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On a Stratification of the Moduli 0 0 of K3 Surfaces 0 2 n a G. van der Geer and T. Katsura J 6 ] Abstract. In this paper we give a characterization of the height of K3 surfaces in G characteristic p > 0. This enables us to calculate the cycle classes of the loci in A families of K3 surfaces where the height is at least h. The formulas for such loci . h can be seen as generalizations of the famous formula of Deuring for the number of at supersingular elliptic curves in characteristic p. In order to describe the tangent m spaces to these loci we study the first cohomology of higher closed forms. [ 2 1. Introduction v Elliptic curves in characteristic p come in two sorts: ordinary and supersingular. The 1 6 distinction can be expressed in terms of the formal group of an elliptic curve. Multipli- 0 cation by p on the formal group takes the form 0 1 9 [p](t) = atph + higher order terms, (1) 9 / h where a 6= 0 and t is a local parameter. The number h satisfies 1 ≤ h ≤ 2 and is called t a the height. By definition, the elliptic curve is ordinary if h = 1 and supersingular if m h = 2. There is a classical formula of Deuring for the number of supersingular elliptic v: curves over an algebraically closed field k of characteristic p: i X 1 p−1 = , r a #Aut(E) 24 X ∼ E supers. / = where the sum is over supersingular elliptic curves over k up to isomorphism. Ifone viewsK3 surfaces asa generalizationofellipticcurves, one canmakeasimilar distinction of K3 surfaces in characteristic p by using the formal Brauer group as Artin showed. The formal Brauer group is a 1-dimensional formal group associated to the second étale cohomology with coefficients in the multiplicative group. Multiplication by p in this formal group has the form (1), but now we have 1 ≤ h ≤ 10 or h = ∞, the latter if multiplication by p is zero. The height can be used to define a stratification of the moduli spaces of K3 surfaces. A generic K3 surface will have h = 1; those with h = ∞ are most special in this respect and called supersingular. In this paper we first express the height of a K3 surface in terms of the action of the Frobenius morphism on the second cohomology group with coefficients in the sheaf 1 W(O ) of Witt vectors of the structure sheaf O . The natural co-filtration W (O ) of X X n X W(O ) induces co-filtrations on the cohomology which correspond to approximations X of the formal group. Using this characterization we can calculate the cycle classes of the strata in the moduli space where the height ≥ h. This is done by interpreting the loci as degeneracy loci of maps between bundles. The resulting formulas can be viewed as a generalization of Deuring’s formula. Generalizations of Deuring’s formula to principally polarized abelian varieties were worked out in joint work of Ekedahl and one of us and can be found in [G]. The supersingular locus comes with a multiplicity. In order to describe the tangent spaces to our strata we use differential forms rather than crystalline cohomology. We calculate the dimensions of cohomology groups H1(Z ) and H1(B ) where the sheaves Z and B are the sheaves of certain closed forms i i i i introduced by Illusie. We study the dimensions of the cohomology groups H1(Z ) and i H1(B ) and of their images in H1(X,Ω1). We think that these spaces are quite helpful i to understand the geometry of surfaces in characteristic p. Acknowledgements. Both authors would like to thank the Max-Planck-Institut in Bonn for excellent working conditions during their visit in 1998/99. The second author would like to thank NWO and the University of Amsterdam for their support and hospitalityduringhisstay inAmsterdam in1997. We wouldliketothank Arthur Ogus for sharing some of his ideas with us. 1. Witt vector cohomology Let X be a non-singular complete variety defined over an algebraically closed field k of characteristic p > 0. We denote by W = W (O ) the sheaf of Witt rings of length n n n X as defined by J.-P. Serre, cf. [S]. The sheaf W (O ) is a coherent sheaf of rings which n X comes with three operators: i) Frobenius F : W (O ) → W (O ), n X n X ii) Verschiebung V : W (O ) → W (O ), n X n+1 X iii) Restriction R : W (O ) → W (O ), n+1 X n X defined by the formulas p p p F(a ,a ,...,a ) = (a ,a ,...,a ), 0 1 n−1 0 1 n−1 V(a ,a ,...,a ) = (0,a ,a ,...,a ), 0 1 n−1 0 1 n−1 R(a ,a ,...,a ) = (a ,a ,...,a ). 0 1 n 0 1 n−1 They satisfy the relations RVF = FRV = RFV = p. The cohomology groups Hi(X,W (O )) are finitely generated W (k)-modules. The n X n projective system {W (O ),R} induces a sequence n X n=1,2,... ... ←− Hi(X,W (O ))←R−Hi(W (O )) ←− ... n X n+1 X so that we can define Hi(X,W(O )) = proj. lim Hi(X,W (O )). X n X 2 This is a W(k)-module, but not necessarily a finitely generated W(k)-module, cf. Sec- tion 3. The semi-linear operators F and V act on it and they satisfy the relations FV = VF = p. 2. Formal Groups Smooth formal Lie groups of dimension 1 over an algebraically closed field k of characteristic 6= 0 are characterized by their height, cf. [H], [Ma]. To a smooth formal Lie group Φ of dimension one one can associate its covariant Dieudonné module M = D(Φ), a free W(k)-module. It possesses two operators F and V with the following properties: the operator F is σ-linear, the operator V is σ−1-linear and topologically nilpotent and they satisfy FV = VF = p. Here σ denotes the Frobenius map on k. Then M is a free W(k)-module with the following properties: a) dim(Φ) = dim (M/VM), k b) height(Φ) = rank (M). W Note that one has the equalities rank (M) = dim (M/pM) = dim (M/FM)+dim (M/VM). W k k k 3. The Formal Brauer Group of Artin-Mazur For a proper variety X/k one may consider the formal completion of the Picard group. The group of S-valued points of Pic(X) fits into the exact sequence 0 −→ Pic(S) −c→ H1(X ×S,G ) −→ H1(X,G ) m m for any local artinian schecme S with residue field k. Here cohomology is étale cohomology. This idea of studying infinitesimal properties of cohomology was generalized to the higher cohomology groups Hr(X,G ) by Artin and Mazur, cf. [A-M]. Their work leads m to contravariant functors Φr:Art → Ab with Φr(S) = kerHr(X ×S,G ) −→ Hr(X,G ), m m which under suitable circumstances are representable by formal Lie groups. For a K3 surface X this is the case and we find for r = 2 the formal Brauer group Φ = Φ = Φ2. X Its tangent space is T = H2(X,O ). Φ X For a K3-surface X we have two possibilities: i) h(Φ) = ∞ and Φ = Gˆ , the formal additive group. The K3-surface is called a supersingular (in the sense of Artin). ii) h(Φ) < ∞. Then Φ is a p-divisible formal group. Moreover, it is known that 1 ≤ h(Φ) ≤ 10. This follows from the following theorem of Artin, cf. [A]. We shall write simply h for h(Φ). 3 (3.1) Theorem. If the formal Brauer group Φ of a K3 surface X is p-divisible then X it satisfies the relation 2h ≤ B − ρ, where B is the second Betti number and ρ the 2 2 rank of the Néron-Severi group. For the proof one combines Theorem (0.1) of [A] with Deligne’s [D] result on lifting K3 surfaces, see also [I]. We give a proof in Section 10. This theorem implies that if ρ = 22 then necessarily we have h = ∞. If h 6= ∞ then it follows that 1 ≤ h ≤ 10. One should view h = 1 as the generic case. It was conjectured by Artin that if h = ∞ then ρ = 22. This is known for elliptic K3 surfaces, see [A]. Note that a surface with ρ = 22 is called supersingular by Shioda, cf. [Sh]. The following result by Artin and Mazur is crucial: (3.2) Theorem. The Dieudonné module of the formal Brauer group Φ is given by X D(Φ ) ∼= H2(X,W(O )). X X For the proof we refer to [A-M]. The point to notice is that D(Φ ) = H2(X,DG ) = H2(X,W(O )). X m X (3.3) Remark. Note that this explains why the Witt vector cohomology is sometimes not finitely generated: if Φ ∼= Gˆ then H2(X,W(O )) is not finitely generated over X a X W(k) because D(Gˆ ) = W(k)[[T]]. a 4. Vanishing of Cohomology We collect a number of results on the vanishing of cohomology groups for K3 surfaces that we need in the sequel. (4.1) Lemma. Let X be a K3 surface. We have H1(X,W (O )) = 0 for all n > 0, n X hence H1(X,W(O )) = 0. X Proof. Since X is a K3 surface we have by definition H1(X,O ) = 0. The lemma is X deduced from this by induction on n. Assume that H1(X,W (O )) = 0. Then the n−1 X exact sequence V Rn−1 0 −→ W (O )−→W (O )−→O −→ 0 n−1 X n X X induces an exact sequence H1(X,W (O ))−V→H1(W (O )) R−n→−1 H1(X,O ). n−1 X n X X This implies that H1(X,W (O )) = 0. (cid:3) n X (4.2) Lemma. For a projective surface X with H1(X,O ) = 0 the induced map X R:H2(X,Wn(OX)) → H2(X,Wn−1(OX)) is surjective with kernel ∼= H2(X,OX). Proof. This follows from the exact sequence R 0 → O −→ W (O )−→W (O ) → 0 X n X n−1 X and the vanishing of H1(X,O ) and of H3(X,O ). (cid:3) X X 4 (4.3) Lemma. In H2(X,W (O )) we have n X RV(H2(X,W (O ))) = V(H2(X,W (O ))). n X n−1 X Proof. The commutativity of the diagram V W (O ) −→ W (O ) n X n+1 X R R  V  Wn−1(OX) −→ Wn(OX). y y gives in cohomology a commutative diagram H2(X,W (O )) −V→ H2(X,W (O )) n X n+1 X R R   H2(X,W (O )) −V→ H2(X,W (O )). yn−1 X yn X The surjectivity of the left hand R, which follows from the preceding lemma, implies the claim. (cid:3) (4.4) Lemma. Assume that for some n > 0 the map F : H2(X,W (O )) −→ n X H2(X,W (O )) vanishes. Then for all 0 ≤ i ≤ n the map F : H2(X,W (O )) −→ n X i X H2(X,W (O )) is zero. Moreover, for all 0 ≤ i ≤ n the module H2(X,W (O )) is a i X i X vector space over k. Proof. The first result follows from the commutativity of the diagram H2(X,W (O )) R−n→−i H2(X,W (O )) n X i X F F H2(X,W (O )) R−n→−i H2(X,W (O )) yn X yi X and Lemma (4.2). The second claim follows from p = FVR and k ∼= W (k)/pW (k). (cid:3) i i (4.5) Lemma. Assume that X is a K3 surface. The following two sequences are exact: 0 → H2(X,W (O )) −V→ H2(X,W (O )) R−n→−1 H2(X,O ) → 0, n−1 X n X X ′ 0 → H2(X,W(O )) −V→ H2(X,W(O )) −R→ H2(X,O ) → 0, X X X where R′ is the map induced by W (O )R−n→−1W (O ) as n → ∞. n X 1 X Proof. The first exact sequence follows from the exact sequence V Rn−1 0 → W (O ) −→ W (O ) −→ O → 0 n−1 X n X X and Lemma (4.2). Because the projective system H2(X,W (O )) satisfies the Mittag- n X Leffler condition we may take the projective limit. (cid:3) 5. Characterization of the Height Let X be a K3 surface and let Φ be its formal Brauer group in the sense of Artin- X Mazur. The isomorphism class of this formal group is determined by its height h. The following theorem expresses this height in terms of Witt vector cohomology. 5 (5.1) Theorem. The height satisfies h(Φ ) ≥ i+1 if and only if the Frobenius map X F:H2(X,W (O )) → H2(X,W (O )) is the zero map. i X i X (5.2) Corollary. We have the following characterization of the height: h(Φ ) = min{i ≥ 1:[F : H2(W (O )) → H2(W (O ))] 6= 0}. X i X i X Proof of the Theorem. “⇐” In case h(Φ ) = ∞ the implication ⇐ is immediate. X So we may consider the case where the height of Φ is finite. Assume that the map X F:H2(X,W (O )) → H2(X,W (O )) is the zero map. We set i X i X M = D(Φ) ∼= H2(X,W(O )), the covariant Dieudonné module. X Since dim (H2(X,W(O ))/VH2(X,W(O )) = 1 by Lemma (4.5), we have by b) in k X X Section 2 dim (H2(X,W(O ))/FH2(X,W(O )) = h−1. k X X The surjectivity of the projection H2(X,W(O )) −→ H2(X,W (O )) implies the sur- X i X jectivity of H2(X,W(O ))/FH2(X,W(O )) −→ H2(X,W (O ))/FH2(X,W (O )). X X i X i X By assumption we have H2(X,W (O ))/FH2(X,W (O )) ∼= H2(X,W (O )) and by i X i X i X Lemma (4.5) we have dim H2(X,W (O )) = i, k i X i.e. we find h−1 ≥ i, or equivalently, h ≥ i+1. Conversely, we now prove “⇒”. If h(Φ ) = ∞ then Φ = Gˆ , the formal additive X X a group of dimension 1. So F acts as zero on D(Gˆ ) = D(Φ ) = H2(X,W(O )). As in a X X Lemma (4.4) we conclude that F acts on H2(X,W (O )) as the zero map. Therefore i X we may assume that h(Φ ) = h < ∞. We thus assume that h(Φ ) ≥ i+1. We set X X H = H2(X,W(O )) X and have Vh−1H ⊂ ... ⊂ V2H ⊂ VH ⊂ H. Under projection this is mapped surjectively to 0 ⊂ Vh−2H2(O ) ⊂ ... ⊂ VH2(W (O )) ⊂ H2(X,W (O )). X h−2 X h−1 X All the inclusions are strict because of Lemma (4.5). Claim. We have Vh−1H2(X,W(O )) = FH2(X,W(O )). X X Proof of the claim. Since our modules are free over W we deduce from Manin’s results [M] (but see also [H] because we use the covariant theory): D(Φ ) ∼= W[F,V]/W[F,V](F −Vh−1). X 6 Note that F − Vh−1 is written on the right. But we can transfer it to the left using FV = p = VF as follows: F( a FiVj) = aσFFiVj = (aσFiVj)F = ij ij ij X X X = (aσFiVj)Vh−1 = Vh−1( aσhFiVj). ij ij X X This together with Theorem (3.2) proves the claim. We now find FH2(W (O )) = 0. By Lemma (4.4) we conclude that F acts on h−1 X H2(W (O )) for i ≤ h−1 as zero. (cid:3) i X (5.3) Corollary. The height of Φ is ∞ if and only if the Frobenius endomorphism X F : H2(X,W (O )) → H2(X,W (O )) is zero. 10 X 10 X Proof. If the height is finite, then we know by Artin and Mazur (see (3.1)) that we have h ≤ 10. (cid:3) (5.4) Corollary. Set H = H2(X,W (O )) and consider the filtration 10 X {0} ⊂ R9V9H ⊂ R8V8H ⊂ ... ⊂ Rh−1Vh−1H ⊂ ... ⊂ H. If h is the height of Φ then F(H) = Rh−1Vh−1(H). X Proof. The (h−1)-th step Vh−1H2(W(O )) in the filtration X V10H2(W(O )) ⊂ V9H2(W(O )) ⊂ ... ⊂ H2(W(O )) X X X maps surjectively to the corresponding step Rh−1Vh−1H of the filtration on H. By our claim we have Vh−1H2(W(O )) = FH2(W(O )). X X This implies the assertion. (cid:3) (5.5) Corollary. If h(Φ ) = h < ∞ and if {ω,Vω,V2ω,...,Vh−1ω} is a W-basis of X H2(X,W(O )) then F acts as zero on H2(X,W (O )) if and only if F(ω¯) = 0, with ω¯ X i X the image of ω in H2(X,W (O )). i X (5.6) Corollary. If h(Φ ) = h < ∞, then dim ker[F : H2(W ) → H2(W )] = X k i i min{i,h−1}. Proof. By Lemma (4.5) and Corollary (5.2), we have dim ker[F : H2(W ) → H2(W )] k i i = i if i ≤ h − 1. Assume i ≥ h. Using the notation in Corollary 5.5, we know that hVi−h+1Ri−h+1ω¯,Vi−h+2Ri−h+2ω¯,Vi−h+3Ri−h+3ω¯,...,Vi−1Ri−1ω¯i is a basis of kerF. (cid:3) The case h ≥ 2 is characterized by the vanishing of Frobenius on H2(O ). We now X formulate in an inductive way a similar characterization of the condition h ≥ n+1. If for a K3 surface X one assumes that F is zero on H2(W ) (i = 1,...n−1) then one has i FH2(W ) ⊂ Vn−1H2(O ) and F vanishes on VH2(W ). Since we have a natural n X n−1 (σ−1)n−1-isomorphism H2(O ) ∼= Vn−1H2(O ), one has an induced homomorphism X X φn:H2(OX) ∼= H2(Wn)/VH2(Wn−1) → Vn−1H2(OX) ∼= H2(OX). (2) This map is σn-linear. The following theorem is clear by the construction of φ . n 7 (5.7) Theorem. Suppose F is zero on H2(W ) for i = 1,...n−1. Then F vanishes i on H2(W ) if and only if φ : H2(O ) → H2(O ) vanishes. n n X X 6. Closed Differential Forms Let F : X → X(p) be the relative Frobenius morphism of a K3 surface X. By means of theCartieroperatorC : Ω• → Ω• wecandefinesheavesB Ω1 ofringsinductively X,closed X i X by B Ω1 = 0, B Ω1 = dO and C−1(B Ω1 ) = B Ω1 . Similarly, we define sheaves 0 X 1 X X i X i+1 X Z Ω1 inductively by Z Ω1 = Ω1 , Z Ω1 = Ω1 , the sheaf of d-closed forms and i X 0 X X 1 X X,closed by setting Z Ω1 := C−1(Z Ω1 ). i+1 X i X Usually we simply write B and Z . The sheaves B and Z can be viewed as locally i i i i free subsheaves of (Fi) Ω1 on X(pi). They were introduced by Illusie in [Il] and can ∗ X be used to provide de Rham-cohomology with a rich structure. The inverse Cartier operator gives rise to an isomorphism C−i : Ω1 −≃→Z /B X(pi) i i or a σ−i-linear isomorphism Ω1 ∼= Z /B . Note that we have the inclusions X i i 0 = B ⊂ B ⊂ ... ⊂ B ⊂ ... ⊂ Z ⊂ ... ⊂ Z ⊂ Z = Ω1 . 0 1 i i 1 0 X We also have an exact sequence 0 → Z −→ Z −−dC−i→dΩ1 → 0. (3) i+1 i X (6.1) Lemma. If X is a K3 surface X we have i) H0(B ) = 0 for all i ≥ 0; ii) the i natural inclusion B → B induces an injective homomorphism H1(B ) → H1(B ). i i+1 i i+1 Proof. i) The natural injection B → Ω1 induces an injection H0(B ) → H0(Ω1 ) and i X i X we know H0(Ω1 ) = 0. ii) This follows from i) and the exact sequence X Ci 0 → B −→ B −→B → 0 i i+1 1 with C the Cartier operator. (cid:3) There is a close relationship between the Witt vector cohomology and the cohomology of B as follows. Serre introduced in [S] a map D : W (O ) −→ Ω1 of sheaves i i i X X in the following way: pi−1−1 p−1 D (a ,a ,...,a ) = a da +...+a da +da . i 0 1 i−1 0 0 i−2 i−2 i−1 It satisfies D V = D , and Serre showed that this induces an injective map of sheaves i+1 i of additive groups D : W (O )/FW (O ) −→ Ω1 i i X i X X inducing an isomorphism ∼ D : W (O )/FW (O )−→B Ω1 (4) i i X i X i X 8 F The exact sequence 0 → W −→W −→ W /FW → 0 gives rise to the exact sequence i i i i 0 → H1(W /FW ) → H2(W )−F→H2(W ) → H2(W /FW ) → 0 (5) i i i i i i and we thus have an isomorphism H1(W /FW ) ∼= ker[F : H2(W ) → H2(W )]. Com- i i i i bining the result on the dimension of the kernel of F on H2(W ) from Section 5 with i (4) we get an interpretation of the height h in terms of the groups H1(B ). i (6.2) Theorem. We have min{i,h−1} if h 6= ∞, dimH1(B ) = i i if h = ∞. n The Verschiebung induces an exact sequence V 0 → W /FW −→W /FW → O /FO → 0 i i i+1 i+1 X X and this gives rise to 0 → H1(W /FW )−V→H1(W /FW ) −→ H1(O /FO ) → ... i i i+1 i+1 X X i.e., Verschiebung induces for all i an injective map. Moreover, it is surjective if and only if h 6= ∞ and i ≥ h−1. We have a commutative diagram (with β the natural map induced by B ⊂ Ω1 ) i i X H1(W /FW ) −D→i H1(Ω1 ) i i X ∼= =   H1(ByiΩ1X) −−β−i→ H1(yΩ1X) We study the kernel of D , equivalently the kernel of the natural map β : H1(B Ω1 ) → i i i X H1(Ω1 ) in Sections 9-11. X (6.3) Lemma. The Euler-Poincaré characteristics of B and Z are given by χ(B ) = 0 i i i and χ(Z ) = −20. i Proof. Since the kernel and the cokernel of F on H2(W ) have the same dimension by i (5) the result for B follows from (4) and (5). The identity χ(B ) + χ(Ω1 ) = χ(Z ) i i X i resulting from the isomorphism Z /B ∼= Ω1 implies the result. (cid:3) i i X 7. De Rham Cohomology The de Rham cohomology of a K3 surface is the hypercohomology of the complex (Ω• ,d). The dimensions hp,q of the graded pieces are given by the Hodge diamond. X 1 0 0 1 20 1 0 0 1 On H2 we have a perfect pairing h, i given by Poincaré duality; cf. [D]. dR 9 The Hodge spectral sequence with Eij = Hj(X,Ωi ) converges to H∗ (X). The 1 X dR second spectral sequence of hypercohomology has as E -term Eij = Hi(Hj(Ω• )) abut- 2 2 X i+j ting to H (X/k). But the Cartier operator yields an isomorphism of sheaves dR C−1 : Ωi −∼→Hi(F (Ω• )), X(p) ∗ X/k so that we can rewrite this as Eij = Hi(X′,Hj(Ω•)) ∼= Hi(X′,Ωj ) ⇒ H∗ (X), 2 X′ dR where X′ = X(p) is the base change of X under Frobenius. We thus get two filtrations on the de Rham cohomology: the Hodge filtration (0) ⊂ F2 ⊂ F1 ⊂ H2 , dR and the so-called conjugate filtration (0) ⊂ G ⊂ G ⊂ H2 . 1 2 dR We have rank(F1) = rank(G ) = 21, rank(F2) = rank(G ) = 1 and 2 1 (F1)⊥ = F2 and G⊥ = G . 1 2 We have also F1/F2 ∼= H1(X,Ω1 ), G /G ∼= H1(X,Ω1 ), X 2 1 X cf. [D]. Moreover, from the description with the second spectral sequence it follows that G istheimageunder FrobeniusofH2 andalsoofH2(X,O ). Theconjugatefiltration 1 dR X is an analogue of the complex conjugate of the Hodge filtration in characteristic zero. The relative position of these two filtrations is an interesting invariant of a K3 surface. We have the three cases a) F1 ∩G = {0}; 1 b) G ⊂ F1; G 6= F2; 1 1 c) G = F2. 1 The first case happens if and only if F : H2(X,O ) → H2 (X) → H2(X,O ) is not X dR X zero, i.e. if h = 1. Such X are called ordinary. The second case happens if h ≥ 2, while the last case is by definition the superspecial case. In this case the two filtrations coincide. It is known that two superspecial K3 surfaces are isomorphic (as unpolarized varieties)(cf. [O]). We have the following result of Ogus (cf. [O]) which provides us with an interpretation of H1(Z ). 1 (7.1) Proposition. We have an isomorphism F1 ∩G ∼= H1(X,Z ). 2 1 Proof. Themap z : H1(Z ) → H2 givenby {f } 7→ (0,{f },0)isinjective. Indeed, if 1 1 dR ij ij {f } represents an element in the kernel, then it is of the form (δh ,dh +ω −ω ,dω ) ij ij ij j i i for a h ∈ C (O ), ω ∈ C (Ω1 ). Then the ω are closed and h defines a cocycle. ij 1 X i 0 X i ij Since H1(O ) = 0 we can write dh = η −η and f is a coboundary. The image is X ij j i ij contained in F1 and is orthogonal to the image G of Frobenius. Indeed, take a class 1 F(a) and consider the cupproduct hF(a),z (f )i. Applying the Cartier operator we see 1 ij 10