ON A RESULT OF ARTIN

AISE JOHAN DE JONG

Let k be an algebraically closed field of characteristic p > 0. Let t be a parameter and set S = Spec(k[[t]]). Let f : X → S be a smooth proper morphism with generic fibre X_{η} and special fibre X_{0}. In Artin’s paper on supersingular K3 surfaces we find a remarkable result on the specialization L_{η} → L_{0} of Néron-Serveri groups of X_{η} and X_{0}. Namely, if the formal Brauer group of the generic fibre X_{η} is annihilated by a power of p and a mild assumption on relative Pic holds, then the cokernel of the map L_{η} → L_{0} is finite, see [Art74, Theorem 1.1 a]. The condition on the formal Brauer group is equivalent to the condition that all the slopes of the Newton polygon on H^{2}(X_{η}) are 1, see [AM77].

In this note we show (under some additional hypotheses)

(1) if the slopes on H^{2}(X_{η}) are all 1, then any crystalline cohomology class on X_{0} lifts to an element of the crystalline cohomology of X, and

(2) given an invertible sheaf L_{0} on X_{0} whose crystalline chern class lifts to a crystalline cohomology class on X, there exists an invertible sheaf L on X whose restriction to the special fibre is a power of L_{0}.

Combined these results imply Artin’s result in some cases, e.g., in the case of a family of K3 surfaces.

Analyzing the proof of Theorem 1 we see that an assertion as in (1) should generalize to higher degree cohomology. Moreover, Theorem 1 may be true without any assumption on torsion in crystalline cohomology. Equally likely some of the assumptions of Theorem 2 can be weakened. In higher degrees we can ask:

Consider an algebraic cycle α_{0} in codimension c on X_{0} whose crystalline cohomology class cl(α_{0}) ∈ H_{cris}^{2c}(X_{0}/Σ) lifts to an element of H_{cris}^{2c}(X/Σ). Is α_{0} the restriction of an algebraic cycle on X?

This question can be viewed as an equi-characteristic p analogue of the p-adic variational Hodge conjecture, see [MP11, Conjecture 9.2].

Throughout this note W is a Cohen ring for k. Set Σ = Spec(W). We denote σ : W [[t]] → W [[t]] the unique lift of frobenius such that σ(t) = t^{p}. The structure sheaf on the crystalline site (X/Σ)_{cris} of X over Σ is denoted O_{X/Σ}. We write H_{cris}^{2}(X/Σ) = H^{2}((X/Σ)_{cris}, O_{X/Σ}) and similarly for other schemes over Σ.

Theorem 1. In the situation above assume (1) the crystalline cohomology of X/S is torsion free^{1}, and

(2) all slopes of frobenius on H^{2}(X_{η}) are 1. Then for any ξ_{0} ∈ H_{cris}^{2}(X_{0}/Σ) there exists an n > 0 and an element ξ ∈ H_{cris}^{2}(X/Σ) such that ξ|_{X_{0}} = p^{n}ξ_{0}.

1This means that each R^{i}f_{∗}O_{X/Σ} is a crystal in finite locally free O-modules on (S/Σ)_{cris}.

Proof. Let M be the value of R^{2}f_{∗}O_{X/Σ} on W [[t]]. The assumption on torsion implies that M is a finite free W [[t]]-module and that M/tM is equal to H_{cris}^{2}(X_{0}/Σ). The module M comes with a connection ∇ : M → Mdt and a horizontal σ-linear map F : M → M. The assumption on slopes of Frobenius implies that M is isogenous to another crystal (M′, ∇′, F) such that F′ is divisible by p, see [Kat79]. Then (M′, ∇′, p^{−1}F′) is a unit root F -crystal over S. Since k is algebraically closed we see that we can find a basis e′_{1}, …, e′_{n} of M′ such that F (e′_{i}) = pe′_{i} (since unit root crystals correspond to representations of the fundamental group – see [BM90] for example). Since M is isogenous to M′ we can find e_{1}, …, e_{n} ∈ M such that ∇(e_{i}) = 0, F (e_{i}) = pe_{i}, and such that M/ ∑ W [[t]]e_{i} is annihilated by a power of p. In particular the elements \overline{e}_{i} = e_{i} mod t generate a W -submodule of H_{cris}^{2}(X_{0}/Σ) of finite colength. Thus we can write

p^{e}ξ_{0} = ∑ λ_{i}\overline{e}_{i}

for some e ≥ 0 and some λ_{i} ∈ W. Take ξ = ∑ λ_{i}e_{i}. Note that ∇(ξ) = 0. Hence ξ determines a gobal section of R^{2}f_{∗}O_{X/Σ} over (S/Σ)_{cris}. Consider the Leray spectral sequence for crystalline cohomology associated to f : X → S with E_{2} page E_{2}^{p,q} = H_{cris}^{p}(S/Σ, R^{q}f_{∗}O_{X/Σ}). The obstructions to lifting ξ to an element of H^{2}(X/Σ) are: first an element d_{2}(ξ) of H_{cris}^{2}(S/Σ, R^{1}f_{∗}O_{X/Σ}) followed by another element of a subquotient of H_{cris}^{3}(S/Σ, R^{0}f_{∗}O_{X/Σ}). Both groups are zero. Namely H_{cris}^{i}(S/Σ, E) = 0 for i > 1 for any crystal E of finite locally free O_{S/Σ}-modules on (S/Σ)_{cris}. This is true because if E corresponds to the triple (N, ∇, F) over W [[t]], then crystalline cohomology of E is computed by the two term complex ∇ : N → Ndt. □

Theorem 2. In the situation above assume (1) the crystalline cohomology of X/S is torsion free,

(2) H^{1}(X_{0}, O_{X_{0}}) = H^{0}(X_{0}, Ω_{X_{0}/k}^{1}) = 0,

(3) L_{0} is an invertible sheaf on X_{0}, and

(4) ξ ∈ H_{cris}^{2}(X/Σ) is an element such that ξ|_{X_{0}} is c_{1}(L_{0}).

Then L_{0}^{⊗p^{2}} is isomorphic to the restriction of an invertible sheaf L on X.

Proof. By Grothendieck’s formal existence theorem it suffices to lift L_{0}^{⊗p^{2}} to the schemes X_{n} = X ×_{S} Spec(k[t]/(t^{n+1}). Recall that the crystalline chern class of an invertible sheaf L_{n} on X_{n} is computed using the short exact sequence

0 → (1 + J_{X_{n}/Σ})^{∗} → O_{X_{n}/Σ}^{∗} → \underline{O_{X_{n}}^{∗}} → 0

on the crystalline site as well as the maps

(1 + J_{X_{n}/Σ})^{∗} ⟶^{log} J_{X_{n}/Σ} → O_{X_{n}/Σ}

In this proof we will work with the maps

c′_{1} : Pic(X_{n}) → H_{cris}^{2}(X_{n}/Σ, J_{X_{n}/Σ})

instead of the usual crystalline chern class. Note that c′_{1}(L_{0}^{⊗p^{2}}) = p^{2}c′_{1}(L_{0}) is equal to the restriction of pξ′ to X_{0} where ξ′ ∈ H_{cris}^{2}(X/Σ, J_{X/Σ}) is the image of ξ via the map p : O_{X/Σ} → J_{X/Σ}. The equality holds because the kernel of the map H_{cris}^{2}(X/Σ, J_{X/Σ}) → H_{cris}^{2}(X/Σ, O_{X/Σ}) is zero by the vanishing assumptions in the theorem.

Suppose we have found an invertible sheaf L_{n} on X_{n} such that c′_{1}(L_{n}) is equal to the restriction of pξ′ in H^{2}(J_{X_{n}/Σ}). We have a first order thickening X_{n} → X_{n+1} whose ideal sheaf I can be endowed with a nilpotent divided power structure (by setting all γ_{n}, n ≥ 2 equal to zero). There is a short exact sequence

0 → (1 + I)^{∗} → O_{X_{n+1}}^{∗} → O_{X_{n}}^{∗} → 0.

Moreover, the obstruction to lifting L_{n} to an invertible sheaf on X_{n+1} is the image ob in H^{2}(X_{n+1}, I) of the class of L_{n} in H^{1}(X_{n}, O_{X_{n}}^{∗}). A computation shows that ob is the image of pξ′ under the canonical map

H_{cris}^{2}(X/Σ, J_{X/Σ}) → H_{cris}^{2}(X_{n}/Σ, J_{X_{n}/Σ}) → H^{2}(X_{n+1}, I).

The arrow on the right uses that (X_{n} → X_{n+1}, γ) is an object of the crystalline site of X_{n} and that J_{X_{n}/Σ} restricts to I on this object. But pξ′ maps to p times the image of ξ′ in H^{2}(X_{n}, I) which is zero. Hence we can lift L_{n} to some invertible sheaf L_{n+1} on X_{n+1}.

Let (M, ∇, F) be as in the proof of Theorem 1. By our assumption that the degree 1 Hodge cohomology groups of X_{0}/k are zero and the assumption on torsion we conclude that R^{1}f_{∗}O_{X/Σ} = 0 and that H_{cris}^{2}(X_{n}/Σ, J_{X_{n}/Σ}) → H_{cris}^{2}(X_{n}/Σ, O_{X_{n}/Σ}) is injective. Hence, using the Leray spectral sequence and the base change theorem in crystalline cohomology we have

H_{cris}^{2}(X_{n}/Σ, J_{X_{n}/Σ}) ⊂ H_{cris}^{2}(X_{n}/Σ, O_{X_{n}/Σ}) = (M ⊗_{W [[t]]} D_{n})^{∇}

where D_{n} is the divided power envelope of the ideal (t^{n}) in the ring W [[t]]. Now D_{n} is p-torsion free and in fact injects into K[[t]] where K = f.f.(W). Since both pξ′ and c′_{1}(L_{n+1}) have the same value modulo t and are horizontal they have to agree. This proves the desired induction step. □

References

[AM77] M. Artin and B. Mazur, Formal groups arising from algebraic varieties, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 1, 87–131. MR 0457458 (56 #15663)

[Art74] M. Artin, Supersingular K3 surfaces, Ann. Sci. École Norm. Sup. (4) 7 (1974), 543–567 (1975). MR 0371899 (51 #8116)

[BM90] Pierre Berthelot and William Messing, Théorie de Dieudonné cristalline. III. Théorèmes d’équivalence et de pleine fidélité, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 173–247. MR 1086886 (92h:14012)

[Kat79] Nicholas M. Katz, Slope filtration of F -crystals, Journées de Géométrie Algébrique de Rennes (Rennes, 1978), Vol. I, Astérisque, vol. 63, Soc. Math. France, Paris, 1979, pp. 113–163. MR 563463 (81i:14014)

[MP11] Davesh Maulik and Bjorn Poonen, Néron-Severi groups under specialization, arXiv:0907.4781v3, 2011.