Calabi-Yau threefolds over finite fields and torsion in cohomologies
CCALABI-YAU THREEFOLDS OVER FINITE FIELDS AND TORSION INCOHOMOLOGIES
YEUK HAY JOSHUA LAM
Abstract.
We study various examples of Calabi-Yau threefolds over finite fields. Inparticular, we provide a counterexample to a conjecture of K. Joshi on lifting Calabi-Yau threefolds to characteristic zero. We also compute the p -adic cohomologies of someCalabi-Yau threefolds constructed by Cynk-van Straten which have remarkable arith-metic properties, as well as those of the Hirokado threefold. These examples and com-putations answer some outstanding questions of B. Bhatt, T. Ekedahl, van der Geer-Katsura and Patakfalvi-Zdanowicz, and shed new light on the Beauville-Bogomolovdecomposition in positive characteristic. Our tools include p -adic Hodge theory as wellas classical algebraic topology. We also give potential examples showing that Hodgenumbers of threefolds in positive characteristic are not derived invariants, contrary tothe case of characteristic zero. Contents
1. Introduction 21.1. A conjecture of K. Joshi 21.2. Beauville-Bogomolov decomposition in positive characteristic 21.3. The CvS Calabi-Yau threefolds 31.4. Questions of Ekedahl and Takayama 41.5. Sketch of the arguments 41.6. Some speculations 51.7. Acknowledgements 52. Godeaux Calabi-Yau threefolds 52.1. Construction 62.2. Joshi’s conjecture 62.3. A question of van der Geer-Katsura 82.4. Beauville-Bogomolov decomposition 82.5. Hodge diamond 113. The CvS Calabi-Yau threefolds 124. Supersingularity 145. Conjugate spectral sequence and liftings 155.1. A criterion for non-degeneration of the conjugate spectral sequence 155.2. The Hirokado threefold 176. Topology 177. The CvS variety in characteristic 3 188. The CvS variety in characteristic 5 199. Further questions 239.1. Hodge numbers and derived equivalences 23
Date : September 22, 2020. a r X i v : . [ m a t h . AG ] S e p YEUK HAY JOSHUA LAM H and Y Introduction
In this paper we investigate various arithmetic properties of Calabi-Yau varieties overfinite fields. The common theme tying together all the results of this paper is that for avariety over a mixed characteristic local ring, the classical topology (including torsion) ofthe generic fiber has control over all the cohomologies of the special fiber; the most generalform of this phenomenon is due to the recent work of Bhatt-Morrow-Scholze, although wewill only need older results of Caruso and Faltings.1.1.
A conjecture of K. Joshi.
In a fascinating work [Jos14] K. Joshi studies manyproperties of surfaces and threefolds over finite fields, and states the following
Conjecture 1.1 (Conjecture 7.7.1 of [Jos14]) . Let X be a smooth proper Calabi-Yau three-fold over a finite field k . Then X lifts to characteristic zero if and only if (1) H ( X, Ω ) = 0 , and (2) X is classical. Here we may take as definition the adjective classical as meaning (for a threefold) H ( X, ¯ Q l ) = 0 (it is a consequence of the results in [Jos14] that this is equivalent to theoriginal definition of classical, which is in terms of Hodge-Witt cohomology; for the originaldefinition we refer the reader to Section 7.4 of loc.cit.). Note that if a Calabi-Yau threefoldis liftable to characteristic zero then it is automatically classical, by Artin comparison andclassical Hodge theory over C . Theorem 1.2.
There exists a counterexample over F to Conjecture 1.1. More precisely,there exists a liftable X/ F with H ( X, Ω ) (cid:54) = 0 . The deformation theory of Calabi-Yau threefolds in positive characteristic is mysteri-ous, and most existing results require liftings to characteristic zero, making the latter animportant problem to study. As mentioned above, Hirokado constructed the first example ofnon-liftable ones, and by now there are several more examples of such; we refer the readerto [Tak17] for an excellent survey. In particular, the results of this paper show that allknown examples of non-liftable Calabi-Yau threefolds satisfy b = 0, i.e. the first conditionin Joshi’s conjecture, and it seems reasonable to ask if this is always the case: Question . Do all non-liftable Calabi-Yau threefolds X/ F p satisfy b ( X ) = 0?1.2. Beauville-Bogomolov decomposition in positive characteristic.
Over the com-plex numbers, varieties with trivial canonical bundle are built from three classes of varieties:abelian, hyperk”ahler, and Calabi-Yau. This important result is known as the Beauville-Bogomolov decomposition; recently Patakfalvi-Zdanowicz [PZ19] have obtained interestingresults towards a decomposition result in positive characteristic. It turns out that theGodeaux Calabi-Yau threefold example sheds new light on this question, and in particular
ALABI-YAU THREEFOLDS OVER FINITE FIELDS AND TORSION IN COHOMOLOGIES 3 answers a question in loc.cit. [PZ19, Question 13.6]. The details of this can be found inSection 2.4.1.3.
The CvS Calabi-Yau threefolds.
We also study some beautiful Calabi-Yau three-folds constructed by Cynk-van Straten, who constructed a Calabi-Yau threefold X / O (here O is the ring of integers of a certain number field) which is rigid in characteristic zero, butadmits (obstructed) deformations in characteristic 5. Numerically what this means is thatthe Hodge number h jumps from being 0 to 1 when we reduce mod 5; see Theorem 1.4for the Hodge diamonds of both the generic and special fibers, and also 3.1 for more detailon these Calabi-Yau threefolds. We will refer to these as the CvS Calabi-Yau threefolds inwhat follows.We already saw an example of a Hodge number jumping in the special fiber in thecounterexample to Joshi’s conjecture above: there it was the Hodge number h instead (andin fact on other components on the moduli space we may also have jumping behavior of h ,or even both h and h ). There the reason was the torsion in the first Betti cohomologyof the generic fiber; in the case of the CvS Calabi-Yau threefold in characteristic 5 we willshow that the jumping behavior is not due to torsion in Betti cohomology, but rather a non-degeneration of the Hodge-de Rham spectral sequence. In fact, for the integral model X / O ,all cohomologies are torsion free except for Hodge cohomologies, which are guaranteed tohave torsion because of the jumping behaviour. Theorem 1.4.
For the CvS Calabi-Yau threefold the following hold: (1)
The variety X / F has torsion free crystalline cohomology; in particular the coho-mology groups H ∗ dR ( X ) of the integral model are torsion free as well. (2) The Hodge cohomology groups H i ( X , Ω j ) are torsion free O -modules, except for H ( X , Ω ) and H ( X , Ω ) , which have non-trivial torsion. (3) The Hodge diamond of X is given by
10 00 39 01 1 1 10 39 00 01 . This should be compared with that of the generic fiber
X/K , which was computedby Cynk-van Straten [CS09, Section 5.2]:
10 00 38 01 0 0 10 38 00 01 . Note in particular that Hodge symmetry holds for X , which is not generally thecase for smooth proper varieties over finite fields. (4) The special fiber X is supersingular, i.e. its Artin-Mazur formal group is isomor-phic to ˆ G a . YEUK HAY JOSHUA LAM
This theorem answers a question posed by Bhatt [Bha17, p.48 2(c)], who asked for anexample of a smooth proper variety over O C p whose ´etale and crystalline cohomologies aretorsion free but whose Hodge cohomologies are not (as well as an example where Hodgecohomology has more torsion than de Rham cohomology). Remark . Note that, for both the generic and special fibers, the Hodge cohomologygroups H (Ω ) , H (Ω ) (as well as H ( O ) which is part of the definition of a strict CY)were already computed by Cynk-van Straten in [CS09]: our contribution for part (3) of theTheorem 1.4 is the computations of H (Ω ) , H (Ω ). On the other hand, it would also beinteresting to figure out the length of torsion in the Hodge cohomology groups H ( X , Ω )and H ( X , Ω ), since we have not been able to do so.1.4. Questions of Ekedahl and Takayama.
Our techniques also allow us to answer somequestions of Ekedahl and Takayama. Since these are addressed using the same technique,we describe them in the same section.As mentioned above, the Hirokado threefold H was the first example of a non-liftableCalabi-Yau threefold. This example was studied in detail by Ekedahl [Eke03] by identifying H as a Deligne-Lusztig variety, from which he computed most cohomology groups of H ,including all Hodge cohomologies (see Theorem 5.3). However, the issue of the precise formof the crystalline cohomology was left open: in the computations of loc.cit. this boils downto the issue of the degeneration of the Hodge-de Rham spectral sequence, which could notbe resolved.On the other hand, Takayama asks [Tak17][Section 4.1] whether the non-liftability of Y can again be explained by the vanishing of b .These two issues are resolved by the following Corollary 1.6.
If a rigid Calabi-Yau threefold
X/k is not liftable to W ( k ) , then the Hodge-de Rham spectral sequence does not degenerate at the E -page, and b = 0 . Indeed, Corollary 1.6 implies that the Hodge-de Rham spectral sequence degeneratesfor H , since H was shown to not lift to W ( k ) by Ekedahl, thus resolving the first questionabove; on the other hand, Cynk-van Straten also showed that Y does not lift to W ( k ),and thus b ( Y ) = 0, resolving the second issue stated above.1.5. Sketch of the arguments.
We comment briefly on the proofs of our results.For the counterexample in Theorem 1.2, X is constructed by taking a free quotient ofa quintic hypersurface in P by a finite group of order 5. Here we remark on why this isa plausible strategy: in fact we will consider the quotient X of a hypersurface Y over Z by a group scheme of order 5, and let X (respectively Y ) denote the special fiber of X (respectively Y ). In characterstic zero, the variety X [1 /
5] has π = Z /
5, and therefore bya result of Caruso we have H dR ( X ) ∼ = F , where the left hand side is the algebraic de Rham cohomology of X . Thus, by the Hodge-deRham spectral sequence, the groups H ( X, Ω ) , H ( X, Ω ) cannot both be zero. There arevarious choices of group schemes of order 5, and it turns out that choosing µ gives H ( X, Ω ) (cid:54) = 0 , H ( X, O ) = 0 , which is what we wanted: the latter equality needed for the variety X to be Calabi-Yau. Forexample, we can check that H ( X, O ) = 0 as follows: since X is a quotient of a hypersurfaceby µ we have Pic( X ) ∼ = Z / , ALABI-YAU THREEFOLDS OVER FINITE FIELDS AND TORSION IN COHOMOLOGIES 5 where this is an equality of group schemes, and hence H ( X, O ) = T e Pic( Z ) = 0 . Remark . We could also have chosen to quotient by Z / α . This example is remi-niscent of the example of Enriques surfaces in characteristic 2, where the moduli space hastwo components according to whether Pic τ (the torsion part of the Picard group) is µ or Z /
2, and intersecting along the locus where Pic τ ∼ = α . Remark . As far as we are aware this example was first considered by Aspinwall-Morrison[AM94] in their study of the discrete torsion phenomenon in string theory, at least for thecase of quotients of the Fermat hypersurface.In order to compute the cohomologies required throughout this paper, we make frequentuse of Caruso’s theorem whenever we have characteristic zero lifts: the idea here is thatBetti cohomologies of complex varieties are often easier to access, and using Caruso’s resultwe can deduce information about the special fibers. This technique is perhaps most notablein the proof of Theorem 1.4, and seems to be the first application of integral p -adic Hodgetheory to actual computations. To calculate Betti cohomlogies of the lifts we use classicaltools such as Mayer-Vietoris and the Serre spectral sequence.1.6. Some speculations.
We end in Section 9 with some more possible applications of themain theme of this paper, namely the comparison between torsion in Betti cohomology ofthe generic fiber and cohomologies of the special fiber.We suggest possible examples showing that Hodge numbers are not derived invariantsfor threefolds in positive characteristic: this question was recently investigated in [AB19]and in characteristic zero it is a theorem of [PS11] that the Hodge numbers of threefoldsare in fact derived invariants. Our idea is that, since torsion in cohomology is not invariant(as shown by Addington [Add17]), and torsion contributes to Hodge cohomologies in thespeical fiber, it seems likely that the Hodge cohomologies will difer in general. In 9.1 we givespecific conjectural Calabi-Yau threefold examples; however, since the examples in questionwere “constructed” physically by the use of string dualities (cf. [Sch06]) and the authoris not aware of rigorous mathematical constructions of these Calabi-Yau threefolds in theliterature, we leave the details to future work.In Section 9.2 we consider the question of torsion in H ( X, Z ) (the group Tors( H ( X, Z ))is referred to as the Brauer group Br( X )) for a Calabi-Yau threefold X . Again, a torsionclass contributes to Hodge cohomologies of the reduction mod p , and if X has a CY reduc-tion then the contribution must be to h , which is also the dimension of the deformationspace of X ; in other words, classes in the Brauer group give extra mod p directions for X to deform in.We consider some particular examples of abelian surface fibered Calabi-Yau threefoldswith non-zero Brauer groups, and suggest how the extra deformations may be realized,analogous to the Moret-Bailly example [Mor81] of a family of mutually isogenous abeliansurfaces in positive characteristic.1.7. Acknowledgements.
To be added.2.
Godeaux Calabi-Yau threefolds
In this section we introduce the Godeaux Calabi-Yau threefolds, using which we givecounterexamples to Joshi’s conjecture, as well as answer a question of van der Geer-Katsura.We also compute the Hodge diamond of these threefolds.
YEUK HAY JOSHUA LAM
Construction.
We make crucial use of some results of Kim-Reid on the
Tate-Oortgroup scheme whose definition we now recall. Throughout this section we fix a prime p .For a group scheme G over a base scheme S and an element f ∈ O ( S ), we denote by G [1 /f ]the group scheme over S [1 /f ]. Definition/Theorem 2.1 (Section 3 of [Rei19]) . Let B denote the ring Z [ S, t ] / ( P ) where P := St p − + p . There is an order p group scheme TO p such that • TO p [1 /t ] is isomorphic to the multiplicative group scheme µ p ; • TO p [1 /S ] is a form of the ´etale group scheme Z /p ; • the fiber of TO p over the point Spec( F p ) = Spec( B/ ( S, t )) is isomorphic to α p .Here as usual, over a base scheme in characteristic p , α p denotes the order p group schemewhich is the kernel of Frobenius on G a . We refer to TO p as the Tate-Oort group scheme. There is the following simple description of TO p over ¯ B := B ⊗ Z F p ∼ = F p [ S, t ] / ( St ).We make the following definitions: G ¯ B := (cid:18) x tx (cid:19) ⊂ GL(2 , ¯ B ) , (1) TO p := V ( x p = Sx ) ⊂ G ¯ B ;(2)i.e. the second equation means we take the closed subscheme of G ¯ B defined by the equation x p = Sx . One has to check first of all that G ¯ B is a group, and secondly that (2) defines asubgroup: we refer the reader to [Rei19, Proposition 3.1] for the details. Note that when S is invertible the (2) indeed defines an ´etale covering of Spec ¯ B ; when t invertible we maydefine the map G m → G ¯ B , (3) λ (cid:55)→ (cid:18) λ − t λ (cid:19) (4)and observe that TO p is precisely the image of µ p ⊂ G m . Finally when t = S = 0 we havethat TO p is by definition the subgroup of G a ∼ = (cid:18) x (cid:19) ⊂ GL(2 , ¯ B ) , satisfying x p = 0, which is also the definition of α p . To summarize this discussion, wehave written down an explicit group scheme ¯ TO p satisfying the three conditions specifiedby Definition/Theorem 2.1 mod p .We record here the following theorem of Kim-Reid (see [Rei19, Section 6], and also[RK]). Theorem 2.2.
There exists a quintic hypersurface Y ⊂ P B with an action of TO suchthat the quotient X is non-singular over the locus S = t = 0 . Joshi’s conjecture.
In [Jos14] Joshi studies many interesting questions about p -adiccohomologies of varieties, and states the following conjectural criterion for lifting Calabi-Yauthreefolds: Conjecture 2.3 (Conjecture 7.7.1 of [Jos14]) . Let X be a smooth proper Calabi-Yau three-fold over k . Then X lifts to characteristic zero if and only if (1) H ( X, Ω ) = 0 , and (2) X is classical. ALABI-YAU THREEFOLDS OVER FINITE FIELDS AND TORSION IN COHOMOLOGIES 7
In the statement of Conjecture 2.3 the adjective classical means one of several equivalentthings, one of which is that the Betti number b (the dimension of H et ( X, Q l ) for any l (cid:54) = p )is non-vanishing; we will take this as our definition in what follows. Note that if X admits alift to characteristic zero ˜ X then it is automatically classical: indeed, by Artin’s comparisontheorem H et ( X, ¯ Q l ) ∼ = H ( ˜ X ( C ) , Q ) ⊗ ¯ Q l , where the right hand side denotes Betti cohomology of the topological space ˜ X ( C ), and isnon-zero by Hodge theory. Theorem 2.4.
There exists a Calabi-Yau threefold
X/k with a lift to to characteristic zeroand such that H ( X, Ω ) ∼ = k. Remark . Note that this only gives a counterexample for one direction of the conjecture.It would be very interesting to investigate the opposite direction, namely the existence of alifting given conditions (1) and (2).
Proof of Theorem 2.4.
Over the ring B [1 /S ], the group scheme TO becomes the ´etalegroup scheme Z /
5, and by the non-singularity of the fiber of X over S = t = 0, we havethat for a generic invertible S , the fiber of X is also non-singular. Pick such a genericchoice of S and let ˜ X = X be the corresponding fiber of X ; also denote by X the specialfiber of ˜ X . Note that X corresponds to some smooth point x ∈ Spec( B )( k ) and we mayassume ˜ X corresponds to a W ( k )-point of Spec( B ).Note that in characteristic zero we have H ( ˜ X ( C ) , Z ) = Z / , since ˜ X ( C ) arises as a free quotient of a hypersurface by Z /
5, and therefore by the universalcoefficients theorem H ( ˜ X ( C ) , Z /
5) = Z / . Also by Deligne-Illusie’s theorem the Hodge-de Rham spectral sequence degenerates for thespecial fiber X : indeed, X lifts to W ( k ) dim X = 3 < −
1. Also, by Theorem 8.1 wehave dim k H ( X k ) = dim F H ( ˜ X ( C ) , Z /
5) = 1and therefore h ( X ) + h ( X ) = 1. Now since h ( X ) = dim T e (Pic( X )) = 0, we concludethat h (cid:54) = 0, and hence H ( X, Ω ) ∼ = k, as required. (cid:3) Remark . It is also possible to explicitly construct a non-vanishing section in H ( X, Ω ),as follows. Note that we have the covering Y → X, which is a µ -cover by construction, and hence inseparable. By the same argument as in[CD89, Proposition 0.1.2], we may produce a section in H ( X, Ω ), whose vanishing locusis precisely the image of the singular locus of Y . We review the construction here: we havethat π ∗ O Y ∼ = O X ⊕ (cid:77) i =1 L ⊗ i YEUK HAY JOSHUA LAM for some line bundle L on X . Now since the map π is purely inseparable, Y is locally givenby the equation y = b for some b ∈ Γ( L ⊗ ). Now L ⊗ ∼ = O , and one may check that db ∈ Γ(Ω X ) glue together togive a global section. See also the discussion in [Rei19, Section 6.2.2] for the constructionof this section.2.3. A question of van der Geer-Katsura.
In their study of heights of Calabi-Yauthreefolds, van der Geer-Katsura asks the following [GK03, Section 7]
Question . Can a Calabi-Yau variety of dimension 3 in positivecharacteristic have non-zero regular 1-forms or regular 2-forms?We observe that Theorem 2.4 gives a positive answer to the first part of this question.It would be interesting to figure out the answer to the second question, namely whether wecan have H ( X, Ω ) (cid:54) = 0. Note that even though the Godeaux Calabi-Yau threefold has anextra class in H , the class lands in H (Ω ) by the following Proposition 2.8.
For the Godeaux Calabi-Yau threefolds, we have H ( X, Ω ) = 0 .Proof. Suppose not, i.e that H ( X, Ω ) (cid:54) = 0. Note that the Godeaux surface S appears asa hyperplane section of X , and it is classical in the sense that, just like X , we havePic τ ( S ) ∼ = Z / . By standard facts about classical surfaces [Lan81], we have h ( S ) = 0, and therefore H ( S, Ω S ) = 0by Serre duality. Also (by the construction of classical Godeaux surfaces in [Lan81], as wellas the construction of Godeaux Calabi-Yau threefolds in Appendix A, both X and S liftto characteristic zero: let us denote these lifts by ˜ X , ˜ S respectively. Now the Lefschetzhyperplane theorem implies that the induced map on Betti cohomology groups H ( ˜ X, Z ) → H ( ˜ S, Z )is injective. Then by Caruso [Car08, Th´eor`eme 1.1] the map H ( X ) → H ( S )is injective also, since the ´etale cohomologies are obtained by applying Breuil’s functor T st(cid:63) .This is a contradiction since H ( S, Ω S ) = 0, and therefore we have H ( X, Ω X ) = 0, asclaimed. (cid:3) Beauville-Bogomolov decomposition.
In an interesting recent paper[PZ19], Patakfalvi-Zdanowicz considered the problem of extending the Beauville-Bogomolov decomposition,which is a powerful tool in studying K -trivial varieties over the complex numbers, to posi-tive characteristic. Along the way, they posed the following Question, which we will answerin the present work. Question . Let k be an algebraically closed field of characteristic p >
0. Does there exist a finite µ p -quotient Y → X such that Y is a singular projectiveGorenstein K -trivial variety and X is a smooth, weakly ordinary, projective K -trivial varietyover k with ˆ q ( X ) = 0? ALABI-YAU THREEFOLDS OVER FINITE FIELDS AND TORSION IN COHOMOLOGIES 9
Here ˆ q denotes the augmented irregularity , namelyˆ q ( X ) := max { dim Alb X (cid:48) | X (cid:48) → X is finite ´etale } , where for a variety X (cid:48) , Alb X (cid:48) denotes its Albanese variety. Remark . The interest in the question above is that a positive answer allows one toconstruct many K -trivial varieties which, for example, have Calabi-Yau and abelian varietiesparts which are “difficult” to separate. For example, one may take a quotient of the form( Y × E ) /µ p with E being an ordinary elliptic curve, and µ p acts on Y as in Question 2.9,and µ p acts on E by translation via the inclusion µ p (cid:44) → E [ p ]. For details see [PZ19].We answer Question 2.9 in the positive: Theorem 2.11.
The pair ( Y, X ) , with Y a generic choice of hypersurface X in AppendixA, X := Y /µ the smooth quotient, satisfies the conditions in Question 2.9.Proof. By Theorem A.1 the threefold X is Calabi-Yau, so in particular K -trivial. Notethat the µ -action is free for a generic choice of Y : that is, Y has a nowhere vanishingvector field. Therefore Y is singular, since otherwise the global vector field implies it hasvanishing Euler characteristic, which is not true since a quintic hypersurface in P has Eulercharacteristic − Y is certainly Gorenstein, being a hypersurface in P . We now come to the augmented irregularity ˆ q ( X ). Note that X is (geometrically)simply connected, since it is homeomorphic to Y , which in turn is simply connected, sinceit is a hypersurface in P . On the other hand, we have H ( X, O ) = 0, and hence ˆ q ( X ) = 0,as required by Question 2.9. Alternatively, for any (cid:96) we have H et ( X, Q (cid:96) ) = 0since X has a characteristic zero lift with fundamental group Z /
5, and therefore Alb X istrivial.It remains to show that X is weakly ordinary; in other words, it suffices to show thatfor a generic choice of X , the action of Frobenius on H ( X, O ) is invertible. Now note thatthe groups H ( Y, O ), H ( X, O ) are both one dimensional, and the pullback map(5) H ( X, O X ) → H ( Y, O Y )is an isomorphism: indeed, the map π is finite and therefore π ∗ is exact, and the above mapis given by H ( X, O X ) → H ( X, π ∗ O Y ) , and since π ∗ O Y ∼ = O X ⊕ (cid:77) i =1 L ⊗ i for some line bundle L on X (in fact, L corresponds to a non-zero element of Pic( X ) τ ,which gives rise to the covering Y ), we have that the map (5) is non-zero, and since bothgroups are one dimensional k -vector spaces, we have it being an isomorphism, as claimed.Now it suffices to check that Frobenius acts non-trivially on the group H ( Y, O ).We recall that we have the µ -action on P where ζ ∈ µ acts via the map( X : X : X : X : X ) (cid:55)→ ( X : ζX : ζ X : ζ X : ζ X ) . The hypersurface Y is defined by a quintic polynomial in the X i ’s which is invariant underthis µ -action. It suffices to exhibit a single such invariant quintic polynomial such that the Frobenius action on the corresponding group H ( Y, O ) is non-trivial, since the condition ofbeing weakly ordinary is an open one. There is the following simple formula for this actiongiven by [Sti87]: see Theorem 2.12 below as well as Lemma 2.13, and the fact that thetangent space to the formal group in Theorem 2.12 is precisely H ( O ). But now we cansimply take the invariant quintic X X X X X , and it is trivial to check, using Theorem2.12, that the Frobenius action on H ( O ) is non-trivial: indeed, by Theorem 2.12, theaction of Frobenius is given by the coefficient X X X X X in ( X X X X X ) , whichis obviously non-zero, as required. (cid:3) Theorem 2.12 ([Sti87, Theorem 1]) . For a hypersurface Y ⊂ P N of degree N + 1 definedby the equation F ( X , · · · , X N ) = 0 , there is a formal group law for H N − ( Y, ˆ G m,Y ) whoselogarithm l ( τ ) is given by l ( τ ) = (cid:88) m ≥ β m τ m m , where β m = coefficient of X m − · · · · · X m − N in F m − . The following lemma relates the p th coefficient of the logarithm of a formal group tothe action of Frobenius on its tangent space, which is certainly well known but for whichwe have not been able to find an adequate reference. Lemma 2.13.
Suppose we have a discrete valuation ring R with maximal ideal p , and aone-dimensional formal group G over R with logarithm l ( τ ) = (cid:88) m ≥ β m τ m m ; the action of Frobenius on the tangent space of G ⊗ R/ p is given by multiplication by β p mod p .Proof. Let F denote the Frobenius operator. Consider the Cartier module CG := lim G ( tR [ t ] /t n ) , which may be identified with tR (cid:74) t (cid:75) .Furthermore on CG We have the Frobenius operator F (we only consider the Frobeniusat p ) acting by the formula F ( γ ( t )) = γ ( ζt /p ) (cid:1) γ ( ζ t /p ) (cid:1) · · · (cid:1) γ ( ζ p t /p ) , where ζ denotes a primitive p th root of unity, the symbol (cid:1) denotes the addition in theformal group G , and γ ( t ) denotes any element of tR (cid:74) t (cid:75) which we have identified with CG (c.f.[Sti87, p.1117]).A straightforward calculation then shows that(6) l ( F τ ) = (cid:88) n ≥ β np τ n n . Since the map l satisfies l ( τ ) ∼ = τ mod τ , differentiating the logarithm gives an isomorphism dl : T e ( G [1 /p ]) ∼ = T e G a,R [1 /p ] , ALABI-YAU THREEFOLDS OVER FINITE FIELDS AND TORSION IN COHOMOLOGIES 11 (where for a group H , we denote by T e H the tangent space at the identity, and G [1 /p ]denotes the base change of G to the generic fiber of Spec( R )) and the calculation (6) showsthat F acts as β p on T e G a,R [1 /p ] . Since the logarithm is Frobenius equivariant, F also actsby β p on T e G [1 /p ]. Therefore F acts by β p mod p on the T e ( G ⊗ R/ p ), as required. (cid:3) Hodge diamond.
We may now compute all the Hodge numbers of the GodeauxCalabi-Yau threefolds which are µ -quotients of quintic hypersurfaces; the computation isstraightforward in characteristic zero while somewhat tricky in characteristic 5.In the following we denote by X the integral model of the Godeaux Calabi-Yau threefold,and X the special fiber. We now compute the Hodge diamond of X ; first we prove thefollowing topological result. Proposition 2.14.
Let X ( C ) denote the complex manifold associated to the generic fiberof X . Then the Betti cohomology group H ( X ( C ) , Z ) is torsion free.Proof. We apply the homology Serre spectral sequence to the fibration X ( C ) → K ( π ( X ( C ))),whose fiber is the universal cover of X ( C ), which we denote by ˜ X ( C ). We remind the readerthat for a fibration F → X → B this E -page spectral sequence is given by E p,q = H p ( B, H q ( F )) ⇒ H p + q ( X ) , where all cohomologies are taken with integral coefficients. In the following we will denoteby d i the differentials on the i th page. In our setup the E -page simplifies to(7) E p,q = H p ( Z / , H q ( ˜ X ( C ))) , where the right hand side of (7) denotes group homology. Since ˜ X ( C ) is simply connected, H ( ˜ X ( C )) = 0, and therefore we have E p, = 0 for all p. Hence the differential d : E , → E , is zero, and we have E ∞ , = E , / Im( d )(8) = H ( Z / , H ( ˜ X ( C ))) /d ( H ( Z / , Z )) . (9)Now recall that ˜ X ( C ) is a hypersurface in P , and therefore H ( ˜ X ( C )) ∼ = Z by the Lefschetz hyperplane theorem. The action of Z / H ( ˜ X ( C )) can only be thetrivial one, and therefore we have H ( Z / , H ( ˜ X ( C ))) ∼ = Z . On the other hand H ( Z / , Z ) ∼ = Z / d ( H ( Z / , Z )) vanishes.Using (8) we conclude that E ∞ , ∼ = H ( ˜ X ( C )) ∼ = Z . The filtration on H ∗ ( X ( C ) , Z ) given by this spectral sequence gives a short exact sequence0 → E ∞ , → H ( X ( C ) , Z ) → E ∞ , → . However, we also have E ∞ , ∼ = E , ∼ = H ( Z / , Z ) = 0 , and therefore H ( ˜ X, Z ) ∼ = Z . We conclude that H ( X ( C ) , Z ) is torsion free by the universal coefficients theorem, asclaimed. (cid:3) This gives immediately the following
Corollary 2.15.
The Hodge diamond of X is given by (10) 11 00 2 01 21 21 10 2 00 11 . Proof.
By Proposition 2.14 and Caruso’s theorem, H ( X ) ∼ = k , H ( X ) ∼ = k. Recall also that we have H ( X, Ω ) ∼ = k, H ( X, O ) ∼ = 0 . Therefore (recall that X denotes an integral lift of X ) H ( X, Ω ) has non-trivial torsion.Hence H ( X, Ω ) is at least two-dimensional. On the other hand the Hodge-de Rhamspectral sequence degenerates at the E -page since X lifts to W , and we conclude that theHodge diamond is as shown in (10). (cid:3) The CvS Calabi-Yau threefolds
We recall the construction of some remarkable Calabi-Yau threefolds given by Cynk-vanStraten [CS09]. The general class to which these Calabi-Yau threefolds belong is knownas the class of “double octic” Calabi-Yau threefolds, and arise as (resolutions of) doublecovers X → P branched along a degree eight hypersurface D . It is straightforward to checkthat eight is precisely the degree for the Calabi-Yau condition K X = 0. The cover of P n issmooth if the divisor D is, and will have singularities whenever D does as well. The latteris the case of interest since D will be taken to be a union of eight hyperplanes in specialpositions. In this case we will have to blow up along double and triple lines (i.e. lines lyingon the intersection of two or three hyperplanes, respectively) and also fourfold and fivefoldpoints (i.e. points lying on the intersection of four or five hyperplanes, respectively).The variety X in characteristic 5 is constructed by taking the hypersurface to be thefollowing union of eight hyperplanes( x − t )( x + t )( y − t )( y + t )( z − t )( z + t )( x + y + Az − At )( x − By − Bz + t ) = 0 , where A, B are the two solutions to the equation x + x − . ALABI-YAU THREEFOLDS OVER FINITE FIELDS AND TORSION IN COHOMOLOGIES 13
These hyperplanes intersect at 28 double lines (all pairwise intersections) and 9 fourfoldpoints (for completeness we list these here: the fourfold points are given by( x, y, z, t ) =(2 A + 1 , − , − , , (1 , , AA − , , (1 , − , , , ( − , − , , , ( − , , , , ( − , , − , , (1 , , , , (0 , , , , (0 , , , , as is easy to verify). Therefore, after blowing up at these 9 fourfold points and the 28double lines, we obtain a smooth Calabi-Yau threefold, defined over Q ( √ π , where π = √
5, and so we obtain a smooth Calabi-Yau threefold X defined over Z [ − √ ] . We will denote the π -adic completion of this ring by O , and its fieldof fractions of K ; by abuse of notation we will denote the base change of X over O as X aswell. The following theorem about this variety is proved by Cynk-van Straten in [CS09] Theorem 3.1 ([CS09]) . (1) The variety X / O is smooth and is a strict Calabi-Yau threefold; in particular H ( X , O ) = H ( X , O ) = 0 , Ω ∼ = O . (2) The Hodge numbers of the generic fiber X K are given by h = 38 , h = 0 , whereas the Hodge numbers of the special fiber X are given by h = 39 , h = 1 . Therefore the threefold is rigid in characteristic zero but not in characteristic 5; (3)
The unique first order deformation of X over F [ (cid:15) ] /(cid:15) is not liftable to F [ (cid:15) ] /(cid:15) . (4) Moreover, X does not lift to W .Remark . We remark that the jump in the cohomology group H upon reduction mod p , or equivalently in H , can be explained by the obstruction class of the deformation inthe direction of the extra class in H . Remark . Note that the cohomology group H ( X K , Z ) has rank 38 since each blow upcentered at the 28 lines and 9 points adds a class in H and we start with P which has H ∼ = Z . In other words, all the classes in H ( X K ) comes from the base of the doublecovering map.There is a similar construction of the variety Y in characteristic 3; the difference withthe previous case is that the Calabi-Yau threefold constructed in characteristic 0 has badreduction at 3, and one must perform a small resolution. The details can be found in [CS09,Section 5.1]. Here we merely summarize the facts we need in what follows: Theorem 3.4 ([CS09]) . There is a strict Calabi-Yau threefold Y over F with no lift toany ring in which (cid:54) = 0 . Its Hodge numbers are given by h = 42 , h = 0 . In particular, the variety Y is rigid and does not lift to W . Supersingularity
In this section we show that both threefolds Y , X constructed by Cynk-van Stratenare supersingular (see Definition 4.2).We first recall the definition of the Artin-Mazur formal group: for any variety X ofdimension n over an algebraically closed field k of positive characteristic, the functor Φ onArtin local k -algebras with residue field k given by A (cid:55)→ Φ( A ) := H n ( X ⊗ k A, G m ) → H n ( X ⊗ k , G m )is representable by a formal group when H i ( X, O ) = 0 for all i (cid:54) = 0 , n . So in particular Φis representable whenever X is a strict Calabi-Yau variety. Definition 4.1.
For X a strict CY, the one-diensional formal group representing the functorΦ is called the Artin-Mazur formal group of X . The height of X , written ht( X ), is definedto be the height of this formal group.Note that the height can be a positive integer or infinity; the latter occurs when theformal group is isomorphic to ˆ G a . Definition 4.2.
We say that a Calabi-Yau threefold X in characteristic p is supersingularif its height is ∞ .Now we recall another notion of height, introduced by [Yob19]. For this recall that wehave the Witt sheaves W m ( O ), as well maps Frobenius maps F : W m ( O ) → F ∗ W m ( O ) andrestriction maps R m − : W m ( O ) → O . Definition 4.3.
The quasi-Frobenius splitting height ht s ( X ) is the smallest postive integer h such that there exists a map φ : F ∗ W m ( O ) → O such that the diagram W m O X F ∗ W m O X O XR m − Fφ commutes. Note that as in the case of the Artin-Mazur height this is a positive integer orinfinity.The following result was proved by Yobuko in [Yob19] Theorem 4.4. (1)
For a Calabi-Yau variety over k , the Artin-Mazur height is equal to the the quasi-Frobenius splitting height, that is, ht( X ) = ht s ( X ) . (2) If the height is finite, then X admits a lift to W ( k ) . By Theorem 4.4 and the fact that neither of the varieties X , X lift to W ( k ), we havethe following result: Theorem 4.5.
The varieties Y , X are both supersingular. ALABI-YAU THREEFOLDS OVER FINITE FIELDS AND TORSION IN COHOMOLOGIES 15
Remark . Theorem 4.5 implies that the first slope of the Newton polygon is at least1: indeed, the part of H ( X/W ) ⊗ K with slopes in the interval [0 ,
1) correspond tothe p -divisible quotient of the Artin-Mazur formal group [AM77, Corollary 3.3]. Since byTheorem 4.5 there is no p -divisible quotient, there must not be any slopes in the interval[0 , Y and X : either havingslopes { , } or slopes { / , / } . It is in fact possible to compute the precise Newtonpolygon by computing that of the corresponding Hilbert modular form, since we know that X is Hilbert-modular and the explicit modular form was found by Cynk-Sch¨utt-van Straten[CSS20]. Remark . It is also possible to give a purely computational proof of the weaker fact thatthe action of Frobenius on the middle cohomologies of X , X has no unit roots. Indeed, bya result of Stienstra, for a double cover of P n branched along a hypersurface with equation W = 0 the trace of Frobenius mod p can be computed as a certain coefficient of a powerof W . Since we have explicit expressions for the polynomials W in each case, this is easilydone with the help of a computer. For example for X we have to compute the coefficientof t x y z in W , and the answer turns out to be 60 A + 85, which is divisible by 5.5. Conjugate spectral sequence and liftings
In this section we recall some facts about the conjugate spectral sequence, prove acriterion for non-degeneration of the conjugate spectral sequence in terms of liftings; as aresult we will be able to answer a question of Ekedahl.5.1.
A criterion for non-degeneration of the conjugate spectral sequence.
Westate a lemma which will be of use later on. First recall the conjugate spectral sequence(see Figure 1), specialized to the case of threefolds. We have only drawn the arrows whichwill be of particular interest to us. H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) Figure 1. E -page of the conjugate spectral sequenceWe record the following lemma, which is a direct consequence of the methods in Deligne-Illusie [DI87], and certainly well known to experts. Lemma 5.1.
Let X be a smooth proper variety over k . Then each differential H i ( X, Ω j ) → H i +2 ( X, Ω j − ) in the conjugate spectral sequence (1) is given by cup product with the class ξ ∈ H ( X, T X ) which is the obstruction class to lifting X over W ( k ) . Proof.
We recall some basic facts about the conjugate spectral sequence. By definition it isthe spectral sequence induced by the filtration on the de Rham complex Ω • given by F p Ω • := τ ≤− p Ω • ;we denote the associated graded sheaves by gr p := F p /F p +1 , which in this case is given bygr p = Ω − p [ p ] . By construction, each differential on the E -page of the spectral sequence (this is before weperform the reindexing which makes the conjugate spectral sequence into an E -spectralsequence) H p + q (gr p ) → H p + q +1 ( gr p +1 )is the connecting homomorphism induced by the sequence0 → gr p +1 → F p /F p +2 → gr p → . Equivalently the differential d is induced by a class in Ext (Ω i +1 , Ω i ), or equivalently a mapΩ i +1 → Ω i [2] . Note that in our case H p + q (gr p ) = H q +2 p (Ω − p ). Now Deligne-Illusie [DI87] shows that inthe case i = 0 this map is induced by the obstruction class ξ : this is the combination ofTheorem 3.5 and Proposition 3.3 of loc.cit.. It remains to show that the same is true forother values of i : that is we would like to compute the map α : Ω i +1 → Ω i [2]. Now notethat we have the multiplication mapΩ [ − i ] ⊗ F /F → F i /F i +2 , and since the differentials for higher i (resp. i = 0) are induced by the extension F i /F i +2 (resp. F /F ), we have that the composition(11) Ω ⊗ Ω i m −→ Ω i +1 α −→ Ω i [2]is given by tensoring the map · ∪ ξ : Ω → Ω [2] by Ω i . Finally the first map m in (11)above is surjective, and so we are done. (cid:3) Lemma 5.2.
If a Calabi-Yau threefold X over a field k of characteristic p has no lift to W ( k ) , then the differenial in the conjugate spectral sequence H (Ω ) → H (Ω ) is non-trivial.Proof. By Lemma 5.1 the differential is given by cupping with the class ξ ∈ H ( T X ) whichis the obstruction to lifting to W ( k ). But now by the Calabi-Yau condition we have ξ ∈ H (Ω ) = H ( T X ) , (we will continue to denote its image in H (Ω ) by ξ ) and consequently the differential inthe spectral sequence is the same as the one induced by cupping with ξ in the map(12) H (Ω ) ⊗ H (Ω ) → H (Ω ) ∼ = H (Ω ) , where the last isomorphism is again by the Calabi-Yau condition. On the other hand bySerre duality the pairing (12) is non-degenerate, and hence the differential in question isnon-trivial as claimed. (cid:3) ALABI-YAU THREEFOLDS OVER FINITE FIELDS AND TORSION IN COHOMOLOGIES 17
The Hirokado threefold.
The Hirokado threefold H was constructed by Hirokado[Hir99] as the first example of a Calabi-Yau variety in positive characteristic which does notadmit a lift to characteristic zero; the p -adic cohomologies (for p = 3, since the Hirokadothreefold is in characteristic three) were further investigated by Ekedahl in [Eke03] (thisvariety was denoted by X by both Hirokado and Ekedahl, but we have opted to denote itby H so as not to confuse it with the other varieties considered in this paper). However inloc.cit. the computation of the crystalline cohomology could not be completed: we statethe following theorem of Ekedahl: Theorem 5.3 (Theorem 4.2 of [Eke03]) . (1) The Hirokado threefold H does not admit a lift to W/ ; (2) the crystalline cohomology of X is given by H ( H /W ) = H ( H /W ) = W,H ( H /W ) = H ( H /W ) = 0 ,H ( H /W ) = W , H ( H /W ) = W/p n W, and H ( H /W ) = W/p n W ⊕ W . (13) Furthermore the Hodge-de Rham spectral sequence degenerates at the E -page pre-cisely when n > . The precise value of n was not determined, however. Here we show that Lemma 5.2implies that n = 0, which completes the computation of all cohomologies of X (includingthe de-Rham Witt cohomology, which was also studied in detail in [Eke03]). Theorem 5.4.
The crystalline cohomology of H is torsion free: that is, the value of n inTheorem 5.3 is zero.Proof. By Theorem 5.3, H does not lift to W/
9, and hence by Lemma 5.2 the differential H (Ω ) → H (Ω ) in the conjugate spectral sequence is non-zero. In particular, the conju-gate spectral sequence does not degenerate at E , and hence the Hodge-de Rham spectralsequence does not degenerate at E . Therefore by the last part of Theorem 5.3 we have n = 0, as claimed. (cid:3) Topology
In this section let X / O denote the Cynk-van Straten threefold over the ring O = Z (cid:2) − √ (cid:3) , and X its generic fiber. The goal of this section is to prove Proposition 6.1below, which will be crucial when we compute the cohomologies of the special fiber X . Proposition 6.1.
The first homology H ( X, Z ) is 2-torsion (which may or may not bevanishing). In particular, w have H ( X, F ) = 0 .Proof. Recall that X is constructed by first blowing up P at 28 double curves and 9fourfold points to get the threefold Y , and then taking the double cover of Y branchedalong the smooth divisor D , which is the strict transform of the original collection of eighthyperplanes.Now H ( Y, Z ) = 0 since it comes from blowing up points and lines so that the ex-ceptional divisors have no H . Now consider the double cover π : X → Y ramified along D ⊂ Y . For each prime l (cid:54) = 2, we will compute H ( X, F l ) and H ( Y, F l ) using the Mayer-Vietorisexact sequence. Since the first Betti number of X is zero, we will have proven the propositionif we can show that H ( X, F l ) = 0.Note that the preimage π − ( D ) is isomorphic to D since the cover is branched along D , so we will abuse notation and write D for this preimage inside X as well. Now for ourapplication of Mayer-Vietoris we write X = ( X \ D ) ∪ D (cid:48) (cid:15) , (14) Y = ( Y \ D ) ∪ D (cid:15) , (15)where D (cid:48) (cid:15) (respectively D (cid:15) ) is a tubular neighborhood of D inside X (respectively Y ). Alsodenote by the intersection of X \ D and D (cid:48) (cid:15) (respectively the intersection of Y \ D and D (cid:15) )by Int (cid:48) (respectively Int). We then have the following exact sequences from Mayer-Vietoris(with F l -coefficients which we omit from the notation):(16) · · · H (Int (cid:48) ) H ( X \ D ) ⊕ H ( D (cid:48) (cid:15) ) H ( X ) H (Int (cid:48) ) · · ·· · · H (Int) H ( Y \ D ) ⊕ H ( D (cid:15) ) H ( Y ) H (Int) · · · α f (cid:48) β g (cid:48) γf g Note that H ( D (cid:48) (cid:15) ) = H ( D (cid:15) ) = 0, since both D (cid:48) (cid:15) and D (cid:15) deformation retract onto D which is a disjoint union of P ’s. Now any torsion in H ( X, Z ) must come from H ( X \ D ),since H (Int (cid:48) ) is torsion free. Note also that H ( Y ) = 0 since it is the blow up of P atpoints and lines, and all the exceptional divisors have vanishing H .So we can restrict attention to the following part of diagram 16:(17) H (Int (cid:48) ) H ( X \ D ) H (Int) H ( Y \ D ) f (cid:48) α βf By construction the map π : X \ D → Y \ D is an unramified double cover and thereforethe maps α and β are both isomorphisms as long as we use homology with F l -coefficientsfor l (cid:54) = 2.Since the bottom horizontal map in 17 is surjective, f (cid:48) is as well, and therefore H ( X, F l ) =0, as required. (cid:3) The CvS variety in characteristic 3
Theorem 7.1.
The Calabi-Yau threefold Y has H crys = 0 , and hence vanishing third Bettinumber.Proof. Recall that for Y the Hodge numbers in degree three are given by h = h = 1 , h = h = 0 . ALABI-YAU THREEFOLDS OVER FINITE FIELDS AND TORSION IN COHOMOLOGIES 19
Now since Y does not lift to W by Theorem 3.4, we may apply Lemma 5.2 to Y , anddeduce that the differentials H ( Y , Ω ) → H ( Y , Ω )(18) H ( Y , Ω ) → H ( Y , Ω )(19)are non-zero; therefore H ( Y ) = 0 since the two classes in H ( Y , Ω ) and H (Ω ) areboth killed by the differentials. Therefore H = 0 by universal coefficients.The vanishing of b now follows from a theorem of Katz-Messing (and does not requireliftability): indeed as a consequence of the Weil conjectures they proved [KM74, Theorem1] that Frobenius on rational cystalline cohomology and l -adic ´etale cohomologies have thesame characteristic polynomials (for any l ), and so in particular H i crys [1 /p ] and H i ´ et havethe same dimensions. (cid:3) Remark . This addresses a question in [Tak15, p.1] about the third Betti number of Y (see “The author is not aware...”). Note that this does not give a new proof of non-liftabilityof Y since we used the non-liftability to deduce the vanishing of b . Question . We note that the Hodge numbers h = 0 and h = 42 of the CvS threefold Y agree with the corresponding Hodge numbers of the Hirokado threefold H (see Theorem[Eke03, Theorem 3.6 (iii)]): in fact, all the Hodge numbers of Y and H agree, with thepossible exception of h and h (and the Serre dual of these, of course). Also, by Theorem7.1 above, the crystalline cohomologies H also agree. Finally, using the same techniquesas Section 6, one may compute all the Betti numbers of Y as well and see that they agreewith those of H (we may compute the Betti numbers of H from the crystalline cohmologies5.3 and the result of Katz-Messing[KM74, Theorem 1]. We therefore find it natural to ask:is Y isomorphic to ˜ F , or at least birational to it?8. The CvS variety in characteristic 5
We first recall the following theorem in integral p -adic Hodge theory due to Caruso[Car08, Th´eor`eme 1.1] and Faltings [Fal99] (see also [Bha, Remark 2.3 (4)]): Theorem 8.1.
For a variety X over a ring of integers O of some p -adic field K with goodreduction, let e be the ramification degree of K/ Q p , and k be the residue field of O . If ie < p − , then we have dim k H i dR ( X k ) = dim F p H i ( X ´ et , F p ) . Lemma 8.2.
For the CvS Calabi-Yau threefold X / F , we have H (Ω ) = 0 . We give two proofs of this.
Proof 1.
We again consider the conjugate spectral sequence. H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω ) H (Ω )If x ∈ H (Ω ) is non-zero, then since by Theorem 3.1, the group H (Ω ) is zero, this class x survives to the E ∞ -page of the spectral sequence, and hence H ( X ) (cid:54) = 0. Now by theintegral comparison theorem of Caruso [Car08] and Faltings [Fal99]dim F H dR ( X ) = dim F H ( X ´ et , F )and therefore H ( X, F p ) (cid:54) = 0. Since H ( X, Z ) vanishes (indeed, it cannot have torsionby the universal coefficients theorem, and H ( X, O ) = 0 implies its free part vanishes also),the class in H ( X, F p ) (cid:54) = 0 comes from H ( X, Z ), and so by the universal coefficientstheorem we have H ( X, F ) (cid:54) = 0, contradicting Proposition 6.1. (cid:3) Proof 2.
Recall that, for the generic fiber, we havedim H ( X, Ω ) = 38 , dim H ( X, Ω ) = 0 , while in the special fiber we havedim H ( X , Ω ) = 39 , dim H ( X , Ω ) = 1 . This implies also that the dimensions of H (Ω ) and H (Ω ) go up by one upon reducingmodulo π . This implies that there is torsion in the integral Hodge cohomology groups.Indeed, by universal coefficients, for all i, j we have an exact sequence0 → H i ( X , Ω j ) /πH i ( X , Ω j ) → H i ( X , Ω j ) → H i +1 ( X , Ω j )[ π ] → → Ω j O · π −→ Ω j O → Ω j → . Here π denotes a uniformizer of O , which in our case can be chosen to be √ H ( X , Ω ) is non-zero. Then by Serre-duality we have H ( X , Ω )nonzero as well, and so there must be torsion in H ( X , Ω ) (since otherwise we wouldhave a class in H ( X , Ω ), which is absurd). Let us call this class ξ . Then ξ contributesto the extra class in H ( X , Ω ).Now consider H ( X , Ω ). Again since the dimension of this group is one bigger thanthat of the generic fiber, there is torsion in either H ( X , Ω ) or H ( X , Ω ). But the lattercannot happen, since this would mean that H ( X , Ω ) has dimension at least 2 more thanthe generic fiber. Hence we have torsion in H ( X , Ω ).But this is the formal deformation space of X , and so this implies that there is a lift ofthe non-trivial deformation of X over F [ (cid:15) ] /(cid:15) to O [ (cid:15) ] /(cid:15) . The deformations are given byequisingular deformations of the hyperplane arrangement D , and so we must find a solutionto the equation x + x − ALABI-YAU THREEFOLDS OVER FINITE FIELDS AND TORSION IN COHOMOLOGIES 21 in O [ (cid:15) ] /(cid:15) lifting the solution 2 + (cid:15) ∈ F [ (cid:15) ]. We now check that this is impossible: indeed ifwe have such a solution λ + µ(cid:15) for λ, µ ∈ O , then reducing modulo (cid:15) we must have that λ = − √ , and so the equation ( λ + µ(cid:15) ) + ( λ + µ(cid:15) ) − λµ + µ = 0which has no solutions. Therefore H ( X , Ω ) cannot have torsion, and so H ( X , Ω ) = 0,as claimed. (cid:3) Theorem 8.3.
The crystalline cohomology of X is torsion free, i.e. H ( X /W ) = W ⊕ , H ( X /W ) = W ⊕ . Moreover the Hodge cohomologies H i ( X , Ω j ) of the integral model are torsion free exceptfor H ( X , Ω ) and H ( X , Ω ) .Remark . Note that this gives a proof of the fact that H ( X, Z ) has no 5-torsion by therecent result of Bhatt-Morrow-Scholze which says that torsion in Betti cohomology forcestorsion in crystalline cohomology. It is probably possible to compute this cohomology groupdirectly and see that it is 5-torsion free, however. Therefore this variety has no 5-torsionin its Brauer group, which is another invariant that is often of interest in the study ofCalabi-Yau manifolds [GP08a]. Proof.
We give two arguments that H is torsion free, one using the fact that X hasobstructed deformation in characteristic 5, and one using the fact that it does not lift to W .For the first argument, a theorem of Ekedahl-Shepherd-Barron says that for a Calabi-Yauthreefold, if dim H n dR = dim (cid:77) i + j = n H i (Ω j ) , (in other words that the differentials in the non-trivial differentials in the Hodge-de Rhamspectral sequence do not touch the middle cohomology, then the deformation ring of X in characteristic 5 is height 1 smooth, which would mean that the non-trivial first orderdeformation lifts to F [ (cid:15) ] /(cid:15) , contradiction. Therefore H ( X ) has dimension 2, and hence H is torsion free, as required.For the second argument, we use the fact that X does not lift to W . Then by Lemma5.2 the differential from H ( X , Ω ) to H ( X , Ω ) in the conjugate spectral sequence is non-trivial, and hence the unique class in H (Ω ) dies in the E ∞ -page, and so again H ( X )has dimension 2, and we argue as above.Now we come to H ( X /W ). Torsion in this group would imply H ( X ) is non-zero,and hence one of H ( X , Ω ) and H ( X , Ω ) has to be non-zero. This contradicts 8.2 andthe fact that H ( X , O ) = 0.Now we must have H ( X , Ω ) = 0 as well arguments similar to either proofs of 8.2:either use the fact that any class in H ( X , Ω ) = 0 must survive to the E ∞ -page of theconjugate spectral sequence, since we have dim F H ( X ) = 2 and we already have non-zero differentials H ( X , Ω ) → H ( X , Ω ) and H ( X , Ω ) → H (Ω ), or use a similarargument to the second proof of Lemma 8.2 by chasing the torsion in the integral Hodgecohomologies. Finally the assertions about torsion in the integral Hodge cohomologies H i ( X , Ω j ) followfrom the fact that H ( X , Ω ) = H ( X , Ω ) = 0. (cid:3) Remark . Note that this gives a counterexample to an expectation stated by Joshi in[Jos14, page 8, paragraph beginning “We expect that...”], namely that for a Calabi-Yauvariety X over a finite field of characteristic ≥
5, if b (cid:54) = 0 (”classical” in the terminologyof [Jos14]) and H ∗ crys ( X/W ) is torsion free then X lifts to W .We may now sum up this section by giving the proof of Theorem 1.4, whose preciseform we now state. Theorem 8.6.
For the CvS Calabi-Yau threefold X / O the following hold: (1) The variety X / F has torsion free crystalline cohomology; in particular the coho-mology groups H ∗ dR ( X ) of the integral model are torsion free as well. More precisely,cohomology groups are given by H ∗ dR ( X / O ) = O , , O , O , O , , O ; H ∗ crys ( X /W ) = W, , W , W , W , , W. Here, we have written the groups H i • in increasing order of from i = 1 to i = 6 .Therefore the de Rham cohomology groups of the special fiber are given by: H ∗ dR ( X ) = F , , F , F , F , , F . (2) The Hodge cohomology groups H i ( X , Ω j ) are torsion free O -modules, except for H ( X , Ω ) and H ( X , Ω ) , which have non-trivial torsion; more precisely they aregiven by H ( X , Ω ) = O /π a , H ( X , Ω ) = O ⊕ O /π b , for some integers a, b > . (3) The Hodge diamond of X is given by
10 00 39 01 1 1 10 39 00 01 . Note in particular that Hodge symmetry holds for X , which is not generally thecase for smooth proper varieties over finite fields. (4) The special fiber X is supersingular, i.e. its Artin-Mazur formal group is isomor-phic to ˆ G a . Even though some of the arguments have already been given in the course of the proofsof various theorems, we give them again here for the convenience of the reader.
Proof of Theorem 8.6.
For the computation of the cohomology groups H ∗ dR ( X ), we havethe E -page of the conjugate spectral sequence ALABI-YAU THREEFOLDS OVER FINITE FIELDS AND TORSION IN COHOMOLOGIES 23 F F F F F F F F , where we have drawn the only non-trivial differentials. This gives H ∗ dR ( X ) = F , , F , F , F , , F as claimed. Since H ∗ crys ( X /W ) is torsion free by Theorem 8.3, H ∗ dR ( X ) is also torsionfree, which proves part (1) of the theorem. By universal coefficients and comparing Hodgecohomologies of the generic fiber and those of the special fiber, we must have a torsion classin H ( X , Ω ) giving rise to extra classes in H ( X , Ω ) and H ( X , Ω ); similarly theremust be a torsion class in H ( X , Ω ) giving extra classes in H ( X , Ω ) and H ( X , Ω ).Finally the torsion parts of H ( X , Ω ) and H ( X , Ω ) are one dimensional upon reducingmod π , so we have proven part (2).We have computed the Hodge diamond of X , and the supersingularity was proved in4.5: this gives parts (3) and (4). This concludes the proof of the theorem. (cid:3) Further questions
In this section we discuss some future directions and possible implications for the maintheme of this paper, namely the relationship between torsion in cohomology of the genericfiber and various cohomologies of the special fiber.9.1.
Hodge numbers and derived equivalences.
The derived category D bcoh ( X ) of avariety X has become a prominent player in recent years in algebraic geometry, partly dueto its role in the homological mirror symmetry conjectures. One of the major questions inthe study of this category is how much information is contained in D bcoh ( X ); for example,one could ask about numerical invariants such as the Hodge numbers of X . For smoothproper varieties of dimension up to 3 in characteristic zero, it is known by work of [PS11]that the Hodge numbers are derived invariants: more precisely, for varieties X, Y for whichthere exists an equivalence of categories D bcoh ( X ) ∼ = D bcoh ( Y ) , we have h ij ( X ) = h ij ( Y ) for all i, j . This question was investigated by Antieau-Bragg in[AB19] for varieties in positive characteristic, where they prove the same result for varietiesup to dimension 2, and for threefolds under some restrictions; in particular for threefoldsthey assume that the crystalline cohomologies are torsion free.Here we suggest that the Hodge numbers should not be derived invariant. We suggestseveral potential counterexamples, and hope to check the details in future work. OUr idea issimply that, as has been exploited several times in this paper, torsion in Betti cohomologyin the generic fiber contributes to Hodge cohomologies in the special fiber. More precisely,for derived equivalent threefolds X and Y , if the torsion in their cohomologies differ, thenit is very likely that their Hodge cohomologies differ as well. More specifically, such examples have been shown to exist by Addington [Add17]. In-deed, in loc.cit. two derived equivalent Calabi-Yau threefolds X and Y are exhibited suchthat π ( X ) ∼ = 0 , Br(X) ∼ = ( Z / ⊕ ,π ( Y ) ∼ = ( Z / ⊕ , Br(Y) ∼ = 0 . (20)Here, Br denotes the Brauer group, which for a Calabi-Yau threefold X simply means H ( X, Z ) tors . The derived equivalence between X and Y was shown by Bak [Bak09] aswell as [Sch12], and Addington found that a certain extension between H and Br, therebyshowing (20).The Calabi-Yau threefolds X and Y were originally studied by Gross-Popescu [GP01]and the Brauer group was computed by Gross-Pavanelli [GP08b]; one feature is that theyare fibered by abelian surfaces. It seems likely that one can construct these threefolds incharacteristic 2, and compute their cohomologies explicitly. For our purposes it is moreconvenient to have primes bigger than 2 in the torsion; for this we will consider analo-gous abelian surfaces fibered Calabi-Yau threefolds studied by Donagi-Gao-Schulz [DGS09](though it is likely that these have also appeared in [GP01]), where they compute torsionin the cohomologies of these threefolds. In particular, the (family of) varieties X , , X , (following the notation of loc.cit.) have H ( X , ) ∼ = 0 , Br( X , ) ∼ = ( Z / ⊕ ,H ( X , ) ∼ = ( Z / ⊕ ; Br( X , ) ∼ = 0 , furthermore it seems likely that they are derived equivalent, as in the case considered byAddington above. We state the following conjecture, which would easily imply that Hodgenumbers are not derived invariant in positive characteristic (using Caruso’s result), and wewill study it in future work. Conjecture 9.1.
The Calabi-Yau threefolds in [DGS09] can be constructed rigorously, and(generically) have good reduction at primes p dividing the torsion in its Betti cohomology.Furthermore, the derived equivalences between these Calabi-Yau threefolds extend to char-acteristic p . Torsion in cohomology and deformations.
As emphasized at several points of thispaper, one of our main points was to leverage the relation between the Betti cohomologyof the generic fiber of a variety and various cohomologies of the special fiber. Thinkingalong these lines, we have the following question which we hope to address in future work.A Calabi-Yau threefold X in characteristic zero sometimes has non-trivial Brauer group,which is to say torsion in H ( X, Z ): this invariant is of interest in algebraic geometry aswell as in physics. Suppose that we have p -torsion in H ( X, Z ), and that X has goodreduction at p ; let us denote its special fiber by X p . Then H ( X p ) will have extra classes,and therefore also in Hodge cohomologies; assuming X p is still CY, the extra classes willappear in H ( X p , Ω ) (as well as in H ( X p , Ω ), by Serre-duality). In other words, thedeformation space in characteristic p now has extra dimensions. What can be said aboutthese extra deformations? Are they obstructed? For examples of Calabi-Yau threefoldswith non-trivial Brauer group, we refer the reader to [GP08b; Add17]. The first example toconsider seems to be the case of the Enriques Calabi-Yau threefold, which has H ∼ = Z / ALABI-YAU THREEFOLDS OVER FINITE FIELDS AND TORSION IN COHOMOLOGIES 25
In fact, we propose the following recipe for constructing these extra directions of defor-mations in some cases. Let us consider again the varieties X , and X , from Section 9.1.In fact, we have X , = X , / ( Z / × Z / , where on the right hand side Z / × Z / Z := X , / Z / , for some choice of Z / (cid:44) → Z / × Z /
3. By the result of Addington [Add17, Proposition onpage 3], we have that Br(Z) = Z / , and so we may expect that h jumps by 1 in the special fiber, giving us an extra directionof deformation. Our speculation is that, in the case that the groups are α × α in thespecial fiber, we may choose a diagonal α (cid:44) → α × α to form the quotient variety Z , and that, just as in the Moret-Bailly [Mor81] example, thischoice of diagonal α has moduli: there is a P -worth of diagonal α ’s with which we canform the quotient X , /α , giving us an extra direction to deform in. One can make similarspeculations for the quotients associated to each of the X m,n ’s considered in Section 9.1.On the other hand, in the example Z just mentioned, there should be extra deformationseven if the group is not α × α in the special fiber: we are not sure what to expect in thesecases.9.3. Modularity.
The rigid variety X/ Q ( √
5) has also been shown by Cynk-van Straten[CSS20] to be modular, in the sense that the two dimensional Galois representation ofGal( Q / Q ( √
5) given by the l -adic ´etale cohomology of X is isomorphic to that of a Hilbertmodular form on GL ( Q ( √ Question . Does the torsion class in H ( X , Ω ) have any arithmetic meaning in termsof this Hilbert modular form or the Galois representation?9.4. Cohomology of the Hirokado variety.
Ekedahl has interpreted the Hirokado vari-ety as a Deligne-Lusztig (which is in turn possibly isomorphic to the CvS variety Y ). The l -adic, mod l , and even crystalline cohomologies of Deligne-Lusztig varieties are known togive representations of finite groups of Lie type. Does the fact that the Hodge-de Rhamspectral sequence does not degenerate have any interesting consequences from this point ofview?9.5. Relations between H and Y . As observed in Section 7, all the known cohomologiesof the Hirokado threefold H and the CvS threefold Y agree. Are these two varieties in factisomorphic, or at least birational? Appendix A. Construction of quotients of quintic hypersurfaces
In this appendix we show the existence of smooth Calabi-Yau threefolds which are freequotients of quintic hypersurfaces in P by µ . The calculation follows essentially that ofLang [Lan81], who constructed similar surfaces (the Godeaux surfaces) as free quoitients ofquintic hypersurfaces in P by µ .Consider the action of µ on P by(21) ( X : X : X : X : X ) (cid:55)→ ( X : ζX : ζ X : ζ X : ζ X ) . We prove the following
Theorem A.1.
The quotient map P → P /µ (where µ acts by the above action) givesan embedding of P into P , and furthermore a generic hyperplane section of P under thisembedding is a smooth Calabi-Yau threefold.Proof. First we compute the dimensions of invariant quintics under the above actions of µ . Note that the invariant polynomials are spanned by invariant monomials. In [Lan81]the computation was done for the µ -action on P given by the action on the last fourcoordinates:(22) ( X : X : X : X ) (cid:55)→ ( ζX : ζ X : ζ X : ζ X ) . The dimensions are encoded in the power series(23) 1 + 2 x + 4 x + 7 x + 12 x + 16 x + 24 x + · · · ;that is, the coefficient of x n is the dimension of the invariant polynomials of degree n . Thepower series above is given by15 (cid:2) (1 − x ) − + 4(1 − ζx ) − (1 − ζ x ) − (1 − ζ x ) − (1 − ζ x ) − (cid:3) , and it is in turn a theorem of Molien that this gives the desired dimensions (cf [Lan81,p.421]).Now we return to our case of interest, which is the µ -action on P given by (21),instead of the P case given by (22); in what follows we will refer to these as the P and P cases respectively. Since the µ -action on the additional variable X is trivial, the invariantpolynomials of degree n in the P is simply given by the invariant polnomials in degree ≤ n in the P case, multiplied by the appropriate power of X to make it of degree n . Thereforethere are 1 + 2 + 4 + 7 + 12 = 26invariant quintics in the P case.As in [Lan81] we show that the map P → P given by these 26 invariant quintics is an embedding. To do this it suffices to show thatthe invariant degree 10 polynomials, degree 15 polynomials, etc, are all generated by theinvariant quintics. It is easier to do this calculation in terms of the invariant polynomials inthe P case; translated to this setting, the statement becomes that the degree ≥ ≤
5. We now explicitly check this. • degree 2: X X , X X ; • degree 3: X X , X X , X X , X X ; • degree 4: X X , X X , X X , X X , ( X X ) , ( X X ) , X X X X .On the other hand, from (23) above we have that the number of invariant degree sixpolynomails is 16 (for the P case).In degree six, none of the powers X i are invariant, and hence each invariant monomialhas at least two variables. Suppose we have such an invariant monimal ψ ; if it is not divisibleby any cube X i , then it is straightforward to check that it must be one of( X X ) X X , X X ( X X ) , ALABI-YAU THREEFOLDS OVER FINITE FIELDS AND TORSION IN COHOMOLOGIES 27 which is certainly generated by lower degree invariants (namely the invariant degree twopolynomials). On the other hand, if the invariant monomial ψ is divisible by X i , say X without loss of generaility, since we have the invariants X X , X X , X X in degrees two, three, four respectively, and ψ contains at least two variables, it is divisibleby a lower degree invariant monomial, as required. This shows that the degree six invariantsare generated by lower degree invariants.For degree seven and eight, again an invariant monomial ψ must have at least twovariable; in degree seven all such ψ are divisible by X i for some i , and we can argue asabove, and in degree eight, if it is not divisible by X i for any i , then it must be( X X X X ) , which is generated by lower degree invariants.Finally, in degree nine and higher, every monomial ψ is divisible by X i for some i , soas long as it contains at least two variables we are done. On the other hand, in degrees n divislbe by 5, we could have the monomials X ni , but these are again generated by theinvariant quintics X i . Therefore we have shown that all invariant polynomials of degree 5 n where n ≥ µ -action in the P -case) are all by the quintic invariants, as required.We may now proceed as in [Lan81]. From the map P → P , and the fact that thefixed points of the µ -action on P are isolated points, a generic hyperplane H will notcontain any fixed points. Therefore the threefold X := P /µ ∩ H is smooth by Bertini’s theorem, and it is the quotient of the quintic hypersurface in P defined by H , on which the µ -action is free.Finally we show that the quotients X have trivial canonical bundle. Claim A.1.
We have Ω X ∼ = O X .Proof. First observe that the above construction works just as well in mixed characteristic;that is, the same construction and arguments as above give varieties over Z p (or perhapssome other local ring-the argument will go through without change), and so for the remain-der of the proof we denote by X this integral model, and X [1 /p ] its generic fiber. We firstshow that Ω is trivial in characteristic zero: that is, on the variety X [1 / . Let Y be the µ -cover of X [1 / Y is smooth and it is a quintic hypersurface in P , so certainlyΩ Y is trivial, and let us denote by σ a nowhere vanishing section. Now by adjunction wehave Ω Y ∼ = Ω P (5) | Y ∼ = O P | Y , and taking global sections we have that σ is invariant under the µ -action, and so descendsto X ; furthermore this descended section is also nowhere vanishing, and hence Ω X [1 /p ] istrivial, which is what we wanted.It remains to show the same for the integral model X . On X we have the exact sequence0 → Ω X · p −→ Ω X → Ω X → X the special fiber of X . Hence we have an injection H ( X, Ω X ) /pH ( X, Ω X ) (cid:44) → H ( X, Ω X ) . Therefore, by taking an appropriate multiple of the section σ above, we see that the linebundle Ω X has a section whose vanishing locus has codimension at least 2, and hence iseverywhere non-zero. Therefore Ω X is the trivial line bundle, as required. (cid:3) This concludes the proof of the theorem. (cid:3)
Remark
A.2 . As a consistency check, we can compute the Hodge numbers of the threefold X . In characteristic zero, according to the proof of Theorem A.1 there are 26 µ -invariantquintics, and quotienting by the subgroup of GL(5) stabilizing these quintics (up to scalars),namely the group of diagonal matrices G m , there are21 = 26 − X . Therefore h = 21; note that h = 1 (the same as the coveringhypersurface Y ), and so the Euler characteristic is χ ( X ) = −
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