary 2.3, there is no symplectic group action G on X A such that a crepant resolution Y of Y = X A /G becomes a mirror partner of X.
Remark 6.2. We also computed the height ht Φ X for the following quasi-diagonal threefolds, but none of them gives a new height beyond the lists of Propositions 5.2 and 6.2.