LECTURES ON SUPERSINGULAR K3 SURFACES AND THELECTURES ON SUPERSINGULAR K3 SURFACES AND THE CRYSTALLINE TORELLI THEOREM CHRISTIAN LIEDTKE Expanded lecture notes of a course given as

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016 LECTURES ON SUPERSINGULAR K3 SURFACES AND THE

CRYSTALLINE TORELLI THEOREM

CHRISTIAN LIEDTKE

Expanded lecture notes of a course given as part of the IRMA master class “Around Torelli’stheorem for K3 surfaces” held in Strasbourg, October 28 - November 1, 2013

ABSTRACT. We survey crystalline cohomology, crystals, and formal group lawswith an emphasis on geometry. We apply these concepts to K3 surfaces, andespecially to supersingular K3 surfaces. In particular, wediscuss stratificationsof the moduli space of polarized K3 surfaces in positive characteristic, Ogus’crystalline Torelli theorem for supersingular K3 surfaces, the Tate conjecture,and the unirationality of K3 surfaces.

INTRODUCTION

In these notes, we cover the following topics

– algebraic de Rham cohomology, crystalline cohomology, and F -crystals,– characteristic-p aspects of K3 surfaces,– Ogus’ crystalline Torelli theorem for supersingular K3 surfaces,– formal group laws, and in particular, the formal Brauer group, and– unirationality and supersingularity of K3 surfaces.

We assume familiarity with algebraic geometry, say, at the level of the textbooksof Hartshorne [Har77] and Griffiths–Harris [G-H78].

One aim of these notes is to convince the reader that crystalsand crystallinecohomology are rather explicit objects, and that they are characteristic-p versionsof Hodge structures and de Rham cohomology, respectively. Oversimplifying andputting it a little bit sloppily, thecrystalline cohomologyof a smooth and propervariety in characteristicp is the de Rham cohomology of a lift to characteristic zero.(Unfortunately, such lifts may not exist, and even if they do, they may not be unique- it was Grothendieck’s insight that something like crystalline cohomology existsnevertheless and that it is well-defined.) Just as complex deRham cohomologycomes with complex conjugation, crystalline cohomology comes with a Frobeniusaction, and this latter leads to the notion of acrystal. Therefore,period mapsin characteristicp should take values in moduli spaces of crystals. For example,Ogus’ crystalline Torelli theoremstates that moduli spaces of certain K3 surfaces,

Date: May 19, 2015.2010Mathematics Subject Classification.14F30, 14J28, 14J10, 14M20.Key words and phrases.Crystalline cohomology, K3 surfaces, moduli spaces, formal groups,

rational and unirational varieties.1

http://arxiv.org/abs/1403.2538v3

2 CHRISTIAN LIEDTKE

namely,supersingular K3 surfaces, can be entirely understood via a period mapto a moduli space of suitably enriched crystals. Conversely, the classification ofcrystals arising from K3 surfaces gives rise to a stratification of the moduli spaceof K3 surfaces in characteristicp: for example, theheight stratificationin terms ofNewton polygonsarises this way.

A second aim of these notes is to introduceformal group laws, and, followingArtin and Mazur, to explain how they arise naturally from algebraic varieties. Forus, the most important examples will be theformal Picard groupand theformalBrauer groupof a smooth and proper variety in characteristicp. Whereas the for-mer arises as formal completion of the Picard scheme along its origin, the latterdoes not have such a description, and is something new. Oversimplifying again,we have a sort of crystal associated to a formal group law, namely, its Cartier–Dieudonne module. For example, the Cartier–Dieudonne modules of the formalPicard group and the formal Brauer group give rise to subcrystals inside first andsecond crystalline cohomology of the variety in question. And despite their ab-stract appearance, these formal group laws do have a geometric interpretation: forexample, for supersingular K3 surfaces, the formal Brauer group controls certainvery special one-parameter deformations,moving torsors, which are characteristic-p versions of twistor space. These deformations are the key tothe proof thatsuper-singular K3 surfaces are unirational.

And finally, a third aim of these notes is to make some of the more abstractconcepts more accessible, which is why we have put an emphasis on computingeverything for K3 surfaces. Even for a reader who cares not somuch for K3 sur-faces, these notes may be interesting, since we show by example, how to performcomputations with crystals and formal Brauer groups.

These notes are organized as follows:

Section 1We start by discussing de Rham cohomology over the complex numbers,and then turn to algebraic de Rham cohomology. After a short detour toℓ-adiccohomology, we introduce the Witt ringW , and survey crystalline cohomology.

Section 2We define K3 surfaces, give examples, and discuss their position withinthe surface classification. Then, we compute their cohomological invariants, andend by introducing polarized moduli spaces.

Section 3Crystalline cohomology takes it values inW -modules, whereW denotesthe Witt ring, and it comes with a Frobenius-action, which leads to the notion of anF -crystal. After discussing the Dieudonne–Manin classification ofF -crystals upto isogeny, we introduce Hodge and Newton polygons ofF -crystals.

Section 4TheF -crystal associated to the second crystalline cohomology group ofa K3 surface comes with a quadratic form arising from Poincare duality, which iscaptured in the notion of a K3 crystal. After discussing supersingular K3 crystalsand the Tate conjecture, we explicitly classify these crystals and construct theirmoduli space.

Section 5Associated to a supersingular K3 surface, we have its supersingular K3crystal, which gives rise to a period map from the moduli space of supersingular

SUPERSINGULAR K3 SURFACES 3

K3 surfaces to the moduli space of supersingular K3 crystals. Equipping super-singular K3 crystals with ample cones, we obtain a new periodmap, which is anisomorphism by Ogus’ crystalline Torelli theorem.

Section 6After defining formal group laws and giving examples, we introduce basicinvariants and state the Cartier–Dieudonne classification. Then, we discuss formalgroup laws arising from algebraic varieties, and relate their Cartier–Dieudonnemodules to crystalline cohomology.

Section 7We show that a K3 surface is unirational if and only if it is supersingular.The key idea is to give a geometric construction that uses theformal Brauer group.

Section 8Finally, we discuss several stratifications of the moduli space of polarizedK3 surfaces, which arise fromF -crystals and formal Brauer groups.

For further reading, we refer the interested reader

– to Wedhorn’s notes [We08] for more on de Rham cohomology andF -zips,as well as to Illusie’s survey [Ill02] for the degeneration of the Frolicherspectral sequence with characteristic-p methods,

– to Chambert-Loir’s survey [CL98] for more on crystalline cohomology,and to Katz’s article [Ka79] for more onF -crystals,

– to my own survey [Li13] for more on the classification of surfaces in posi-tive characteristic,

– to Huybrechts’ lecture notes [Hu15] for more on K3 surfaces,– to Ogus’ original articles [Og79] and [Og83] for more on supersingular K3

crystals and the Torelli theorem, and finally,– to the books of Hazewinkel [Ha78] and Zink [Zi84] for more onformal

group laws and their applications.

Of course, this list is by no means complete, but merely represents a small part ofthe literature for some of the topics touched upon in these notes.

Acknowledgements.These notes grew out of a lecture series I gave as part ofthe master class “Around Torelli’s theorem for K3 surfaces”at the Institut deRecherche Mathematique Avancee (IRMA) in Strasbourg from October 28 to No-vember 1, 2013, which was organized by Christian Lehn, Gianluca Pacienza, andPierre Py. I thank the organizers for the invitation and hospitality. It was a greatpleasure visiting the IRMA and giving these lectures. I thank Nicolas Adding-ton, Gerard van der Geer, Christopher Hacon, Eike Lau, and the referee for manyhelpful comments on earlier versions of these notes.

1. CRYSTALLINE COHOMOLOGY

1.1. Complex Geometry. LetX be a smooth and projective variety over the fieldC of complex numbers. In this subsection we briefly recall how the topological,differentiable and holomorphic structure onX give rise to extra structure on thede Rham cohomology groupsH∗

dR(X/C), and how these structures are related.First, we considerX as a differentiable manifold, where differentiable shall

always mean differentiable of classC∞ without further mentioning it. LetAnX be

the sheaf ofC-valued differentiablen-forms with respect to the analytic topology,

4 CHRISTIAN LIEDTKE

and in particular,A0X is the sheaf ofC-valued differentiable functions onX. Then,

by the Poincare lemma, thedifferentiable de Rham complex

0 → A0X

d−→ A1

Xd−→ A2

Xd−→ ...

is a fine resolution of the constant sheafC. Taking global sections, we obtain acomplex, whose cohomology is by definition de Rham cohomology. And sincethe differentiable de Rham complex is a fine resolution ofC, we obtain a naturalisomorphism

Hn(X,C) ∼= HndR(X/C)

for all n, and refer to [Wa71, Chapter 5] for details and proofs.Next, we considerX as an algebraic variety with the Zariski topology. Let

ΩnX/C be the coherentOX -module of Kahlern-forms, and in particular,Ω0

X/C isthe structure sheafOX . As in the differentiable setting, we obtain a complex ofsheaves

0 → OXd−→ Ω1

X/Cd−→ Ω2

X/Cd−→ ...,

the algebraic de Rham-complex. Now, theΩnX/C are locally freeOX-modules,

but not acyclic in general. Therefore, we choose injective resolutions of theΩnX/C

that are compatible with the differentials of the algebraicde Rham complex, andeventually, we obtain a double complex. Then, we take globalsections of theinjective sheaves of this double complex and take cohomology, which definesal-gebraic de Rham cohomology. In particular, algebraic de Rham cohomology arisesas hypercohomology of the algebraic de Rham complex. Since the columns of thisdouble complex compute the coherent cohomology groupsHj(X,Ωj

X/C), there isa spectral sequence to algebraic de Rham cohomology

Ei,j1 = Hj(X,Ωi

X/C) ⇒ H i+jdR,alg(X/C),

theFrolicher spectral sequence. We refer to [Ill02] or [We08] for details.Of course, we can also considerX as a complex manifold with the analytic

topology. Then, we can define the holomorphic de Rham complex, which givesrise to a holomorphic Frolicher spectral sequence, see [G-H78, Chapter 3.5] or[Vo02, Chapitre II.8]. By the holomorphic Poincare lemma,the holomorphic de-Rham complex is a resolution (although not acyclic) of the constant sheafC, whichis why the hypercohomology of the holomorphic de Rham complex is canonicallyisomorphic to the differentiable de Rham cohomology, see loc. cit. By Serre’sGAGA-theorems [Se55], the algebraic and holomorphicEp,q

1 ’s are canonically iso-morphic, and thus, also the hypercohomologies of the holomorphic and algebraicde Rham complex are canonically isomorphic. In particular,the differentiable,holomorphic, and algebraic de Rham cohomologies are mutually and canonicallyisomorphic, and there is only one Frolicher spectral sequence. We will thereforeomit the subscriptalg from algebraic de Rham cohomology in the sequel. Apartfrom the already given references, we refer the interested reader to [Gr66] for moreon this subject.


For complex projective varieties, and even for compact Kahler manifolds, theFrolicher spectral sequence degenerates atE1 by the Hodge decomposition theo-rem, see [G-H78, Chapter 0.7] or [Vo02, Theoreme 8.28], aswell as Exercise 1.1below. Next, the Frolicher spectral sequence gives rise toa filtration

0 = Fn+1 ⊆ ... ⊆ F i ⊆ ... ⊆ F 0 = HndR(X/C),

theHodge filtration. Since the Frolicher spectral sequence degenerates atE1, weobtain canonical isomorphisms

F i/F i+1 ∼= Ei,n−i1 = Hn−i(X,Ωi

X) for all i, n with 0 ≤ i ≤ n.

Next, we considerX again only as a differentiable manifold with its analytictopology. Then, sheaf cohomology of the constant sheafZ is isomorphic to singu-lar cohomology

Hn(X,Z) ∼= Hnsing(X,Z),

see, for example, [Wa71, Chapter 5]. In particular, the inclusionZ ⊂ C gives riseto an isomorphism

Hnsing(X,Z)⊗Z C ∼= Hn

dR(X/C).

and thus, to an integral structure on de Rham cohomology. Similarly, the inclusionR ⊂ C gives rise to a real structure on de Rham cohomology. In particular, we ob-tain a complex conjugation onHn

dR(X/C), which we apply to the Hodge filtrationto define a second filtration, the so-calledconjugate filtration, by setting

F icon := Fn−i .

It follows from the Hodge decomposition theorem that the Hodge and its conjugatefiltration satisfy

F i ∩ Fn−i ∼= Ei,n−i1 = Hn−i(X,Ωi

X/C),

see [G-H78, Chapter 0.6] or [Vo02, Remarque 8.29]. From this, we deduce

Hn−i(X,ΩiX/C) = Ei,n−i

1∼= En−i,i

1 = H i(X,Ωn−iX/C),

for all i, n with 0 ≤ i ≤ n. We stress that this isomorphism is induced fromcomplex conjugation, making it a priori not algebraic.

Putting all the results and observations made so far together, we see that eachde Rham cohomology groupHn

dR(X/C) is a finite dimensionalC-vector spacetogether with the following data

(1) an integral structure,(2) a real structure, and in particular, a complex conjugation −, and(3) two filtrationsF • andF •

con.

Again, we refer to [Ill02] and [We08] for details and furtherreferences.

6 CHRISTIAN LIEDTKE

1.2. Algebraic de Rham cohomology.LetX be a smooth and proper variety, butnow over a fieldk of arbitrary characteristic. The definition of algebraic deRhamcohomology and the Frolicher spectral sequence (but not its degeneration atE1)in the previous section is purely algebraic, and thus, we have these cohomologygroups and spectral sequences also overk. Let us now discussE1-degeneration ofthe Frolicher spectral sequence, as well as extra structures on de Rham cohomologyin this purely algebraic setting. Although many aspects of this section are discussedin greater detail in [We08], let us run through the main points needed later on forthe reader’s convenience.

Exercise 1.1.Let X be a smooth and proper variety over a fieldk. Show thatalready the existence of the Frolicher spectral sequence implies the inequalities

∑

i+j=n

hi,j(X) ≥ hndR(X) for all n ≥ 1,

wherehi,j(X) = dimkHj(X,Ωi

X/k) andhndR(X) = dimkHndR(X/k). More-

over, show that equality for alln is equivalent to the degeneration of the Frolicherspectral sequence atE1.

If k is of characteristic zero, then the Frolicher spectral sequence ofX degener-ates atE1 by the following line of reasoning, which is an instance of the so-calledLefschetz principle: namely,X can be defined over a subfieldk0 ⊆ k that is finitelygenerated overQ, and then,k0 can be embedded intoC. Since cohomology doesnot change under flat base change, and field extensions are flat, it suffices to proveE1-degeneration fork = C, where it holds by the results of Section 1.1.

In arbitrary characteristic,E1-degeneration of the Frolicher spectral sequenceholds, for example, for curves, Abelian varieties, K3 surfaces, and complete inter-sections, see [We08, (1.5)]. For example, for curves it follows from Theorem 1.2below, and for K3 surfaces, we will show it in Proposition 2.5.

On the other hand, Mumford [Mu61] gave explicit examples of smooth andprojective surfacesX in positive characteristicp with non-closed global1-forms.This means that the exterior derivatived : H0(X,Ω1

X/k)→ H0(X,Ω2X/k) is non-

zero, which gives rise to a non-zero differential in the Frolicher spectral sequence,which implies that it doesnot degenerate atE1.

In Section 1.4 below we will recall and construct the ringW (k) of Witt vectorsassociated to a perfect fieldk of positive characteristic, which is a local, complete,and discrete valuation of ring of characteristic zero with residue fieldk, whoseunique maximal ideal is generated byp. In particular, the truncated Witt ringWn(k) :=W (k)/(pn) is a flatZ/pnZ-algebra. For a schemeX overk, a lift of XtoWn(k) is a flat schemeX → SpecWn(k) such thatX ×Spec Wn(k) Speck

∼= X.Such lifts may not exist. However, if they exist, we have the following fundamentalresult concerning the degeneration of the Frolicher spectral sequence:

Theorem 1.2 (Deligne–Illusie). Let X be a smooth and proper variety over aperfect fieldk of characteristicp ≥ dim(X), and assume thatX admits a lift toW2(k). Then, the Frolicher spectral sequence ofX degenerates atE1.


This is the main result of [D-I87], and we refer the interested reader to [Ill02] foran expanded version with lots of background information. Infact, this theorem canbe used to obtain a purely algebraic proof of theE1-degeneration of the Frolicherspectral sequence in characteristic zero [Ill02, Theorem 6.9]. This theorem alsoshows that the above mentioned examples of Mumford of surfaces with non-closed1-forms do not lift toW2(k). We refer the interested reader to [Li13, Section 11]for more about liftings, and liftable, as well as non-liftable varieties.

In the previous section, where all varieties were defined over the complex num-bers, we applied complex conjugation to the Hodge filtrationto obtain a new fil-tration, the conjugate filtration. Over fields of positive characteristic, there is nocomplex conjugation. However, since algebraic de Rham theory is the hyperco-homology of the de Rham complex, there exists a second spectral sequence, theconjugate spectral sequence(this is only a name in analogy with complex geome-try: there is no complex conjugation in positive characteristic)

Ei,j2 := H i(X,Hj(Ω•

X/k)) ⇒ H i+jdR (X/k),

see [G-H78, Chapter 3.5] or [We08, Section 1]. Ifk = C, and when consideringX as a complex manifold and holomorphic differential forms, it follows from theholomorphic Poincare lemma that the cohomology sheavesHj(Ωi

X/C) are zero forall j ≥ 1, and thus, the conjugate spectral sequence induces only a trivial filtrationon de Rham cohomology.

On the other hand, ifk is of positive characteristic, then the cohomology sheavesHj(Ωi

X/k) are usually non-trivial forq ≥ 1. More precisely, letF : X → X ′ bethek-linear Frobenius morphism. Then, there exists a canonicalisomorphism

C−1 : ΩnX′/k

∼=−→ Hn(F∗Ω

•X/k)

for all n ≥ 0, theCartier isomorphism, and we refer to [Ill02, Section 3] or [We08,(1.6)] for details, definitions, and further references. Because of this isomorphism,the conjugate spectral sequence is usually non-trivial in positive characteristic, andgives rise to an interesting second filtration on de Rham cohomology, called theconjugate Hodge filtration. The data of de Rham-cohomology, the Frobenius ac-tion, and the two Hodge filtrations are captured in the following structure.

Definition 1.3. An F -zip over a schemeS of positive characteristicp is a tuple(M,C•,D•, ϕ•), whereM is a locally freeOS-module of finite rank,C• is adescending filtration onM ,D• is an ascending filtration onM , andϕ• is a familyof OS-linear isomorphisms

ϕn : (grnC)(p) ∼=−→ grnD,

wheregr denotes the graded quotient modules, and(p) denotes Frobenius pullback.The function

τ : Z → Z≥0

n 7→ rankOSgrnC

is called the(filtration) typeof theF -zip.

8 CHRISTIAN LIEDTKE

The category ofF -zips overFp-schemes with only isomorphisms as morphismsforms a smooth Artin-stackF overFp, andF -zips of typeτ form an open, closed,and quasi-compact substackFτ ⊆ F , see [M-W04, Proposition 1.7]. Despitethe similarity with period domains for Hodge structures, this moduli spaceF isa rather discrete object, and more of a modp reduction of such a period domain.More precisely, ifk is an algebraically closed field of positive characteristic, thenthe set ofk-rational points ofFτ is finite, and we refer to [M-W04, Theorem 4.4]and [We08, Theorem 3.6] for precise statements.

Thus, F -zips capture discrete invariants arising from de Rham cohomologyof smooth and proper varieties in positive characteristic.For example, anF -zip(M,C•,D•, ϕ•) is calledordinary if the filtrationsC• andD• are in opposition,that is, if the rankCi ∩ Dj is as small as possible for alli, j ∈ Z. We refer to[M-W04] and [We08] for details, examples, proofs, and further references. More-over, one can also considerF -zips with additional structure, such as orthogonaland symplectic forms, see [M-W04, Section 5], as well as [P-W-Z15] for furthergeneralizations.

Finally, we would like to mention that Vasiu independently developed in [Va06],[Va10] a theory similar to theF -zips of Moonen and Wedhorn [M-W04] within theframework of ShimuraF -crystals modulop and truncated Dieudonne modules. Werefer the interested reader to [Va06, Sections 3.2.1 and 3.2.9], [Va10], and [X14,Section 2.22] for details, examples, and applications.

1.3. ℓ-adic cohomology. The singular cohomology of a smooth and complex pro-jective variety coincides with cohomology in the locally constant sheafZ (withrespect to the analytic topology). Now, letX be a smooth variety over a fieldkof characteristicp ≥ 0. The Zariski topology onX is too coarse to give a goodcohomology theory for locally constant sheaves. However, for locally constantsheaves of finite Abelian groups, it turns out that etale topology, which is a purelyalgebraically defined Grothendieck topology that is finer than the Zariski topology,has the desired properties, and we shall equipX with this topology. It goes with-out saying that then, being locally constant is then meant with respect to the etaletopology. We refer to [Del77] and [Mil80] for precise definitions.

Let ℓ be a prime number, and then, we defineℓ-adic cohomology

Hnet(X,Zℓ) := lim←−H

net(X,Z/ℓ

mZ).

(When considering locally constant sheaves, etale cohomology works best for finiteAbelian groups. So rather than trying to define something likeHn

et(−,Z) directly,we take the inverse limit overHn

et(−,Z/ℓmZ), which then results in coefficients

in theℓ-adic numbersZℓ, see also Section 1.6 below. To deal directly with infinitegroups, Bhatt and Scholze [B-S15] introduced the pro-etale topology, but we willnot pursue this here.) Next, we define

Hnet(X,Qℓ) := Hn

et(X,Zℓ)⊗ZℓQℓ.

If X is smooth and proper over an algebraically closed fieldk, then this cohomol-ogy theory has the following properties:


(1) Hnet(X,Qℓ) is a contravariant functor inX. The cohomology groups are

finite dimensionalQℓ-vector spaces, and zero ifn < 0 or n > 2 dim(X).(2) There is a cup-product structure

∪i,j : H iet(X,Qℓ) × Hj

et(X,Qℓ) → H i+jet (X,Qℓ).

Moreover,H2 dim(X)et (X,Qℓ) is 1-dimensional, and∪n,2dim(X)−n induces

a perfect pairing, calledPoincare duality.(3) Hn

et(X,Zℓ) defines an integral structure onHnet(X,Qℓ),

(4) If k = C, one can choose an inclusionQℓ ⊂ C (such inclusions existusing cardinality arguments and the axiom of choice, but they are neithercanonical nor compatible with the topologies on these fields), and then,there exist isomorphisms for alln

Hnet(X,Qℓ)⊗Qℓ

C ∼= Hn(X,C),

where we considerX as a differentiable manifold on the right hand sideandC is locally constant with respect to the analytic topology. This com-parison isomorphism connectsℓ-adic cohomology to de Rham-, singular,and constant sheaf cohomology ifk = C.

(5) If char(k) = p > 0 andℓ 6= p, then the dimensiondimQℓHn

et(X,Qℓ) isindependent ofℓ (see [K-M74]). Thus,

bn(X) := dimQℓHn

et(X,Qℓ)

is well-defined forℓ 6= p and it is called thenth Betti number.(6) Finally, there exists a Lefschetz fixed point formula, there are base change

formulas, there exist cycle classes inH2qet (X,Qℓ) for codimensionq sub-

varieties,...

We refer to [Har77, Appendix C] for an overview, and to [Mil80] or [Del77] fora thorough treatment. The following example shows that the assumptionℓ 6= p inproperty (5) above is crucial, and gives a hint of the subtleties involved.

Example 1.4. Let A be ag-dimensional Abelian variety over an algebraicallyclosed fieldk of positive characteristicp. For a primeℓ, we define theℓ-torsionsubgroup schemeA[ℓ] to be the kernel of multiplication byℓ : A → A. TheschemeA[ℓ] is a finite flat group scheme of lengthℓ2g overk, whereas the groupof its k-rational points depends onℓ:

A[ℓ](k) ∼=

(Z/ℓZ)2g if ℓ 6= p, and(Z/pZ)r for some0 ≤ r ≤ g if ℓ = p.

This integerr is called thep-rank of A, and we have

dimQℓH1(A,Qℓ) =

2g if ℓ 6= p, andr if ℓ = p.

In particular, the assumptionℓ 6= p in property (5) is crucial, since we haver < 2gin any case. The group schemeA[p] is of rank p2g (although only rankpr canbe “seen” viak-rational points), which should be reflected in the “correct” p-adic

10 CHRISTIAN LIEDTKE

cohomology theory. Anticipating crystalline cohomology,which we will introducein Section 1.5 below, there exists an isomorphism (see [Ill79a, Theoreme II.5.2])

H1et(A,Qp)⊗Qp K

∼=(H1

cris(A/W )⊗W K)[0]⊂ H1

cris(A/W )⊗W K,

where the subscript[0] denotes the slope zero sub-F -isocrystal (W is the ring ofWitt vectors ofk, K is its field of fractions, and we refer to Section 3 for details).SinceH1

cris(A/W ) is of rank2g, it gives the “correct” answer, and even the factthatH1

et(A,Qp) is “too small” can be explained using crystalline cohomology. Infact, we will see in Section 1.6 that there exists no “correct” p-adic cohomologytheory withQp-coefficients.

1.4. The ring of Witt vectors. In the next section, we will introduce crystallinecohomology. Since these cohomology groups are modules overthe ring of Wittvectors, let us shortly digress on this ring.

Let k be a perfect field of positive characteristicp. For example,k could be afinite field or algebraically closed. Associated tok, there exists a ringW (k), calledthering of Witt vectors(or simply,Witt ring) of k, such that

(1) W (k) is a discrete valuation ring of characteristic zero,(2) the unique maximal idealm of W (k) is generated byp, and the residue

fieldW (k)/m is isomorphic tok,(3) W (k) is complete with respect to them-adic topology,(4) everym-adically complete discrete valuation ring of characteristic zero

with residue fieldk containsW (k) as subring,(5) the Witt ringW (k) is functorial ink.

Note that property (4) shows thatW (k) is unique up to isomorphism. There isalso a more intrinsic characterization of the functork 7→ W (k) as left adjoint toreduction modulop, but we shall not pursue this here.

Instead, let us quickly run through an explicit construction of W (k). We referto [Se68, Chapitre II.6] and [Ha78, Section 17] for details,proofs, and generaliza-tions. Also, we refer to [C-D14] and [C-D15] for a completelydifferent approachto the ring of Witt vectors. Letp be a prime. Then, we define theWitt polynomials(with respect top, which is understood from the context and omitted) to be thefollowing polynomials with coefficients inZ:

W0(x0) := x0W1(x0, x1) := xp0 + px1

...

Wn(x0, ..., xn) :=∑n

i=0 pixp

n−i

i = xpn

0 + pxpn−1

1 + ...+ pnxn

Then, one can show that there exist unique polynomialsSn andPn in 2n + 2variables with coefficients inZ such that the following holds true:

Wn(x0, ..., xn) + Wn(y0, ..., yn) = Wn(Sn(x0, ..., xn, y0, ..., yn))Wn(x0, ..., xn) · Wn(y0, ..., yn) = Wn(Pn(x0, ..., xn, y0, ..., yn))


For an arbitrary ringR, which is not necessarily of characteristicp, we define thetruncated Witt ringWn(R) to be the setRn, whose ring structure is defined to be

(x0, ..., xn−1) ⊕ (y0, ..., yn−1):= (S0(x0, y0), ..., Sn−1(x0, ..., xn−1, y0, ..., yn−1))

(x0, ..., xn−1) ⊙ (y0, ..., yn−1):= (P0(x0, y0), ..., Pn−1(x0, ..., xn−1, y0, ..., yn−1))

It turns out thatWn(R) is indeed a ring with zero0 = (0, ..., 0) and unit1 =(1, 0, ..., 0). For example, we haveS0(x0, y0) = x0 + y0 andP0(x0, y0) = x0 · y0,and thus,W1(R) is justR with its usual addition and multiplication. Next, ifR ispositive characteristicp, we define

V : (x0, ..., xn−1) 7→ (0, x0, ..., xn−2)σ : (x0, ..., xn−1) 7→ (xp0, ..., x

pn−1).

Then,V is calledVerschiebung(German for “shift”), and it is an additive map.Next, σ is calledFrobenius, and it is a ring homomorphism. (In order to avoid aclash of notations when dealing withF -crystals, see Section 3 below, it is custom-ary to denote the Frobenius onW (k) by σ rather thanF .) The mapsV andσ arerelated to multiplication byp onWn(R) by

σ V = V σ = p · idWn(R)

We haveW1(R) = R as rings, and for alln ≥ 2 the projectionWn(R) →Wn−1(R) onto the first(n − 1) components is a surjective ring homomorphism.Then, by definition, thering of Witt vectorsW (R) is the inverse limit

W (R) := lim←−

Wn(R),

or, equivalently, the previous construction with respect to the infinite productRN.

Exercise 1.5.For the finite fieldFp, show thatWn(Fp) ∼= Z/pnZ and thus,

W (Fp) ∼= lim←−

Z/pnZ

is isomorphic toZp, the ring ofp-adic integers. Show thatσ is the identity andVis multiplication byp in W (Fp).

Exercise 1.6. If k is a field of positive characteristic, show thatW (k) is a ring ofcharacteristic zero with residue fieldk. Setm := ker(W (k)→ k) and show that ifk is perfect, thenm is generated byp, thatW (k) is m-adically complete, and thatW (k) is a DVR. On the other hand ifk is not perfect, and if moreoverk/kp is nota finite field extension, show thatm is not finitely generated and thatWn(k) is notNoetherian for alln ≥ 2.

If X is a scheme, we can also sheafify the construction ofWn(R) to obtainsheaves of ringsWnOX andWOX , respectively. The cohomology groups

H i(X,WnOX) and H i(X,WOX ) := lim←−

H i(X,WnOX)

were introduced and studied by Serre [Se58], they are calledSerre’s Witt vectorcohomologygroups, and we will come back to them in Section 6. However, letus already note at this point that the torsion of theW (k)-moduleH i(X,WOX)


may not be finitely generated (for example, this is the case ifi = 2 andX isa supersingular K3 surface), which is rather unpleasant. Let us finally mentionthat theWnOX are just the zeroth step of thede Rham–Witt complex(WnΩ

jX , d)

introduced by Illusie in [Ill79a], and we refer to [Ill79b] for an overview.

1.5. Crystalline cohomology. To a complex projective varietyX, we have itsde Rham cohomologyH∗

dR(X/C) and showed in Section 1.1 that it comes withextra structure. Now, letX be a smooth and proper variety over a perfect fieldk ofpositive characteristicp. Let us shortly summarize what we achieved so far in thealgebraic setting:

(1) In Section 1.2, we associated toX its algebraic de Rham-cohomology,which is ak-vector spaceHn

dR(X/k) together with a Frobenius action andtwo filtrations, which is captured in the notion of anF -zip (Definition 1.3).

On the other hand, there is no integral structure. Another drawbackis the following: although there exists a Chern mapc1 : Pic(X) →H2

dR(X/k), we have for everyL ∈ Pic(X)

c1(L⊗p) = p · c1(L) = 0 in H2

dR(X/k) ,

giving a zero Chern class even for some very ample line bundles. Also,counting fixed points via Lefschetz fixed point formulas (an importanttechnique when dealing with varieties over finite fields) gives us these num-bers as traces ink, and thus, we obtain the number of fixed points only asa congruence modulop.

These observations suggest to look for cohomology theorieswhose groupsare modules over rings of characteristic zero.

(2) In Section 1.3, we discussedℓ-adic cohomologyHnet(X,Qℓ), which comes

with an integral structure fromHnet(X,Zℓ), but we have no Hodge filtra-

tions.Moreover, ifℓ = p, then we have seen in Example 1.4 thatHn

et(X,Qp)does not always give the desired answer. Even worse, Serre showed thatthere does not exist a “well-behaved” cohomology withQp-coefficients,and we refer to Section 1.6 for a precise statement.

On the other hand, ifℓ 6= p, then the fieldsk andQℓ usually have littlein common, making comparison theorems between de Rham- andℓ-adiccohomology even difficult to conjecture.

It is here, that Witt vectors enter the picture. As we shall now see, crystallinecohomology has all desired features and provides an answer to all problems justraised.

To explain crystalline cohomology, let us assume for a moment thatX is smoothand projective overk and that there exists a projectivelift of X to W := W (k),that is, a smooth projective schemeX → Spec W such that its special fiberX ×Spec W Speck is isomorphic toX. Then, for eachn, the de Rham-cohomologygroupHn

dR(X/W ) is a finitely generatedW -module. It was Grothendieck’s in-sight [Gr68b] that these cohomology groups are independentof choice of liftX ofX. In fact, these cohomology groups can even be defined in caseX does not admit


a lift to W . The construction is quite involved, but we refer to [CL98, Section 1.3]for motivation and to [CL98, Section 2] for a detailed introduction.

More precisely, for everym ≥ 1, we have cohomology groupsH∗cris(X/Wm(k)),

all of which are finitely generatedWm(k)-modules. Form = 1, we obtain de Rham-cohomology

HndR(X/k)

∼= Hncris(X/W1(k)) = Hn

cris(X/k) for all n ≥ 0,

and, by definition, the limit

Hncris(X/W ) := Hn

cris(X/W (k)) := lim←−Hncris(X/Wm(k))

is calledcrystalline cohomology. The origin of the name is as follows: althoughX may not lift toW (k), its cohomology “grows” locally overW . One can makethese growths “rigid”, so to glue and to obtain a well-definedcohomology theoryoverW (k). And thus, growing and being rigid, it is natural to call suchan objecta “crystal”.

If K denotes the field of fractions ofW , then it has the following properties:

(1) Hncris(X/W ) is a contravariant functor inX. These groups are finitely

generatedW -modules, and zero ifn < 0 or n > 2 dim(X).(2) There is a cup-product structure

∪i,j : H icris(X/W )/torsion × Hj

cris(X/W )/torsion → H i+jcris (X/W )/torsion

Moreover,H2 dim(X)cris (X/W ) ∼= W , and∪n,2dim(X)−n induces a perfect

pairing, calledPoincare duality.(3) Hn

cris(X/W ) defines an integral structure onHncris(X/W )⊗W K,

(4) If ℓ is a prime different fromp, then (see [K-M74])

bn(X)def= dimQℓ

Hnet(X,Qℓ) = rankW Hn

cris(X/W ),

showing that crystalline cohomology computesℓ-adic Betti numbers.(5) If X lifts to W , then crystalline cohomology is isomorphic to de Rham

cohomology of a lift, from which we deduce a universal coefficient formula

0 → Hncris(X/W )⊗W k → Hn

dR(X/k) → TorW1 (Hn+1cris (X/W ), k) → 0,

for all n ≥ 0. This formula also holds true ifX does not lift. In any case,this shows that crystalline cohomology computes de Rham cohomology.

(6) Finally, there exists a Lefschetz fixed point formula, there are base changeformulas, there exist cycle classes inH2q

cris(X/W ) for codimensionq sub-varieties,...

By functoriality, the absolute Frobenius morphismF : X → X induces aσ-linearmorphismϕ : Hn

cris(X/W ) → Hncris(X/W ) of W -modules. Ignoring torsion,

this motivates to consider freeW -modules together with injectiveσ-linear maps,which leads to the notion of anF -crystal, to which we come back in Section 3.

We refer the interested reader to [CL98] for a much more detailed introduction tocrystalline cohomology, to [Gr68b], [Be74] and [B-O78] forproofs and technicaldetails, as well as to [Ill79a] and [Ill79b] for the connection with the de Rham–Wittcomplex.


Exercise 1.7.LetX be a smooth and proper variety over a perfect fieldk of posi-tive characteristicp, and assume that the Frolicher spectral sequence degenerates atE1. Using only the properties of crystalline cohomology mentioned above, showthat the following are equivalent

(1) For alln ≥ 0, theW -moduleHncris(X/W ) is torsion-free.

(2) We have

dimQℓHn

et(X,Qℓ) = dimkHndR(X/k)

for all n ≥ 0 and all primesℓ 6= p.

Thus, thep-torsion of crystalline cohomology measures the deviationbetweenℓ-adic Betti numbers to dimensions of de Rham-cohomology.

Examples 1.8.Let us give a two fundamental examples.

(1) LetA be an Abelian variety of dimensiong. Then, allHncris(A/W ) are

torsion-freeW -modules. More precisely,H1cris(A/W ) is free of rank2g

and for alln ≥ 2 there are isomorphisms

Hncris(A/W ) ∼= ΛnH1

cris(A/W ).

Let us mention the following connection (for those familiarwith p-divisiblegroups and Dieudonne modules), which we will not need in thesequel: letA[pn] be the kernel of multiplication bypn : A→ A, which is a finite flatgroup scheme of rankp2gn. By definition, the limit

A[p∞] := lim−→

A[pn]

is the p-divisible groupassociated toA. Then, theDieudonne-moduleassociated toA[p∞] is isomorphic toH1

cris(A/W ), compatible with theFrobenius-actions on both sides, see, for example, [Ill79a, Section II.7.1].We will come back to the Frobenius action onH1

cris(A/W ) in Section 3.(2) For a smooth and proper varietyX, letα : X → Alb(X) be its Albanese

morphism. Then,α induces an isomorphism

H1cris(X/W ) ∼= H1

cris(Alb(X)/W ).

In particular,H1cris(X/W ) is always torsion-free. From this, we can com-

pute the crystalline cohomology of curves via their Jacobians. We referto [Ill79a, Section II.5 and Section II.6] for connections of p-torsion ofH2

cris(X/W ) with Oda’s subspace ofH1dR(X/k), the non-reducedness of

the Picard scheme ofX, as well as non-closed1-forms onX.

In Section 2, we will compute the crystalline cohomology of aK3 surface.

We already mentioned that Illusie constructed a complex(WmΩjX/k, d), the

de Rham–Witt complex, and that it coincides with the de Rham complex ifm = 1.This complex gives rise to spectral sequences for allm ≥ 1

Ei,j1 := Hj(X,WmΩi

X/k) ⇒ H i+jcris (X/Wm(k)).

Form = 1, this is the Frolicher spectral sequence. In the limitm → ∞, this be-comes theslope spectral sequencefrom Hodge–Witt cohomologyHj(X,WΩi

X/k)


to crystalline cohomology. Whereas the Frolicher spectral sequence ofX may ormay not degenerate atE1 if k is of positive characteristic, the slope spectral se-quence modulo torsion always degenerates atE1. Moreover, the slope spectralsequence (including torsion) degenerates atE1 if and only if thep-torsion of allHj(X,WΩi

X/k) is finitely generated. Fori = 0, this gives a conceptional frame-work for finite generation of Serre’s Witt vector cohomology. We refer to [Ill79a]for details.

Finally, reduction modulop gives a map

πn : Hncris(X/W ) → Hn

dR(X/k),

which, by the universal coefficient formula, is onto if and only if Hn+1cris (X/W )

has nop-torsion. Thus, if all crystalline cohomology groups are torsion-freeW -modules, then de Rham-cohomology is crystalline cohomology modulop. Next,by functoriality, the Frobenius ofX induces aσ-linear mapϕ : H∗

cris(X/W ) →H∗

cris(X/W ). Under suitable hypotheses onX, the Frobenius action determinesthe Hodge- and the conjugate filtration on de Rham-cohomology. More precisely,we have the following result of Mazur, and refer to [B-O78, Section 8] details,proofs, and further references.

Theorem 1.9(Mazur). LetX be a smooth and proper variety over a perfect fieldk of positive characteristicp. Assume thatH∗

cris(X/W ) has nop-torsion, and thatthe Frolicher spectral sequence ofX degenerates atE1. Then,

πn maps ϕ−1(piHn

cris(X/W ))

onto F i

πn p−i maps Im(ϕ) ∩

(piHn

cris(X/W ))

onto Fn−icon ,

whereF i andFn−icon denote the Hodge- and its conjugate filtration onHn

dR(X/k),respectively.

1.6. Serre’s observation. In this section, we have discussedℓ-adic and crystallinecohomology, whose groups areQℓ-vector spaces andW (k)-modules, respectively.One might wonder, whether crystalline cohomology arises asbase change froma cohomology theory, whose groups areZp-modules, or even, whether all of theabove cohomology theories arise from a cohomology theory, whose groups areZ-modules orQ-vector spaces. Now, cohomology theories that satisfy the “usual”properties discussed in this section are examples of so-called Weil cohomologytheories, and we refer to [Har77, Appendix C.3] for axioms and discussion.

Serre observed that there exists no Weil cohomology theory in positive charac-teristic that take values inQ-, Qp-, or R-vector spaces. In particular, the abovequestion has a negative answer. Here is the sketch of a counter-example: there ex-ist supersingular elliptic curvesE overFp2 such thatEnd(E) ⊗Q is a quaternionalgebra that is ramified atp and∞. By functoriality, we obtain a non-trivial rep-resentation ofEnd(E) onH1(E), which, being a Weil cohomology theory, mustbe 2-dimensional. In particular, we would obtain a non-trivialrepresentation ofEnd(E)⊗Q in a2-dimensionalQp- orR- vector space, a contradiction. We referto [Gr68b, p. 315] or [CL98, Section I.1.3] for details.


2. K3 SURFACES

2.1. Definition and examples. In this section, we turn to K3 surfaces, and willcompute the various cohomology groups discussed in the previous section forthem. Let us first discuss their position within the classification of surfaces: letX be a smooth and projective surface over an algebraically closed fieldk of char-acteristicp ≥ 0. Moreover, assume thatωX is numerically trivial, that is,ωX haszero-intersection with every curve onX. In particular,X is a minimal surface ofKodaira dimension zero. By the Kodaira–Enriques classification (if p = 0) and re-sults of Bombieri and Mumford (ifp > 0), thenX belongs to one of the followingclasses:

(1) Abelian surfaces, that is, Abelian varieties of dimension 2.(2) (Quasi-)hyperelliptic surfaces.(3) K3 surfaces.(4) Enriques surfaces.

We refer to [BHPV, Chapter VI] for the surface classificationoverC, and to [Li13]for an overview in positive characteristic. Ifp 6= 2, 3, the only surfaces withωX∼= OX are Abelian surfaces and K3 surfaces. We refer the interested reader

to [Li13, Section 7] for some classes of Enriques surfaces inp = 2, as well asquasi-hyperelliptic surfaces inp = 2, 3 that have trivial canonical sheaves, and to[B-M2] and [B-M3] for a detailed analysis of these surfaces.Here, we are mainlyinterested in K3 surfaces, and recall the following definition, which holds in anycharacteristic.

Definition 2.1. A K3 surfaceis a smooth and projective surfaceX over a fieldsuch that

ωX∼= OX and h1(X,OX) = 0.

Examples 2.2.Let k be an algebraically closed field of characteristicp ≥ 0.

(1) If X is a smooth quartic surface inP3k, thenωX

∼= OX by the adjunctionformula, and taking cohomology in the short exact sequence

0 → OP3k(−4) → OP3

k→ OX → 0

we findh1(OX) = 0. In particular,X is a K3 surface.(2) Similarly, smooth complete intersections of quadric and cubic hypersur-

faces inP4k, as well as smooth complete intersections of three quadric hy-

persurfaces inP5k give examples of K3 surfaces.

(3) If p 6= 2 andA is an Abelian surface overk, then the quotientA/ ± idhas16 singularities of typeA1, and its minimal resolutionKm(A) of sin-gularities is a K3 surface, theKummer surface associated toA. (We referthe interested to reader to [Sh74b] and [K78] to learn what goes wrong ifp = 2, and to [Sch07] how to remedy this.)

We note that these three classes differ in size: the three example classes in (1)and (2) form19-dimensional families, whereas the Kummer surfaces in (3) form a3-dimensional family.


2.2. Cohomological invariants. Let us now compute theℓ-adic Betti numbers,the Hodge numbers, and the crystalline cohomology groups ofa K3 surface. Wewill give all details so that the interested reader can see where the characteristic-pproofs are more difficult than the ones in characteristic zero.

Proposition 2.3. Theℓ-adic Betti numbers of a K3 surface are as follows

i 0 1 2 3 4bi(X) 1 0 22 0 1

In particular, we havec2(X) =∑4

i=0(−1)ibi(X) = 24.

PROOF. SinceX is a surface, we haveb0 = b4 = 1. By elementary deformationtheory of invertible sheaves,H1(OX) is the Zariski tangent space ofPic0X/k at

the origin, see [Se06, Section 3.3], for example. Sinceh1(OX) = 0 by definitionof a K3 surface, it follows thatPic0X/k is trivial. Thus, also the Albanese varietyAlb(X), which is the dual of the reduced Picard scheme, is trivial, and we findb1(X) = 2dimAlb(X) = 0. By Poincare duality, we haveb1 = b3 = 0. Next,from Noether’s formula for surfaces

12χ(OX ) = c1(X)2 + c2(X),

we computec2(X) = 24, which, together with the known Betti numbers, impliesb2(X) = 22.

Next, we recall that theHodge diamondof a smooth projective varietyY isgiven by ordering the dimensionshi,j(Y ) = hj(Y,Ωi

Y/k) in a rhombus.

Proposition 2.4. The Hodge diamond of a K3 surface is as follows:

h0,0

h1,0 h0,1

h2,0 h1,1 h0,2

h2,1 h1,2

h2,2

=

10 0

1 20 10 01

PROOF. We haveh0,0 = h2,2 = 1 sinceX is a surface, andh0,1 = 0 by thedefinition of a K3 surface. Next, Serre duality givesh0,1 = h2,1 andh1,0 = h1,2.If k = C, then complex conjugation induces the Hodge symmetryh1,0 = h0,1.However, in positive characteristic, this Hodge symmetry may fail in general (see[Se58] and [Li08] for examples), and thus, we have to computeh1,0(X) anotherway: using the isomorphismE∨ ∼= E⊗det(E), which holds for locally free sheavesof rank2 (see [Har77, Exercise II.5.16], for example), we findTX ∼= Ω1

X/k for aK3 surface, and thus

H1,0(X)def= H0(Ω1

X/k)∼= H0(TX) .

Now, by a theorem of Rudakov and Shafarevich [R-S76], a K3 surface has no non-zero global vector fields, and thus, these cohomology groupsare zero. Finally, we


use the Grothendieck–Hirzebruch–Riemann–Roch theorem tocompute

χ(Ω1X/k)

= rank(Ω1X/k) · χ(OX) + 1

2

(c1(Ω

1X/k) · (c1(Ω

1X/k)−KX)

)− c2(Ω

1X/k)

= 4 + 0 − 24 = −20,

which impliesh1(Ω1X/k) = 20.

As a consequence of this proposition, together with the Rudakov–Shafarevichtheorem on non-existence of global vector fields on K3 surfaces, we obtain

Proposition 2.5. For a K3 surfaceX, the Frolicher spectral sequence

Ei,j1 = Hj(X,Ωi

X/k) ⇒ H i+jdR (X/k)

degenerates atE1. Moreover,Hncris(X/W ) is a freeW -module of rankbn(X) for

all n ≥ 0.

PROOF. By Proposition 2.4 and using the isomorphismΩ1X∼= TX (seen in the

proof of Proposition 2.4), we haveH2(TX) ∼= H2(Ω1X) = H1,2 = 0. Since

H2(TX) = 0, deformations ofX are unobstructed, and thus,X lifts to W2(k) (seealso the discussion in Section 2.3 and Theorem 2.7 below), and thus, degeneracy ofthe Frolicher spectral sequence atE1 follows from Theorem 1.2. From Proposition2.4 and Exercise 1.1 we compute the dimensions of the de Rham cohomologygroups, which then turn out to be the same as theℓ-adic Betti numbers given inProposition 2.3. Thus, by Exercise 1.7, the crystalline cohomology groups are freeW -modules of the stated rank.

Remark 2.6. For a smooth and proper varietyX over a perfect fieldk, the slopespectral sequence from Hodge–Witt to crystalline cohomology degenerates atE1

if and only if all theW -modulesHj(X,WΩiX/k) are finitely generated [Ill79a,

Theoreme II.3.7]. For a K3 surface, this is the case if and only if it is not super-singular - we refer to Section 4 for definition of supersingularity and to [Ill79a,Section II.7.2] for details.

2.3. Deformation theory. Infinitesimal and formal deformations of a smooth andproper varietyX over a fieldk can be controlled by a tangent–obstruction theoryarising from thek-vector spacesH i(TX), i = 0, 1, 2, see [Se06, Chapter 2] or[F-G05, Chapter 6] for a reader-friendly introduction.

Let us recall the most convenient case: ifH2(TX) = 0, then every infinitesimaldeformation of ordern can be extended to one of ordern+ 1, and then, the set ofall such extensions is an affine space underH1(TX). In particular, this applies tolifting problems: ifk is perfect of positive characteristic, then the Witt ringW (k)is a limit of ringsWn(k), see Section 1.4. Since the kernel ofWn+1(k)→ Wn(k)is the ideal generated bypn−1 and(pn−1)2 = 0, it is a small extension, and thus,a smooth and proper varietyX over a perfect fieldk of positive characteristic withH2(TX) = 0 admits a formal lift toW (k), and we refer to [Ill05, Chapter 8.5] and[Li13, Section 11.2] for details and further references. Since this most convenientcase applies to K3 surfaces, we have the following result.


Theorem 2.7. Let X be a K3 surface over a perfect fieldk of positive charac-teristic. Then, the formal deformation spaceDef(X) of X is smooth of relativedimension20 overW (k), that is,

Def(X) ∼= SpfW (k)[[t1, ..., t20]] .

In particular,X formally lifts overW (k).

PROOF. By Proposition 2.4 and using the isomorphismΩ1X∼= TX (seen in the

proof of Proposition 2.4), we find

h0(TX) = h2(TX) = 0 and h1(TX) = 20,

from which all assertions follow from standard results of deformation theory, see[Se06, Chapter 2] or [F-G05, Chapter 6], for example.

If X is a K3 surface over a perfect fieldk of positive characteristic, then the pre-vious theorem implies that there exists a compatible systemXn → SpecWn(k)nof algebraicschemesXn, each flat overWn(k), and with special fiberX1 = X.Now, the limit of this system is aformalscheme [Har77, Section II.9], and it is notclear whether it is algebraizable, that is, we do not know, whether this limit arisesas completion of a scheme overW (k) along its special fiber (in fact, it is not truein general, see Section 2.4 below).

By Grothendieck’s existence theorem (see [Ill05, Theorem 8.4.10], for exam-ple), algebraization of formal schemes holds, for example,if one is able to equipXn with a compatible systemLn of invertible sheaves onXn such thatL1 is am-ple onX1 = X. This poses the question whether a given formal deformationcanbe equipped with such a compatible system of invertible sheaves. The obstructionto deforming an invertible sheaf to a small extension lies inH2(OX ), which is1-dimensional for a K3 surface. We thus expect that this should impose one non-trivial equation toDef(X), which is true and made precise by the following resultsof Deligne, [Del81a, Proposition 1.5] and [Del81a, Theorème 1.6].

Theorem 2.8(Deligne). LetX be a K3 surface over a perfect fieldk of positivecharacteristic, and letL be a non-trivial invertible sheaf onX. Then, the spaceDef(X,L) of formal deformations of the pair(X,L) is a formal Cartier divisorinsideDef(X), that is,

Def(X,L) ⊂ Def(X),

is a formal subscheme defined by one equation. Moreover,Def(X,L) is flat and ofrelative dimension19 overW (k).

Unfortunately, it is not clear whetherDef(X,L) is smooth overW (k), and werefer to [Og79,§2] for an analysis of its singularities. In particular, if wepick anample invertible sheafL onX in order to construct a formal lift of the pair(X,L)toW (k) (in order to apply Grothendieck’s existence theorem), thenit could happenthatDef(X,L) is flat, but not smooth overW (k). Thus, a priori, we only have analgebraic lift ofX to some finite extension ringR ⊇ W (k). However, thanks to arefinement of Ogus [Og79, Corollary 2.3] of Deligne’s result, we have


Theorem 2.9(Deligne, Ogus). LetX be a K3 surface over an algebraically closedfield of odd characteristic. Then, there exists a projectivelift of X toW (k).

PROOF. By [Og79, Corollary 2.3], any nonsuperspecial K3 surface can be liftedprojectively toW (k), and we refer to [Og79, Example 1.10] for the notion ofsuperspecial K3 surfaces. Since the Tate-conjecture holdsfor K3 surfaces in oddcharacteristic (see Theorem 4.6 below), the only nonsuperspecial K3 surface is thesupersingular K3 surface with Artin invariantσ0 = 1, see [Og79, Remark 2.4].However, this latter surface is the Kummer surface associated to the self-productof a supersingular elliptic curve by [Og79, Corollary 7.14]and can be lifted “byhand” projectively toW (k).

2.4. Moduli spaces. By Theorem 2.7, the formal deformation spaceDef(X) of aK3 surfaceX is formally smooth and20-dimensional overW (k). However, it isnot clear (and in fact, not true) whether all formal deformations are algebraizable.By a theorem of Zariski and Goodman (see [Ba01, Theorem 1.28], for example), asmooth and proper algebraic surface is automatically projective, which applies inparticular to K3 surfaces. Thus, associated to an ample invertible sheafL on analgebraic K3 surfaceX, there is a formal Cartier divisorDef(X,L) ⊂ Def(X)by Theorem 2.8, along whichL extends. Since formal and polarized deformationsare algebraizable by Grothendieck’s existence theorem (see Section 2.3), we canalgebraize the19-dimensional formal family overDef(X,L). Using Artin’s ap-proximation theorems, this latter family can be descended to a polarized familyof K3 surfaces that is19-dimensional and of finite type overW (k), and one maythink of it as an etale neighborhood of(X,L) inside a moduli space of suitablypolarized K3 surfaces. In fact, this can be made precise to give a rigorous algebraicconstruction for moduli spaces of polarized K3 surfaces, and we refer to [Ar74b,Section 5] and [Ri06] for technical details. These moduli spaces of polarized K3surfaces are19-dimensional, whereas the unpolarized formal deformationspace is20-dimensional.

Now, before proceeding, let us shortly leave the algebraic world: over the com-plex numbers, there exists a20-dimensional analytic moduli space for compactKahler surfaces that are of type K3, and most of which are notalgebraic. More-over, this moduli space is smooth, but not Hausdorff. Insideit, the set of algebraicK3 surfaces is a countable union of analytic divisors. In fact, these divisors corre-spond to moduli spaces of algebraic and polarized K3 surfaces. We refer to [BHPV,Chapter VIII] for details and further references. We mention this to convince thereader that it is not possible to obtain a20-dimensional moduli space of algebraicK3 surfaces, even over the complex numbers.

Therefore, when considering moduli of algebraic K3 surfaces, one usually looksat moduli spaces of (primitively) polarized surfaces. Here, an invertible sheafL ona varietyX is calledprimitive if it is not of the formM⊗k for somek ≥ 2. Then,


for a ringR, we consider the functorM2d,R

(schemesoverR

)→ (groupoids)

S 7→

flat morphisms of algebraic spaces(X ,L)→ S,all of whose geometric fibers are K3 surfaces, andsuch thatL restricts to a primitive polarization ofdegree (=self-intersection)2d on each fiber

In view of the above discussion it should not be surprising that this functor isrepresentable by a separated Deligne–Mumford stack [Ri06,Theorem 4.3.3].

Combining [Ri06, Proposition 4.3.11], [Ma14, Section 5], and [MP15, Corol-lary 4.16], we even have the following results on the global geometry of thesemoduli spaces. Before stating them, let us mention that (2) and (3) are neither ob-vious nor straight forward, and partly rest on the Kuga–Satake construction, whichallows to obtain results about moduli spaces of K3 surfaces from the correspondingresults for moduli spaces of Abelian varieties.

Theorem 2.10(Madapusi-Pera, Maulik, Rizov). The Deligne–Mumford stack

(1) M

2d,Z[ 1

2d]is smooth overZ[ 1

2d ].

(2) M2d,Fp

is quasi-projective overFp if p ≥ 5 andp ∤ d.

(3) M2d,Fp

is geometrically irreducible overFp if p ≥ 3 andp2 ∤ d.

In Section 8 below, we will introduce and discuss a stratification ofM2d,Fp

,which only exists in positive characteristic. It would be interesting to understandthe geometry and the singularities (if there are any) ofM2d,Fp

if p divides2d.

3. F-CRYSTALS

In Section 1.5, we introduced crystalline cohomology, and we computed it forK3 surfaces in Section 2.2. Just as Hodge structures abstractly capture the linearalgebra data coming from de Rham-cohomology of a smooth complex projectivevariety (see the end of Section 1.1),F -crystalscapture the semi-linear data com-ing from crystalline cohomology. In this section, after introducingF -crystals, weassociate to them two polygons: theHodge polygonand theNewton polygon. Un-der the assumptions of Theorem 1.9 (which hold, for example,for K3 surfaces),theF -crystal associated toH∗

cris(X/W ) not only computes de Rham-cohomology,but also the Hodge and its conjugate filtration. On the other hand, by a result ofDieudonne and Manin, we can classifyF -crystals up toisogenyin terms ofslopes,which gives rise to a second polygon, the Newton polygon, which always lies onor above the Hodge polygon. The deviation between Hodge and Newton polygongives rise to new discrete invariants of varieties in positive characteristic that haveno analog in characteristic zero. In Section 8, we will use these discrete invariantsto stratify the moduli space of K3 surfaces.


3.1. F-crystals. In Section 1.4, we introduced and discussed the Witt ringW :=W (k) for a perfect fieldk of positive characteristicp. We denote byK its field offractions. We also recall that the Frobenius morphismx 7→ xp of k induces a ringhomomorphismσ :W →W by functoriality, and that there exists an additive mapV : W →W such thatp = σ V = V σ. In particular,σ is injective.

Definition 3.1. An F -crystal(M,ϕM ) overk is a freeW -moduleM of finite ranktogether with an injective andσ-linear mapϕM :M →M , that is,ϕM is additive,injective, and satisfies

ϕM (r ·m) = σ(r) · ϕM (m) for all r ∈W, m ∈M.

An F -isocrystal(V, ϕV ) is a finite dimensionalK-vector spaceV together with aninjective andσ-linear mapϕV : V → V .

A morphismu : (M,ϕM ) → (N,ϕN ) of F -crystals (resp.,F -isocrystals) is aW -linear (resp.,K-linear) mapM → N such thatϕN u = u ϕM . An isogenyof F -crystals is a morphismu : (M,ϕM )→ (N,ϕN ) of F -crystals, such that theinduced mapM ⊗W K → N ⊗W K is an isomorphism ofF -isocrystals.

Let us give two examples ofF -crystals, one arising from geometry (and beingthe prototype of such an object), the other one purely algebraic (and being crucialfor the isogeny classification later on).

Example 3.2.LetX be a smooth and proper variety overk. Then, for everyn ≥ 0,

Hn := Hncris(X/W )/torsion

is a freeW -module of finite rank. By functoriality, the absolute Frobenius mor-phismF : X → X induces aσ-linear mapϕ : Hn → Hn. Next, Poincare dualityinduces a perfect pairing

〈−,−〉 : Hn ×H2 dim(X)−n → H2 dim(X)(X/W ) ∼=W,

which satisfies the following compatibility with Frobenius

〈ϕ(x), ϕ(y)〉 = pdim(X) · σ〈x, y〉.

Sinceσ is injective onW , it follows that alsoϕ : Hn → Hn is injective, and thus,(Hn, ϕ) is anF -crystal.

Example 3.3. LetWσ〈T 〉 be the non-commutative polynomial ring in the variableT overW subject to the relations

T · r = σ(r) · T for all r ∈W.

Let α = r/s ∈ Q≥0, wherer, s are non-negative and coprime integers. Then,

Mα := Wσ〈T 〉/(Ts − pr)

together withϕ : m 7→ T ·m defines anF -crystal(Mα, ϕ) of ranks. The rationalnumberα is called theslopeof (Mα, ϕ).

The importance of the previous example comes from the following result, whichclassifiesF -crystals over algebraically closed fields up to isogeny:


Theorem 3.4(Dieudonne–Manin). Let k be an algebraically closed field of posi-tive characteristic. Then, the category ofF -crystals overk up to isogeny is semi-simple and the simple objects are the(Mα, ϕ), α ∈ Q≥0 from Example 3.3.

We note in passing that not everyF -isocrystal is of the formM ⊗W (k) K forsomeF -crystal(M,ϕ). ThoseF -isocrystals that do are calledeffective.

Definition 3.5. Let (M,ϕ) be anF -crystal over an algebraically closed fieldk ofpositive characteristic, and let

(M,ϕ) ∼⊕

α∈Q≥0

Mnαα

be its decomposition up to isogeny according to Theorem 3.4.Then, the elementsin the set

α ∈ Q≥0 |nα 6= 0

are called theslopesof (M,ϕ). For every slopeα of (M,ϕ), the integer

λα := nα · rankW Mα

is called themultiplicity of the slopeα. In case(M,ϕ) is anF -crystal over a perfectfieldk, we define its slopes and multiplicities to be the ones of(M,ϕ)⊗W (k)W (k),wherek is an algebraic closure ofk.

3.2. Newton and Hodge polygons.Let (M,ϕ) be anF -crystal. We order itsslopes in ascending order

0 ≤ α1 < α2 < ... < αt

and denote byλ1, ..., λt the respective multiplicities.Then, theNewton polygonof (M,ϕ) is defined to be the graph of the piece-

wise linear functionNwtM from the interval[0, rankM ] ⊂ R to R, such thatNwtM (0) = 0 and whose graph has the following slopes

slope α1 if 0 ≤ t < λ1,slope α2 if λ1 ≤ t < λ1 + λ2,

...

Since we ordered the slopes in ascending order, this polygonis convex. Next, itfollows easily from the definitions that the vertices of thispolygon have integralcoordinates. Clearly, since the Newton polygon is built from slopes, it only de-pends on the isogeny class of(M,ϕ). Conversely, we can read off all slopes andmultiplicities from the Newton polygon, and thus, the Newton polygon actuallydetermines theF -crystal up to isogeny.

Next, we define theHodge polygonof (M,ϕ), whose definition is motivatedby Theorem 1.9, but see also Theorem 3.8 below. Sinceϕ is injective,M/ϕ(M)is an ArtinianW -module, and thus, there exist non-negative integershi and anisomorphism

M/ϕ(M) ∼=⊕

i≥1

(W/piW )hi .


Moreover, we define

h0 := rankM −∑

i≥1

hi.

Then, the Hodge polygon of(M,ϕ) is defined to be the graph of the piecewise lin-ear functionHdgM from the interval[0, rankM ] ⊂ R toR, such thatHdgM (0) =0 and whose graph has the following slopes

slope 0 if 0 ≤ t < h0,slope 1 if h0 ≤ t < h0 + h1,

...

As above with the Newton polygon, the Hodge polygon is convexand its verticeshave integral coordinates.

Example 3.6. LetMα with α = r/s ∈ Q≥0 be theF -crystal from Example 3.3.Then,

Mα/ϕ(Mα) ∼= W/prW,

we obtainh0 = s− 1, hr = 1, andhi = 0 for i 6= 0, r.Now, we assume that0 < α < 1, that is,0 < r < s, and we define aW -module

Nα together with an embedding intoMα ⊗W K as follows:

Nα := W [T,U ]/(TU − p, T s−r − U r) → Mα ⊗W KT 7→ TU 7→ pT−1 = p1−rT s−1.

Then,Nα inherits the structure of anF -crystal fromMα⊗WK, and it is isogenoustoMα. Since

Nα/ϕ(Nα) ∼= (W/pW )r,

we obtainh0 = s − r, h1 = r, andhi = 0 for i 6= 0, 1. In particular, we find thefollowing Hodge polygons (solid lines) and Newton polygons(dotted lines):

0 s− 1

Mα

0 s− r

Nα

This example shows that the Hodge polygon, unlike the Newtonpolygon, is ingeneralnot an isogeny invariant of theF -crystal (M,ϕ). However, the isogenyclass of anF -crystal, that is, its Newton polygon, puts restrictions onthe possibleHodge polygons:

Proposition 3.7. Let (M,ϕ) be anF -crystal. Then, its Newton-polygon lies on orabove its Hodge polygon, and both have the same startpoint and endpoint:

NwtM (t) ≥ HdgM (t) for all t ∈ [0, rankM ], andNwtM (t) = HdgM (t) for t0 = 0 andt = rankM.


3.3. F-crystals arising from geometry. Now, we link the Hodge polygon of anF -crystal that arises from crystalline cohomology of a variety to the Hodge num-bers of that variety, which justifies the terminology: letX be a smooth and propervariety overk and fix an integern ≥ 0. Then, we consider the Hodge numbers

hi := hi,n−i = dimkHn−i(X,Ωi

X/k) for all 0 ≤ i ≤ n

and, as before with the Hodge polygon, we construct from these integers a piece-wise linear function

Hdgn

X :[0, hndR

]→ R, where hndR =

n∑

i=0

hi,

and whose associated convex polygon is called thegeometric Hodge polygon.The following deep and important result shows that under extra hypotheses the

F -crystal associated to the crystalline cohomology of a smooth and proper varietydetects its Hodge numbers. In fact, part of this result just rephrases Theorem 1.9 interms of Hodge polygons.

Theorem 3.8(Mazur, Nygaard, Ogus). LetX be a smooth and proper variety overa perfect fieldk of positive characteristic. Fix an integern ≥ 0 and let

Hn := (Hncris(X/W )/torsion, ϕ)

be the associatedF -crystal. Then

(1) For all t ∈ [0, rankHn], we have

NwtHn(t) ≥ Hdgn

X(t).

(2) If Hncris(X/W ) is torsion-free, and if the Frolicher spectral sequence ofX

degenerates atE1, then for allt ∈ [0, rankHn], we have

HdgHn(t) = Hdgn

X(t).

In particular,Hn computes all Hodge numbershi,n−i ofX.

The following exercise shows that there are restrictions onthe slopes ofF -crystals that arise as crystalline cohomology of varieties.

Exercise 3.9.LetX be smooth and proper variety of dimensiond, and let(Hn, ϕ),n = 0, ..., 2d be theF -crystalsHn

cris(X/W )/torsion as above.

(1) Using Poincare duality, show thatϕ ϕ∨ = pd · id for all n ≥ 0, anddeduce that the slopes ofHn lie inside the interval[0, d].

(2) Use the hard Lefschetz theorem together with Poincare duality to show thatthe slopes ofHn lie inside the interval

[0, n] if 0 ≤ n ≤ d[n− d, d] if d ≤ n ≤ 2d .

We refer to [Ka79] for more about crystals and their slopes.


3.4. Abelian varieties. LetA be an Abelian variety of dimensiong over an alge-braically closed fieldk of positive characteristic. As already mentioned in Exam-ples 1.8, there exist isomorphisms

Hncris(A/W ) ∼= ΛnH1

cris(A/W ) for all n ≥ 0.

In fact, these isomorphisms are compatible with Frobenius actions on both sides,and thus, are isomorphisms ofF -crystals. In particular, it suffices to understand theF -crystalH1

cris(A/W ), which is a freeW -module of rank2g. Thus,H∗cris(A/W )

is torsion-free and since the Frolicher spectral sequenceof A degenerates atE1

(see Section 1.2), the assumptions of Theorem 3.8 are fulfilled. Let us now discussthe two casesg = 1 andg = 2 in greater detail.

3.4.1. Elliptic curves. If A is an elliptic curve, that is,g = 1, then its Hodgepolygon is given by the solid polygon

0 1 2

For the Newton polygon, there are two possibilities:

(1) The Newton polygon is equal to the Hodge polygon, and in this case,Ais calledordinary. It follows from the results of Examples 1.8 thatA isordinary if and only ifA[p](k) ∼= Z/pZ.

(2) The Newton polygon is equal to the dotted line, and in thiscase,A is calledsupersingular. This case is equivalent toA[p](k) = 0.

By a result of Deuring, there are roughlyp/12 supersingular elliptic curves overan algebraically closed field of positive characteristicp, whereas all the other onesare ordinary (see Theorem 8.3 for a similar count for K3 surfaces). In particular,a generic elliptic curve in positive characteristic is ordinary, that is, Newton andHodge polygon ofH1

cris(A/W ) coincide. We refer the interested reader to [Har77,Chapter IV.4] and [Si86, Chapter V] for more results, reformulations, and back-ground information on ordinary and supersingular ellipticcurves.

3.4.2. Abelian surfaces.If A is an Abelian surface, that is,g = 2, then its Hodgepolygon is given by the solid polygon

0 2 4

For the Newton polygon, there are now three possibilities:

(1) The Newton polygon is equal to the Hodge polygon, that is,A is ordinary,or, equivalently,A[p](k) ∼= (Z/pZ)2.

(2) The Newton polygon has three slopes (lower dotted line),or, equivalently,A[p](k) ∼= Z/pZ.

(3) The Newton polygon has only one slope (upper dotted line), that is,A issupersingular, or, equivalently,A[p](k) = 0.

We refer to [Ill79a, Exemples II.7.1] for details and further results.


3.5. K3 surfaces. LetX be a K3 surface overk. In Section 2.2, we computed thecohomology groups of a K3 surface. In particular, the only interesting crystallinecohomology group isH2

cris(X/W ), which is free of rank22. Moreover, in loc. cit.we also computed the Hodge numbers

h0 := h0,2 = 1, h1 := h1,1 = 20, and h2 := h2,0 = 1

from which we obtain the geometric Hodge polygon. In Section2.2, we havealso seen that the crystalline cohomology groups ofX have nop-torsion, and thatthe Frolicher spectral sequence degenerates atE1. Thus, by Theorem 3.8, thegeometric Hodge polygon ofX coincides with the Hodge polygon of theF -crystalH2

cris(X/W ).

Exercise 3.10.For a K3 surfaceX, show that there are12 possibilities for theNewton polygon of theF -crystalH2

cris(X/W ):

(1) The Newton polygon has three slopes and multiplicities as follows:

slope 1− 1h 1 1 + 1

hmultiplicity h 22 − 2h h

whereh is an integer with1 ≤ h ≤ 11. In caseh = 1, Hodge and Newtonpolygon coincide, and then,X is calledordinary.

(2) The Newton polygon is of slope1 only (upper dotted line), and then,X iscalledsupersingular. In this case we seth =∞.

A discussion and details can be found in [Ill79a, Section II.7.2].

SinceX is projective, there exists an ample line bundleL ∈ Pic(X) and wewill see in Section 4.2 that theW -module generated by the Chern classc1(L)insideH2

cris(X/W ) gives rise to anF -crystal of slope1 (see also Exercise 6.18).In particular, this shows that the caseh = 11 in (1) of Exercise 3.10 cannot occur.

In Section 6.2, we will define the formal Brauer group of a K3 surface. In Propo-sition 6.17, we will see that the parameterh from Exercise 3.10 can be interpretedas the height of the formal Brauer group. And eventually, in Section 8, we will seethath gives rise to a stratification of the moduli spacesM

2d,Fpfrom Section 2.4,

and that this stratification can also be interpreted in termsof F -zips (see Definition1.3).

4. SUPERSINGULARK3 SURFACES

For a complex K3 surfaceX, the de Rham cohomology groupH2dR(X/C)

comes with an integral and a real structure, as well as two filtrations (Section1.1). Moreover, Poincare duality equipsH2

dR(X/C) with a non-degenerate bi-linear form. This linear algebra data is captured in the notion of apolarized Hodgestructure of weight2, and such data is parametrized by theirperiod domain, whichis an open subset (with respect to the analytic topology) inside some Zariski-closedset of some Graßmannian.

In the previous section, we associated to a K3 surfaceX over a perfect fieldk of positive characteristic theF -crystal (H,ϕ) arising fromH2

cris(X/W ). Thisis a module overW = W (k), which may be thought of as an integral structure.


By Theorems 1.9 and 3.8, the two Hodge filtrations ofH2dR(X/k)

∼= H/pH areencoded in(H,ϕ). Poincare duality, which also exists for crystalline cohomology(Section 1.5), equips(H,ϕ) with a non-degenerate bilinear form. This resultingstructure(H,ϕ, 〈−,−〉) is called aK3 crystal, and should be thought of as thecharacteristic-p version of a polarized Hodge structure of weight2 arising from aK3 surface.

Following Ogus [Og79], we will only construct a period domain for supersingu-lar K3 crystals. One crucial technical point is that this will bea Zariski closed sub-set of some Graßmannian and thus, projective overk. In general, I would expectperiod domains for non-supersingular K3 crystals to be opensubsets of Zariski-closed sets of Graßmannians, where open might also be in the sense of Tate’s rigidanalytic spaces or Berkovich spaces.

4.1. K3 crystals. We start by introducing K3 crystals and their Tate modules, andshortly digress on the Tate conjecture.

Definition 4.1 (Ogus). Let k be a perfect field of positive characteristicp and letW = W (k) be its Witt ring. AK3 crystal of rankn overk is a freeW -moduleHof rankn together with aσ-linear injective mapϕ : H → H (that is,(H,ϕ) is anF -crystal), and a symmetric bilinear form

〈−,−〉 : H ⊗W H → W

such that

(1) p2H ⊆ im(ϕ),(2) ϕ⊗W k has rank1,(3) 〈−,−〉 is a perfect pairing,(4) 〈ϕ(x), ϕ(y)〉 = p2σ〈x, y〉.

The K3 crystal is calledsupersingular, if moreover

(5) theF -crystal(H,ϕ) is purely of slope1.

Example 4.2. LetX be a K3 surface overk. By Example 3.2,H := H2cris(X/W )

with Frobeniusϕ is anF -crystal. By the results of Section 2.2, it is of rank22.

(1) By Exercise 3.9, all slopes ofH lie in the interval[0, 2], which can alsobe seen from the detailed classification in Exercise 3.10. This implies thatcondition (1) of Definition 4.1 is fulfilled.

(2) Poincare duality equipsH with a symmetric bilinear pairing〈−,−〉, whichsatisfies conditions (3) and (4) of Definition 4.1 by general properties ofPoincare duality.

(3) SinceX is a K3 surface, we haveh0 := h2(OX ) = 1. By Theorem 3.8,we haveh0 = 1 for the Hodge polygon ofH. This implies that condition(2) of Definition 4.1 holds true.

Thus,(H,ϕ, 〈−,−〉) is a K3 crystal of rank22. It is supersingular if and only ifits Newton polygon is a straight line of slope one, which corresponds to case (2) inExercise 3.10, that is,h =∞.


Exercise 4.3.Let A be Abelian surface, that is, an Abelian variety of dimension2. Show that Frobenius and Poincare duality turnH2

cris(A/W ) into a K3 crystalof rank 6. We refer the interested reader to [Og79, Section 6], where crystalsarising from (supersingular) Abelian varieties are discussed in general. For Abeliansurfaces, it turns out that these crystals are closely related to K3 crystals of rank6,see [Og79, Proposition 6.9].

4.2. The Tate module. If X is a smooth and proper variety overk, then thereexists a crystalline Chern class map

c1 : Pic(X) → H2cris(X/W ) .

Being a homomorphism of Abelian groups,c1 satisfies for allL ∈ Pic(X)

c1(F∗(L)) = c1(L

⊗p) = pc1(L),

whereF : X → X denotes the absolute Frobenius morphism. In particular,c1(Pic(X)) is contained in the Abelian subgroup (in fact,Zp-submodule) of theF -crystal (H2

cris(X/W ), ϕ) of those elementsx that satisfyϕ(x) = px. Thisobservation motivates the following definition.

Definition 4.4. Let (H,ϕ, 〈−,−〉) be a K3 crystal. Then, theTate moduleof H isdefined to be theZp-module

TH := x ∈ H |ϕ(x) = px .

Thus, by our computation above, we havec1(NS(X)) ⊆ TH , and it is natural toask whether this inclusion is in fact an equality, or, at least up top-torsion. IfX isdefined over a finite field, this is the content of theTate conjecture.

Conjecture 4.5(Tate [Ta65]). LetX be a smooth and proper surface over a finitefield Fq of characteristicp. Then, the following statements hold true:

(1) The first Chern class induces an isomorphism

c1 : NS(X) ⊗Z Qp∼=−→ TH ⊗Zp Qp .

(2) For every primeℓ 6= p, the first Chern class induces an isomorphism

c1 : NS(X) ⊗Z Qℓ∼=−→ H2

et

(X ×Fq Fq, Qℓ(1)

)Gal(Fq/Fq),

where the right hand side denotes invariants under the Galois action.(3) The rank ofNS(X) is the pole order of the zeta functionZ(X/Fq, T ) at

T = q−1.

The equivalences of (1), (2), and (3) follow from the Weil conjectures, moreprecisely, from the Riemann hypothesis, which relates the zeta function toℓ-adicand crystalline cohomology, see [Har77, Appendix C] and [Li13, Section 9.10]. In[Ta66, Theorem 4], Tate proved this conjecture for Abelian varieties, as well as forproducts of curves. For K3 surfaces, it was established in several steps dependingon the slopes of theF -crystalH2

cris - in terms of the notations of Exercise 3.10: forh = 1 by Nygaard [Ny83a], forh < ∞ by Ogus and Nygaard [N-O85], and ingeneral by Charles [Ch13], Madapusi Pera [MP15], and Maulik[Ma14].


Theorem 4.6(Nygaard, Nygaard–Ogus, Charles, Madapusi-Pera, Maulik). Tate’sconjecture holds for K3 surfaces over finite fields of odd characteristic.

Let us mention the following, somewhat curious corollary: namely, Swinnerton-Dyer observed (see [Ar74a]) that Tate’s conjecture for K3 surfaces implies that theNeron–Severi rank of a K3 surface overFp is even. This was used in [B-H-T11]and [L-L12] to show that every K3 surface ofodd Neron–Severi rank containsinfinitely many rational curves, and we refer to [Li13] and [Be14] for an overview.

4.3. Supersingular K3 surfaces. Let us now discuss supersingular K3 crystals ingreater detail, that is, K3 crystals that are of slope1 only. It turns out that they arelargely determined by their Tate-modules. In case a supersingular K3 crystal arisesasH2

cris of a K3 surface, the Tate conjecture predicts that the surface has Picardrank22, that is, the K3 surface isShioda-supersingular.

First, let us recall a couple of facts on quadratic forms and their classification,and we refer to [Se70, Chapitre IV] for details and proofs: let R be a ring andΛ afreeR-module of finite rank together with a symmetric bilinear form

〈−,−〉 : Λ⊗R Λ → Λ .

We choose a basise1, ..., en of Λ, form the matrixG := (gij := 〈ei, ej〉)i,j , anddefine itsdiscriminantto bedet(G). A different choice of basis ofΛ changes itby an element ofR×2, and thus, the classd(Λ) of det(G) in R/(R×2) does notdepend on the choice of basis. The discriminant is zero if andonly if the form isdegenerate, that is, if there exists a0 6= v ∈ Λ such that〈v,w〉 = 0 for all w ∈ Λ.Next, we letΛ∨ := HomR(Λ, R) be the dualR-module. Viav 7→ 〈v,−〉, weobtain a natural mapΛ → Λ∨, which is injective if and only if the form is non-degenerate. In case this map is an isomorphism, which is the case if and only if thediscriminant is a unit, the form is calledperfect.

Let us now assume thatR is a DVR, say, with valuationν. The example wehave in mind is the ring ofp-adic integersZp or, more generally, the ringW (k) ofWitt vectors of a perfect fieldk of positive characteristicp, together with itsp-adicvaluationordp. Then, since units have valuation zero,

ordp(Λ) := ν (d(Λ))

is a well-defined integer, and the form is perfect if and only if ordp(Λ) = 0.Finally, we note that quadratic forms overQp are classified by their rank, their

discriminant, and their so-called Hasse invariant, see [Se70, Chapitre IV] for proofsand details. These results are the key to the following classification of Tate modulesof supersingular K3 crystals from [Og79].

Proposition 4.7(Ogus). Let (H,ϕ, 〈−,−〉) be a supersingular K3 crystal and letTH be its Tate module. Then,

rankW H = rankZp TH

and the bilinear form(H, 〈−,−〉) restricted toTH induces a non-degenerate form

TH ⊗Zp TH → Zp,

which is not perfect. More precisely,


(1) ordp(TH) = 2σ0 > 0 for some integerσ0, called theArtin invariant.(2) (TH , 〈−,−〉) is determined up to isometry byσ0.(3) rankW H ≥ 2σ0.(4) There exists an orthogonal decomposition

(TH , 〈−,−〉) ∼= (T0, p〈−,−〉) ⊥ (T1, 〈−,−〉)

whereT0 andT1 areZp-lattices, whose bilinear forms are perfect, and ofranksrankT0 = 2σ0 andrankT1 = rankW H − 2σ0.

Combining this proposition with the Tate conjecture (Theorem 4.6), we obtaina characterization of those K3 surfaces whose associated K3crystal is supersingu-lar. Namely, let us recall from Section 2.2 that the second crystalline cohomologygroup of a K3 surface is a freeW -module of rank22. Using that the first crystallineChern map is injective, this shows that the rank of the Neron–Severi group of a K3surface can be at most22. This said, we have the following result.

Theorem 4.8. Let X be a K3 surface over an algebraically closed field of oddcharacteristic. Then, the following are equivalent

(1) The K3 crystalH2cris(X/W ) is supersingular.

(2) The Neron–Severi groupNS(X) has rank22.

PROOF. If NS(X) has rank22, then c1(NS(X)) ⊗Z W is a sub-F -crystal ofH2

cris(X/W ) of slope1, thereby establishing(2) ⇒ (1). Conversely, assume thatH := H2

cris(X/W ) is a supersingularF -crystal. By Proposition 4.7, the Tate mod-ule ofH has rank22. If X can be defined over a finite field of odd characteristicthen Theorem 4.6 implies thatNS(X) is of rank22. If X is not definable over afinite field, then there exists some varietyB over some finite fieldFq, such thatXis definable over the function field ofB. Spreading outX overB and passing toan open and dense subset ofB if necessary, we may assume thatX is the genericfiber of a smooth and projective familyX → B of K3 surfaces overFq. SinceHis supersingular, all fibers in this family also have supersingularH2

cris by [Ar74a,Section 1], and since the Neron–Severi rank is constant in families of K3 surfaceswith supersingularH2

cris by [Ar74a, Theorem 1.1], this establishes the conversedirection(1)⇒ (2).

Remark 4.9. K3 surfaces satisfying (1) are calledArtin-supersingular, see [Ar74a],where it is formulated in terms of formal Brauer groups, a point of view that wewill discuss in Section 6 below. K3 surfaces satisfying (2) are calledShioda-supersingular, see [Sh74a]. In view of the theorem, a K3 surface in odd char-acteristic satisfying (1) or (2) is simply calledsupersingular.

Examples 4.10.Let us give examples of supersingular K3 surfaces.

(1) LetA be a supersingular Abelian surface in odd characteristic (see Section3.4). Then, the Kummer surfaceX := Km(A) of A is a supersingularK3 surface. Letσ0 be the Artin invariant of the Tate moduleTH of the


supersingular K3 crystalH := H2cris(X/W ). Then,

σ0(TH) =

1 if A = E × E, whereE is a supersingularelliptic curve (the isomorphism class ofAdoes not depend on the choice ofE),

2 else,

see [Og79, Theorem 7.1 and Corollary 7.14], or [Sh79, Proposition 3.7 andTheorem 4.3]. Conversely, by loc. cit, every supersingularK3 surface inodd characteristic withσ0 ≤ 2 is the Kummer surface of a supersingularAbelian surface.

(2) The Fermat quartic

X4 := x40 + x41 + x42 + x43 = 0 ⊂ P3k

defines a K3 surface in characteristicp 6= 2 and it is supersingular if andonly if p ≡ 3 mod 4 by [Sh74a, Corollary to Proposition 1]. Moreover, ifX4 is supersingular, then it hasσ0(TH) = 1 by [Sh79, Example 5.2], andthus, it is a Kummer surface by the previous example.

We note that supersingular Kummer surfaces form a1-dimensional family, whereasall supersingular K3 surfaces form a9-dimensional family - we refer to Section 5for moduli spaces.

In view of Theorem 4.8, we now identify the Neron–Severi lattices arising fromsupersingular K3 surfaces abstractly, and classify them interms of discriminants,which gives rise to the Artin invariant of such a lattice.

Definition 4.11. A supersingular K3 latticeis a free Abelian groupN of rank22with an even symmetric bilinear form〈−,−〉 with the following properties

(1) The discriminantd(N ⊗Z Q) is−1 in Q∗/Q∗2.(2) The signature of(N ⊗Z R) is (1, 21).(3) The cokernel ofN → N∨ is annihilated byp.

We note that we follow [Og83, Definition 1.6], which is slightly different from[Og79, Definition 3.17] (in the latter article, it is stated for Zp-modules rather thanZ-modules). Let us shortly collect the facts: by [Og83, (1.6)] and the referencesgiven there, the Neron–Severi lattice of a supersingular K3 surface is a supersin-gular K3 lattice in the sense of Definition 4.11 (for example,condition (2) followsfrom the Hodge index theorem). Next, ifN is a supersingular K3 lattice, thenits discriminantd(N), which is an integer, is equal to−p2σ0 for some integer1 ≤ σ0 ≤ 10.

Definition 4.12. . The integerσ0 associated to a supersingular K3 lattice is calledtheArtin invariant of the lattice. IfX is a supersingular K3 surface, we define itsArtin invariant to be the Artin invariant of its Neron–Severi lattice.

This invariant was introduced in [Ar74a], and an important result is the follow-ing theorem, see [R-S76, Section 1] and [Og79, Section 3].

Theorem 4.13(Rudakov–Shafarevich). The Artin invariant determines a super-singular K3 lattice up to isometry.


We refer the interested reader to [R-S78, Section 1] for explicit descriptions ofthese lattices, which do exist for all values1 ≤ σ0 ≤ 10.

Before proceeding, let us shortly digress on quadratic forms over finite fields:let V be a2n-dimensional vector space over a finite fieldFq of odd characteristic.Let 〈−,−〉 : V × V → Fq be a non-degenerate quadratic form. Two-dimensionalexamples are the hyperbolic planeU , as well asFq2 with the quadratic form arisingfrom the norm. By the classification of quadratic forms over finite fields,V is iso-metric tonU or to (n− 1)U ⊥ Fq2. The form〈−,−〉 is callednon-neutralif thereexists non-dimensional isotropic subspace insideV . By the classification resultjust mentioned, there is precisely one non-neutral quadratic space of dimension2noverFq, namely,(n − 1)U ⊥ Fq2 .

Next, for a supersingular K3 lattice(N, 〈−,−〉), we setN1 := N/pN∨. Then,N1 is a (22 − 2σ0)-dimensionalFp-vector space and〈−,−〉 induces a quadraticform onN1, which is non-degenerate and non-neutral. The form〈−,−〉 onpN∨ ⊆N is divisible byp and dividing it bypwe obtain a non-degenerate and non-neutralbilinear form on the2σ0-dimensionalFp-vector spaceN0 := pN∨/pN . We referto [Og83, (1.6)] for details. In Section 4.4 below, we will use theseFp-vectorspaces to classify supersingular K3 crystals explicitly aswell as to construct theirmoduli spaces - the point is that it is easier to deal withFp-vector spaces ratherthanZ- or Zp- lattices.

Finally, for a supersingular K3 latticeN , we setΓ := N ⊗Z Zp and denote theinduced bilinear form onΓ again by〈−,−〉. Then, we haveordp(Γ) = 2σ0. By[Og79, Lemma 3.15], non-neutrality of the form induced onN0 is equivalent to theHasse invariant ofΓ being equal to−1. Moreover, since the cokernel ofN → N∨

is annihilated byp, the same is true forΓ→ Γ∨, and thus, by [Og79, Lemma 3.14],we obtain an orthogonal decomposition

(Γ, 〈−,−〉) ∼= (Γ0, p〈−,−〉) ⊥ (Γ1, 〈−,−〉),

whereΓ0 andΓ1 are perfectZp-lattices of ranks2σ0 and22 − 2σ0, respectively.In particular,Γ satisfies the conditions of a supersingular K3 lattice overZp asdefined in [Og79, Definition 3.17]. We refer to [Og79, Corollary 3.18] for detailsabout the classification of supersingular K3 lattices overZp up to isogeny and upto isomorphism.

4.4. Characteristic subspaces.In order to classify supersingular K3 crystals, wenow describe them in terms of so-called characteristic subspaces, and then, classifythese latter ones. For a supersingular K3 surface, this characteristic subspace arisesfrom the kernel of the de Rham Chern classc1 : NS(X)→ H2

dR(X/k). (Note thatin characteristic zero,c1 is injective modulo torsion.) These considerations stressyet again the close relation between crystals and de Rham cohomology.

Definition 4.14. Let σ0 ≥ 1 be an integer, letV be a2σ0-dimensionalFp-vectorspace withp 6= 2, and let

〈−,−〉 : V × V → Fp


be a non-degenerate and non-neutral quadratic form. Next, let k be a perfect fieldof characteristicp and setϕ := idV ⊗ Fk : V ⊗Fp k → V ⊗Fp k. A subspaceK ⊂ V ⊗Fp k is calledcharacteristicif

(1) K is totally isotropic of dimensionσ0, and(2) K + ϕ(K) is of dimensionσ0 + 1.

Moreover, a characteristic subspaceK is strictly characteristicif moreover

(3)

V ⊗Fp k =

∞∑

i=0

ϕi(K)

holds true.

For a perfect fieldk of odd characteristic, we define the categories

K3 (k) :=

category of supersingular K3 crystalswith only isomorphisms as morphisms

.

and

C3 (k) :=

category of pairs(T,K), whereT is a supersingularK3 lattice overZp, and whereK ⊂ T0 ⊗Zp k is astrictly characteristic subspace,

with only isomorphisms as morphisms

.

Finally, we defineC3(k)σ0to be the subcategory ofC3(k), whose characteristic

subspaces areσ0-dimensional.

Theorem 4.15(Ogus). Letk be an algebraically closed field of odd characteristic.Then, the assignment

K3 (k)γ−→ C3 (k)

(H,ϕ, 〈−,−〉) 7→(TH , ker(TH ⊗Zp k → H ⊗Zp k) ⊂ T0 ⊗Zp k

)

defines an equivalence of categories.

The mapγ from the theorem has the following geometric origin, and we referto [Og83, Section 2] for details: ifX is a supersingular K3 surface, thenH :=H2

cris(X/W ) is a supersingular K3 crystal. Moreover, the Tate moduleTH is asupersingular K3 lattice overZp, the first Chern classc1 identifiesNS(X) ⊗Z Zp

with TH , and the characteristic subspace associated toH arises from the kernel ofc1 : NS(X)⊗Z k → H2

dR(X/k).

Now, we describe and classify characteristic subspaces over an algebraicallyclosed fieldk of odd characteristicp explicitly, and we refer to [Og79, p. 33-34]for technical details: letV be a2σ0-dimensionalFp-vector space with a non-neutralform 〈−,−〉, let ϕ = id ⊗ Fk : V ⊗Fp k → V ⊗Fp k, and letK ⊂ V ⊗Fp k be astrictly characteristic subspace. Then,

ℓK := K ∩ ϕ(K) ∩ ... ∩ ϕσ0−1(K)

is a line insideV ⊗Fp k. We choose a basis element0 6= e ∈ ℓK and set

ei := ϕi−1(e) for i = 1, ..., 2σ0.


Then, theei form a basis ofV ⊗Fp k. We have〈e, eσ0+1〉 6= 0, and changinge bya scalar if necessary (here, we use thatk is algebraically closed), we may assume〈e, eσ0+1〉 = 1. We note that this normalization makese unique up to a(pσ0+1).throot of unity. Then, we define

ai := ai(e, V,K) := 〈e, eσ0+1+i〉 for i = 1, ..., σ0 − 1 .

If ζ is a (pσ0 + 1).th root of unity, then, replacinge by ζe, transforms theai asai 7→ ζ1−piai. This said, we denote byµn the group scheme ofn.th roots of unity,and then, we have the following classification result.

Theorem 4.16(Ogus). Letk be an algebraically closed field of odd characteristic.Then, there exists a bijection

C3 (k)σ0→ Aσ0−1

k (k)/µpσ0+1(k)K 7→ (a1, ..., aσ0−1)

where theai := ai(e, V,K) are as defined above.

Having described characteristic subspaces over algebraically closed fields, wenow study them in families.

Definition 4.17. Let (V, 〈−,−〉) be a2σ0-dimensionalFp-vector space with a non-neutral quadratic form. IfA is anFp-algebra, ageneatrixof V ⊗Fp A is a directsummand

K ⊂ V ⊗Fp A

of rankσ0 such that〈−,−〉 restricted toK vanishes identically. We define the setof geneatrices

GenV (A) :=

generatrices ofV ⊗Fp A

as well as

MV (A) := K ∈ GenV (A),K + F ∗A(K) is a direct summand of rankσ0 + 1 ,

which is the set of characteristic generatrices.

Proposition 4.18(Ogus). The functor fromFp-algebras to sets given by

A 7→ MV (A)

is representable by a schemeMV , which is smooth, projective, and of dimensionσ0 − 1 overFp.

Let N be a supersingular K3 lattice with Artin invariantσ0. At the end ofSection 4.3 we setN0 := pN∨/pN and noted that it is a2σ0-dimensionalFp-vector space that inherits a non-degenerate and non-neutral bilinear form fromN .We setMN :=MN0

.

Definition 4.19.MN is called themoduli space ofN -rigidified K3 crystals.

Examples 4.20.If V is 2σ0-dimensional, then

(1) If σ0 = 1, thenMV∼= Spec Fp2.

(2) If σ0 = 2, thenMV∼= P1

Fp2

.


(3) If σ0 = 3, thenMV is isomorphic to the Fermat surface of degreep+ 1 inP3Fp2

.

We refer to [Og79, Examples 4.7] for details, as well as to Theorem 7.9 for ageneralization to higher dimensionalV ’s.

Anticipating the crystalline Torelli theorem in Section 5,let us comment on theσ0 = 1-case and give a geometric interpretation: then, we haveMV

∼= Spec Fp2.By Theorem 5.5 or Examples 4.10, there exists precisely one supersingular K3surface withσ0 = 1 up to isomorphism over algebraically closed fields of oddcharacteristic. More precisely, this surface is the KummersurfaceKm(E × E),whereE is a supersingular elliptic curve. Although this surface can be definedoverFp, there is no modelX overFp such that all classes ofNS(XFp

) are alreadydefined overFp. Models with full Neron–Severi group do exist overFp2 - but then,there is a non-trivial Galois-action ofGal(Fp2/Fp) on NS(XF

p2). This explains

(via the crystalline Torelli theorem) the Galois action onMV , as well as the factthatMV ×Fp Fp consists of two points, whereas it corresponds to only one surface.

5. OGUS’ CRYSTALLINE TORELLI THEOREM

We now come to the period map and the crystalline Torelli theorem for super-singular K3 surfaces. To state it, we fix a primep ≥ 5 and a supersingular K3latticeN as in Definition 4.11. Then, there exists a moduli spaceSN of N -markedsupersingular K3 surfaces, which is a scheme that is locallyof finite type, almostproper, and smooth of dimensionσ0(N)− 1 overFp.

Associating to anN -marked supersingular K3 surfaceX theF -zip associatedtoH2

dR(X/k) yields a morphism

πmod pN : SN → F

τ

(notation as at the end of Section 1.2). However,Fτ is a rather discrete object, itessentially only remembers the Artin invariantσ0(X), and thus,πmodp

N is more ofa “modp shadow” of the saught-after period map.

Associating to anN -marked supersingular K3 surface theN -rigidified K3 crys-tal associated toH2

cris(X/W ) yields a morphism

πN : SN → MN

(see Definition 4.19), which is locally of finite type, etale, and surjective, but it isnot an isomorphism.

If we equipN -rigidified K3 crystals with ample cones, we obtain a new modulispacePN that comes with a forgetful morphismPN →MN , and then,πN lifts toa morphism

πN : SN → PN ,

the period map. By Ogus’ crystalline Torelli theoremfor supersingular K3 sur-faces, the period map is an isomorphism.


5.1. Moduli of marked supersingular K3 surfaces. LetN be a supersingular K3lattice in characteristicp as in Definition 4.11, and letσ0 be its Artin invariant as inDefinition 4.12. For an algebraic spaceS overFp, we denote byNS the constantgroup algebraic space defined byN overS. Then, we consider the functorSN ofN -marked K3 surfaces(

Algebraic spacesoverFp

)→ (Sets)

S 7→

smooth and proper morphismsf : X → Sof algebraic spaces, each of whose geometricfibers is a K3 surface, together with amorphism of group spacesNS → PicX/S

compatible with intersection forms.

Since anN -marked K3 surfaceX → S has no non-trivial automorphisms [Og83,Lemma 2.2], it is technical, yet straight forward to prove that this functor can berepresented by an algebraic space [Og83, Theorem 2.7]. It follows a posteriorifrom the crystalline Torelli theorem [Og83, Theorem III’] that this algebraic spaceis a scheme.

Theorem 5.1(Ogus). The functorSN is represented by a schemeSN , which islocally of finite type, almost proper, and smooth of dimension σ0(N)− 1 overFp.

Here, a scheme is calledalmost properif satisfies the surjectivity part of thevaluative criterion with DVR’s as test rings. However, thismoduli space isnotseparated. This non-separatedness arises from elementarytransformations, whichis analogous to degenerations of complex Kahler K3 surfaces. We refer to [B-R75,Section 7], [Mo83], and [Og83, page 380] for details and further discussion.

5.2. The period map. Let (H,ϕ) be the K3 crystal associated toH2cris(X/W ) of

a K3 surfaceX over a perfect fieldk of positive characteristicp. Moreover, ifXisN -marked, then the inclusionN → NS(X) composed with the first crystallineChern map yields a mapN → TH , whereTH denotes the Tate module of the K3crystal. Thus, anN -marked K3 surface gives rise to anN -rigidified supersingularK3 crystal, which gives rise to a morphism of schemes

π : SN → MN

(we refer to [Og79, Section 5] for families of crystals). Although this morphism isetale and surjective by [Og83, Proposition 1.16], it is notan isomorphism. In orderto obtain an isomorphism (the period map), we have to enlargeMN by consideringN -marked supersingular K3 crystals together withample cones.

Definition 5.2. Let N be a supersingular K3 lattice. Then, we define itsroots tobe the set

∆N := δ ∈ N | δ2 = −2 .

For a rootδ ∈ ∆N , we define thereflection in δ to be the automorphism ofNdefined by

rδ : x 7→ x+ 〈x, δ〉 · δ for all x ∈ N.


We denote byRN the subgroup ofAut(N) generated by allrδ, δ ∈ ∆N . Wedenote by±RN the subgroup ofAut(N) generated byRN and±id. Finally, wedefine the set

VN :=x ∈ N ⊗ R |x2 > 0 and〈x, δ〉 6= 0 for all δ ∈ ∆N

⊂ N ⊗ R.

Then, the subsetVN ⊂ N ⊗ R is open, and each of its connected componentsmeetsN . A connected component ofVN is called anample cone, and we denoteby CN the set of ample cones. Moreover, the group±RN operates simply andtransitively onCN . We refer to [Og83, Proposition 1.10] for details and proof.

Definition 5.3. LetN be a supersingular K3 lattice, and letS be an algebraic spaceoverFp. For a characteristic geneatrixK ∈ MN (S), that is, a local direct factorK ⊂ OS ⊗N0 as in Definition 4.17, we set for each points ∈ S

Λ(s) := N0 ∩K(s)N(s) := x ∈ N ⊗Q | px ∈ N andpx ∈ Λ(s)∆(s) := δ ∈ N(s) | δ2 = −2

An ample conefor K is an element

α ∈∏

s∈S

CN(s)

such thatα(s) ⊆ α(t) whenevers is a specialization oft.

Having introduced these definitions, we consider the functor PN(Algebraic spaces

overFp

)→ (Sets)

S 7→

characteristic spacesK ∈ MN (S)together with ample cones

There is a natural forgetful mapPN →MN , given by forgetting the ample cones.Then, we have the following result, and refer to [Og83, Proposition 1.16] for detailsand proof.

Theorem 5.4(Ogus). The functorPN is represented by a schemePN , which islocally of finite type, almost proper, and smooth of dimension σ0(N) − 1 overFp.The natural map

PN → MN

is etale, surjective, and locally of finite type.

We repeat that the morphismPN →MN is neither of finite type nor separated,whereasMN is smooth, projective and of finite type overFp by Proposition 4.18.

Now, for an algebraic spaceB over Fp and a familyX → B of N -markedK3 surfaces, that is, an element ofSN (B), we have an associated family ofN -rigidified supersingular K3 crystals, that is, an element ofMN (B). This gives riseto a morphism

πN : SN → MN ,

which is surjective, but not an isomorphism. Now, for every point b ∈ B, there isa unique connected component ofVNS(Xb)⊗R that contains the classes of all ample


invertible sheaves ofXb, thereby equipping the family of K3 crystals with amplecones. This induces a morphism

πN : SN −→ PN

that liftsπN from above, and is called theperiod map. By [Og83, Theorem III], itis an isomorphism.

Theorem 5.5(Ogus’ Crystalline Torelli Theorem). LetN be a supersingular K3lattice in characteristicp ≥ 5. Then, the period mapπN is an isomorphism.

Since ample cones are sometimes inconvenient to handle, andsince they arealso responsible forPN being neither separated nor of finite type, let us note thefollowing useful application of the crystalline Torelli theorem: if two supersingularK3 surfaces have isomorphic K3 crystals, then they correspond via πN to pointsin the same fiber ofPN → MN . In particular, the two surfaces are abstractlyisomorphic, and we obtain the following result, and refer to[Og83, Theorem 1] fordetails.

Corollary 5.6. Two supersingular K3 surfaces in characteristicp ≥ 5 are isomor-phic if and only if their associated K3 crystals are isomorphic.

Theorem 5.5 is the main result of [Og83]. Let us roughly sketch its proof: Theexistence of the period mapπN is clear. Separatedness ofπN follows from atheorem of Matsusaka and Mumford [M-M64]. Properness ofπN follows froma theorem of Rudakov and Shafarevich [R-S82] that supersingular K3 surfaceshave potential good reduction, that is, given a supersingular K3 surfaceX overK := k((t)), there exists a finite extensionR′ ⊇ R := k[[t]], say with field offractionsK ′, and a smooth model ofX ×K K ′ overR′ (this result uses thatXis supersingular – in general, K3 surfaces do not have potential good reduction).Next, πN is etale, which eventually follows from the fact thatπN : SN →MN isetale [Og79, Theorem 5.6], which in turn rests on the description of its derivative[Og79, Corollary 5.4]. Finally, to prove thatπN is an isomorphism, it suffices tofind one pointζ ∈ PN such thatπ−1

N (ζ) consists of a single point - this is doneby takingζ to be the supersingular K3 surface that is the Kummer surfacefor theself-product of a supersingular elliptic curve. We refer to[Og83, Section 3] fordetails.

6. FORMAL GROUP LAWS

In this section, we introduce formal group laws, which, at first sight, looks ratherindependent from what we studied so far. Before explaining,why this is not so,let us first give the prototype of such an object: letG be a group scheme, say, offinite type and smooth over a fieldk. If (OG,O,m) is the local ring at the neutralelementO ∈ G, then(m/m2)∨ yields the Lie algebrag, which captures first orderinfinitesimal information ofG aroundO. Using the group structure, the formalcompletion ofG alongO

G := Spf lim←−OG,O/m

n


becomes a group object in the category of formal schemes, andthis formal grouplaw lies somewhere betweeng andG. We will see that commutative formal grouplaws can be classified via theirCartier–Dieudonne modules, which areW (k)-modules that resembleF -crystals.

For example, ifX is a smooth and proper variety overk, then the Picard schemePicX/k is a group scheme overk, whose formal completion along the origin iscalled theformal Picard scheme. If k is perfect of positive characteristic, then theCartier–Dieudonne module of the formal Picard scheme determinesH1

cris(X/W ).As shown by Artin and Mazur [A-M77], the formal Picard group is just then =

1-case of a whole series of formal group lawsΦnX/k that arise from cohomological

deformation functors, and we refer to Section 6.2 for precise definitions.For us,Φ2

X/k, which is called theformal Brauer group, will be most relevant

in the sequel: its Cartier–Dieudonne module controls the part ofH2cris(X/W ) that

is of slope less than1. What makes this formal group law so fascinating is thatdespite its appearance it does in generalnot arise as formal completion of somegroup scheme associated toX. Later, in Section 7, we will use the formal Brauergroup to construct non-trivial deformations of supersingular K3 surfaces, whichultimately proves their unirationality.

6.1. Formal group laws. We start with a short introduction to commutative for-mal group laws and their classification, and refer to [Ha78] for the general theoryof formal group laws, and especially to [Zi84] for the theoryof Cartier–Dieudonnemodules.

Definition 6.1. An n-dimensionalformal group lawover a ringk consists ofnpower series~F = (F1, ..., Fn)

Fi(x1, ..., xn, y1, ...yn) ∈ k[[x1, ..., xn, y1, ..., yn]], i = 1, ..., n

such that for alli = 1, ..., n

Fi(~x, ~y) ≡ xi + yi modulo terms of degree≥ 2, andFi(~x, Fi(~y, ~z)) = Fi(Fi(~x, ~y), ~z),

where we use the notation~x to denote(x1, ..., xn), etc. A formal group law~F iscalledcommutativeif

Fi(~x, ~y) = Fi(~y, ~x) for all i = 1, ..., n.

A homomorphism~α : ~F → ~G from ann-dimensional formal group law~F toanm-dimensional formal group law~G consists ofm formal power series~α =(α1, ..., αm) in n variables such thatαi(~x) ≡ 0 mod degree1 and

αi(~x) ≡ 0 modulo terms of degree≥ 1, and~α(~F (~x, ~y)) = ~G(~α(~x), ~α(~y)).

It is called anisomorphismif there exists a homomorphism~β : ~G → ~F such that~α(~β(~x)) = ~x and~β(~α(~y)) = ~y. An isomorphism~α is calledstrict if αi(~x) ≡ ximodulo terms of degree≥ 2 for all i = 1, ..., n.


For example, for an integern ≥ 1, we definemultiplication byn to be

[n] (~x) := ~F (~F (..., ~x), ~x)︸︷︷︸n times

∈ k[[~x]] .

If ~F is a commutative formal group law, then[n] : ~F → ~F is a homomorphismof formal group laws. The group laws arising from algebraic varieties, which wewill introduce in Section 6.2 below, will all be commutative. Let us also mentionthe following result, which will not need in the sequel, and refer to [Ha78, Section6.1] for details.

Theorem 6.2. A 1-dimensional formal group law over a reduced ring is commu-tative.

Examples 6.3.Here are two basic examples of1-dimensional formal group laws.

(1) Theformal additive groupGa is defined byF (x, y) = x+ y.(2) Theformal multiplicative groupGm is defined byF (x, y) = x+ y + xy.

Both are commutative, and their names will be explained in Example 6.10.

OverQ-algebras, all commutative formal group laws of the same dimension aremutually isomorphic:

Theorem 6.4. Let ~F be ann-dimensional commutative formal group law over aQ-algebra. Then, there exists a unique strict isomorphism

log ~F : ~F (~x, ~y) → Gna ,

called thelogarithmof ~F .

Example 6.5. The logarithm of the formal multiplicative groupGm over aQ-algebrak is explicitly given by

logGm

: x 7→∞∑

n=1

(−1)n+1xn

n

which also motivates the name. Note that this power series only makes sense inrings that contain1n for all integersn ≥ 1, that is, the base ring must be aQ-algebra.

On the other hand, ifk is anFp-algebra, then Theorem 6.4 no longer holds true.In fact, there exist many1-dimensional commutative formal group laws over alge-braically closed fields of positive characteristic that arenot isomorphic toGa. Thefollowing discrete invariant is crucial - we will only defineit for 1-dimensionalformal group laws and refer to [Ha78, (18.3.8)] for its definition for higher dimen-sional formal group laws.

Definition 6.6. Let F = F (x, y) be a1-dimensional commutative formal grouplaw over a fieldk of positive characteristicp, and let[p] : F → F be multiplicationby p. Then, theheighth = h(F ) of F is defined to be

- h :=∞ in case[p] = 0, and one also says thatF is unipotent.


- Else, there exists an integers ≥ 1 and a0 6= a ∈ k, such that

[p](x) = a · xps

+ higher order terms,

in which case we seth := s. If F is of finite height, one also says that it isp-divisible.

Remark 6.7. Let us give more characterizations of the height of a1-dimensionalformal group lawF = F (x, y) over a fieldk of positive characteristicp. To do so,we defineF (p) = F (p)(x, y) to be the formal group law obtained fromF by raisingeach of the coefficients to thepth power. Then,z 7→ zp defines a homomorphismof formal group laws overk

σ : F → F (p),

which is calledFrobenius.

(1) If finite, the height is characterized as being the largest integerh such thatthere exists a power seriesβ ∈ k[[z]] with

[p](z) = β(zph

).

The seriesβ defines a homomorphismF (ph) → F , which leads to thefollowing reformulation:

(2) If finite, the height is characterized as being the largest integerh such thatthere exists a factorization

Fσh

//

[p]

22F (ph) ∃// F,

whereσh denotes theh-fold composition ofσ with itself.(3) Finally, if the height is finite, then[p] is anisogeny, and then, there exists an

integerm ≥ 0 and a homomorphism of formal group lawsψ : F → F (pm)

such thatψ [p] = σm . In particular, if finite, the height is characterizedas being the smallest integerh such that there exists a factorization

F[p]

//

σh

22F∃// F (ph).

We refer to [Ha78, Section 18.3] for details and generalizations.

Exercise 6.8.Over fields of positive characteristic, show that

h(Gm) = 1 and h(Ga) =∞.

In particular, they are not isomorphic.

The importance of the height lies in the following classifcation result.

Theorem 6.9(Lazard). Let k be an algebraically closed field of positive charac-teristic.

(1) For every integerh ≥ 1 or h = ∞ there exists a1-dimensional formalgroup law of heighth overk.


(2) Two1-dimensional formal group laws overk are isomorphic if and only ifthey have the same height.

Example 6.10. Let us show how formal group laws arise from group schemes,which also justifies some of the terminology introduced above. Thus, letG be asmooth (commutative) group scheme of dimensionn over a fieldk. Let (OG,O,m)be the local ring at the neutral elementO ∈ G. Using smoothness, we compute them-adic completion to be

OG,O := lim←−OG,O/mm ∼= k[[x1, ..., xn]] = k[[~x]] ,

and note thatG := Spf OG,O

is theformal completion ofG alongO. The multiplicationµ : G×G→ G inducesa morphism of formal schemesµ : G× G→ G that turnsG into a (commutative)group object in the category of formal schemes. Explicitly,µ corresponds to ahomomorphism ofk-algebras

µ# : k[[~x]] → k[[~y]] ⊗ k[[~z]] ∼= k[[~y, ~z]] ,

where⊗ denotes the completed tensor product. Clearly,µ# is uniquely determinedby the images of the generatorsxi, that is, by then formal power series

~G := (G1, ..., Gn) := µ#(x1, ..., xn).

It is not difficult to see that~G is ann-dimensional (commutative) formal grouplaw, and that it encodesG. Thus, G carries the information of all infinitesimalneighborhoods ofO in G, and in particular, of the tangent space atO, that is, theLie algebrag of G. Put a little bit sloppily,G lies betweenG andg. Here are somestandard examples

(1) The completion of the multiplicative group schemeGm∼= Spec k[x, x−1]

is the formal multiplicative group lawGm.(2) The completion of the additive group schemeGa

∼= Speck[x] is the formaladditive group lawGa.

(3) If E is an elliptic curve over a fieldk, then the completionE is a com-mutative1-dimensional formal group law. Ifk is of positive characteristic,then its height is equal to

h(E) =

1 if E is ordinary, and2 if E is supersingular.

Thus, if k is algebraically closed andE is ordinary, thenE ∼= Gm. Werefer to [Si86, Chapter IV] for more about this formal group law.

In order to classify1-dimensional commutative formal group laws over perfectfieldsk of positive characteristic that are not algebraically closed, or even higherdimensional commutative formal group laws overk, the height is not sufficient. Tostate the general classification result, which is in terms ofmodules over some non-commutative ringCart(k), we first have to define this ring. Be definition,Cart(k)


is the non-commutative ringW (k)〈〈V 〉〉〈F 〉 of power series inV and polynomialsin F modulo the relations

FV = p, V rF = V (r), F r = σ(r)F, rV = V σ(r) for all r ∈W (k),

whereσ(r), V (r) ∈W (k) denote Frobenius and Verschiebung ofW (k).

Theorem 6.11(Cartier–Dieudonne). Letk be a perfect field of positive character-istic. Then, there exists a covariant equivalence of categories between

(1) The category of commutative formal group laws overk.(2) The category of leftCart(k)-modulesM such that

(a) V is injective,(b) ∩iV i(M) = 0, that is,M is V -adically separated, and(c) M/VM is a finite-dimensionalk-vector space.

The leftCart(k)-module associated to a formal group~G under this equivalence iscalled theDieudonne–Cartier moduleof ~G, and is denotedD~G.

Following [Mu69b, Section 1], let us sketch the direction(1) → (2) of thisequivalence: let~G = (G1, ..., Gn) be ann-dimensional commutative formal grouplaw overk. Then,~G defines a functor

Φ ~G: (k-algebras) → (Abelian groups)

R 7→ (x1, ..., xn) ∈ Rn | eachxi nilpotent

where we define the group structure on the right by setting~x ⊕Φ~G~y := ~G(~x, ~y).

Similarly, we define the functor

Φ ~W : (k-algebras) → (Abelian groups)

R 7→

(x0, x1, ...)

∣∣∣∣xi ∈ R, eachxi nilpotent,and almost allxi = 0

using ~W := (W0,W1, ...) to define the group structure, where theWi are theWitt polynomials from Section 1.4. We note that~W is an example of an infinitedimensional formal group law. Next, we define

D~G := Homgroup functors/k(Φ ~W ,Φ ~G).

Multiplication by elements ofW (k) gives rise to endomorphisms ofΦ ~W , and onecan define Frobenius and Verschiebung forΦ ~W . Equipped with these operations,Hom(Φ ~W

,Φ ~W) becomes a non-commutativeW (k)-algebra that is isomorphic to

the opposite ring ofCart(k) defined above. In particular, this turnsD~G into a leftCart(k)-module, which turns out to satisfy the conditions in (2) of Theorem 6.11.We refer to [Mu69b, Section 1] and [Ha78, Chapter V] for details, generalizations,as well as different approaches.

Examples 6.12.Let k be an algebraically closed field of positive characteristic,and letG = G(x, y) be a commutative1-dimensional formal group law overk.

(1) If G is of finite heighth, thenDG ∼= Cart(k)/(F − V h−1), which is afreeW (k)-module of rankh.


(2) If G is of infinite height, thenG ∼= Ga, and thenDGa∼= k[[x]] with F = 0

andV (xn) = xn+1.

In particular,h is equal to the minimal number of generators ofDG asW (k)-module.

Exercise 6.13.LetG be the1-dimensional commutative formal group law of finiteheighth over an algebraically closed fieldk of positive characteristic. Show thatDG, considered as aW (k)-module with theσ-linear action defined byϕ := F isanF -crystal. Show that there exists an isomorphism ofF -crystals

DG ∼= Nα with α = (h− 1)/h,

whereNα is as in Example 3.6. In particular, it is of rankh, of slopeα = 1 − 1h ,

and has Hodge numbersh0 = 1, h1 = h− 1, andhi = 0 if i 6= 0, 1.

6.2. Formal groups arising from algebraic varieties. We now explain how Artinand Mazur [A-M77] associated formal group laws to algebraicvarieties: letX bea smooth and proper variety over a fieldk, and letn ≥ 1 be an integer. Then, weconsider the following functor from the categoryArtk of local Artiniank-algebraswith residue fieldk to Abelian groups:

ΦnX : Artk → (Abelian groups)

R 7→ ker(Hn

et(X ×k R,O×X×kR

)res−→ Hn

et(X,O×X )

)

whereO×X denotes the sheaf of invertible elements ofOX with respect to multipli-

cation (we could also writeGm). The pro-representability of this functor is studiedin [A-M77], and there, also a tangent-obstruction theory for it with tangent spaceHn(X,OX) and obstruction spaceHn+1(X,OX ) is established.

Example 6.14. The casen = 1 is easy to explain: LetR ∈ Artk. We identifyH1

et(X ×k R,O×X×kR

) with the group of invertible sheaves ofX ×k R, and then,Φ1X(R) becomes the group of invertible sheaves onX ×k R, whose restriction to

X is trivial. Thus, elements ofΦ1X(R) are in bijection to morphismsSpec R →

PicX/k, such that the closed point ofSpecRmaps to zero, that is, the class ofOX .

Thus, ifPicX/k denotes the completion of the Picard schemePicX/k along its zeroas in Example 6.10, then we obtain an isomorphism of functors

PicX/k∼= Φ1

X .

In particular,Φ1X is pro-representable by a commutative formal group law if and

only if Pic0X/k is smooth overk, that is, if and only ifPic0X/k is an Abelian variety.

In this case, it is called theformal Picard group, and it is of dimensionh1(OX).

Let us now turn toΦ2X , which classifies deformations of0 ∈ H2

et(X,O×X ). The

groupH2et(X,O

×X ) is called the(cohomological) Brauer groupof X, and we refer

to [G-S06] for its algebraic aspects, and to [Gr68a] for the more scheme-theoreticside of this group. Unlike the Picard group, there is in general no Brauer scheme,whose points parametrize elements of the Brauer group ofX. In particular,Φ2

X ,


unlike Φ1X , doesnot seem to arise as completion of some group scheme asso-

ciated toX. Nevertheless, we can still study the functorΦ2X . For example, if

h1(OX ) = h3(OX ) = 0 (this holds, for example, for K3 surfaces), thenΦ2X is

pro-representable by a commutative formal group law of dimensionh2(OX), theformal Brauer group, which is denoted

BrX := Φ2X .

In Section 7, we will use this formal group law to construct non-trivial 1-dimensionaldeformations of supersingular K3 surfaces, which is the keyto proving their unira-tionality.

For n ≥ 3, the functorsΦnX are far less understood. However, we refer the

interested reader to [G-K03] for an analysis ofΦnX in caseX is ann-dimensional

Calabi–Yau variety.

6.3. The connection to Witt vector and crystalline cohomology.In Section 1.4,we introduced Serre’s Witt vector cohomology groupsHn(WOX), which, byfunctoriality of the Witt vector construction, carry actions of Frobenius and Ver-schiebung. In particular, all these cohomology groups are left Cart(k)-modules.The following result from [A-M77, Proposition (II.2.13)] and [A-M77, Corollary(II.4.3)] links thisCart(k)-module structure to the Cartier–Dieudonne modules ofcommutative formal group laws associated to theΦn

X .

Proposition 6.15(Artin–Mazur). LetX be a proper variety over a perfect fieldkand assume thatΦn

X is pro-representable by a formal group law~F (for example,this holds true ifhn−1(OX) = hn+1(OX) = 0). Then, there exists an isomorphismof leftCart(k)-modules

D~F ∼= Hn(X,WOX ).

To link the formal group lawΦnX to crystalline cohomology, we use the slope

spectral sequence from Section 1.5. As mentioned there, it degenerates atE1 ifand only if the torsion of the Hodge–Witt cohomology groups is finitely generated.However, combining the previous proposition with Examples6.12, we see that ifΦnX∼= Ga (for example, ifn = 2 andX is a supersingular K3 surface), then

Hn(WOX) will not be finitely generated and the slope spectral sequence will notdegenerate atE1, see [Ill79a, Theoreme II.2.3.7]. On the other hand, the slopespectral sequence always degenerates atE1 modulo torsion, and from this, weobtain an isomorphism ofF -isocrystals

Hn(X,WOX)⊗W K ∼= (Hncris(X/W )⊗W K)[0,1[,

where the right hand denotes the direct sum of sub-F -isocrystals of slope strictlyless than1. (Here, a little bit of background: the point is that theHj(WΩi

X)⊗WKare finite-dimensionalK vector spaces and their sets of slopes are disjoint. Theslope spectral sequence degenerates atE1 after tensoring withK, and since itcan be made compatible with the Frobenius actions on both sides, the isogenydecomposition of theF -isocrystalHn

cris(X/W ) ⊗W K can be read off from theisogeny decomposition of theF -isocrystalsHj(WΩi

X)⊗W K, wherei+ j = n.


From this, it is not so difficult to see that allF -isocrystals of slope strictly lessthan1 in Hn

cris(X/W )⊗W K arise fromHn(WOX)⊗W K via the slope spectralsequence.)

Example 6.16. Assume thatΦnX is pro-representable by a1-dimensional formal

group law offinite height h. By Exercise 6.13 and Proposition 6.15, we haveisomorphisms

N(h−1)/h ⊗W K ∼= DΦnX ⊗W K ∼= (Hn

cris(X/W )⊗W K)[0,1[.

In particular,1− 1h is the only slope ofHn

cris(X/W ) less than1.

This example applies to the case whereX is a K3 surface, andn = 2, that is,ΦnX is the formal Brauer group. In this special case, we can say more. Before

doing so, we remind the reader that we classified the possibleslopes and Newtonpolygons ofH2

cris(X/W ) in Exercise 3.10 in terms of some parameterh.

Proposition 6.17. LetX be a K3 surface over a perfect fieldk of positive charac-teristic and leth be the height of its formal Brauer group.

(1) If h <∞, then the slopes and multiplicities ofH2cris(X/W ) are as follows

slope 1− 1h 1 1 + 1

hmultiplicity h 22 − 2h h

(2) If h =∞, thenH2cris(X/W ) is of slope1 with multiplicity 22.

In particular, h determines the Newton polygon ofH2cris(X/W ).

PROOF. By Proposition 6.15,Φ2X is pro-representable by a1-dimensional formal

group lawBrX , and leth be its height.If h =∞, thenDBrX⊗W K = 0 by Examples 6.12, and so, there are no slopes

ofH2cris(X/W ) less than1. Thus, by Exercise 3.10,H2

cris(X/W ) is of slope1 withmultiplicity 22.

If h <∞, then the considerations of Example 6.16 show that the only slope lessthan1 of H2

cris(X/W ) is equal to1− 1h . By Exercise 3.10, the Newton polygon of

H2cris(X/W ) has the stated slopes and multiplicities.

Exercise 6.18.LetX be a K3 surface over an algebraically closed fieldk of pos-itive characteristic, leth be the height of its formal Brauer group, and letρ be therank of its Neron–Severi group.

(1) In caseh < ∞ use the previous proposition and the fact that the image ofthe crystalline Chern class has slope1 to deduce theIgusa–Artin–Mazurinequality

ρ ≤ 22 − 2h.

(In fact, this inequality can be generalized to arbitrary smooth projectivevarieties, see [Ill79a, Proposition (II.5.12)].) SinceX is projective, wehaveρ ≥ 1, and therefore,h = 11 is impossible.

(2) In caseρ = 22 show thath =∞. In fact, the Tate conjecture (see Conjec-ture 4.5 and Theorem 4.8) predicts the equivalence ofρ = 22 andh =∞.


(3) Assuming the Tate conjecture, show that there exist no K3surfaces withρ = 21 (this observation is due to Swinnerton-Dyer, see [Ar74a]).

We refer to [Ha78, Appendix B] for more results on formal group laws arisingin algebraic geometry, as well as further references.

7. UNIRATIONAL K3 SURFACES

Over algebraically closed fields, a curve is rational if and only if it is unirationalby Luroth’s theorem. By a theorem of Castelnuovo, this is also true for surfaces incharacteristic zero. On the other hand, Zariski constructed examples of unirationalsurfaces that are not rational over algebraically closed fields of positive character-stic. Since the characterization of unirational surfaces in positive characteristic isstill unclear, K3 surfaces provide an interesting testing ground.

The first examples of unirational K3 surfaces were constructed by Rudakov andShafarevich in characteristic2 and3, as well as by Shioda in arbitrary characteris-tic. Also, Artin and Shioda showed that unirational K3 surfaces are supersingular(using different notions of supersingularity, now known tobe equivalent by theestablished Tate-conjecture). Conversely, Artin, Rudakov, Shafarevich, and Sh-ioda conjectured that supersingular K3 surfaces are unirational, which would givea cohomological characterization of unirationality.

In this section, we will use the formal Brauer group to associate to a supersin-gular K3 surfaceX over an algebraically closed fieldk of positive characteristica 1-dimensional familyX → Spec k[[t]] of supersingular K3 surfaces with spe-cial fiberX, such that generic and special fiber are related by apurely inseparableisogeny. In particular, the generic fiber is unirational if and only if the special fiberis. Then, we fill up the moduli space of supersingular K3 surfaces with such fami-lies, which implies that every two supersingular K3 surfaces are purely inseparablyisogenous. Since Shioda established the existence of some unirational K3 surfacesin every positive characteristic, the existence of these isogenies implies that all ofthem are, thereby establishing the Artin–Rudakov–Shafarevich–Shioda conjecture.

7.1. The Luroth problem. We start by recalling some definitions and classicalfacts concerning the (uni-)rationality of curves and surfaces. In this section, theground fieldk is always assumed to be algebraically closed, in order to avoid thedistinction between (uni-)rationality overk andk.

Definition 7.1. An n-dimensional varietyX over an algebraically closed fieldk iscalledunirational, if there exists a dominant and rational map

Pnk 99K X .

Moreover,X is calledrational if this map can be chosen to be birational.

Equivalently,X is rational if and only ifk(X) ∼= k(t1, ..., tn), andX is uni-rational if and only ifk(X) ⊆ k(t1, ....tn) of finite index. In particular, rationalvarieties are unirational, which motivates the following question.

Question 7.2(Luroth). Are unirational varieties rational?


Luroth showed that this is true for curves, which nowadays is an easy con-sequence of the Riemann–Hurwitz formula, see for example, [Har77, ExampleIV.2.5.5]. Next, Castelnuovo (characteristic zero) and Zariski (positive character-istic) showed that a surface is rational if and only ifh1(OX) = h0(ω⊗2

X ) = 0, thatis, we have a cohomological criterion for rationality. Using this, one can show thatLuroth’s question also has a positive answer for surfaces in characteristic zero. Werefer to [BHPV, Theorem VI.3.5] or [Be96, Chapter V] for details and proofs incharacteristic zero, and to [Li13, Section 9] for a discussion in positive character-istic. In particular, a K3 surface in characteristic zero cannot be unirational.

On the other hand, Zariski [Za58] gave examples of unirational surfaces overalgebraically closed fields of positive characteristic that are not rational, see also[Li13, Section 9] for more examples and discussion, as well as [L-S09] for re-sults that show that unirationality is quite common even among simply connectedsurfaces of general type in positive characteristic.

Finally, there are3-dimensional unirational varieties over algebraically closedfields of characteristic zero that are not rational by results of Iskovskih and Manin[I-M71], Clemens and Griffiths [C-G72], as well as Artin and Mumford [A-M72].

7.2. Unirational and supersingular surfaces. By Castelnuovo’s theorem, a sur-face in characteristic zero is rational if and only if it is unirational. Although thisis not true in positive characteristic, we have at least the following necessary con-dition for unirationality.

Theorem 7.3(Shioda+ε). Let X be a smooth, proper, and unirational surfaceover an algebraically closed fieldk of positive characteristic.

(1) The Picard rankρ is equal to the second Betti numberb2, that is,X isShioda-supersingular.

(2) The crystalH2cris(X/W ) is of slope1, that is,X is Artin-supersingular.

PROOF. Assertion (1) is shown in [Sh74a, Section 2]. Assertion (2)followsfrom (1), sincec1(NS(X)) ⊗ W defines anF -crystal of slope1 and rankρ in-sideH2

cris(X/W ), which is of rankb2, see the discussion in Section 4.2 and theproof of Theorem 4.8.

The proof also shows that we have the following implicationsfor surfaces inpositive characteristic

unirational ⇒ Shioda-supersingular⇒ Artin-supersingular.

For K3 surfaces in odd characteristic, both notions of supersingularity are equiva-lent by the Tate conjecture, see Theorem 4.8. Thus, it is natural to ask whether allconverse implications hold, at least, for K3 surfaces.

Conjecture 7.4(Artin, Rudakov, Shafarevich, Shioda). A K3 surface is unirationalif and only if it is supersingular.

Let us note that Shioda [Sh77a , Proposition 5] gave an example of a Shioda-supersingular Godeaux surface that is not unirational. However, his examples havea fundamental group of order5 and the non-unirationality of them is related to


congruences of the characteristic modulo5. Therefore, in loc. cit. he asks whethersimply connected and supersingular surfaces are unirational, which is wide open.Coming back to K3 surfaces, Conjecture 7.4 holds in the following cases:

Theorem 7.5. LetX be a K3 surface characteristicp > 0. Then,X is unirationalin the following cases:

(1) X is the Kummer surfaceKm(A) for a supersingular Abelian surface andp ≥ 3 by Shioda[Sh77b].

(2) X is Shioda-supersingular and(a) p = 2 by Rudakov and Shafarevich[R-S78].(b) p = 3 andσ0 ≤ 6 by Rudakov and Shafarevich[R-S78].(c) p = 5 andσ0 ≤ 3 by Pho and Shimada[P-S06].(d) p ≥ 3 andσ0 ≤ 2 by (1), since these surfaces are precisely Kummer

surfaces for supersingular Abelian surfaces (see Examples4.10).

In particular, we have examples in every positive characteristic that support Con-jecture 7.4. Let us comment on the methods of proof:

(1) Shioda showed (1) by dominating Kummer surfaces by Fermat surfaces,that is, surfaces of the formxn0 + ...+ xn3 = 0 ⊂ P3

k, and explicitly con-structed unirational parametrizations of these latter surfaces in case thereexists aν such thatpν ≡ −1 mod n, see [Sh74a].

(2) Rudakov and Shafarevich [R-S78] showed their unirationality result usingquasi-elliptic fibrations, which can exist only if2 ≤ p ≤ 3.

We refer to [Li13, Section 9] for more details and further references.

7.3. Moving torsors. An interesting feature of supersingular K3 surfaces is thatall of them come with elliptic fibrations. In this section we will show that thosethat admit an elliptic fibration with section admit very particular1-dimensional de-formations: namely, the generic and special fiber are related by purely inseparableisogenies, which implies that one is unirational if and onlyif the other one is.

Definition 7.6. A genus-1 fibration from a surface is a proper morphism

f : X → B

from a normal surfaceX onto a normal curveB with f∗OX∼= OB such that the

generic fiber is integral of arithmetic genus1. In case the geometric generic fiber issmooth, the fibration is calledelliptic, otherwise it is calledquasi-elliptic. In bothcases, the fibration is calledJacobianif it admits a secion.

If X → B is a K3 surface together with a fibration, thenB ∼= P1 for otherwise,the Albanese map ofX would be non-trivial, contradictingb1(X) = 0. Moreover,if F is a fiber, thenF 2 = 0, and sinceωX

∼= OX , the adjunction formula yields2pa(F )− 2 = F 2+KXF = 0, that is, the fibration is of genus1. If the geometricgeneric fiber is singular, then, by Tate’s theorem [Ta52], the characteristicp of theground fieldk satisfies2 ≤ p ≤ 3. In particular, ifp ≥ 5, then every genus-1fibration is generically smooth, that is, elliptic.


Theorem 7.7. Let X be a supersingular K3 surface in odd characteristicp, or,Shioda-supersingular ifp = 2.

(1) X admits an elliptic fibration.(2) If p = 2 or p = 3 andσ0 ≤ 6, thenX admits a quasi-elliptic fibration.(3) If p ≥ 5 andσ0 ≤ 9, thenX admits a Jacobian elliptic fibration.

PROOF. Since indefinite lattices of rank≥ 5 contain non-trivial isotropic vectors,the Neron–Severi latticeNS(X) of a supersingular K3 surface contains a class0 6= E with E2 = 0. By Riemann–Roch on K3 surfaces,E or−E is effective, sayE. Then, the Stein factorization ofX 99K Proj

⊕nH

0(X,OX (nE)) eventuallygives rise to a (quasi-)elliptic fibration, see, for example, [R-S81, Section 3] or theproof of [Ar74a, Proposition 1.5] for details. Moreover, ifX admits a quasi-ellipticfibration, it also admits an elliptic fibration, see [R-S81, Section 4].

Assertion (2) follows from the explicit classification of Néron–Severi lattices ofsupersingular K3 surfaces and numerical criteria for the existence of a quasi-ellipticfibration, see [R-S81, Section 5].

Assertion (3) follows again from the explicit classification of Neron–Severi lat-tices of supersingular K3 surfaces, see [Li15, Proposition3.9] and [Li15, Remark3.11].

Let us now explain how to deform Jacobian elliptic fibrationson supersingularK3 surfaces to non-Jacobian elliptic fibrations: thus, letX → P1 be a Jacobianelliptic fibration from a K3 surface. Contracting the components of the fibers thatdo not meet the zero section, we obtain theWeierstraß modelX ′ → P1. Thisfibration has irreducible fibers,X ′ has at worst rational double point singularities,and we letA → P1 be the smooth locus ofX ′ → P1. Then,A → P1 is a relativegroup scheme - more precisely, it is the identity component of the Neron model ofX → P1. Now, one can ask for commutative diagrams of the form

X ⊇ A → A↓ ↓ ↓P1k = P1

k → P1S

↓ ↓ ↓Spec k = Spec k → S

whereS is the (formal) spectrum of a local complete and Noetheriank-algebrawith residue fieldk, the right squares are Cartesian, whereA → P1

S is a familyof elliptic fibrations with special fiberA → P1

k, and where all elliptic fibrationsare torsors (principal homogeneous spaces) under the Jacobian elliptic fibrationA → P1

k. To bound the situation and in order to algebraize, we assumethat thesemoving torsorscome with a relative invertible sheaf onA → P1

S that is of somefixed degreen. In [Li15, Section 3.1], we showed that such families correspondto n-torsion elements in the formal Brauer group, that is, elements in the Abeliangroup

BrX(S)[n] .

To obtain non-trivial moving torsors overS := Spec k[[t]], this group must benon-zero, and for this to be the case, it simply follows from the fact thatBrX is


a 1-dimensional formal group law, thatk must be of positive characteristicp, thatBrX must be isomorphic toGa, and thatp has to dividen, see [Li15, Lemma 3.3].In particular,X must be a supersingular K3 surface. In casep = n, one can evenfind a degree-p multisectionD → YS, that is purely inseparable over the baseP1

S ,see [Li15, Proposition 3.5]. Next, we use these considerations to construct a familyof supersingular K3 surfaces.

Theorem 7.8. LetX be a supersingular K3 surface in characteristicp ≥ 5.

(1) If σ0 ≤ 9, thenX admits a Jacobian elliptic fibration.(2) Associated to a Jacbobian elliptic fibration onX, there exists a smooth

projective family of supersingular K3 surfaces

X → X↓ ↓

Spec k → Spec k[[t]].

This family has the following properties:(a) The Artin invariants of special and geometric generic fiber satisfy

σ0(Xη) = σ0(X) + 1 .

In particular, this family has non-trivial moduli.(b) There exist dominant and rational maps

X(1/p)η 99K X ×k η 99K X

(p)η ,

both of which are purely inseparable of degreep2.

The idea is to compactify the moving torsor associated to theJacobian ellipticfibrationX → P1 that arises from a nontrivialp-torsion element ofBrX(S) withS = Spec k[[t]]. After resolving the rational double point singularities in families,we obtain a familyX → P1

S of supersingular K3 surfaces with special fiberX.Specialization induces an injection of Neron–Severi groupsNS(Xη) → NS(X),whose cokernel is generated by the class of the zero-sectionof X → P1. Fromthis, the assertion on Artin invariants follows, and we refer to [Li15, Theorem 3.6]for details. By [Li15, Proposition 3.5], the elliptic fibration on the generic fiberadmits a degree-p multisectionD that is purely inseparable over the base. Then,the assertions on dominant and rational maps follow from base-changing the familyXη → P1

η toD → P1η, which trivializes the moving torsor, see [Li15, Theorem 3.6].

Spreading out the familyX → S of Theorem 7.8 to some curve of finite typeoverk, and using the theorem of Rudakov and Shafarevich on potential good reduc-tion of supersingular K3 surfaces (see the remarks after Corollary 5.6), we obtaina smooth and projective familyY → B, whereB is a smooth and projective curveoverk, with X as some fiber, andX as fiber over the completionOB,η . Associat-ing to such a family their rigidified K3 cystals, we obtain thefollowing statementabout moduli spaces of rigidified K3 crystals.

Theorem 7.9. LetN andN+ be the supersingular K3 lattices in characteristicp ≥ 5 of Artin invariantsσ0 andσ0 +1, respectively. Then, there exists a fibration


of moduli spaces of rigidified K3 crystals

MN+−→ MN ,

whose geometric fibers are rational curves.

We note that the fibers of this fibration correspond to the moving torsor families(at least, an open and dense subset does), and refer to [Li15,Theorem 4.3] and[Li15, Theorem 4.5] for details.

As explained in and after Examples 4.20, there is only one supersingular K3 sur-face with Artin invariantσ0 = 1. In particular, by Theorem 7.8 and Theorem 7.9together with an induction on Artin invariants, we can first relate every supersin-gular K3 surface to the unique supersingular K3 surface withσ0 = 1 via dominantrational maps, and ultimately obtain the following structure result for supersingularK3 surfaces.

Theorem 7.10.LetX andY be supersingular K3 surfaces in characteristicp ≥ 5.Then, there exist dominant and rational maps

X 99K Y 99K X

which are purely inseparable, that is, the surfaces arepurely inseparably isoge-nous.

By a theorem of Shioda [Sh77b], supersingular Kummer surfaces in odd charac-teristic are unirational (see also Theorem 7.5 above), and combining this with theprevious theorem, this implies Conjecture 7.4.

Theorem 7.11.Supersingular K3 surfaces in characteristicp ≥ 5 are unirational.

In particular, a K3 surface in characteristicp ≥ 5 is supersingular in the senseof Artin if and only if it is supersingular in the sense of Shioda if and only if it isunirational.

7.4. Unirationality of moduli spaces. It follows from Theorem 4.16 (see [Og79,Proposition 4.10] for details), as well as Theorem 7.9, thatthe moduli space ofN -rigidified K3-crystalsMN is unirational. Thus, also the moduli spacePN ofN -rigidified K3-crystals together with ample cones is in somesense unirational(this space is neither separated nor of finite type, but obtained by glueing openpieces ofMN ). Moduli spaces of polarized K3 surfaces are much better behaved,see Theorem 2.10. In fact, constructing families of supersingular K3 surfaces usingthe formal Brauer group (similar to the moving torsor construction above, but moregeneral), the supersingular loci inside moduli spaces of polarized K3 surfaces arerationally connected - this is a forthcoming result of Lieblich, see [Lieb13, Section9] for announcements of some of these results.

8. BEYOND THE SUPERSINGULARLOCUS

In Section 2.4, we introduced and discussed the moduli stackM2d,Fp

of degree-2d primitively polarized K3 surfaces overFp. In the previous sections, we focusedon supersingular K3 surfaces, that is, K3 surfaces, whose formal Brauer groups are


of infinite height. In this final section, we collect and survey a couple of results onthis moduli space beyond the supersingular locus. We stressthat these results arejust a small outlook, as well as deliberately a little bit sketchy.

8.1. Stratification. Associated to a K3 in positive characteristicp, we associatedthe following discrete invariants:

(1) The heighth of the formal Brauer group, which satisfies1 ≤ h ≤ 10 orh =∞, see Proposition 6.17 and Exercise 6.18.

(2) If h = ∞ thendisc(NS(X)) = −p2σ0 for some integer1 ≤ σ0 ≤ 10, theArtin-invariant, see Definition 4.12.

These invariants allow us to define the following loci (just as a set of points, we donot care about scheme structures at the moment) inside the moduli spaceM

2d,Fp

of degree-2d polarized K3 surfaces

Mi := surfaces withh ≥ i M∞,i := surfaces withh =∞ andσ0 ≤ i .

Thus, at least on the set-theoretical level, we obtain inclusions

M2d,Fp

= M1 ⊃ ... ⊃ M10 ⊃ M∞

||M∞,10 ⊃ M∞,9 ⊃ ... ⊃ M∞,1.

This stratification was introduced by Artin [Ar74a], and studied in detail by Ekedahl,van der Geer, Katsura [G-K00], [G-K01], [E-G15], and Ogus [Og01]. It turns outthat eachMi+1 is a closed subset ofMi and thatM∞,i is closed inM∞,i+1. Forexample, the first closedness assertion can be deduced easily from the followingresult [G-K00, Theorem 5.1].

Proposition 8.1 (van der Geer–Katsura). Let X be a K3 surface over an alge-braically closed field of positive characteristicp. Then

h = minn ≥ 1 :

(F : H2(X,Wn(OX))→ H2(X,Wn(OX))

)6= 0

.

In fact, this generalizes to higher dimensions: ifX is a Calabi–Yau variety ofdimensionn, then the height of the one-dimensional formal group law associated toΦnX (notation as in Section 6.2) can be characterized as in the previous proposition,

and we refer to [G-K03, Theorem 2.1] for details.

8.2. Stratification via Newton polygon. By Proposition 6.17, the heighth of theformal Brauer group of a K3 surface determines the smallest slope of the Newtonpolygon of theF -crystalH2

cris. Moreover, by Exercise 3.10, the smallest slopedetermines this Newton polygon completely. In particular,the height stratification(the first part of the stratification introduced above)

M2d,Fp

= M1 ⊃ ... ⊃ M10 ⊃ M∞

coincides with the stratification by the Newton polygon associated to theF -crystalH2

cris. This stratification also illustrates Grothendieck’s theorem that the Newtonpolygon goes up under specialization.


8.3. Stratification via F-zips. Let (X,L) be a primitively polarized K3 surfaceover an algebraically closed fieldk of positive characteristicp such thatp does notdivide L2 =: 2d. The cup product induces a non-degenerate quadratic form onH2

dR(X/k). (Since we assumedp ∤ 2d, we havep 6= 2 and thus, we do not haveto deal with subtleties of quadratic forms in characteristic 2.) Then, we define theprimitive cohomologyto be

M := c1(L)⊥ ⊂ H2

dR(X/k).

We note that the conditionp ∤ 2d ensures thatc1(L) is non-zero, and thus,Mis a 21-dimensionalk-vector space. The Hodge and its conjugate filtration onH2

dR(X/k) give rise to two filtrationsC• andD• onM , and the Cartier isomor-phism induces isomorphismsϕn : (grnC)

(p) → grnD, see Section 1.2. Next, thequadratic form onH2

dR(X/k) induces a non-degenerate quadratic formψ onM ,and it turns out that the filtrations are orthogonal with respect toψ. Putting thisdata together, we obtain anorthogonalF -zip

(M, C•, D•, ϕ•, ψ )

of filtration typeτ with τ(0) = τ(2) = 1 andτ(1) = 19, see Definition 1.3. Werefer to [M-W04, Section 5] or [P-W-Z15] forF -zips with additional structure.

As already mentioned in Section 1.2 and made precise by [M-W04, Theorem4.4],F -zips of a fixed filtration type form an Artin stack that has only a finite num-ber of points. More precisely, orthogonalF -zips of typeτ as above are discussedin detail in [M-W04, Example (6.18)]. Let us sketch their results: if (V, ψ) is anorthogonal space of dimension21 overFp, thenSO(V, ψ) has a root system oftypeB10. After a convenient choice of roots, and with appropriate identifications,the Weyl groupW of SO(V, ψ) becomes a subgroup of the symmetric groupS21

as follows

W ∼= ρ ∈ S21 | ρ(j) + ρ(22 − j) = 22 for all j .

We setWJ := ρ ∈ W | ρ(1) = 1 and it is easy to see that the set of cosetsWJ\W consists of20 elements. As shown in [M-W04, Example (6.18)], thereexists a bijection between the set of isomorphism classes oforthogonalF -zips oftypeτ overFp andWJ\W . Moreover, the Bruhat order onW induces atotal orderon this set of cosets, that is, we can find representativesw1 > ... > w20. Usingthis bijection and the representatives, we define

M(i) := surfaces whose associated orthogonalF -zip corresponds towi .

This gives a decomposition ofM2d,Fp

into 20 disjoint subsets. This decompositionis related to the stratification from Section 8.1 as follows

Mi \Mi+1 = M(i) for 1 ≤ i ≤ 10

M∞,21−i \M∞,20−i = M(i) for 11 ≤ i ≤ 20,

where it is convenient to setM11 := M∞, andM∞,0 := ∅. In particular, bothdecompositions eventually give rise to the same stratification of the moduli space.Again, we refer to [M-W04, Example (6.18)] for details.


8.4. Singularities of the strata. In [G-K00] and [Og01], Katsura, van der Geer,and Ogus found a beautiful description of the singularitiesof the height strataMi,which, by Section 8.2 coincide with the Newton-strata.

Theorem 8.2(van der Geer–Katsura, Ogus). Let Mii≥1 be the height stratifi-cation ofM

2d,Fp. Still assumingp ∤ 2d, we have an equality of sets

(Mi)sing = M∞,i−1 for all 1 ≤ i ≤ 10,

where−sing denotes the singular locus of the corresponding height stratum.

8.5. Cycle classes.To state the next result, we letπ : X →M2d,Fp

be the univer-sal polarized K3 surface, and we still assumep ∤ 2d. Then, we define theHodgeclassto be the first Chern class

λ1 := c1

(π∗Ω

2X/M

2d,Fp

).

Let us recall thatM2d,Fp

is 19-dimensional, and that each stratum of our strati-fication is of codimension1 inside the next larger stratum. The following result[E-G15, Theorem A] describes the cycle classes of these strata in terms of theHodge class. We note that the moduli spacesM2d,Fp

are non-complete, but thatthe formulas still make sense on an appropriate compactification.

Theorem 8.3(Ekedahl–van der Geer). In terms of the Hodge classλ1, the cycleclasses of the strata insideM2d,Fp

are as follows

[Mi] = (p− 1)(p2 − 1) · · · (pi−1 − 1)λi−11

[M∞] = 12(p− 1)(p2 − 1) · · · (p10 − 1)λ101

[M∞,i] =1

2

(p2(11−i) − 1)(p2(12−i) − 1) · · · (p20 − 1)

(p + 1)(p2 + 1) · · · (pi + 1)λ20−i1

for 1 ≤ i ≤ 10.

The appearance of the factor1/2 is related to the fact that the formulas of[G-K00, Theorem 14.2 and Section 15] count the supersingular stratum doubly,see also [G-K01] and [E-G15].

Theorem 8.3 measures the “size” of these strata, and can be thought of as ageneralization of a theorem of Deuring for elliptic curves:namely, in characteristicp, elliptic curves form a1-dimensional moduli space overFp. The formal Picardgroup of an elliptic curve either has height1 (ordinary elliptic curve) or height2 (supersingular elliptic curve), see Example 6.10. Ordinary elliptic curves forman open and dense set, and thus, the number of supersingular elliptic curves in afixed characteristicp is finite. Now, theorem of Deuring gives a precise answer:classically, it is phrased by saying that there are[p/12] + εp supersingular ellipticcurves for some0 ≤ εp ≤ 2 depending on the congruence class ofp modulo12, see, for example, [Si86, Theorem V.4.1]. However, if we count supersingularelliptic curves and weight each one of them with respect to their automorphismgroup (which can be thought of as counting them on the moduli stack rather than


the coarse moduli space) we obtain the following, much more beautiful formula∑

E supersingular/∼=

1

#Aut(E)=

p− 1

24.

Theorem 8.3 is a generalization of this way of counting to K3 surfaces.We refer to [E-G15] for more about the singularities of the strata, as well as to

[G13] for irreducibility results of the strata.

8.6. A Torelli theorem via Shimura varieties. We end our survey with a verysketchy discussion of a Torelli theorem for K3 surfaces in positive characteristicbeyond the supersingular ones.

For curves, Abelian varieties, K3 surfaces,... the classical period map associatesto such a variety some sort of linear algebra data as explained at the end of Section1.1. Over the complex numbers, this linear algebra data is parametrized as pointsinside some Hermitian symmetric domain modulo automorphisms. Thus, the pe-riod map can be interpreted as a morphism from the moduli space of these varietiesto a Hermitian symmetric domain. In this setting, a Torelli theorem is the statementthat this period map is an immersion, or, at least etale.

First, we set up some Shimura varieties, which will serve as the Hermitian sym-metric domain modulo automorphisms: letU be the hyperbolic plane overZ, andsetN := U⊕3⊕E⊕2

8 . Thus, abstractly,N is isometric toH2(X,Z) of a K3 surfacewith the cup-product pairing (Poincare duality), also known as theK3 lattice. Lete, f be a basis for the first copy ofU in N . Then, ford ≥ 1, we define

Ld := 〈e− df〉⊥ ⊆ N ,

which is modelled on the primitive cohomologyP 2(X) := c1(L)⊥ ⊂ H2(X,Z)

of a polarizationL of self-intersection2d. We note thatGd := SO(Ld) is a semi-simple algebraic group overQ. Next, letKLd

⊂ Gd(Af ) be the largest subgroupof SO(Ld)(Z) that acts trivially on the discriminantdisc(Ld) := L∨

d /Ld. Finally,let YLd

be the space of oriented negative definite planes inLd ⊗ R. Associated tothis data, we have the Shimura varietySh(Ld). It is a smooth Deligne–Mumfordstack overQ such that, as complex orbifolds, itsC-valued points are given by thedouble quotient

Sh(Ld)(C) = GLd(Q)\ (YLd

×Gd(Af )) /KLd,

and we refer to [MP15, Section 4.1] for details.Let us now return to K3 surfaces: in Section 2.4 we introducedthe moduli space

M

2d,Z[ 1

2d]of degree-2d primitively polarized K3 surfaces overZ[ 1

2d ]. Rather than

working with this moduli space, we will addspin structuresfirst: letL be a primi-tive polarization withL2 = 2d. LetP 2

ℓ (X) be the primitiveℓ-adic cohomology ofX, that is, the orthogonal complement ofc1(L) insideH2

et(X,Zℓ). For us, a spinstructure is a choice of isometric isomorphism

det(Ld)⊗ Z2∼=−→ det(P 2

2 (X)),


and we denote byM

2d,Z[ 1

2d]the moduli space of primitively polarized K3 surfaces

together with a choice of spin structure. Forgetting the spin structure induces amorphism

M

2d,Z[ 1

2d]→ M

2d,Z[ 1

2d],

which is etale of degree2, and we refer to [MP15, Section 4.1] and [Ri06, Section6] for details and precise definitions.

Now, over the complex numbers, there classical period map can be interpretedas a morphism

ıC : M2d,C → Sh(Ld)C.

Obviously, the left side can be defined overQ by considering families of K3 surfacewith polarization and spin structures overQ. The right side possesses a canonicalmodel overQ. And then, as shown by Rizov [Ri05], the period mapıC descendsto a mapıQ overQ.

Thus, by what we have said above, the following result [MP15,Theorem 5] is aTorelli type theorem for K3 surfaces in positive and mixed characteristic

Theorem 8.4(Madapusi Pera). There exists a regular integral modelS(Ld) forSh(Ld) overZ[12 ] such thatıQ extends to anetale map

ıZ[ 12] : M

2d,Z[ 12]→ S(Ld) .

When adding level structures, one can even achieve a period map that is anopen immersion [MP15, Corollary 4.15]. As explained in [MP15, Section 1], theconstruction of this map overZ[ 1

2d ] is essentially due to Rizov [Ri10], and an-other construction is due to Vasiu [Va]. Finally, let us alsomention that Nygaard[Ny83b] proved a Torelli-type theorem for ordinary K3 surfaces using the theory ofcanonical lifts for such surfaces and then applying the Kuga–Satake construction.

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LECTURES ON SUPERSINGULAR K3 SURFACES AND THELECTURES ON SUPERSINGULAR K3 SURFACES AND THE CRYSTALLINE TORELLI THEOREM CHRISTIAN LIEDTKE Expanded lecture notes of a course given as