Raynaud-Mukai construction and Calabi-Yau Threefolds in Positive Characteristic

In this article, we study the possibility of producing a Calabi-Yau threefold in positive characteristic which is a counter-example to Kodaira vanishing. The only known method to construct the counter-example is so called inductive method such as the Raynaud-Mukai construction or Russel construction. We consider Mukai's method and its modification. Finally, as an application of Shepherd-Barron vanishing theorem of Fano threefolds, we compute $H^1(X, H^{-1})$ for any ample line bundle $H$ on a Calabi-Yau threefold $X$ on which Kodaira vanishing fails.


Introduction
Although every K3 surface in positive characteristic can be lifted to characteristic 0 [2], there are some non-liftable Calabi-Yau threefolds, namely a smooth threefold X with trivial canonical bundle and H 1 (X, O X ) = H 2 (X, O X ) = 0. If a Calabi-Yau polarized threefold (X, L) over the field k of char(k) = p ≥ 3 is a counter-example to Kodaira vanishing, i.e., H i (X, L −1 ) = 0 for i = 1 or i = 2, X is non-liftable to the second Witt vector ring W 2 (K) (and the Witt vector ring W (k)) by the cerebrated Raynaud-Deligne-Illusies version of Kodaira vanishing theorem [3]. But this does not necessarily imply that X cannot be liftable to characteristic 0. Moreover, a non-liftable variety is not necessarily a counter-example to Kodaira vanishing and as far as the author is aware, it is not known whether Kodaira vanishing holds for the non-liftable Calabi-Yau threefolds [6,7,8,16,4,1] that have been found so far. We do not even know whether Kodaira vanishing holds for all Calabi-Yau threefolds. Thus Kodaira type vanishing for Calabi-Yau threefolds is an interesting problem, which is independent from but seems to be closely related to the lifting problem.
A counter-example to Kodaira vanishing has been given by M. Raynaud, which is a surface over a curve [14]. This example was extended to arbitrary dimension by S. Mukai [11,12], which we will call the Raynaud-Mukai construction or, simply, Mukai construction.
The idea is, so to say, an inductive construction. Namely, we start from a polarized smooth curve (C, D). The ample divisor D satisfies a special condition, which is a sufficient condition for the non-vanishing H 1 (X, O X (−D)) = 0, and called a (pre-)Tango structure. Then we give an algorithm to construct from a variety X with a (pre-)Tango structure D a new varietyX with a higher dimensional (pre-)Tango structureD such that dimX = dim X + 1, using cyclic cover technique. There is another way of constructing counter-examples using quotient of p-closed differential forms [15,19]). But this is also an inductive construction and the obtained varieties are the same as the Raynaud-Mukai construction [19]. As far as the author is aware, non-inductive construction of higher dimensional counter-examples is not yet found.
In this paper, we consider the problem of whether we can construct a Calabi-Yau threefold with Kodaira non-vanishing by Mukai construction or by its modification. Section 2 presents the Raynaud-Mukai construction. For p ≥ 5, Raynaud-Mukai varieties are of general type so that the only possibility resides in the cases of p = 2, 3. Then in section 3, we will see that Mukai construction does not produce any K3 surfaces or Calabi-Yau threefolds (Corollary 9 and Corollary 10). Then we consider possible modifications of the Raynaud-Mukai construction: we keep the inductive construction but give up obtaining a (pre-)Tango structure. We show that if there exists a surface X of general type together with a (pre-)Tango structure D satisfying some property (this is not obtained by Mukai construction), we can construct a Calabi-Yau threefoldX with a (pre-)Tango structureD (Corollary 11) and describe the cohomology H 1 (X, OX) in certain situations (Proposition 13). Unfortunately, we could not prove or disprove existence of such a polarized surface (X, D).
Finally, in section 3 we show that if Kodaira non-vanishing H 1 (X, L −1 ) = 0 holds for a polarized Calabi-Yau threefold (X, L) over the field k of char k = p ≥ 5 satisfying the condition that L ℓ is a Tango-structure for some ℓ ≥ 1, we compute the cohomology H 1 (X, H −1 ) for any ample line bundle H of X (Theorem 18, Corollary 19).

The Raynaud-Mukai construction
In this section, we present the Raynaud-Mukai construction. Although [12] is available now, we prefer to use the version described in [11], which is slightly different from the 2005 version. As 1979 version is only available in Japanese, we present some details for the readers convenience.
The idea is to construct from a counter-example to Kodaira vanishing, i.e., a polarized variety (X, L) with H 1 (X, L −1 ) = 0 a new counter-example (X,L) with dimX = dim X +1. This inductive construction starts from a polarized curve (X, L) called a Tango-Raynaud curve.
Definition 1 (pre-Tango structure). Let X be a smooth projective variety. Then an ample divisor D, or an ample line bundle L = O X (D), is called a pre-Tango structure if there exists an element η ∈ k(X)\k(X) p , where k(X) denotes the function field of X, such that the Kähler differential is dη ∈ Ω X (−pD), which will be simply denoted as (dη) ≥ pD. In this paper, the element η will be called a justification of the pre-Tango structure.
Existence of a pre-Tango structure implies Kodaira non-vanishing. In fact, consider the absolute Frobenius morphism Thus, if there exists a pre-Tango structure D and dim X ≥ 2, then we have Kodaira non-vanishing: Notice that the inclusion H 0 (X, B X (−D)) ⊂ H 1 (X, O X (−D)) may be strict, so that there is a possibility that a non pre-Tango structure L causes a Kodaira nonvanishing. However, since the iterated Frobenius map is trivial for e ≫ 0, L n is a pre-Tango structure for sufficiently large n ∈ N.
Pre-Tango structure for curves are characterized by the Tango-invariant [21,20]. Let C be a smooth projective curve of genus g (≥ 2). Then the Tango-invariant is defined as where [· · · ] denotes the round up. We easily know that Then, C has a pre-Tango structure D if n(C) > 0. We just set D = (df ) p and then D is ample on C such that (df ) ≥ pD.
In the following, we will call the pair (X, L) in Definition 1 a pre-Tango polarization. The Raynaud-Mukai construction is an algorithm to make a new pre-Tango polarization from a pre-Tango polarization whose dimension is lower by one.
2.2. purely inseparable cover. From a pre-Tango polarized variety (X, L) we can construct a reduced and irreducible purely inseparable cover τ : G −→ X of degree p. Conversely, existence of such a cover implies existence of a pre-Tango polarization.
2.2.1. Construction and characterization. Given a pre-Tango polarized variety (X, L = O X (D)), choose an element (0 =)η ∈ H 0 (X, B X (−D))(= Ker F ). Then we have a corresponding non-split short exact sequence where E is a rank 2 vector bundle on X. Taking the Frobenius pull-back, we obtain an exact sequence Notice that the new sequence corresponds to F (η) = 0 so that it splits and by using the split maps, we obtain the sequence with the reverse arrows Tensoring by L (p) −1 over O X , we finally obtain the sequence Now we consider the P 1 -fibration together with the canonical section F ⊂ P , which is defined by the image of 1 ∈ O X in E, and π (p) : (1) and (2). Moreover, we consider the relative Frobenius morphism ψ :

Then we can show
We can show that existence of such a G characterizes pre-Tango structure. To summarize, we have Theorem 4 (See Proposition 1.1 in [12]). Let X be a smooth projective variety of characteristic p > 0 and L be an ample line bundle. Then the following are equivalent: (1) L is a pre-Tango structure.
(2) There exists a P 1 -bundle π : P −→ X and a reduced irreducible effective divisor G ⊂ P such that (a) ρ : G −→ X is a purely inseparable cover of degree p smoothness. For smoothness of the purely inseparable cover G, we have Theorem 5 (S. Mukai [12]). Let (X, D) be a pre-Tango polarized variety over the field of characteristic p > 0 and G is the purely inseparable cover constructed from a justification (0 =)η ∈ k(X)\k(X) p . Then G is smooth if and only if (dη) = pD. This means that for the multiplication by dη Coker dη is locally free at every x ∈ X.
Proof. For a proof in the case of dim X = 2, see Theorem 3 [18].
Definition 6 (Tango structure). Let X be a smooth projective variety with a pre-Tango structure L = O X (D). Then, D, or L, is called a Tango structure if and only if a justification η ∈ k(X)\k(X) p satisfies (dη) = pD. In this case, the pre-Tango polarization (X, L) or (X, D) will be called a Tango polarization.
A smooth projective curve X of genus g ≥ 2 with a Tango structure D is called a Tango-Raynaud curve. For examples of Tango-Raynaud curves, see for example [14,11,12].
2.3. cyclic cover. Let (X, D) be a pre-Tango polarization and D is divided by k ∈ N with (p, k) = 1 and we have D = kD ′ . If X is a curve, we can divide D by any natural number k dividing deg D using the theory of Jacobian variety (cf. page 62 of [13]). But the condition (p, k) = 1 is necessary for the covering to be cyclic. Now we construct a kth cyclic cover of the P 1 -fibration π : P −→ X ramified over F + G, which means that π is ramified at the reduced preimage of F + G. There are at least two well-known constructions.
The first one is rather explicit and is suitable for computing cohomologies (cf. [18]). We first choose m ∈ N such that k|(p + m) and set i+j ≥ k where we choose a non-trivial element ξ ∈ O P (mF +G) such that mF +G is the zero locus of ξ. Now we consider the affine morphism X ′ := Spec k−1 i=0 M ⊗i → P and this is the cyclic cover ramified over mF + G. Since X is smooth, F ∼ = X is also smooth. Moreover if D is a Tango structure and G is smooth by Theorem 5, then X ′ is smooth if and only if m = 1; if m > 1 then X ′ is singular along F , which may cause non-normality of X ′ . Normalization of X ′ , if necessary, is carried out by Esnault-Viehwegs method (see § 3 of [5]).
and this is smooth if D is Tango. We note that this normalization procedure highly depends on the condition (p, k) = 1 since we use the kth root of unity. Then we set the natural morphism ϕ :X −→ X ′ −→ P . The second construction uses normalization. Since we have liner equivalence G ∼ pF − pπ * (D) there exist a function R ∈ k(P ) such that (R) = G − (pF − pπ * (D)) = G − (pF − pkπ * (D ′ )). Then letX be the normalization of P in the finite extension k(P )(R 1/k ) of k(P ) and ϕ :X −→ P be the normalization morphism. Then we set f = π • ϕ. Now if we work locally we know that there exist divisorsG andF oñ X such that ϕ * F = kF and ϕ * G = kG. Moreover, we haveG ∼ pF − pf * (D ′ ) oñ X. We note that the condition (p, k) = 1 is necessary to assure the existence ofF , division of F by k. Otherwise, if k = p ℓ r with ℓ ≥ 1 and (p, r) = 1 we haveF ⊂X such that ϕ * F = k ′F with k ′ = p ℓ−1 r = k/p.X is smooth if D is Tango.
Now we set f := π • ϕ :X −→ X, which is actually a fibration of rational curves with moving singularities, i.e., rational curves with cusp singularity of type x p = y t atG.
This result is stated in [11] without proof and in the case of k ≡ 1 mod p a proof using Maruyama's elementary transformation [10] is given in [12]. We give here a proof of general case.
Proof. Letη = R 1/k ∈ k(X). Since (η) =G − pF + pf * (D ′ ),η is locally described as η = g(δφ −1 ) p where g, φ and δ are local equations definingG,F and f * (D ′ ). Then its Kähler differential is Now we consider dg. As a Cartier divisor we describe D = {(U i , g i )} i for an open cover X = i U i and g i ∈ k(X). Since D is a pre-Tango structure, there exists a justification η ∈ k(X) such that (dη) ≥ pD, which locally means that we have η| U i = g p i c i for some c i ∈ O U i so that we have (dη)| U i = g p i dc i . Then, as in Proposition 1 [18], G ⊂ P is locally described as where x is the (local) coordinate corresponding to the canonical section F of π : P −→ X. Hence the local defining equation of G ⊂ P is c i x p + y p , and since ϕ * F = kF and ϕ * G = kG, the defining equation ofG is g = c i Z kp + W kp , where Z is the local coordinate ofX corresponding toF , namely Z = εφ with some local unit ε. Thus we have (4) dg = ε pk φ pk dc i . (3) and (4) we obtain dη = (δφ −1 ) p dg = ε pk (φ k−1 δ) p dc i so that

Thus by
where the equality holds if (dη) = pD, i.e., if D is a Tango structure.

Calabi-Yau threefolds and the Raynaud-Mukai construction
3.1. Raynaud-Mukai varieties cannot be Calabi-Yau. The aim of this section is to show that Mukai construction does not produce K3 surfaces or Calabi-Yau threefolds. Notice that Raynaud-Mukai variety is always of general type for p ≥ 5 (cf. Prop. 7 [11] or Prop. 2.6 [12]) so that the only possibility is the case p = 2, 3. Now let (X, D), D = kD ′ (k ∈ N), π : P −→ X, F, G ⊂ P , (X,D), ϕ :X −→ P and f :X −→ X be as in the previous section. The canonical divisor of X will be simply denoted by K. Now we have Proposition 8 (cf. Prop. 7 [11]). LetK be the canonical divisor ofX. Then we haveK Proof. Since the finite morphism ϕ :X −→ P is ramified atF = (ϕ * (F )) red and G = (ϕ * (G)) red with the same ramification index k and F + G ∼ (p + 1)F − pkπ * D ′ , we computeK Moreover, since E is the rank 2 vector bundle satisfying we have K P ∼ −2F + π * (K + kD ′ ). Then we obtain the required formula.
We notice that since Pic P ∼ = Z · [F ] ⊕ π * Pic X and ϕ is finite, we have PicX ∼ = Z · [F ] ⊕ f * Pic X. This fact will be used implicitly in the following discussion. Proof. Assuming dimX = 2, we have only to show that we never haveK ∼ 0. Assume that we haveK ∼ (pk − p − k − 1)F + f * (K − (pk − p − k)D ′ ) ∼ 0, from which have two relations pk − p − k − 1 = 0 and K − (pk − p − k)D ′ = 0. By the first relation, we have k = p+1 p−1 ∈ N, so that we must have p = 2 and k = 3 or p = 3 and k = 2. This implies K = D ′ by the second relation. However, since (X, D) is a (pre-)Tango polarized curve, we have (dη) ≥ pD for some justification η ∈ k(X), namely D ′ = K ≥ pD = pkD ′ , which is impossible unless pk = 1.
By a similar discussion to the proof of Corollary 9, we can also show Corollary 10. A Raynaud-Mukai threefold can never be Calabi-Yau.
Proof. LetX be a Mukai threefold obtained from a Mukai surface X with a (pre-)Tango structure D = kD ′ as a kth cyclic cover of the P 1 -fibration P and assume thatK ∼ 0. Then as in the proof of Corollary 9 we have (p, k) = (2, 3) or (3, 2) and Now we will consider the situation whose dimensions are all lower by one. Namely, let the surface X be constructed from a (pre-)Tango polarized curve (X 1 , D 1 ) with We have the k 1 th cyclic cover ϕ 1 : X −→ P 1 of the P 1 -fibering π 1 : P 1 −→ X 1 ramified over F 1 + G 1 andF 1 = (ϕ * 1 (F 1 )) red andG 1 = (ϕ * 1 (G 1 )) red have the same ramified index k 1 . We set f 1 = π 1 • ϕ 1 . Then by Proposition 8, we have by definition, the condition (5) entails Then the coefficient ofF 1 must be 0 so that we have But since we must have k 1 ∈ N and (k 1 , p) = 1, these values of k 1 are not allowed.

a modification of the Raynaud-Mukai construction. The Raynaud-Mukai
construction is an algorithm to construct from a given (pre-)Tango polarization (X, D) with D = kD ′ a new (pre-)Tango polarization (X,D) with dim X = dimX − 1 by taking a kth cyclic cover. We apply this procedure inductively starting from a (pre-)Tango polarized curve. We have seen in the previous subsection that the essential reason that the Raynaud-Mukai construction does not produce Calabi-Yau threefolds is that we cannot find the degree k cyclic covers with (p, k) = 1 in all inductive steps. Now we will consider some modification of the Raynaud-Mukai construction. There are following two possibilities.
(I) Let (X, D) be a (pre-)Tango polarized surface obtained by a method other than Mukai construction. Then apply the Raynaud-Mukai construction to obtain a (pre-)Tango polarized threefold (X,D). (II) Let (X, D) be a (pre-)Tango polarized surface by the Raynaud-Mukai construction. Then we construct a Calabi-Yau threefold in a similar way to Mukai construction. Namely, we do not assume the condition (p, k) = 1 for the degree k of "cyclic cover ". The Calabi-Yau threefolds obtained by (I) are counter-examples to Kodaira vanishing. The surface X required in (I) is precisely as follows: Corollary 11. Let (X, D) a (pre-)Tango polarized surface with D = kD ′ for some k ∈ N. Then the Raynaud-Mukai construction gives a polarized Calabi-Yau threefold (X,D) by a kth cyclic cover if and only if (i) (p, k) = (2, 3) or (3, 2), and (ii) D = kD ′ for some ample D ′ and K X ∼ D ′ . In particular, X is a surface of general type.
Proof. By the same discussion as in the proof of Corollary 9 and 10.
Unfortunately we do not know how to construct a polarized surface (X, D) as in Corollary 11. But Theorem 12(i) below seems to indicate a possibility.
Theorem 12 (S. Mukai [11]). Let X be a (smooth) surface over the field k of char k = p > 0. Assume that Kodaira vanishing fails on X. Then we have (i) X is of general type or quasi-elliptic surface with Kodaira dimension 1 (if p = 2, 3). (ii) There exists a surface X ′ birationally equivalent with X such that there is a morphism g : X ′ −→ C to a curve C whose fibers are all connected and singular.
It is proved that, in the case of surfaces, Kodaira (non-)vanishing is preserved in birational equivalence (see Corollary 8 [20]). Thus by Theorem 12(ii) it seems to be reasonable to consider a fibration ρ : X −→ C to a curve.
For a Calabi-Yau threefold, we often assume simple connectedness which implies H 1 (X, OX) = 0 for our example. For this property, we have the following.
Proposition 13. Assume that the surface X in Corollary 11 has a fibration over a curve C: g : X −→ C and set h : Proof. Consider the Leray spectral sequence . Then by the 5-term exact sequence we have where the last term H 2 (C, h * OX ) vanishes since dim C < 2. Thus we have On the other hand, we have (p, k) = (2, 3) or (3, 2) by Corollary 11 and the explicit construction of the cyclic cover gives Now since π * O P = O X and π * O P (−i) = 0 for i > 0 we obtain h * OX = g * O X .
Remark 14. Using another spectral sequence and 5-term exact sequence we can show the inclusion H 0 (C, R 1 g * (f * OX )) ⊂ H 0 (C, R 1 h * OX ) but the equality does not hold in general.
Next we consider the construction (II), whose algorithm is as follows: Given a (pre-)Tango curve, we make a (pre-)Tango polarized surface (X, D) and a P 1 -bundle π : P −→ X with the canonical section F ⊂ P together with a purely inseparable cover π| G : G −→ X of degree p corresponding to D. Then choose k = p ℓ r with (p, r) = 1 and ℓ ≥ 1 and let ϕ :X −→ P be the normalization of P in k(P )(R 1/k ) where R ∈ K(P ) is such that (R) = G − (pF − pπ * (D)). Proof. Let (X, D) be a (pre-)Tango polarized surface by the Raynaud-Mukai construction. Then we obtain a P 1 -bundle π : P −→ X together with the canonical section F and the purely inseparable cover G → X of degree p (see Theorem 4).

Cohomology of Calabi-Yau threefold with Tango-structure
In this section, we compute the cohomology H 1 (X, H −1 ) for arbitrary ample H under the assumption that X is a Calabi-Yau threefold on which Kodaira vanishing fails.
Theorem 17 (N. Shepherd-Barron [17]). Let X be a normal locally complete intersection Fano threefold over the field k of char k = p ≥ 5 and L be an ample line bundle on X. Then we have H 1 (X, L −1 ) = 0.
Recall that, for a polarized smooth variety (X, L), Kodaira non-vanishing H 1 (X, L −1 ) = 0 does not necessarily imply L is a (pre-)Tango structure. But by Enriques-Severi-Zariski's theorem, there exists ℓ > 0 such that we have H 1 (X, L −p ℓ+1 ) = 0 but H 1 (X, L −p ℓ ) = 0. Then such L ℓ is at least a pre-Tango structure. Now based on these observations, we obtain Theorem 18. Let (X, L) be a smooth Calabi-Yau threefold over a field k of char k = p ≥ 5 with Kodaira non-vanishing H 1 (X, L −1 ) = 0. If L ℓ is a Tango structure for some ℓ ≥ 1, then we have for every ample line bundle H on X, where ρ : Y −→ X is a purely inseparable cover of degree p corresponding to the Tango structure as in Theorem 4.
Recall that for a purely inseparable cover p : Y −→ X of degree p there exists a p-closed rational vector field D on X such that (ρ * O Y ) D := {f ∈ ρ * O Y : D(f ) = 0} = O X (cf. [15]). Thus we have Corollary 19. Under the same assumption as Theorem 18, we have where D is a p-closed rational vector field on X corresponding to the purely inseparable cover ρ.