NÉRON-SEVERI GROUPS UNDER SPECIALIZATION

DAVESH MAULIK AND BJORN POONEN

Abstract. André used Hodge-theoretic methods to show that in a smooth proper family X → B of varieties over an algebraically closed field k of characteristic 0, there exists a closed fiber having the same Picard number as the geometric generic fiber, even if k is countable. We give a completely different approach to André’s theorem, which also proves the following refinement: in a family of varieties with good reduction at p, the locus on the base where the Picard number jumps is p-adically nowhere dense. Our proof uses the “p-adic Lefschetz (1, 1) theorem” of Berthelot and Ogus, combined with an analysis of p-adic power series. We prove analogous statements for cycles of higher codimension, assuming a p-adic analogue of the variational Hodge conjecture, and prove that this analogue implies the usual variational Hodge conjecture. Applications are given to abelian schemes and to proper families of projective varieties.

1. Introduction

1.1. The jumping locus. For a smooth proper variety X over an algebraically closed field, let NS X be its Néron-Severi group, and let ρ(X) be the rank of NS X. (A variety is a separated scheme of finite type over a field, possibly non-reduced or reducible. See Sections 2 and 3 for further definitions and basic facts.)

Now suppose that we have a smooth proper morphism X → B, where B is an irreducible variety over an algebraically closed field k of characteristic 0. If b ∈ B(k), then choices lead to an injection of the Néron-Severi group NS Xη̄ of the geometric generic fiber into the Néron-Severi group NS X_{b} of the fiber above b, so ρ(X_{b}) ≥ ρ(Xη̄): see Proposition 3.6. The jumping locus

B(k)_{jumping} := {b ∈ B(k) : ρ(X_{b}) > ρ(Xη̄)}

is a countable union of lower-dimensional subvarieties of B. If k is uncountable, it follows that B(k)_{jumping} ≠ B(k).

This article concerns the general case, in which k may be countable. Our goal is to present a p-adic proof of the following theorem, first proved by Y. André [And96, Théorème 5.2(3)]:

Theorem 1.1. Let k be an algebraically closed field of characteristic 0. Let B be an irreducible variety over k. Let X → B be a smooth proper morphism. Then there exists b ∈ B(k) such that ρ(X_{b}) = ρ(Xη̄).

Remark 1.2. In fact, Y. André’s result is more general, stated in terms of variation of the motivic Galois group in the context of his theory of “motivated cycles”. On the other hand, our techniques, which are completely different, give new information about the jumping locus.

Date: November 23, 2011.

2010 Mathematics Subject Classification. Primary 14C25; Secondary 14D07, 14F25, 14F30.

Key words and phrases. Néron-Severi group, Picard number, jumping locus, specialization, convergent isocrystal, geometric monodromy, variational Hodge conjecture.

D.M. is supported by a Clay Research Fellowship. B.P. is partially supported by NSF grants DMS-0841321 and DMS-1069236.

Special cases were proved earlier by T. Shioda [Shi81] and most notably T. Terasoma [Ter85]. The arguments of Terasoma and André involve, among other ingredients, an application of a version of Hilbert irreducibility for infinite algebraic extensions associated to ℓ-adic representations. We will say more about their methods and their relationship with ours in Section 8.

Remark 1.3. The condition ρ(X_{b}) = ρ(Xη̄) is equivalent to the condition that the specialization map NS Xη̄ → NS X_{b} is an isomorphism: see Proposition 3.6.

Remark 1.4. Theorem 1.1 can be trivially extended to an arbitrary ground field k of characteristic 0, to assert the existence of a closed point b ∈ B such that the geometric Picard number of X_{b} equals ρ(Xη̄). Similarly, one could relax the assumption on B and allow it to be any irreducible scheme of finite type over k.

Remark 1.5. For explicit nontrivial examples of families X → B over \overline{ℚ} with b ∈ B(\overline{ℚ}) such that ρ(X_{b}) = ρ(Xη̄), see [Shi81] and [vL07].

1.2. The p-adic approach. For our proof of Theorem 1.1, we embed a suitable finitely generated subfield of k in a p-adic field (see Section 6) and apply Theorem 1.7 below, which states that for a “family of varieties with good reduction” (in the strong sense of having a smooth proper family as in Setup 1.6, and not only the existence of models fiber-by-fiber), the jumping locus is p-adically nowhere dense.

Setup 1.6. Let K be a field that is complete with respect to a nontrivial discrete valuation, and let k be the residue field. For any valued field L, let O_{L} denote its valuation ring. Assume that char K = 0 and char k = p > 0, and that k is perfect. Let C be the completion of an algebraic closure of K; then C also is algebraically closed (see [Kür13, §46] or [Rib99, p. 142]). Let B be an irreducible separated finite-type O_{K}-scheme, and let f: X → B be a smooth proper morphism.

Theorem 1.7. Assume Setup 1.6. For b ∈ B(O_{C}) ⊆ B(C), let X_{b} be the C-variety above b. Then the set

B(O_{C})_{jumping} := {b ∈ B(O_{C}) : ρ(X_{b}) > ρ(Xη̄)}

is nowhere dense in B(O_{C}) for the analytic topology.

To prove Theorem 1.7, we apply a “p-adic Lefschetz (1, 1) theorem” of P. Berthelot and A. Ogus [BO83, Theorem 3.8] to obtain a down-to-earth local analytic description (Lemma 4.2) of the jumping locus in B(O_{C}). This eventually reduces the problem to a peculiar statement (Proposition 5.1) about linear independence of values of linearly independent p-adic power series.

Remark 1.8. It is well known (cf. [BLR90, p. 235]) that the archimedean analogue of Theorem 1.7 is false. For example, let B be an irreducible ℂ-variety, let E → B be a family of elliptic curves such that the j-invariant map j: B → 𝔸^{1} is dominant, and let X = E ×_{B} E. For an elliptic curve E over an algebraically closed field, ρ(E × E) = 2 + rk End E (cf. the Rosati involution comment in the proof of Proposition 1.13). So B(ℂ)_{jumping} is the set of CM points in B(ℂ). In the j-line, the set of CM points is the image of {z ∈ ℂ : im(z) > 0 and [ℚ(z) : ℚ] = 2} under the usual analytic uniformization by the upper half plane. This image is dense in 𝔸^{1}(ℂ), so its preimage under j is dense in B(ℂ).

Remark 1.9. Remark 1.8 is a particular case of a general topological density theorem of Mark Green [Voi03, Proposition 5.20], as we now explain. In the setting of Theorem 1.1 over k = ℂ, the Lefschetz (1, 1) theorem and [Voi03, Lemma 5.13] let us define countably many closed complex analytic subspaces Z_{i} of the analytification B^{an} such that the union of the associated subsets of B(ℂ) equals B(ℂ)_{jumping}. If some Z_{i} is reduced and of the “expected codimension” h^{2,0}, then [Voi03, Proposition 5.20] implies that B(ℂ)_{jumping} is dense in B(ℂ) with respect to the complex topology. The argument ultimately relies on the topological density of ℚ inside ℝ.

Remark 1.10. We can give a heuristic explanation of the difference between ℂ and a field like C = ℂ_{p}. Namely, [ℂ : ℝ] = 2, but the analogous p-adic quantity [ℂ_{p} : ℚ_{p}] is infinite (in fact, equal to 2^{ℵ_{0}} [Lam86]). So a subvariety in B(ℂ_{p}) of positive codimension can be thought of as having infinite ℚ_{p}-codimension. This makes it less surprising that a countable union of such subvarieties could be p-adically nowhere dense.

Remark 1.11. G. Yamashita, in response to an earlier version of this article, has generalized the p-adic Lefschetz (1, 1) theorem from the smooth case to the semistable case, and has used our method to extend Theorem 1.7 to the case where X → B is semistable [Yam11].

Remark 1.12. If X → B is as in Remark 1.8, but over an algebraic closure k of a finite field 𝔽_{p}, then again we have ρ(Xη̄) = 3, but now ρ(X_{b}) ≥ 4 for all b ∈ B(k) since every elliptic curve E over k has endomorphism ring larger than ℤ. Thus the characteristic p analogue of Theorem 1.1 fails. On the other hand, it seems likely that it holds for any algebraically closed field k that is not algebraic over a finite field.

1.3. Applications to abelian varieties. J.-P. Serre [Ser00, pp. 1–17] and R. Noot [Noo95, Corollary 1.5] used something like Terasoma’s method, combined with G. Faltings’ proof of the Tate conjecture for homomorphisms between abelian varieties, to prove that in a family of abelian varieties over a finitely generated field of characteristic 0, there exists a geometric closed fiber whose endomorphism ring equals that of the geometric generic fiber. Independently at around the same time, D. Masser [Mas96] used methods of transcendence theory to give a different proof, one that gives quantitative estimates of the number of fibers where the endomorphism ring jumps.

Theorem 1.1 reproves the existence result without Faltings’ work or transcendence theory, and Theorem 1.7 strengthens this by showing that in the p-adic setting, the corresponding jumping locus is p-adically nowhere dense in the good reduction locus:

Proposition 1.13. Assume Setup 1.6, and assume moreover that X → B is an abelian scheme. Then

{b ∈ B(O_{C}) : End Xη̄ ↪ End X_{b} is not an isomorphism}

is nowhere dense in B(O_{C}) for the analytic topology.

Proof. Choose a polarization on X_{η}, and replace B by a dense open subvariety to assume that it extends to a polarization of X → B. For a polarized abelian variety A over an algebraically closed field, let ι be the Rosati involution on (End A)_{ℚ} (where the subscript denotes ⊗ℚ). Then (NS A)_{ℚ} is isomorphic to the fixed subspace (End A)_{ℚ}^{ι}: see [Mum70, p. 190], for instance. This implies

ρ(A) = dim(End A)_{ℚ}^{ι}
ρ(A × A) = 2ρ(A) + dim(End A)_{ℚ}.

If in a family, dim(End A)_{ℚ} jumps, then so will ρ(A×A); conversely, if dim(End A)_{ℚ} does not jump, then neither does ρ(A) (since the Rosati involution respects specialization), so ρ(A×A) also does not jump. Thus the (End A)_{ℚ} jumping locus for X → B equals the Picard number jumping locus for X ×_{B} X → B. Apply Theorem 1.7 to X ×_{B} X. Finally, (End Xη̄)_{ℚ} ↪ (End X_{b})_{ℚ} is an isomorphism if and only if End Xη̄ ↪ End X_{b} is an isomorphism, as one sees by considering the action on torsion points (this uses characteristic 0). □

Remark 1.14. Theorem 1.7 of [Noo95] states that for any algebraic group G arising as a Mumford-Tate group of a complex abelian variety, there exists an abelian variety A over a number field F such that the Mumford-Tate group of A equals G and such that moreover the Mumford-Tate conjecture holds; i.e., the action of Gal(\overline{F} /F) on a Tate module T_{ℓ}A′ gives an open subgroup in G(ℚ_{ℓ}). A specialization result for the Mumford-Tate group follows easily from [And96, Théorème 5.2(3)] too.

It would be natural to conjecture a “nowhere dense” analogue, i.e., that the locus in a family of abelian varieties where the dimension of the Mumford-Tate group drops is p-adically nowhere dense in the good reduction locus. But we know how to prove this only if we assume Conjecture 9.2 from Section 9.

A proof similar to that of Proposition 1.13 yields another application of Theorem 1.1:

Proposition 1.15. Let k be an algebraically closed field of characteristic 0. Let A be an abelian variety defined over k. Let B be an irreducible k-variety. Let X → B be an abelian scheme such that X_{b} is isogenous to A for all b ∈ B(k). Then Xη̄ is isogenous to Aη̄ := A×_{k} η̄.

Sketch of proof. Let A ∼ ∏_{i=1}^{r} A_{i}^{n_{i}} be a decomposition of A up to isogeny into simple factors. Applying Theorem 1.1 to X × A_{i} → B shows that the multiplicity of (A_{i})η̄ in the decomposition of Xη̄ equals n_{i}. Since the relative dimension of X → B equals dim A, this accounts for all simple factors of Xη̄. □

Remark 1.16. At least when B is integral and X → B is projective (which is automatic if B is normal [FC90, 1.10(a)]), the conclusion of Proposition 1.15 implies that X → B becomes constant after a finite étale base change B′ → B. This can be proved as follows. The kernel of an isogeny Aη̄ → Xη̄ is the base extension of a finite group scheme G over k, since k is algebraically closed of characteristic 0. Replacing A by A/G, we may assume that Aη̄ ≃ Xη̄. Projectivity of X → B yields a polarization on X, and the corresponding polarization on Aη̄ comes from a polarization defined over k (cf. Proposition 3.1). Choose ℓ ≥ 3, and replace B by a finite étale cover such that X [ℓ] ≃ (ℤ/ℓℤ)_{B}^{2g}. This lets us choose level-ℓ structures so that Aη̄ → Xη̄ becomes an isomorphism of polarized abelian varieties with level-ℓ structure. Let M be the moduli scheme over k of polarized abelian varieties with level-ℓ structure. Then X gives rise to a k-morphism B → M mapping η̄ to a k-point. Since B is integral, B → M is constant.

Remark 1.17. Under the appropriate hypotheses on k and X → B, Theorem 1.7 proves the analogous strengthening of Proposition 1.15. Namely, assuming Setup 1.6, if X → B is an abelian scheme and Xη̄ is not isogenous to Aη̄, then the set of b ∈ B(O_{C}) such that X_{b} is isogenous to A is p-adically nowhere dense.

1.4. Outline of the article. After introducing some notation in Section 2, we review some standard facts about Néron-Severi groups and specialization maps in Section 3. The next three sections prove Theorem 1.7 and use it to prove Theorem 1.1: Section 4 discusses some basic properties of crystalline cohomology and convergent isocrystals, and applies them to give a local description of the jumping locus; Section 5 proves the key p-adic power series proposition to be applied to understand this local description. Section 6 completes the proofs of Theorems 1.7 and 1.1.

Section 7 gives an application of Theorem 1.1: if all closed fibers in a smooth proper family are projective, then there exists a dense open subvariety of the base over which the family is projective, assuming that the base is a variety in characteristic 0. Section 8, which uses only étale and Betti cohomology, and some Hodge theory, sketches André’s approach to Theorem 1.1, and compares the information it provides on the jumping locus to what is obtained from the p-adic approach.

Finally, Section 9 explains conditional generalizations of our results to cycles of higher codimension. The generalization of Theorem 1.7 is proved assuming a p-adic version of the variational Hodge conjecture (Conjecture 9.2). We also prove that the p-adic variational Hodge conjecture implies the classical variational Hodge conjecture.

2. Notation

If A is a commutative domain, let Frac(A) denote its fraction field. If A → B is a ring homomorphism, and M is an A-module, let M_{B} denote the B-module M ⊗_{A} B. If k is a field, then \overline{k} denotes an algebraic closure, chosen consistently whenever possible. Given a prime number p, let ℤ_{p} be the ring of p-adic integers, let ℚ_{p} = Frac(ℤ_{p}), choose algebraic closures \overline{ℚ} ⊆ \overline{ℚ}_{p}, and let ℂ_{p} denote the completion of \overline{ℚ}_{p}.

For any S-schemes X and T, let X_{T} be the T -scheme X ×_{S} T. For a commutative ring R, we may write R as an abbreviation for Spec R. If B is an irreducible scheme, let η denote its generic point. If b ∈ B, let κ(b) be its residue field and let b̄ = Spec \overline{κ(b)}. For example, if X → B is a morphism, then Xη̄ is called the geometric generic fiber. Also let κ(B) be the function field κ(η). If B is a variety over a field F, let |B| be the set of closed points of B; also choose an algebraic closure \overline{F} and for all b ∈ |B| view \overline{κ(b)} as a subfield of \overline{F}. If X is a variety over a field equipped with an embedding in ℂ, then X^{an} denotes the associated complex analytic space.

If 𝒳 is a complex analytic space and i is a nonnegative integer, then we have the Betti cohomology H^{i}(𝒳 , F) for any field F. If X is a variety over a field k, and i and j are integers with i ≥ 0, and ℓ is a prime not divisible by the characteristic of k, then we have the étale cohomology H_{ét}^{i}(X\overline{k}, ℚ_{ℓ}(j)), which is equipped with a Gal(\overline{k}/k)-action (replace \overline{k} by a separable closure if k is not perfect).

3. Basic facts on Néron-Severi groups

3.1. Picard groups and Néron-Severi groups. For a scheme or formal scheme X, let Pic X be its Picard group. If X is a smooth proper variety over an algebraically closed field, let Pic^{0} X be the subgroup consisting of isomorphism classes of line bundles algebraically equivalent to 0 (i.e., to 𝒪_{X}), and define the Néron-Severi group NS X := Pic X/ Pic^{0} X. The abelian group NS X is finitely generated [Nér52, p. 145, Théorème 2] (see [SGA 6, XIII.5.1] for another proof), and its rank is called the Picard number ρ(X).

Proposition 3.1. If k ⊆ k′ are algebraically closed fields, and X is a smooth proper kvariety, then the natural homomorphism NS X → NS X_{k′} is an isomorphism.

Proof. The Picard scheme Pic_{X/k} is a group scheme that is locally of finite type over k (this holds more generally for any proper scheme over a field: see [Mur64, II.15], which uses [Oor62]). Then Pic^{0} X is the set of k-points of the identity component of Pic_{X/k} [Kle05, 9.5.10]. So NS X is the group of components of Pic_{X/k}. Thus NS X is unchanged by algebraically closed base extension. □

Remark 3.2. The Nakai-Moishezon criterion [Deb01, Theorem 1.21] implies that ampleness of a Cartier divisor on a proper scheme X over any field K depends only on its class in NS X_{L} for any algebraically closed field L containing K.

3.2. Specialization of Néron-Severi groups.

Proposition 3.3 (cf. [SGA 6, X App 7]). Let R be a discrete valuation ring with fraction field K and residue field k. Fix an algebraic closure \overline{K} of K. Choose a nonzero prime ideal 𝔭 of the integral closure \overline{R} of R in \overline{K}, so \overline{k} := \overline{R}/𝔭 is an algebraic closure of k. Let X be a smooth proper R-scheme. Then there is a natural homomorphism

sp_{\overline{K},\overline{k}}: NS X\overline{K} → NS X\overline{k}.

depending only on the choices above. Moreover, if sp_{\overline{K},\overline{k}} maps a class [ℒ] to an ample class, then ℒ is ample.

Proof. As in [SGA 6, X App 7.8] or the proof of [BLR90, §8.4, Theorem 3], we have

(3.4)  Pic X_{K} ←^{∼} Pic X → Pic X_{k}.

If ℒ is a line bundle on X_{K} whose image in Pic X_{k} is ample, then the corresponding line bundle on X is ample relative to Spec R by [EGA III_{1}, 4.7.1], so ℒ is ample too.

For each finite extension L of K in \overline{K}, the integral closure R_{L} of R in L is a Dedekind ring by the Krull-Akizuki theorem [Bou98, VII.2.§5, Proposition 5], and localizing at 𝔭 ∩ R_{L} gives a discrete valuation ring R′_{L}. Take the direct limit over L of the analogue of (3.4) for R′_{L} to get Pic X\overline{K} → Pic X\overline{k} (cf. [SGA 6, X App 7.13.3]).

This induces NS X\overline{K} → NS X\overline{k} (cf. [SGA 6, X App 7.12.1]); a sketch of an alternative argument for this is as follows: First we can pass from R to its completion to reduce to the case that R is excellent. It suffices to show the following (after replacing R and K by finite extensions): Given a smooth proper geometrically connected K-curve C_{K} and a line bundle ℒ_{K} on X_{K} ×C_{K}, any two fibers above points in C_{K}(K) specialize to algebraically equivalent line bundles on X_{k}. By Lipman’s resolution of singularities for 2-dimensional excellent schemes [Lip78], C_{K} extends to a regular proper flat R-scheme C, and C_{k} is geometrically connected by Stein factorization (cf. [EGA III_{1}, 4.3.12]). The two specialized line bundles are fibers above points of C(k) of an extension of ℒ_{K} to the regular scheme X ×_{R} C, so they are algebraically equivalent.

The ampleness claim follows from Remark 3.2 and the statement for Pic already discussed.

□

Remark 3.5. In Proposition 3.3, if R is complete, or more generally henselian, then there is only one choice of 𝔭.

Proposition 3.6. Let B be a noetherian scheme. Let s, t ∈ B be such that s is a specialization of t (i.e., s is in the closure of {t}). Let p = char κ(s). Let X → B be a smooth proper morphism. Then it is possible to choose a homomorphism

sp_{t̄,s̄}: NS Xt̄ → NS Xs̄

with the following properties:

(a) If p = 0, then sp_{t̄,s̄} is injective and coker(sp_{t̄,s̄}) is torsion-free.

(b) If p > 0, then (a) holds after tensoring with ℤ[1/p].

(c) If sp_{t̄,s̄} maps a class [ℒ] to an ample class, then ℒ is ample. Moreover:

(i) In all cases, ρ(Xs̄) ≥ ρ(Xt̄).

Proof. A construction of sp_{t̄,s̄} is explained at the beginning of [SGA 6, X App 7.17]: the idea is to choose a discrete valuation ring R with a morphism Spec R = {s′, t′} → B mapping s′ to s and t′ to t, to obtain

NS Xt̄ →^{∼} NS Xt̄′ ⟶^{sp_{t̄′,s̄′}} NS X_{}_{s̄′} ←^{∼} NS Xs̄,

with the outer isomorphisms coming from Proposition 3.1.

For any prime ℓ ≠ p, there is a commutative diagram

(3.7)  [figure: NS Xt̄ ⊗ ℤ_{ℓ}; _{sp_{t̄,s̄}}; H_{ét}^{2}(Xt̄, ℤ_{ℓ}(1)); NS Xs̄ ⊗ ℤ_{ℓ}; H_{ét}^{2}(Xs̄, ℤ_{ℓ}(1))]

(cf. [SGA 6, 7.13.10]: there everything is tensored with ℚ, but the explanation shows that in our setting we need only tensor with ℤ[1/(i − 1)!] with i = 1). This proves the injectivity in (a) and (b). By (3.7), coker(sp_{t̄,s̄}) ⊗ ℤ_{ℓ} is contained in coker (NS Xt̄ ⊗ ℤ_{ℓ} → H_{ét}^{2}(Xt̄, ℤ_{ℓ}(1))). Using the Kummer sequence, one shows [Mil80, V.3.29(d)] that the latter is T_{ℓ} Br Xt̄ := \underleftarrow{lim}_{n}(Br Xt̄)[ℓ^{n}], which is automatically torsion-free; this proves the torsion-freeness in (a) and (b). Finally, (c) follows from the corresponding part of Proposition 3.3, and (i) follows from (a) and (b). □

Proposition 3.8. Let B be a noetherian scheme. For a smooth proper morphism X → B and a nonnegative integer n, define

B_{≥n} := {b ∈ B : ρ(Xb̄) ≥ n}.

(a) The set B_{≥n} is a countable union of Zariski closed subsets of B.

(b) If we base change by a morphism ι: B′ → B of noetherian schemes, then B′_{≥n} = ι^{−1}(B_{≥n}).

Proof. Proposition 3.1 proves (b).

Now we prove (a). Proposition 3.6(i) says that B_{≥n} contains the closure of any point in B_{≥n}. So if B = Spec A for some finitely generated ℤ-algebra A, then B_{≥n} is the (countable) union over b ∈ B_{≥n} of the closure of {b}. Combining this with (b) proves (a) for any noetherian affine scheme. Finally, if B is any noetherian scheme, write B = ⋃_{i=1}^{n} B_{i} with B_{i} affine, let C_{i} be the union of the closures in B of the generic points of all the irreducible components of the closed subsets of B_{i} appearing in the countable union for (B_{i})_{≥n}, and let C = ⋃_{i=1}^{n} C_{i}. Then B_{≥n} = ⋃_{i=1}^{n}(B_{i})_{≥n} ⊆ C and the opposite inclusion follows using Proposition 3.6(i) again as above. □

Corollary 3.9. Let k ⊆ k′ be algebraically closed fields. Let B be an irreducible k-variety. For a smooth proper morphism X → B, the jumping locus

B(k′)_{jumping} := {b ∈ B(k′) : ρ(X_{b}) > ρ(Xη̄)}

is the union of Z(k′) where Z ranges over a countable collection of closed k-subvarieties of B.

Proof. Proposition 3.8(a) yields subvarieties Z for the case k′ = k. The same subvarieties work for larger k′ by Proposition 3.8(b). □

3.3. Pathological behavior in positive characteristic. The material in this section is not needed elsewhere in this article. Let R be a discrete valuation ring, and define K, k, \overline{K}, \overline{k} as in Section 3.2. The two examples below show that sp_{\overline{K},\overline{k}} is not always injective.

Example 3.10. There exist R of equicharacteristic 2 and a smooth proper morphism X → Spec R such that X\overline{K} and X\overline{k} are Enriques surfaces of type ℤ/2ℤ and α_{2}, respectively [BM76, p. 222]. (The type refers to the isomorphism class of the scheme Pic^{τ} parametrizing line bundles numerically equivalent to 0.) In this case NS X\overline{K} → NS X\overline{k} has a nontrivial kernel, generated by the canonical class of X\overline{K}, an element of order 2.

Example 3.11. There exist R of mixed characteristic (0, 2) and a smooth proper morphism X → Spec R such that X\overline{K} and X\overline{k} are Enriques surfaces of type ℤ/2ℤ and μ_{2}, respectively [Lan83, Theorem 1.3], so again we have a nontrivial kernel.

Next, we give an example showing that coker(sp_{\overline{K},\overline{k}}) is not always torsion-free.

Example 3.12. Let O be the maximal order of an imaginary quadratic field in which p splits. Let O′ be the order of conductor p in O. Over a finite extension R of ℤ_{p}, there exists a pisogeny ψ: E → E′ between elliptic curves over R such that End E\overline{K} ≃ O and End E′ ; \overline{K} ≃ O′. Since p splits, E has good ordinary reduction and End E\overline{k} ≃ O. But ψ must reduce to either Frobenius or Verschiebung, so End E′ ; \overline{k} ≃ O too. Using that coker (End E′ ; \overline{K} → End E′ ; \overline{k}) is of order p, one can show that the cokernel of NS((E′ × E′)\overline{K}) → NS((E′ × E′)\overline{k}) contains nonzero elements of order p.

4. Convergent isocrystals and de Rham cohomology

We now begin work toward the p-adic proof of Theorem 1.1.

4.1. Goal of this section.

Definition 4.1. Assume Setup 1.6. Let d = dim B_{K}. Let b be a smooth \overline{K}-point on B_{K}. If B is a closed subscheme of 𝔸^{n}, a polydisk neighborhood of b is a neighborhood U of b in B(\overline{K}) in the analytic topology equipped with, for some ϵ > 0, a bijection

Δ_{d,ϵ} := \{(z_{1}, …, z_{d}) ∈ \overline{K}^{d} : |z_{i}| ≤ ϵ\} → U

defined by an n-tuple of power series in z_{1}, …, z_{d} with coefficients in some finite extension of K. (Such neighborhoods exist by the implicit function theorem. If we replace the embedding B ↪ 𝔸^{n}, by a different one, B ↪ 𝔸^{n′}, the notion of polydisk neighborhood of b changes, but the new system of polydisk neighborhoods of b is cofinal with the original one.) A polydisk neighborhood of b in an arbitrary B is a polydisk neighborhood of b in some affine open subscheme of B. Let ℋ (U) be the subring of \overline{K}[[z_{1}, …, z_{d}]] consisting of power series g with coefficients in some finite extension of K such that g converges on Δ_{d,ϵ}.

The goal of Section 4 is to prove the following:

Lemma 4.2. Assume Setup 1.6. Let b_{0} ∈ B(O\overline{K}) ⊆ B(\overline{K}) be such that B\overline{K} is smooth at b_{0}. Then there exists a polydisk neighborhood U of b_{0} contained in B(O\overline{K}) and a finitely generated ℤ-submodule Λ ⊆ ℋ (U)^{n} for some n such that

\{b ∈ U : ρ(X_{b}) > ρ(Xη̄)\} = ⋃_{\begin{matrix}λ∈Λ \\ λ≠0 \end{matrix}}(zeros of λ in U).

Its proof will be completed in Section 4.7.

Remark 4.3. The analogue over ℂ is a well-known consequence of the Lefschetz (1, 1) theorem, together with [Voi03, Lemma 5.13] and its proof. But the union will often be dense in B(ℂ), so this analogue is not useful for our purposes.

4.2. Coherent sheaves on formal schemes. Assume that K, O_{K}, and k are as in Setup 1.6. In this section, we work with noetherian formal schemes over Spf O_{K}. If X is a noetherian O_{K}-scheme, let X̂ be its completion with respect to the ideal sheaf p𝒪_{X}. Recall “formal GAGA”:

Theorem 4.4.

(a) If X is proper over O_{K}, then the functor Coh(X) → Coh(X̂) carrying ℱ to its p-adic completion ℱ̂is an equivalence of categories [EGA III_{1}, Corollaire 5.1.6].

(b) Under this equivalence, line bundles on X correspond to line bundles on X̂.

(c) If f: X → B is a proper morphism of noetherian O_{K}-schemes, f̂: X̂ → B̂ is the induced morphism of formal schemes, ℱ ∈ Coh(X), and q ≥ 0, then the natural morphism { \atop R^{q}f_{∗}ℱ} → R^{q}f̂_{∗}ℱ̂ in Coh(B̂) is an isomorphism [EGA III_{1}, Théorème 4.1.5]. (For a construction of this morphism in a more general context, see Section 4.5.3.)

We write K ⊗ (·) as an abbreviation for K ⊗_{O_{K}} (·). Similarly, \overline{K} ⊗ (·) means \overline{K} ⊗_{O_{K}} (·).

Definition 4.5 (cf. [Ogu84, Definition 1.1]). For any noetherian formal scheme T over Spf O_{K}, let Coh(K ⊗ 𝒪_{T}) denote the full subcategory of (K ⊗ 𝒪_{T})-modules isomorphic to K ⊗ℱ for some coherent 𝒪_{T} -module ℱ. Equivalently, we could consider the category whose objects are coherent 𝒪_{T} -modules but whose set of morphisms from ℱ to 𝒢 is K⊗Hom(ℱ , 𝒢).

Definition 4.6. Similarly, if B is a noetherian O_{K}-scheme, define Coh(K ⊗ 𝒪_{B}) to be the category whose objects are coherent 𝒪_{B}-modules and whose set of morphisms from ℱ to 𝒢 is K ⊗ Hom(ℱ , 𝒢).

Proposition 4.7. Let B be a noetherian O_{K}-scheme.

(a) The functor Coh(K ⊗ 𝒪_{B}) → Coh(B_{K}) sending ℱ to ℱ |_{B_{K}} is an equivalence of categories.

(b) There is a functor Coh(K ⊗ 𝒪_{B}) → Coh(K ⊗ 𝒪B̂) sending ℱ to ℱ̂.

(c) The resulting functor Coh(B_{K}) → Coh(K ⊗ 𝒪B̂) is compatible with pullback by an O_{K}- morphism B′ → B on each side.

(d) If B = Spec O_{K}, then the functor in (c) is an equivalence of categories.

Proof. This is all straightforward. In (d), both categories are equivalent to the category of finite-dimensional K-vector spaces. □

Remark 4.8. Let B be a separated finite-type O_{K}-scheme. The rigid-analytic generic fiber of the formal scheme B̂ is open in the rigid analytification (B_{K})^{an} of B_{K}. So, given a coherent 𝒪_{B}-module ℱ and b_{0} ∈ B(O_{K}) ⊆ B(K) such that B_{K} is smooth at b_{0}, we have two routes to construct the “restriction” of ℱ to a sheaf on a sufficiently small rigid-analytic neighborhood N of b_{0}: one route goes through ℱ̂, and the other goes through ℱ_{K} on B_{K}. The resulting sheaves are canonically isomorphic on a sufficiently small N. In particular, if ℱ_{K} is locally free of rank n on B_{K}, and U is a sufficiently small polydisk neighborhood of b_{0}, then a choice of local trivialization of ℱ_{K} lets us “restrict” global sections of \overline{K} ⊗ ℱ̂to obtain elements of ℋ (U)^{n}. (Although we have used some language of rigid geometry in this remark, it is not needed anywhere else in this article.)

4.3. Definition of convergent isocrystal. In this subsection, we recall some key notions of [Ogu84], specialized slightly to the case we need. Assume Setup 1.6, but without the requirement that B is irreducible.

Definition 4.9 (cf. [Ogu84, §1]). A p-adic formal scheme over O_{K} is a noetherian formal scheme T of finite type over Spf O_{K}. (See [EGA I, §10] for basic definitions regarding formal schemes.)

Given a p-adic formal O_{K}-scheme T, let T_{1} be the closed subscheme defined by the ideal sheaf p𝒪_{T}, and let T_{0} be the associated reduced subscheme (T_{1})_{red}.

Definition 4.10 (cf. [Ogu84, Definition 2.1]). An enlargement of B_{k} is a p-adic formal O_{K}- scheme T equipped with a k-morphism z: T_{0} → B_{k}. A morphism of enlargements (T′, z′) → (T, z) is an O_{K}-morphism T′ → T such that the induced k-morphism T′_{0} → T_{0} followed by z equals z′.

Example 4.11. Given s ∈ B(k), let ⌈s⌉ denote the enlargement (Spf O_{K}, Spec k →^{s} B_{k}) of B_{k}.

Example 4.12. If γ: B′ → B is a morphism of O_{K}-schemes of finite type, then we view B̂′ as an enlargement of B_{k} by equipping it with the k-morphism \binom{}{B′}_{0} = (B′_{k})_{red} → B_{k} induced by γ.

Definition 4.13 (cf. [Ogu84, Definition 2.7]). A convergent isocrystal on B_{k} consists of the following data:

(a) For every enlargement (T, z) of B_{k}, a sheaf E_{T} ∈ Coh(K ⊗ 𝒪_{T}).

(b) For every morphism of enlargements g: (T′, z′) → (T, z), an isomorphism θ_{g}: g^{∗}E_{T} → E_{T′} in Coh(K ⊗𝒪_{T}). If h: (T′′, z′′) → (T′, z′) is another, the cocycle condition θ_{h}∘h^{∗}θ_{g} = θ_{g∘h} is required, and θ_{id} = id.

4.4. Crystalline cohomology.

Definition 4.14 (cf. [Gro68, §7] and [Ber74, III.1.1]). Let K and k be as in Setup 1.6, and let W be the Witt ring of k, so O_{K} is finite as a W -module. Given a smooth proper k-variety X and q ∈ ℤ_{≥0}, we have the crystalline cohomology H_{cris}^{q}(X/W), which is a finite W -module. Define

H_{cris}^{q}(X/K) := K ⊗_{W} H_{cris}^{q}(X/W).

There is a Chern class homomorphism [Gro68, §7.4]

c_{1}^{cris} : Pic X → H_{cris}^{2}(X/K).

Remark 4.15. The fact that crystalline cohomology is a Weil cohomology [GM87] implies [Kle68, 1.2.1] that c_{1}^{cris}(ℒ) depends only on the image of ℒ in NS X\overline{k}.

If instead of a single k-variety we have a family, then the K-vector space H_{cris}^{2}(X/K) is replaced by a compatible system of sheaves, i.e., a convergent isocrystal:

Theorem 4.16 (cf. [Ogu84, Theorems 3.1 and 3.7]). Assume Setup 1.6, but without the requirement that B is irreducible. For each q ∈ ℤ_{≥0}, there exists a convergent isocrystal E on B_{k} with an isomorphism of K-vector spaces E_{⌈s⌉} ≃ H_{cris}^{q}(X_{s}/K) for each s ∈ B(k).

Following [Ogu84], we write R_{cris}^{q}f_{∗}𝒪_{X /K} for the convergent isocrystal E whose construction is suggested by Theorem 4.16. (We added the subscript _{cris} for extra clarity.) Despite the notation, it depends only on the morphism X_{k} → B_{k} (cf. [Ogu84, 3.9.3]).

4.5. De Rham cohomology.

4.5.1. Algebraic de Rham cohomology.

Definition 4.17. Let f: X → B be a smooth proper morphism of noetherian schemes, and let q ∈ ℤ_{≥0}. The q^{th} de Rham cohomology is defined as the coherent 𝒪_{B}-module ℝ^{q}f_{∗}Ω_{X /B}^{•} (cf. [Gro66]). It has a Hodge filtration in Coh(B) given by

Fil^{p}(ℝ^{q}f_{∗}Ω_{X /B}^{•}) := im (ℝ^{q}f_{∗}Ω_{X /B}^{≥p} → ℝ^{q}f_{∗}Ω_{X /B}^{•}) .

where Ω^{≥p} is obtained from Ω^{•} by replacing terms in degrees less than p by 0. Also define the coherent 𝒪_{B}-module

H^{02}(X) := \frac{ℝ^{2}f_{∗}Ω_{X /B}^{•}}{Fil^{1}(ℝ^{2}f_{∗}Ω_{X /B}^{•})}.

In the case B = Spec K for a field K, these 𝒪_{B}-modules are K-vector spaces (we may then write H^{02} instead of H^{02}), and there is a Chern class homomorphism (cf. [Har75, II.7.7])

c_{1}^{dR} : Pic X → H_{dR}^{2}(X) := ℍ^{2}(X , Ω_{X /K}^{•}).

De Rham cohomology over varieties behaves well under pullback in characteristic 0:

Proposition 4.18. Let K be a field of characteristic 0. Let B be a noetherian K-scheme. Let f: X → B be a smooth proper morphism.

(a) The 𝒪_{B}-module ℝ^{q}f_{∗}Ω_{X /B}^{•} is locally free, and Fil^{p}(ℝ^{q}f_{∗}Ω_{X /B}^{•}) is a subbundle for each p. In particular, H^{02}(X) is locally free.

(b) Let α: B′ → B be a morphism of noetherian K-schemes. Let f′: X′ → B′ be the base extension of f by α. The natural map

α^{∗}ℝ^{q}f_{∗}Ω_{X /B}^{•} → ℝ^{q}f′_{∗}Ω_{X′/B′}^{•}

is an isomorphism. Moreover, it sends α^{∗} Fil^{p}(ℝ^{q}f_{∗}Ω_{X /B}^{•}) isomorphically to Fil^{p}(ℝ^{q}f′_{∗}Ω_{X′/B′}^{•}). In particular, taking quotients yields

α^{∗}H^{02}(X) ≃ H^{02}(X′).

Proof. See [Del68, Théorème 5.5]. □

4.5.2. Formal de Rham cohomology. There is an analogous definition of de Rham cohomology for formal schemes:

Definition 4.19. Let g: Y → T be a smooth proper morphism of noetherian formal schemes over Spf O_{K} and let q ∈ ℤ_{≥0}. The sheaf K ⊗ℝ^{q}g_{∗}Ω̂_{Y/T}^{•} ∈ Coh(K ⊗𝒪_{T}) has a Hodge filtration in Coh(K ⊗ 𝒪_{T}) given by

Fil^{p}(K ⊗ ℝ^{q}g_{∗}Ω̂_{Y/T}^{•}) := im (K ⊗ ℝ^{q}g_{∗}Ω̂_{Y/T}^{≥p} → K ⊗ ℝ^{q}g_{∗}Ω̂_{Y/T}^{•} ) .

Also define the sheaf

H^{02}(Y/K) := \frac{K ⊗ ℝ^{2}g_{∗}Ω̂_{Y/T}^{•}}{Fil^{1}(K ⊗ ℝ^{2}g_{∗}Ω̂_{Y/T}^{•})}.

In the case T = Spf O_{K}, we define H_{dR}^{q}(Y/K) := K ⊗ ℝ^{q}g_{∗}Ω̂_{Y/T}^{•} and we have

c_{1}^{dR} : Pic Y → H_{dR}^{2}(Y/K).

4.5.3. Comparing algebraic and formal de Rham cohomology. Proposition 4.22 below shows that p-adically completing algebraically de Rham cohomology yields formal de Rham cohomology. In fact, the construction of the implied isomorphism uses very little about the de Rham complex. We thank Brian Conrad for suggesting the argument below.

Let f: X → B be a smooth proper morphism of noetherian O_{K}-schemes. Let ℱ^{•} be a bounded-below f^{−1}𝒪_{B}-linear complex of coherent 𝒪_{X} -modules (i.e., the morphisms ℱ^{i} → ℱ^{i+1} are f^{−1}𝒪_{B}-linear but not necessarily 𝒪_{X} -linear). Completing everything produces an f̂^{−1}𝒪B̂-linear complex ℱ̂^{•} of coherent 𝒪X̂-modules on the formal scheme X̂ over B̂. For example, if ℱ^{•} = Ω_{X /B}^{•}, then ℱ̂^{•} ≃ Ω̂_{X̂/B̂}^{•} naturally. Let i: X̂ → X and j: B̂ → B be the canonical morphisms of ringed spaces. In the derived category D^{+}(f^{−1}𝒪_{B}), we have morphisms

ℱ^{•} → i_{∗}ℱ̂^{•} → ℝi_{∗}ℱ̂^{•}.

Apply ℝf_{∗} to the composition and use fi = jf̂ to obtain a morphism

(4.20)  ℝf_{∗}ℱ^{•} → ℝj_{∗}ℝf̂_{∗}ℱ̂^{•}.

in D^{+}(𝒪_{B}). Since j is flat, ℝj_{∗} is adjoint to the flat pullback j^{∗} on bounded-below derived categories, so (4.20) yields a morphism

j^{∗}ℝf_{∗}ℱ^{•} → ℝf̂_{∗}ℱ̂^{•}

in D^{+}(𝒪B̂). Since j^{∗}𝒢 ≃ 𝒢̂ for any coherent 𝒪_{B}-module 𝒢, passing to homology sheaves yields 𝒪B̂-module morphisms

(4.21)  { \atop ℝ^{q}f_{∗}ℱ^{•}} → ℝ^{q}f̂_{∗}ℱ̂^{•}

for q ≥ 0. The whole construction is functorial with respect to f^{−1}𝒪_{B}-linear morphisms 𝒢^{•} → ℱ^{•}. In particular, (4.21) for the truncation ℱ^{≥p} maps to (4.21) for ℱ^{•}; taking the images on each side for all p makes (4.21) a morphism of filtered 𝒪B̂-modules.

Proposition 4.22. The natural map (4.21) is an isomorphism of filtered 𝒪B̂-modules. In particular, for ℱ^{•} = Ω_{X /B}^{•}, the p-adic completion functor Coh(B) → Coh(B̂) of Theorem 4.4(a) sends ℝ^{q}f_{∗}Ω_{X /B}^{•} to ℝ^{q}f̂_{∗}Ω̂_{X̂/B̂}^{•}.

Proof. It suffices to prove that (4.21) is an isomorphism of 𝒪B̂-modules, since then functoriality with respect to ℱ^{≥p} → ℱ^{•} gives the compatibility with filtrations. We can use functoriality and degree-shifting arguments to reduce to the case in which ℱ^{•} is a single coherent sheaf in degree 0. This case follows from Theorem 4.4(c). □

Corollary 4.23. Under the hypotheses of Proposition 4.22, the functor of Proposition 4.7(c) sends ℝ^{q}f_{∗}Ω_{X_{K}/B_{K}}^{•} to K ⊗ ℝ^{q}f̂_{∗}Ω̂_{X̂/B̂}^{•}, and respects the Hodge filtrations.

Proof. Apply K ⊗ (·) to the conclusion of Proposition 4.22. □

4.5.4. Fibers of de Rham cohomology. For families arising as the p-adic completion of a smooth proper morphism of noetherian O_{K}-schemes, we show that taking de Rham cohomology commutes with restriction to fibers:

Proposition 4.24. Let f: X → B be a smooth proper morphism of noetherian O_{K}-schemes. Let b ∈ B(O_{K}), let X_{b} be the pullback of X by Spec O_{K} →^{b} B, and let X_{b,K} = X_{b} × K.

(a) For each q ≥ 0, there are isomorphisms of filtered K-vector spaces

K ⊗ ℝ^{q}f̂_{∗}Ω̂_{X̂/B̂}^{•}|_{b} → H_{dR}^{q}(X̂_{b}/K).

(b) There is an isomorphism of K-vector spaces

H^{02}(X̂/K)|_{b} → H^{02}(X̂_{b}/K).

Proof.

(a) The algebraic analogue of this isomorphism, namely

(4.25)  (ℝ^{q}f_{∗}Ω_{X_{K}/B_{K}}^{•})|_{b_{K}} → H_{dR}^{q}(X_{b,K}),

is an isomorphism of filtered K-vector spaces by Proposition 4.18(b) with α = (b)_{K}: Spec K → B_{K}. Apply the functor in Proposition 4.7(d) to both sides of (4.25): on the right side we obtain H_{dR}^{q}(X̂_{b}/K), by Corollary 4.23 for X_{b} → Spec O_{K}; on the left side we may apply the functor before restricting to b (by Proposition 4.7(c)), and then Corollary 4.23 for X → B shows that we obtain K ⊗ ℝ^{q}f̂_{∗}Ω̂_{X̂/B̂}^{•}|_{b}.

(b) This follows from (a) for q = 2.

□

4.6. Comparison and the p-adic Lefschetz (1, 1) theorem. The following result identifies crystalline and de Rham cohomologies, even in the family setting.

Theorem 4.26. Assume Setup 1.6, but without the requirement that B is irreducible. Let (T, z) be an enlargement of B_{k}. Let f_{0} : X_{0} → T_{0} be obtained from f: X → B by base change along z: T_{0} → B_{k} ↪ B. Let g: Y → T be a smooth proper lifting of f_{0}. Then for each q ∈ ℤ_{≥0} there is a canonical isomorphism

(4.27)  σ_{cris,T} : K ⊗ ℝ^{q}g_{∗}Ω̂_{Y/T}^{•} → (R_{cris}^{q}f_{∗}𝒪_{X /K})_{T} .

Moreover, if t ∈ T (O_{K}) and s = z(t(Spec k)) ∈ B(k), and g arises as the p-adic completion of a smooth proper morphism of finite-type O_{K}-schemes, then the isomorphism σ_{cris,t} induced by σ_{cris,T} on the fibers above t fits in a commutative diagram

(4.28)  \xymatrix{Pic Y_{t} \ar[d]_{c_{1}^{dR}} \ar[r] & Pic X_{s} \ar[d]^{c_{1}^{cris}} \\ H_{dR}^{2}(Y_{t}/K) \ar[r]^{σ_{cris,t}} & H_{cris}^{2}(X_{s}/K).}

Proof. For (4.27), see [Ogu84, Theorem 3.8.2]. The bottom row of (4.28) is the case T = ⌈s⌉ of (4.27), with the identifications provided by Proposition 4.24(a) and Theorem 4.16. For the commutativity of (4.28), see [BI70, 2.3] and [BO83, 3.4]. □

Finally, we have what one might call a p-adic analogue of the Lefschetz (1, 1) theorem:

Theorem 4.29 (cf. [BO83, Theorem 3.8]). Let X →^{g} Spf O_{K} be a smooth proper p-adic formal scheme. Let ℒ_{k} be a line bundle on X_{k}. Then the following are equivalent:

(a) There exists m such that ℒ_{k}^{⊗p^{m}} lifts to a line bundle on X.

(b) The element of H_{dR}^{2}(X/K) := K ⊗ ℝ^{q}g_{∗}Ω̂_{X/O_{K}}^{•} corresponding under σ_{cris,O_{K}} to c_{1}(ℒ_{k}) ∈ H_{cris}^{2}(X_{k}/K) maps to 0 in the quotient H^{02}(X/K).

All the above definitions and results are compatible with respect to base change from O_{K} to O_{L} for a finite extension L of K [Ogu84, 3.6, 3.9.1, 3.11.1].

4.7. Proof of Lemma 4.2. By smoothness, there is a unique irreducible component of B\overline{K} containing the point of B(\overline{K}) corresponding to b_{0}. Replace K by a finite extension so that b_{0} and this component are defined over K, and replace B by the closure of this component. Then replace B by an open subscheme to assume that B is a closed subscheme of 𝔸^{r} = Spec O_{K}[x_{1}, …, x_{r}] for some r. Translate so that b_{0} is the origin in 𝔸^{r}. Let s ∈ B(k) be the reduction of b_{0}, the origin in 𝔸^{r}(k).

Let ℘: 𝔸^{r} → 𝔸^{r} be the morphism induced by the O_{K}-algebra homomorphism mapping each variable x_{i} to px_{i}. Let B′ = ℘^{−1}(B). Let b′_{0} ∈ B′(O_{K}) be the origin, so ℘(b′_{0}) = b_{0}. Let T = B̂′. Pulling back f: X → B yields a morphism of O_{K}-schemes X′ → B′. Completing yields a morphism of p-adic formal O_{K}-schemes X_{T} → T. We write f for any of these.

Let E be the convergent isocrystal R_{cris}^{2}f_{∗}𝒪_{X /K} on B_{k} given by Theorem 4.16. Because the special fiber of T maps to s ∈ B(k), we have a morphism of enlargements T → ⌈s⌉, so the definition of convergent isocrystal gives an identification

E_{T} ≃ E_{⌈s⌉} ⊗_{O_{K}} 𝒪_{T} ≃ H_{cris}^{2}(X_{s}/K) ⊗_{O_{K}} 𝒪_{T} ,

and the latter is a globally free sheaf in Coh(K ⊗ 𝒪_{T}).

Let ℒ_{k} be a line bundle on X_{s}. Then c_{1}^{cris}(ℒ_{k}) ∈ H_{cris}^{2}(X_{s}/K) gives rise to a constant section γ_{cris}(ℒ_{k}) := c_{1}^{cris}(ℒ_{k}) ⊗ 1 ∈ H_{cris}^{2}(X_{s}/K) ⊗_{O_{K}} 𝒪_{T} ≃ E_{T}. Applying σ_{cris,T}^{−1} yields a section γ_{dR}(ℒ_{k}) of K ⊗ ℝ^{2}f_{∗}Ω̂_{X_{T} /T}^{•}, which can be mapped to a section γ_{02}(ℒ_{k}) of the quotient sheaf H^{02}(X_{T} /K).

Let b′ ∈ B′(O_{K}). Let X_{b′} be the O_{K}-scheme obtained by pulling back X → B by the composition Spec O_{K} →^{b′} B′ → B. Let X_{b′,K} = X_{b′} × K. We can “evaluate” γ_{cris}(ℒ_{k}), γ_{dR}(ℒ_{k}), and γ_{02}(ℒ_{k}) at b′ by pulling back via Spf O_{K} →^{b′} T to obtain values in K-vector spaces

γ_{cris}(ℒ_{k}, b′) ∈ H_{cris}^{2}(X_{s}/K),
γ_{dR}(ℒ_{k}, b′) ∈ H_{dR}^{2}(X̂_{b′}/K), and
γ_{02}(ℒ_{k}, b′) ∈ H^{02}(X̂_{b′}/K).

Because the composition of enlargements b̂′ → T → ⌈s⌉ is the identity, the cocycle condition in Definition 4.13 yields γ_{cris}(ℒ_{k}, b′) = c_{1}^{cris}(ℒ_{k}).

Everything so far has been compatible with base extension from O_{K} to O_{L} for a finite extension L of K, and we may take direct limits to adapt the definitions and results above to O\overline{K}.

Proposition 3.6(b) gives an injective homomorphism

sp_{b̄′,s̄}: (NS X_{b′,\overline{K}})_{ℚ} ↪ (NS Xs̄)_{ℚ}.

Claim 4.30. The class [ℒ_{k}] ∈ (NS Xs̄)_{ℚ} is in the image of sp_{b̄′,s̄} if and only if γ_{02}(ℒ_{k}, b′) = 0.

Proof. Suppose that [ℒ_{k}] is in the image of sp_{b̄′,s̄}. After replacing ℒ_{k} by a tensor power, replacing K by a finite extension, and tensoring ℒ_{k} with a line bundle algebraically equivalent to 0 (which, by Remark 4.15, does not change any of the sections and values γ_{·}(·)), we may assume that the isomorphism class of ℒ_{k} in Pic X_{s} is the specialization of the isomorphism class of some line bundle ℒ_{K} on X_{b′,K}. Lift ℒ_{K} to a line bundle ℒ on the O_{K}-scheme X_{b′}. Completing yields ℒ̂∈ Pic X̂_{b′}. The commutative diagram in Theorem 4.26 shows that the element c_{1}^{dR}(ℒ̂) ∈ H_{dR}^{2}(X̂_{b′}/K) is mapped by σ_{cris,b′} to c_{1}^{cris}(ℒ_{k}). Then Theorem 4.29 applied to X̂_{b′} shows that γ_{02}(ℒ_{k}, b′) = 0.

Conversely, suppose that γ_{02}(ℒ_{k}, b′) = 0. Theorem 4.29 applied to X̂_{b′} shows that after raising ℒ_{k} to a power of p, we have that ℒ_{k} comes from some ℒ̂on X̂_{b′}. By Theorem 4.4(b), ℒ̂comes from some ℒ on X_{b′}. Then [ℒ_{k}] = sp_{b̄′,s̄}([\overline{K} ⊗ ℒ]). This completes the proof of Claim 4.30. □

Because of Remark 4.15, γ_{02} on Pic Xs̄ induces a homomorphism from NS Xs̄ to the space of sections of the sheaf \overline{K} ⊗_{K} H^{02}(X_{T} /K) on T. Let Λ_{T} be the image, so Λ_{T} is a finitely generated ℤ-module. For any b′ ∈ B′(O\overline{K}), evaluation at b′ as in Proposition 4.24(b) defines a homomorphism ev_{b′} from Λ_{T} (or (Λ_{T})_{ℚ}) to H^{02}(X̂_{b′}/\overline{K}).

Applying Claim 4.30 over all finite extensions of O_{K} yields

Corollary 4.31.

(a) For any b′ ∈ B′(O\overline{K}), ρ(X_{b′}) is the rank of the kernel of the composition

\xymatrix{(NS Xs̄)_{ℚ} \ar@{->>}[r]^{γ_{02}} \ar[rr]_{γ_{02}(−,b′)} & (Λ_{T})_{ℚ} \ar[r]^{^{ev^{b′}}} & H^{02}(X̂_{b′}/\overline{K}).}

(b) In particular,

(4.32)  ρ(X_{b′}) ≥ rk ker γ_{02},

with equality if and only if ev_{b′}: Λ_{T} → H^{02}(X̂_{b′}/\overline{K}) is injective.

Proposition 4.18(a) lets us apply Remark 4.8 to ℱ := H^{02}(X′) on B′ to obtain a polydisk neighborhood U′ of b′_{0} in B′(O\overline{K}) such that the subgroup Λ_{T} of global sections of \overline{K} ⊗ H^{02}(X_{T} /K) is expressed on U′ as a subgroup Λ′ of ℋ (U′)^{n}: in fact, if K is enlarged so that all elements of NS(Xs̄) are defined over the residue field k, then the coefficients of the elements in Λ′ lie in K. For b′ ∈ U′, we may interpret ev_{b′} concretely in terms of evaluation of power series in Λ′.

We will prove

(4.33)  rk ker γ_{02} = ρ(Xη̄)

by comparing both with ρ(X_{β}) for a “very general” β ∈ B′(𝒪\overline{K}).

Lemma 4.34. Let Z be a finite-type K-scheme that is smooth of pure dimension n. Fix z_{0} ∈ Z(K). Then no countable union of subschemes of Z of dimension less than n can contain a neighborhood of z_{0} in Z(K).

Proof. Shrink Z so that there is an étale morphism π: Z → 𝔸^{n}. By the definition of étale morphism given in [Mum99, III.§5, Definition 1] and the nonarchimedean implicit function theorem [Igu00, Theorem 2.1.1], π maps any sufficiently small neighborhood of z_{0} in Z(K) bijectively to a neighborhood of π(z_{0}) in 𝔸^{n}(K). Also, the scheme-theoretic image in 𝔸^{n} of any subscheme in Z of dimension less than n is of dimension less than n. So we may reduce to the case Z = 𝔸^{n}. This case follows by induction on n by projecting onto 𝔸^{n−1} and using the uncountability of K. □

Corollary 3.9 and Lemma 4.34 show that there exists β ∈ B′(𝒪\overline{K}) near b′_{0} such that ρ(X_{β}) = ρ(Xη̄). For any b′ ∈ B′(𝒪\overline{K}), we have

(4.35)  rk ker γ_{02} ≤ ρ(X_{β}) = ρ(Xη̄) ≤ ρ(X_{b′})

by (4.32), the choice of β, and Proposition 3.6(i), respectively.

Claim 4.36. For some b′ ∈ B′(𝒪\overline{K}), we have rk ker γ_{02} = ρ(X_{b′}).

Proof. Applying Proposition 5.1^{1} to Λ′ gives a nonempty open subset V of the polydisk in C^{d} such that ev_{u} is injective for u ∈ V. Since \overline{K} is dense in C, the set V contains a point in \overline{K}^{d}, and we let b′ be its image in B′(O\overline{K}), so ev_{b′} is injective. Apply Corollary 4.31(b) to b′. □

1The proofs in Section 5 do not rely on any results in this section.

Applying (4.35) with b′ as in Claim 4.36 proves (4.33).

Next, ℘ maps U′ isomorphically to a polydisk neighborhood U of b_{0} in B(O\overline{K}), and Λ′ corresponds to some Λ ⊆ ℋ (U)^{n}. Substituting (4.33) into Corollary 4.31(b), expressed on U in terms of Λ, shows that for b ∈ U,

ρ(X_{b}) ≥ ρ(Xη̄),

with equality if and only if λ(b) ≠ 0 for every nonzero λ ∈ Λ. This completes the proof of Lemma 4.2.

Remark 4.37. We conjecture that Lemma 4.2 holds with B(O\overline{K}) replaced by B(\overline{K}), but crystalline methods do not suffice to prove this. This would imply that Theorem 1.7 holds with B(C) in place of B(O_{C}).

5. Unions of zero loci of power series

To understand the nature of the following statement, the reader is urged to consider the r = 1 case first.

Proposition 5.1. Let C be as in Setup 1.6. Let D = {(z_{1}, …, z_{d}) ∈ C^{d} : v(z_{i}) ≤ ϵ for all i} for some ϵ > 0. Let R be the subring of C[[z_{1}, …, z_{d}]] consisting of power series that converge on D. Let r be a nonnegative integer. Let Λ be a finite-dimensional ℚ_{p}-subspace of R^{r}. Then there exists a nonempty analytic open subset U of D such that for all u ∈ U, the evaluationat-u map

ev_{u}: Λ → C^{r}
(f_{1}, …, f_{r}) ↦ (f_{1}(u), …, f_{r}(u))

is injective.

Remark 5.2. The archimedean analogue of Proposition 5.1 is false. For example, if Λ is the ℝ-span of 1, x, x^{2}, then there is no u ∈ ℂ such that the evaluation-at-u map Λ → ℂ is injective. Even if we consider only the ℤ-span of 1, x, x^{2}, the evaluation-at-u map fails to be injective on a dense subset of ℂ.

The rest of this section is devoted to the proof of Proposition 5.1. The proof is by induction on r. Because the base case r = 1 is rather involved, we begin by explaining the inductive step.

Suppose that r > 1, and that Proposition 5.1 is known for r′ < r. Let π: R^{r} → R^{r−1} be the projection to the first r − 1 coordinates. Let Λ^{(r)} and Λ_{r−1} be the kernel and image of π|_{Λ}. View Λ^{(r)} as a subgroup of R. For any u ∈ D, we have a commutative diagram with exact rows

(5.3)  \xymatrix{0 \ar[r] & Λ^{(r)} \ar[r] \ar[d]^{_{ev_{u}}} & Λ \ar[r] \ar[d]^{_{ev_{u}}} & Λ_{r−1} \ar[r] \ar[d]^{_{ev_{u}}} & 0 \\ 0 \ar[r] & C \ar[r] & C^{r} \ar[r] & C^{r−1} \ar[r] & 0.}

The inductive hypothesis applied to Λ_{r−1} ⊆ R^{r−1} gives a closed polydisk D′ ⊆ D such that the right vertical map in (5.3) is injective for all u ∈ D′. The inductive hypothesis applied to Λ^{(r)} ⊆ R gives a closed polydisk D′′ ⊆ D′ such that the left vertical map in (5.3) is injective for all u ∈ D′′. Then for u ∈ D′′, the middle vertical map in (5.3) is injective. This completes the proof of the inductive step.

Before discussing the base case r = 1, we prove another lemma. Let v: C → ℚ ∪ {+∞} be the valuation on C, normalized by v(p) = 1. If t⃗ = (t_{1}, …, t_{s}) ∈ C^{s} for some s, define v(t⃗) := min_{j} v(t_{j}).

Lemma 5.4.

(a) If t⃗_{1}, …, t⃗_{n} ∈ C^{s} are ℚ_{p}-independent, then {v (∑ a_{i}t⃗_{i}) : (a_{1}, …, a_{n}) ∈ (ℤ_{p})^{n} − (pℤ_{p})^{n}} is finite.

(b) If t_{1}, …, t_{n} ∈ C, then {v(t) : t = ∑ a_{i}t_{i} ≠ 0 for some a_{i} ∈ ℚ_{p}} has finite image in ℚ/ℤ.

Proof.

(a) The function

(ℤ_{p})^{n} − (pℤ_{p})^{n} → ℚ
(a_{1}, …, a_{n}) ↦ v (∑ a_{i}t⃗_{i})

is continuous for the p-adic topology on the left and the discrete topology on the right, so by compactness its image is finite.

(b) By replacing the t_{i} with a basis for their ℚ_{p}-span, we reduce to the s = 1 case of (a). □

From now on, we assume r = 1; i.e., Λ ⊆ R. Choose a ℚ_{p}-basis f_{1}, …, f_{n} of Λ. We may assume that D is the unit polydisk, so the coefficients of each f_{i} tend to 0. Multiply all the f_{i} by a single power of p to assume that f_{i} ∈ O_{C}[[z_{1}, …, z_{d}]]. For some m, the images of f_{i} in C[[z_{1}, …, z_{d}]]/(z_{1}, …, z_{d})^{m} are ℚ_{p}-independent, because a descending sequence of vector spaces in Λ with zero intersection must be eventually zero. Fix such an m.

Let M be the set of monomials μ in the z_{i} whose total degree deg μ is less than m. For each μ, let c_{i}^{μ} ∈ C be the coefficient of μ in f_{i}. For each μ ∈ M, apply Lemma 5.4(b) to c_{1}^{μ}, …, c_{n}^{μ} to obtain a finite subset S_{μ} of ℚ/ℤ. Let S = ⋃_{μ∈M} S_{μ}. Let q_{1}, …, q_{d} be distinct primes greater than m that do not appear in the denominators of rational numbers representing elements of S. For i = 1, …, n, let t⃗_{i} ∈ C^{#M} be the vector whose coordinates are the c_{i}^{μ} for μ ∈ M. By choice of m, the t⃗_{i} are ℚ_{p}-independent. By Lemma 5.4(a), the set

{v (∑ a_{i}t⃗_{i}) : (a_{1}, …, a_{n}) ∈ (ℤ_{p})^{n} − (pℤ_{p})^{n}}

is finite; choose A ∈ ℚ larger than all its elements. Choose a positive integer N such that

(5.5)  mN > (m − 1)(N + 1/q_{i}) + A

for all i. Let

U := {(z_{1}, …, z_{n}) ∈ C^{n} : v(z_{i}) = N + 1/q_{i} for all i},

so U is open in D.

Consider an arbitrary nonzero element f = ∑_{all μ} c^{μ}μ of Λ. So f = ∑_{i=1}^{n} a_{i}f_{i} for some (a_{1}, …, a_{n}) ∈ ℚ_{p}^{n} − {0}. Let u = (u_{1}, …, u_{d}) ∈ U. It remains to show that f(u) ≠ 0.

By multiplying f by a power of p, we may assume that (a_{1}, …, a_{n}) ∈ (ℤ_{p})^{n} − (pℤ_{p})^{n}. If μ = z_{1}^{e_{1}} ⋯ z_{d}^{e_{d}} and deg μ ≥ m, then

(5.6)  v(c^{μ}μ(u)) ≥ 0 + e_{1}N + ⋯ + e_{d}N ≥ mN

On the other hand, the definition of A yields ξ = z_{1}^{e_{1}} ⋯ z_{d}^{e_{d}} ∈ M such that v(c^{ξ}) < A, so that

(5.7)  v(c^{ξ}ξ(u)) < A + ∑_{i=1}^{d} e_{i}(N + 1/q_{i}) ≤ mN

by (5.5).

To show that f(u) ≠ 0, it remains to show that the valuations v(c^{μ}μ(u)) for μ ∈ M such that c^{μ} ≠ 0 are distinct, since then the minimum of these is finite and equals v(f(u)), by (5.6) and (5.7). In fact, if μ = z_{1}^{e_{1}} ⋯ z_{d}^{e_{d}} and deg μ < m and c^{μ} ≠ 0, then for each i the choice of the q_{i} implies v(c^{μ}) ∈ ℤ_{(q_{i})} (i.e., q_{i} does not divide the denominator of v(c^{μ})), so

v(c^{μ}μ(u)) ∈ \frac{e_{i}}{q_{i}} + ℤ_{(q_{i})};

moreover e_{i} ≤ deg μ < m < q_{i}, so e_{i} is determined by v(c^{μ}μ(u)) whenever c^{μ} ≠ 0. This completes the proof of Proposition 5.1.

6. Proof of Theorems 1.7 and 1.1

Lemma 6.1. Let k ⊆ k′ be an extension of algebraically closed valued fields such that k is dense in k′. Then for any finite-type k-scheme B, the set B(k) is dense in B(k′) with respect to the analytic topology.

Proof. Let b′ ∈ B(k′) be a point to be approximated by k-points. We may replace B by the Zariski closure of the image of b′ under B_{k′} → B. Then b′ is a smooth point. We may shrink B to assume that there is a finite étale morphism π: B → U of some degree d ≥ 1 for some open subscheme U of 𝔸^{n} for some n. Since k is dense in k′, we can approximate π(b′) ∈ (k′)^{n} arbitrarily well by points u ∈ U(k) ⊂ k^{n}. By “continuity of the set of roots of a polynomial as a function of the coefficients”, above each u we can choose one geometric point b of π^{−1}(u) such that b → b′ as u → π(b′). Since k is algebraically closed, b ∈ B(k). □

Proof of Theorem 1.7. Let b_{0} ∈ B(O_{C}). We need to show that any neighborhood U_{0} of b_{0} in B(O_{C}) contains a nonempty open set V that does not meet B(O_{C})_{jumping}. By Lemma 6.1 we may assume that b_{0} ∈ B(O\overline{K}). Lemma 4.2 gives a smaller neighborhood U_{1} of b_{0} in B(O_{C}) in which the jumping locus is described explicitly in terms of power series, and Proposition 5.1 gives a nonempty open subset V of U_{1} such that ρ(X_{b}) = ρ(Xη̄) for all b ∈ V ∩ B(O\overline{K}).

Suppose that b ∈ V ∩ B(O_{C})_{jumping}. By Corollary 3.9, b is contained in a \overline{K}-variety Z such that Z(C) ⊆ B(C)_{jumping}. Lemma 6.1 gives b′ ∈ V ∩ Z(\overline{K}) ⊆ B(𝒪\overline{K})_{jumping}, which contradicts the definition of V. □

Proof that Theorem 1.7 implies Theorem 1.1. Assume that k and X → B are as in Theorem 1.1. Replacing B by a dense open subscheme, we may assume that B is smooth over k. Choose a finitely generated ℤ-algebra A in k such that X → B is the base extension of a morphism X_{A} → B_{A} of A-schemes. By localizing A, we may assume that X_{A} → B_{A} is a smooth proper morphism, and that B_{A} → Spec A is smooth with geometrically irreducible fibers [EGA IV_{3}, 8.10.5(xii), 9.7.7(i)], [EGA IV_{4}, 17.7.8(ii)]. Because of Proposition 3.1, we may replace k by the algebraic closure of Frac(A) in k.

By [Cas86, Chapter 5, Theorem 1.1], there exists an embedding A ↪ ℤ_{p} for some prime p. Extend it to an embedding k ↪ ℂ_{p}. Base extend by A ↪ ℤ_{p} to obtain X_{ℤ_{p}} → B_{ℤ_{p}}. Since B_{ℤ_{p}} → Spec ℤ_{p} is smooth with geometrically irreducible special fiber, the set B_{ℤ_{p}}(O_{ℂ_{p}}) is nonempty. Apply Theorem 1.7 to X_{ℤ_{p}} → B_{ℤ_{p}} to find a nonempty open subset U of B_{ℤ_{p}}(O_{ℂ_{p}}) ⊂ B_{ℤ_{p}}(ℂ_{p}) = B(ℂ_{p}) in the analytic topology such that ρ(X_{b}) = ρ(Xη̄) for all b ∈ U. The field k is dense in ℂ_{p} since even \overline{ℚ} is dense in ℂ_{p}, so Lemma 6.1 shows that U contains a point b of B(k); this b is as required in Theorem 1.1, because of Proposition 3.1. □

7. Projective vs. proper

Theorem 7.1. Let B be a variety over a field k of characteristic 0. Let X → B be a smooth proper morphism such that all closed fibers are projective. Then there exists a Zariski dense open subvariety U of B such that X_{U} → U is projective.

Proof. We may assume that B is irreducible, say with generic point η. Theorem 1.1 yields a closed point b ∈ B such that a specialization map NS Xη̄ → NS Xb̄ is an isomorphism. By assumption, X_{b} is projective, so we may choose an ample line bundle ℒb̄ on Xb̄. By construction of b, the class [ℒb̄] ∈ NS Xb̄ comes from the class in NS Xη̄ of some line bundle ℒ on Xη̄. By Proposition 3.6(c), ℒ is ample. Then ℒ comes from a line bundle on X_{L} for some finite extension L of κ(B), so X_{L} is projective, so X_{η} is projective [EGA II, 6.6.5]. Finally, spreading out shows that X_{U} → U is projective for some dense open subscheme U of B [EGA IV_{3}, 8.10.5(xiii)]. □

Remark 7.2. Under the hypotheses of Theorem 7.1, we may also deduce that every fiber of X → B is projective: just apply Theorem 7.1 to each irreducible subvariety of B. But we cannot deduce that X → B is projective, or even that B can be covered by open sets U such that X_{U} → U is projective, as the following example of Raynaud shows.

Example 7.3 ([Ray70, XII.2]). Let A be a nonzero abelian variety over a field k. Let σ be the automorphism (x, y) ↦ (x, x + y) of A × A. Let B be a rational curve whose singular locus is a single node, and let B̃ be its normalization. Then one can construct an abelian scheme A → B whose base extension to B̃ is simply (A × A) × B̃ with the fibers above the two preimages of the node identified via σ. Moreover, for any neighborhood U of the node, A_{U} → U is not projective.

Question 7.4. Can one construct a counterexample to Theorem 7.1 over k = \overline{𝔽}_{p}?

Example 7.5. With notation as in Example 7.3, one can show that the first projection A × A × B̃ → A factors through a morphism A → A whose fiber above a given a ∈ A(k) is projective if and only if a is a torsion point. This provides a counterexample over \overline{𝔽}_{p} except for the fact that A → A is not smooth.

8. Comparison with the Hodge-theoretic approach

Both André’s Hodge-theoretic proof of Theorem 1.1 and our proof give information about the jumping locus beyond the mere existence of a non-jumping point. In this section, we compare these refinements.

8.1. Refinement in terms of sparse sets. To state the more precise information about the jumping locus that André’s approach yields, we introduce the notion of “sparse”, which is a version for closed points of the notion “thin” defined in [Ser97, §9.1] in the context of Hilbert irreducibility.

Definition 8.1. Let F be a finitely generated field over ℚ. Let B be an F -variety. Call a subset S of |B| sparse if there exists a dominant and generically finite morphism π: B′ → B of irreducible F -varieties such that for each s ∈ S, the fiber π^{−1}(s) is either empty or contains more than one closed point.

Remark 8.2. In the context of Definition 8.1, call a subset of B(\overline{F}) sparse if and only if the associated set of closed points is sparse.

André’s approach yields the following:

Theorem 8.3. Let F be a field that is finitely generated over ℚ. Let B be an irreducible F -variety, and let f: X → B be a smooth projective morphism. Then the set

B_{jumping} := {b ∈ |B| : ρ(Xb̄) > ρ(Xη̄)}.

is sparse.

Remark 8.4. Although Theorem 8.3 has a restriction on F and assumes that f is projective (not just proper), an easy argument shows that it still implies the general case of Theorem 1.1. The extension to the proper case uses Chow’s lemma and weak factorization of birational maps in characteristic 0 (see the first sentence of Remark 2 following Theorem 0.3.1 in [AKMW02]).

The strategy of the proof of Theorem 8.3 is to use Deligne’s global invariant cycle theorem ([Del71] or [Voi03, 4.3.3]) and the semisimplicity of the category of polarized Hodge structures to decompose the Hodge structure on the Betti cohomology H^{2}(X_{b}^{an}, ℚ) as H ⊕ H^{⊥}, where H consists of the classes invariant under some finite-index subgroup of the geometric monodromy group. The space H carries a Hodge structure independent of b; after an étale base change, H comes from the cohomology of a smooth completion of the total space of the family. On the other hand, the “essentially varying part” H^{⊥} is susceptible to Terasoma’s argument ([Ter85] and [GGP04, Lemma 8]), which shows that the corresponding subspace of étale cohomology has no nonzero Tate classes for b outside a sparse set.

For a complete proof, see [And96, §5.2]. For an exposition of the Hodge-theoretic aspects of the proof, see also [Voi10].

8.2. Sparse versus p-adically nowhere dense. The purpose of this subsection is to develop basic properties of sparse sets, and to discuss the difference between the two notions of smallness appearing in Theorems 8.3 and 1.7.

Proposition 8.5.

(a) For any dense open subscheme U of B and any S ⊆ |B|, the following are equivalent: (a) S is sparse,

(b) S ∩ U is sparse as a subset of |B|,

(c) S ∩ U is sparse as a subset of |U|.

In particular, for any closed subscheme Z ⊊ B, the subset |Z| is sparse in |B|.

(b) A finite union of sparse sets is sparse.

(c) A dominant and generically finite morphism of irreducible varieties maps sparse sets to sparse sets.

(d) If S is a sparse subset of |𝔸^{n}|, then S ∩ 𝔸^{n}(F) is a thin set.

(e) If S ⊆ |B| is sparse, then |B| − S is Zariski dense in B.

Proof.

(a) Take B′ = B − Z.

(b) Let V be an irreducible component dominating B of the fiber product of the varieties B′ over B. By (a), we may replace B by a dense open subscheme U, and the other schemes by the corresponding preimages of U. This lets us assume that each morphism V → B′ and B′ → B is surjective. If s is in the union of the sparse sets, then its preimage in some B′ contains more than one closed point, and so does its preimage in V.

(c) Let f: C → B be the morphism, and let S be a sparse subset of |C|, witnessed by C′ → C. By (a), we may shrink B to assume that the morphisms C′ → C → B are surjective. If s ∈ S, then the preimage of f(s) in C′ contains more than one closed point.

(d) By definition.

(e) If not, then by (a) there is a dense open subscheme U of B such that |U| is sparse in |U|. After shrinking U, there is a finite surjective morphism from U to a nonempty open subscheme V of some 𝔸^{n}. Then |V | is sparse in |𝔸^{n}| by (c), so V (F) is a thin set by (d). This contradicts the fact that a finitely generated field F over ℚ is Hilbertian. □

The following proposition shows that neither of the properties of being sparse or p-adically nowhere dense implies the other.

Proposition 8.6.

(i) Let F be a number field. Fix embeddings F ⊆ \overline{ℚ} ⊆ ℂ_{p}. Let π: B′ → B be any dominant and generically finite morphism of irreducible F -varieties such that dim B > 0 and deg π > 1. Let S be the preimage of

{b ∈ |B| : π^{−1}(b) is not a single closed point}.

in B(\overline{ℚ}). Then S is dense in B(ℂ_{p}). (ii) There exists a non-sparse subset S of |𝔸_{ℚ}^{1}| such that the associated subset of 𝔸^{1}(\overline{ℚ}) is

p-adically nowhere dense in 𝔸^{1}(ℂ_{p}).

Proof.

(i) Let T := \{b′ ∈ B′(\overline{ℚ}) : F (b′) = F (π(b′))\}. By replacing B′ and B by dense open subschemes, we may assume that B′ → B is finite étale, so B′(ℂ_{p}) → B(ℂ_{p}) is surjective. Then it suffices to prove that T is p-adically dense in B′(ℂ_{p}), or equivalently, p-adically dense in B′(\overline{ℚ}). To approximate a point P in B′(\overline{ℚ}) by a point in T, choose an irreducible F -curve C in B through π(P) (maybe not geometrically irreducible), and let C′ be the irreducible component of π^{−1}(C) containing P. By replacing B′ → B by C′ → C, we may assume that dim B = 1.

After replacing B′ and B by dense open subschemes, we can compose with a finite étale morphism B → 𝔸^{1} to reduce to the case that B is a dense open subscheme of 𝔸^{1}. By the primitive element theorem, after shrinking again, we may find a dominant morphism ρ: B′ × 𝔸^{1} such that (π, ρ): B′ → B × 𝔸^{1} is an immersion.

Let u_{0} ∈ B′(\overline{ℚ}). We need to p-adically approximate u_{0} by some u ∈ T. Let E := F (u_{0}). Let d := [E : F]. Let σ_{1}, …, σ_{d} be the F -embeddings E → \overline{ℚ}. Let Z be the Zariski closure of the subscheme of (𝔸^{1})^{d} whose geometric points are (Q_{1}, …, Q_{d}) such that for some i ≠ j, we have π(ρ^{−1}(Q_{i}))∩π(ρ^{−1}(Q_{j})) ≠ ∅. Since Z ≠ (𝔸^{1})^{d}, Lemma 8.7 below gives Q ∈ 𝔸^{1}(E) p-adically close to ρ(u_{0}) such that (σ_{1}(Q), …, σ_{d}(Q)) ∉ Z.

For sufficiently close Q, there exists u ∈ ρ^{−1}(Q) ⊆ B′(\overline{ℚ}) p-adically close to u_{0}. By definition of Z, the Gal(\overline{ℚ}/F)-conjugates of u map to distinct points in B(\overline{ℚ}), so u ∈ T.

(ii) The set ℚ is p-adically nowhere dense in ℂ_{p}. On the other hand, ℚ is not thin in ℚ,

so Proposition 8.5(d) proves that ℚ is not sparse.

□

Lemma 8.7. Let E ⊇ F be an extension of number fields. Fix inclusions F ⊆ E ⊆ \overline{ℚ} ⊆ ℂ_{p}. Let σ_{1}, …, σ_{d} be the F -embeddings E → \overline{ℚ}. If Z ⊊ (𝔸1 ; \overline{ℚ})^{d} is a closed subscheme, then the set W := \{Q ∈ 𝔸^{1}(E) : (σ_{1}(Q), …, σ_{d}(Q)) ∉ Z(\overline{ℚ})\} is p-adically dense in 𝔸^{1}(E).

Proof. Let R be the restriction of scalars Res_{E/F} 𝔸^{1}, which is isomorphic to 𝔸_{F}^{d}. The bijection ι: R(F) → 𝔸^{1}(E) is p-adically continuous for the topologies induced by F ⊆ E ⊆ ℂ_{p} (though its inverse need not be, since the topology on 𝔸^{1}(E) is induced by just one of the valuations on E extending the p-adic valuation on F). The embeddings σ_{1}, …, σ_{d} define an isomorphism s: R\overline{ℚ} → (𝔸^{1})d ; \overline{ℚ}. The complement of s^{−1}(Z) in R(F) is p-adically dense in R(F), so its image under ι is p-adically dense in 𝔸^{1}(E). □

Remark 8.8. In the setting of Theorem 1.1, but over a field k of characteristic 0 that need not be algebraically closed, either Theorem 8.3 or Theorem 1.7 can be used to show that there exists d ≥ 1 such that

{b ∈ B | [κ(b) : k] ≤ d and ρ(Xb̄) = ρ(Xη̄)}

is Zariski dense in B. In fact, Theorem 8.3 (with Proposition 8.5(c,d)) proves the stronger result that the degree of any generically finite rational map B −−→ ℙ^{n} can serve as d.

9. Cycles of higher codimension

Many of our arguments apply to specialization of cycles of codimension greater than 1, but the results are conditional. André obtains a generalization at the cost of replacing algebraic cycles by “motivated cycles”, which are the same if the Hodge conjecture is true. In the padic approach, we would need a higher-codimension analogue (Conjecture 9.2) of the p-adic Lefschetz (1, 1) theorem (Theorem 4.29).

9.1. Cycle class maps. To state the generalizations, we recall some standard definitions. For a smooth proper variety X over a field, let 𝒵^{r}(X)_{ℚ} be the vector space of codimension-r cycles with ℚ coefficients.

Definition 9.1. Let X be a smooth proper variety over an algebraically closed field k of characteristic 0. Fix a prime ℓ. Define ρ^{r}(X) as the dimension of the ℚ_{ℓ}-span of the image of the ℓ-adic cycle class map

cl_{ét} : 𝒵^{r}(X)_{ℚ} → H_{ét}^{2r}(X, ℚ_{ℓ}(r)).

The homomorphism cl_{ét} factors through the vector space 𝒵^{r}(X)_{ℚ}/alg of cycles modulo algebraic equivalence. Suppose that k′ is an algebraically closed field extension of k. Then every element of 𝒵^{r}(X_{k′})_{ℚ}/alg comes from a cycle over κ(V) ⊆ k′ for some k-variety V, and can be spread out over some dense open subvariety in V, and hence is algebraically equivalent to the k′-base extension of the resulting cycle above any k-point of V. So 𝒵^{r}(X)_{ℚ}/alg → 𝒵^{r}(X_{k′})_{ℚ}/alg is surjective (it is injective too), and ρ^{r}(X) = ρ^{r}(X_{k′}). This, together with standard comparison theorems, shows that ρ^{r}(X) equals its analogue defined using the Betti cycle class map

cl_{Betti} : 𝒵^{r}(X)_{ℚ} → H^{2r}(X^{an}, ℚ)

when k = ℂ, and also its analogue defined using de Rham cohomology (see [Har75, II.7.8]). The comparison with Betti cohomology shows that ρ^{r}(X) equals dim_{ℚ} cl_{Betti}(𝒵^{r}(X)_{ℚ}), which is independent of ℓ.

As follows from [SGA 6, X App 7, especially §7.14], all the facts in Section 3.2 (other than facts not needed for the proofs of Theorem 1.1, namely the claims about ampleness and about the cokernel of specialization being torsion-free) have analogues with NS X replaced by cl_{ét}(𝒵^{r}(X)_{ℚ}). In particular, in a smooth proper family, ρ^{r}(X) can only increase under specialization, and the jumping locus for ρ^{r} is a countable union of Zariski closed subsets.

9.2. The p-adic approach. Now assume that K, O_{K}, k are as in Setup 1.6. Let X → Spec O_{K} be a smooth proper morphism. We have a specialization map sp, cycle class maps cl_{dR} and cl_{cris} ([Har75, II.7.8] and [GM87], respectively), and a comparison isomorphism σ_{cris} (cf. [Ber74, V.2.3.2]) making the following diagram commute:

\xymatrix{𝒵^{r}(X_{K})_{ℚ} \ar[r]^{sp} \ar[d]_{_{cl_{dR}}} & 𝒵^{r}(X_{k})_{ℚ} \ar[d]^{_{cl_{cris}}} \\ H_{dR}^{2r}(X_{K}/K) \ar[r]^{σ_{cris}} & H_{cris}^{2r}(X_{k}/K).}

Conjecture 9.2 (p-adic variational Hodge conjecture). Let notation be as above. Let Z_{k} ∈ 𝒵^{r}(X_{k})_{ℚ}. If σ_{cris}^{−1}(cl_{cris}(Z_{k})) ∈ Fil^{r} H_{dR}^{2r}(X_{K}/K), then there exists Z ∈ 𝒵^{r}(X_{K})_{ℚ} such that cl_{dR}(Z) = σ_{cris}^{−1}(cl_{cris}(Z_{k})).

Remark 9.3. Conjecture 9.2 seems to be well-known to experts; it is a variant of [Eme, Conjecture 2.2], which is where we learned about it.

Remark 9.4. The r = 1 case of Conjecture 9.2 is a weak form of Theorem 4.29.

Repeating the p-adic proof of Theorem 1.7 yields the following.

Theorem 9.5. Assume Conjecture 9.2. Let notation be as in Setup 1.6. Let r be a nonnegative integer. Then the set

B(O_{C})_{jumping} := {b ∈ B(O_{C}) : ρ^{r}(X_{b}) > ρ^{r}(Xη̄)}

is nowhere dense in B(O_{C}) for the analytic topology.

9.3. The Hodge-theoretic approach. To obtain a result using André’s approach requires either replacing algebraic cycles with motivated cycles as in [And96, §5.2], or else assuming the variational Hodge conjecture, which we now state:

Conjecture 9.6 (Variational Hodge conjecture). Let f: X → B be a smooth projective morphism between smooth quasi-projective complex varieties with B irreducible, and let r be a nonnegative integer. Let b ∈ B(ℂ) and let α_{b} ∈ cl_{Betti}(𝒵^{r}(X_{b})_{ℚ}). If α_{b} is the restriction of a class α ∈ H^{2r}(X^{an}, ℚ) (or equivalently, by Deligne’s global invariant cycle theorem, α_{b} is invariant under the monodromy action of π_{1}(B^{an}, b)), then there is a cycle class α′ ∈ cl_{Betti}(𝒵^{r}(X)_{ℚ}) such that α_{b} is the restriction of α′.

Remark 9.7. The variational Hodge conjecture is a consequence of the Hodge conjecture because Deligne’s global invariant cycles theorem ([Del71] or [Voi03, 4.3.3]) shows that the class α_{b} above is the restriction of a Hodge class \overline{α} on some smooth completion of X. On the other hand, it is a priori a much weaker statement, since the Hodge class \overline{α} is an “absolute Hodge class” in the sense of [DMOS82, p. 28].

Assuming Conjecture 9.6, André’s argument yields the following extension of Theorem 8.3:

Theorem 9.8. Assume that k is an algebraic closure of a field k_{f} that is finitely generated over ℚ. Let B be an irreducible k_{f}-variety. Let X → B be a smooth proper morphism. Let r be a nonnegative integer. If Conjecture 9.6 holds for r, then the set of b ∈ B(k) such that ρ^{r}(X_{b}) > ρ^{r}(Xη̄) is sparse.

Remark 9.9. In fact, André works with motivic Galois groups instead of spaces of cycle classes, and hence proves more, namely that one can consider not only X → B but also all its fibered powers simultaneously.

Remark 9.10.

9.4. An implication. We introduced above two different variational forms of the Hodge conjecture, in order to extend the results on Néron-Severi groups to higher-degree classes. We now show that the p-adic version is stronger.

Theorem 9.11. The p-adic variational Hodge conjecture (Conjecture 9.2) implies the variational Hodge conjecture (Conjecture 9.6).

The proof will use the following statement.

Lemma 9.12. Let K be any field of characteristic 0. Let B be a K-variety. Let f: X → B be a smooth projective morphism. Let α be a relative de Rham class, i.e., an element of H^{0}(B, ℝ^{2r}f_{∗}Ω_{X /B}^{•}). Then there exists a countable set of closed subschemes V_{i} of B such that for any field extension L ⊇ K and any b ∈ B(L), the restriction α_{b} ∈ H_{dR}^{2r}(X_{b}/L) is algebraic (i.e., in cl_{dR}(𝒵^{r}(X_{b})_{ℚ})) if and only if b ∈ ⋃_{i} V_{i}(L).

Proof. This is a standard consequence of the fact that the codimension-r subschemes in the fibers of f are parametrized by countably many relative Hilbert schemes H_{j}, each of which is projective over B. Let E be ℝ^{2r}f_{∗}Ω_{X /B}^{•} viewed as a geometric vector bundle on B. The relative cycle class map is a B-morphism H_{j} → Fil^{r} E; its image I_{j} is projective over B and points on I_{j} over an algebraically closed field parametrize the classes of effective cycles given by H_{j}. Taking ℚ-linear combinations of these classes yields countably many closed subvarieties W_{i} of Fil^{r} E that together parametrize all de Rham cycle classes in fibers of f. The desired subschemes V_{i} are the inverse images of the W_{i} under the section B → Fil^{r} E given by α. □

Proof of Theorem 9.11. Let X, B, f, b, α_{b}, and α be as in Conjecture 9.6. Let Z ∈ 𝒵^{r}(X_{b})_{ℚ} be such that cl_{Betti}(Z) = α_{b}. Let \overline{X} be a smooth projective completion of X. As in Remark 9.7, we note that α_{b} = ᾱ|_{X_{b}}, where ᾱ ∈ H^{2r}(\overline{X}^{an}, ℚ) is a Hodge class. In the Leray spectral sequence, α maps to a relative Betti class in H^{0}(B^{an}, R^{2r}f_{∗}ℂ), and the relative Grothendieck comparison isomorphism identifies this with a relative de Rham class α_{dR} ∈ H^{0}(B, E), where E is the sheaf ℝ^{2r}f_{∗}Ω_{X /B}^{•} equipped with the Hodge filtration Fil^{•} and Gauss-Manin connection. Since ᾱ is Hodge and restricts to the same section of E as α, α_{dR} is a horizontal section lying in H^{0}(B, Fil^{r} E).

There is a finitely generated subring A ⊂ ℂ such that X, B, f, b, Z, E, and α_{dR} are obtained via base change from corresponding objects over A, which we next base extend by an embedding A ↪ ℤ_{p} provided by [Cas86, Chapter 5, Theorem 1.1]. To ease notation, from now on we use the notation of Setup 1.6 with O_{K} := ℤ_{p}, and reuse the symbols above to denote the corresponding objects over O_{K}. Let s ∈ B(k) be the reduction of b ∈ B(O_{K}). We may assume that the constituents of Z are flat over O_{K}, so that the special fiber Z_{s} is the specialization of Z_{K}.

As in the proof of Lemma 4.2, we use cl_{cris}(Z_{s}) ∈ H_{cris}^{2r}(X_{s}/K) to construct a horizontal section γ_{dR}(Z_{s}) of E|_{D} for some closed polydisk neighborhood D of b inside the residue disk in B(O_{K}) corresponding to s. (This time we need only the O_{K}-points, not the O\overline{K}-points.) Since γ_{dR}(Z_{s}) and α_{dR,K}|_{D} are represented by convergent power series on D, it suffices to check that they are in equal in the formal completion of E at b. The corresponding formal sections are horizontal and take the same value at b (namely, cl_{dR}(Z_{K})), so they are equal. In particular, for any b′ ∈ D ⊂ B(O_{K}), the value of γ_{dR}(Z_{s}) at the generic point b′_{K} of b′ is in Fil^{r} E_{b′_{K}}.

So Conjecture 9.2 applied to the O_{K}-morphism X_{b′} → B_{b′} implies that there exists Z′ ∈ 𝒵^{r}(X_{b′_{K}})_{ℚ} such that

σ_{cris}(cl_{dR}(Z′)) = cl_{cris}(Z_{s}).

The isomorphism σ_{cris} also maps α_{dR,K}|_{b′_{K}} = γ_{dR}(Z_{s})|_{b′_{K}} to cl_{cris}(Z_{s}) ∈ H_{cris}^{2}(X_{s}/K), so

cl_{dR}(Z′) = α_{dR,K}|_{b′_{K}}.

So the value of α_{dR,K} on any K-point in D is an algebraic class.

Applying Lemma 9.12 to X_{K} → B_{K} and α_{dR,K} yields a countable set of closed subschemes V_{i} of B_{K}. The previous paragraph shows that D ⊆ ⋃ V_{i}(K). By Lemma 4.34, we have dim V_{i} = dim B_{K} for some i. This V_{i} contains the generic point η of B_{K}. So α_{dR,K}|_{η} = cl_{dR}(Y_{η}) for some Y_{η} ∈ 𝒵^{r}(X_{η})_{ℚ}. Taking the closure in X_{K} of the constituents of Y_{η} defines some Y_{K} ∈ 𝒵^{r}(X_{K})_{ℚ}. To Y_{K} we associate a relative de Rham class α′_{dR,K} ∈ H^{0}(B_{K}, ℝ^{2r}f_{∗}Ω_{X_{K}/B_{K}}^{•}) and a Betti class α′ := cl_{Betti}((Y_{K})^{an}) ∈ H^{2r}(X^{an}, ℚ), where the complex analytic spaces are obtained by fixing an A-algebra homomorphism K ↪ ℂ. The set {q ∈ B_{K} : α′_{dR,K}|_{q} = α_{dR,K}|_{q}} is closed and contains η, so it contains b, which we now view as a K-point. Under the comparison isomorphisms, the equal de Rham classes α′_{dR,K}|_{b} and α_{dR,K}|_{b} correspond to the Betti classes α′_{b} and α_{b}, so α′_{b} = α_{b}. □

Acknowledgements

We thank Claire Voisin for explaining the complex-analytic approach of Section 8 to us, and for many discussions regarding other parts of the article. We thank the referees for many insightful suggestions. We also thank Ekaterina Amerik, Yves André, Daniel Bertrand, Pete Clark, Brian Conrad, Jordan Ellenberg, Robin Hartshorne, Franz-Viktor Kuhlmann, Max Lieblich, Ben Moonen, Arthur Ogus, and Richard Thomas, for useful discussions and comments.

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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA

E-mail address: dmaulik@math.mit.edu

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139-4307, USA

E-mail address: poonen@math.mit.edu
URL: http://math.mit.edu/~poonen/