On a Positive Equicharacteristic

Variant of the p-Curvature Conjecture

Hélène Esnault, Adrian Langer

Received: October 26, 2012

Communicated by Takeshi Saito

Abstract. Our aim is to formulate and prove a weak form in equal characteristic p > 0 of the p-curvature conjecture. We also show the existence of a counterexample to a strong form of it.

2010 Mathematics Subject Classification: 14D05, 14E20, 14F35, 11G10, 11G99 Keywords and Phrases: varieties in positive characteristic, stratified bundles, étale trivializable bundles, monodromy group, abelian varieties

Introduction

If (E,∇) is a vector bundle with an algebraic integrable connection over a smooth complex variety X, then it is defined over a smooth scheme S over Specℤ[\frac{1}{N}] for some positive integer N, so (E,∇) = (E_{S},∇_{S}) ⊗_{S}ℂ over X = X_{S}⊗_{S}ℂ for a geometric generic point ℚ(S) ⊂ ℂ. Grothendieck-Katz’s p-curvature conjecture predicts that if for all closed points s of some non-trivial open U ⊂ S, the p-curvature of (E_{S},∇_{S})×_{S}s is zero, then (E,∇) is trivialized by a finite étale cover of X (see e.g. [An, Conj.3.3.3]). Little is known about it. N. Katz proved it for Gauß-Manin connections [Ka], for S finite over Specℤ[\frac{1}{N}] (i.e., if X can be defined over a number field), D. V. Chudnovsky and G. V. Chudnovsky in [CC] proved it in the rank 1 case and Y. André in [An] proved it in case the Galois differential Lie algebra of (E,∇) at the generic point of S is solvable (and for extensions of connections satisfying the conjecture). More recently, B. Farb and M. Kisin [FK] proved it for certain locally symmetric varieties X. In general, one is lacking methods to think of the problem.

The first author is supported by the SFB/TR45 and the ERC Advanced Grant 226257. The second author is supported by the Bessel Award of the Humboldt Foundation and a Polish MNiSW grant (contract number N N201 420639).

Documenta Mathematica 18 (2013) 23–50

Y. André in [An, II] and E. Hrushovsky in [Hr, V] formulated the following equal characteristic 0 analog of the conjecture: if X → S is a smooth morphism of smooth connected varieties defined over a characteristic 0 field k, then if (E_{S},∇_{S}) is a relative integrable connection such that for all closed points s of some non-trivial open U ⊂ S, (E_{S},∇_{S}) ×_{S} s is trivialized by a finite étale cover of X ×_{S} s, then (E,∇)|_{Xη̄} should be trivialized by a finite étale cover, where η̄ is a geometric generic point and Xη̄ = X ×_{S} η̄. So the characteristic 0 analogy to integrable connections is simply integrable connections, and to the p-curvature condition is the trivialization of the connection by a finite étale cover. André proved it [An, Prop. 7.1.1], using Jordan’s theorem and Simpson’s moduli of flat connections, while Hrushovsky [Hr, p.116] suggested a proof using model theory.

It is tempting to formulate an equal characteristic p > 0 analog of Y. André’s theorem. A main feature of integrable connections over a field k of characteristic 0 is that they form an abelian, rigid, k-linear tensor category. In characteristic p > 0, the category of bundles with an integrable connection is only O_{X^{(1)}}-linear, where X^{(1)} is the relative Frobenius twist of X, and the notion is too weak. On the other hand, in characteristic 0, the category of bundles with a flat connection is the same as the category of O_{X}-coherent D_{X}-modules. In characteristic p > 0, O_{X}-coherent D_{X}-modules over a smooth variety X defined over a field k form an abelian, rigid, k-linear tensor category (see [Gi]). It is equivalent to the category of stratified bundles. It bears strong analogies with the category of bundles with an integrable connection in characteristic 0. For example, if X is projective smooth over an algebraically closed field, the triviality of the étale fundamental group forces all such O_{X}-coherent D_{X}-modules to be trivial ([EM]).

So we raise the question 1: let f : X → S be a smooth projective morphism of smooth connected varieties, defined over an algebraically closed characteristic p > 0 field, let (E,∇) be a stratified bundle relative to S, such that for all closed point s of some non-trivial open U ⊂ S, the stratified bundle (E,∇)|_{X_{s}} is trivialized by a finite étale cover of X_{s} := X ×_{S}s. Is it the case that the stratified bundle (E,∇)|_{Xη̄} is trivialized by a finite étale cover of Xη̄?.

In this form, this is not true. Y. Laszlo [Ls] constructed a one dimensional non-trivial family of bundles over a curve over 𝔽_{2} which is fixed by the square of Frobenius, as a (negative) answer to a question of J. de Jong concerning the behavior of representations of the étale fundamental group over a finite field 𝔽_{q}, q = p^{a}, with values in GL(r,𝔽((t))), where 𝔽 ⊃ 𝔽_{2} is a finite extension. In fact, Laszlo’s example yields also a counter-example to the question as stated above. We explain this in Sections 1 and 4 (see Corollary 4.3). We remark that if E is a bundle on X, such that the bundle E|_{X_{s}} is stable, numerically flat (see Definition 3.2) and moves in the moduli, then Eη̄ cannot be trivialized by a finite étale cover (see Proposition 4.2). In contrast, we show that if the family X → S is trivial (as it is in Laszlo’s example), thus X = Y ×_{k}S, if k is algebraically closed, and if (F_{Y}^{n}× identity_{s})^{∗}(E)|_{Y ×_{k}s} ≅ E|_{Y ×_{k}s} for all closed points s of some non-trivial open in S and some fixed natural number n, then the moduli points of E|_{Y ×_{k}s} are constant (see Proposition 4.4). Here F_{Y}: Y → Y is the absolute Frobenius of Y. In Laszlo’s example, one does have (F_{Y}^{2}× identity_{s})^{∗}(E)|_{Y ×_{k}s} ≅ E|_{Y ×_{k}s} but only over k = 𝔽_{2} (i.e., S is also defined over 𝔽_{2}). When one extends the family to the algebraic closure of 𝔽_{2}, to go from the absolute Frobenius over 𝔽_{2}, that is the relative Frobenius over k, to the absolute one, one needs to replace the power 2 with a higher power n(s), which depends on the field of definition of s, and is not bounded.

So we modify question 1 in question 2: let f : X → S be a smooth projective morphism of smooth connected varieties, defined over an algebraically closed characteristic field k of characteristic p > 0, let E be a bundle such that for all closed points s of some non-trivial open U ⊂ S, the bundle E|_{X_{s}} is trivialized by a finite Galois étale cover of X_{s} := X ×_{S}s of order prime to p. Is it the case that the bundle E|_{Xη̄} is trivialized by a finite étale cover of Xη̄?.

The answer is nearly yes: it is the case if k is not algebraic over its prime field (Theorem 5.1 2)). If k = 𝔽̄_{p}, it might be wrong (Remarks 5.4 2), but what remains true is that there exists a finite étale cover of Xη̄ over which the pull-back of E is a direct sum of line bundles (Theorem 5.1 1)). The idea of the proof is borrowed from the proof of Y. André’s theorem [An, Thm 7.2.2]. The assumption on the degrees of the Galois covers of X_{s} trivializing E|_{X_{s}} is necessary (as follows from Laszlo’s example) and it allows us to apply Brauer-Feit’s theorem [BF, Theorem] in place of Jordan’s theorem used by André. However, there is no direct substitute for Simpson’s moduli spaces of flat bundles. Instead, we use the moduli spaces constructed in [La1] and we carefully analyze subloci containing the points of interest, that is the numerically flat bundles. The necessary material needed on moduli is gathered in Section 3.

Finally we raise the general question 3: let f : X → S be a smooth projective morphism of smooth connected varieties, defined over an algebraically closed characteristic p > 0 field, let (E,∇) be a stratified bundle relative to S, such that for all closed points s of some non-trivial open U ⊂ S, the stratified bundle (E,∇)|_{X_{s}} is trivialized by a finite Galois étale cover of X_{s} := X ×_{S}s of order prime to p. Is it the case that the bundle (E,∇)|_{Xη̄} is trivialized by a finite étale cover of Xη̄?

We give the following not quite complete answer. If the rank of E is 1, (in which case the assumption on the degrees of the Galois covers is automatically fulfilled), then the answer is yes provided S is projective, and for any s ∈ U, Pic^{τ}(X_{s}) is reduced (see Theorem 7.1). The proof relies on (a variant of) an idea of M. Raynaud [Ra], using the height function associated to a symmetric line bundle (that is the reason for our assumption on S) on the abelian scheme and its dual, to show that an infinite Verschiebung-divisible point has height equal to 0 (Theorem 6.2). If E has any rank, then the answer is yes if k is not 𝔽̄_{p} (Theorem 7.2 2)). In general, there is a prime to p-order Galois cover of Xη̄ such that the pull-back of E becomes a sum of stratified line bundles (Theorem 7.2 1)).

Acknowledgements: The first author thanks Michel Raynaud for the fruitful discussions in November 2009, which are reflected in [Ra] and in Section 6. The first author thanks Johan de Jong for a beautiful discussion in November 2010 on the content of [EM], where she suggested question 1 to him, and where he replied that Laszlo’s example should contradict this, and that this should be better understood. The second author would like to thank Stefan Schröer for destroying his naive hopes concerning Néron models of Frobenius twists of an abelian variety. We thank Damian Rössler for discussions on p-torsion on abelian schemes over functions fields. We thank the referee of a first version of the article. He/she explained to us that the dichotomy in Theorem 5.1 2) and in Theorem 7.2 2) should be 𝔽̄_{p} or not rather that countable or not, thereby improving our result.

1 Preliminaries on relative stratified sheaves

Let S be a scheme of characteristic p (i.e., O_{S} is an 𝔽_{p}-algebra). By F_{S}^{r} : S → S we denote the r-th absolute Frobenius morphism of S which corresponds to the p^{r}-th power mapping on O_{S}. If X is an S-scheme, we denote by X_{S}^{(r)} the fiber product of X and S over the r-th Frobenius morphism of S. If it is clear with respect to which structure X is considered, we simplify the notation to X^{(r)}. Then the r-th absolute Frobenius morphism of X induces the relative Frobenius morphism F_{X/S}^{r}: X → X^{(r)}. In particular, we have the following commutative diagram:

\xymatrix{X \ar[dr] \ar[rr]^{F_{X}^{r}} \ar[r]_{F{r \atop X/S}} & X^{(r)} \ar[d] \ar[r]_{W_{X}^{r}} & X \ar[d] \\  & S \ar[r]_{F_{S}^{r}} & S}

which defines W_{X/S}^{r}: X^{(r)} → X. Making r = 1 and replacing X by X^{(i)}, this induces the similar diagram

\xymatrix{X^{(i)} \ar[dr] \ar[rr]^{F_{X^{(i)}}} \ar[r]_{F_{X^{(i)}/S}} & X^{(i+1)} \ar[d] \ar[r]_{W_{X^{(i)}}} & X^{(i)} \ar[d] \\  & S \ar[r]_{F_{S}} & S}

We assume that X/S is smooth. A relative stratified sheaf on X/S is a sequence {E_{i},σ_{i}}_{i∈ℕ} of locally free coherent O_{X^{(i)}}-modules E_{i} on X^{(i)} and isomorphisms σ_{i} : F_{X^{(i)}/S}^{∗}E_{i+1} → E_{i} of O_{X^{(i)}}-modules. A morphism of relative stratified sheaves {α_{i}} : {E_{i},σ_{i}} → {E′_{i},σ′_{i}} is a sequence of O_{X^{(i)}}-linear maps α_{i} : E_{i} → E′_{i}compatible with the σ_{i}, that is such that σ′_{i}∘ F_{X^{(i)}/S}^{∗}α_{i+1} = α_{i}∘ σ_{i}. This forms a category Strat(X/S), which is contravariant for morphisms φ : T → S: to {E_{i},σ_{i}} ∈ Start(X/S) one assigns φ^{∗}{E_{i},σ_{i}} ∈ Strat(X ×_{S}T /T) in the obvious way: φ induces 1_{X^{(i)}}× φ : X^{(i)} ×_{S}T → X^{(i)} and (φ^{∗}{E_{i},σ_{i}})_{i} = {(1_{X^{(i)}}× φ)^{∗}E_{i},(1_{X^{(i)}}× φ)^{∗}(σ_{i})}.

If S = Speck where k is a field, Strat(X/k) is an abelian, rigid, tensor category. Giving a rational point x ∈ X(k) defines a fiber functor via ω_{x} : Strat(X/k) → Vec_{k}, ω_{x}({E_{i},σ_{i}}) = (E_{0})|_{x} in the category of finite dimensional vector spaces over k, thus a k-group scheme π(Strat(X/k),ω_{x}) = Aut^{⊗} (ω_{x}). Tannaka duality implies that Strat(X/k) is equivalent via ω_{x} to the representation category of π(Strat(X/k),ω_{x}) with values in Vec_{k}. For any object 𝔼 := {E_{i},σ_{i}} ∈ Strat(X/k), we define its monodromy group to be the k-affine group scheme π(⟨𝔼⟩,ω_{x}), where ⟨𝔼⟩ ⊂ Strat(X/k) is the full subcategory spanned by 𝔼. This is the image of π(Strat(X/k),ω_{x}) in GL(ω_{x}(𝔼)) ([DM, Proposition 2.21 a)]). We denote by 𝕀_{X/k}∈ Strat(X/k) the trivial object, with E^{i} = O_{X^{(i)}} and σ_{i} = Identity.

Lemma 1.1. With the notation above

1) If h : Y → X is a finite étale cover such that h^{∗}𝔼 is trivial, then h_{∗}𝕀_{Y /k} has finite monodromy group and one has a faithfully flat homomorphism π(⟨h_{∗}𝕀_{Y /k}⟩,ω_{x}) → π(⟨𝔼⟩,ω_{x}). Thus in particular, 𝔼 has finite monodromy group as well.

2) If 𝔼 ∈ Strat(X/k) has finite monodromy group, then there exists a π(⟨𝔼⟩,ω_{x})- torsor h: Y → X such that h^{∗}𝔼 is trivial in Strat(Y /k). Moreover, one has an

isomorphism π(⟨h_{∗}𝕀_{Y /k}⟩,ω_{x})⟶^{≅} π(⟨𝔼⟩,ω_{x}).

Proof. We first prove 2). Assume π(⟨𝔼⟩,ω_{x}) =: G is a finite group scheme over k. One applies Nori’s method [No, Chapter I, II]: the regular representation of G on the affine k-algebra k[G] of regular function defines the Artin k-algebra k[G] as a k-algebra object of the representation category of G on finite dimensional k-vector spaces, (such that k ⊂ k[G] is the maximal trivial subobject). Thus by Tannaka duality, there is an object 𝔸 = (A^{i},τ_{i}) ∈ Strat(X/k), which is an 𝕀_{X/k}-algebra object, (such that 𝕀_{X/k}⊂ 𝔸 is the maximal trivial subobject). We define h_{i} : Y_{i} = Spec_{X^{(i)}}A^{i} → X^{(i)}. Then the

isomorphism τ_{i} yields an O_{X^{(i)}}-isomorphism between Y^{(i)} ⟶^{h^{(i)}} X^{(i)} and Y_{i} ⟶^{h_{i}} X^{(i)}, (see, e.g., [SGA5, Exposé XV, § 1, Proposition 2]), and via this isomorphism, 𝔸 is isomorphic to h_{∗}𝕀_{Y /k}. On the other hand, ω_{x}(𝔼) is a sub G-representation of k[G]^{⊕n} for some n ∈ ℕ, thus 𝔼 ⊂ 𝔸^{⊕n} in Strat(X/k), thus there is an inclusion 𝔼 ⊂ (h_{∗}𝕀_{Y /k})^{⊕n} in Strat(X/k), thus h^{∗}𝔼 ⊂ (h^{∗}h_{∗}𝕀_{Y /k})^{⊕n} in Strat(Y /k). Since (h^{∗}h_{∗}𝕀_{Y /k}) is isomorphic to ⊕_{length_{k}k[G]}𝕀_{Y /k}in Strat(Y /k) (recall that by [dS, Proposition 13], G is an étale group scheme), then h^{∗}𝔼 is isomorphic to ⊕_{r}𝕀_{Y /k}, where r is the rank of 𝔼. This shows the first part of the statement, and shows the second part as well: indeed, 𝔼 is then a subobject of ⊕_{r}h_{∗}𝕀_{Y /k}, thus ⟨𝔼⟩ ⊂ ⟨h_{∗}𝕀_{Y /K}⟩ is a full subcategory. One applies [DM, Proposition 2.21 a)] to show that the induced homomorphism π(⟨h_{∗}𝕀_{Y /k}⟩,ω_{x}) → π(⟨𝔼⟩,ω_{x}) = G is faithfully flat. So π(⟨h_{∗}𝕀_{Y /k}⟩,ω_{x}) acts on ω_{x}(h_{∗}𝕀_{Y}) = k[G] via its quotient G and the regular representation G ⊂ GL(k[G]). Thus the homomorphism is an isomorphism. We show 1). Assume that there is a finite étale cover h : Y → X such that h^{∗}𝔼 is isomorphic in Strat(Y /k) to ⊕_{r}𝕀_{Y /k}where r is the rank of 𝔼. Then 𝔼 ⊂ ⊕_{r}h_{∗}𝕀_{Y /k}, thus π(⟨h_{∗}𝕀_{Y /k}⟩,ω_{x}) → π(⟨𝔼⟩,ω_{x}) is faithfully flat [DM, loc. cit.], so we are reduced to showing that ⟨h_{∗}𝕀_{Y /k}⟩ has finite monodromy. But, by the same argument as on 𝔼, any of its objects of rank r′lies in ⊕_{r′}h_{∗}𝕀_{Y /k}. So we apply [DM, Proposition 2.20 a)] to conclude that the monodromy of h_{∗}𝕀_{Y /k}is finite.

Corollary 1.2. With the notations as in 1.1, if 𝔼 ∈ Strat(X/k) has finite monodromy group, then for any field extension K ⊃ k, 𝔼 ⊗ K ∈ Strat(X ⊗ K/K) has finite monodromy group.

Let E be an O_{X}-module. We say that E has a stratification relative to S if there exists a relative stratified sheaf {E_{i},σ_{i}} such that E_{0} = E. Let us consider the special case S = Speck, where k is a perfect field, and X/k is smooth. An (absolute) stratified sheaf on X is a sequence {E_{i},σ_{i}}_{i∈ℕ} of coherent O_{X}-modules E_{i} on X and isomorphisms σ_{i} : F_{X}^{∗}E_{i+1} → E_{i} of O_{X}-modules. As k is perfect, the W_{X^{(i)}} are isomorphisms, thus giving an absolute stratified sheaf is equivalent to giving a stratified sheaf relative to Speck. We now go back to the general case and we assume that S is an integral k-scheme, where k is a field. Let us set K = k(S) and let η : SpecK → S be the generic point of S. Let us fix an algebraic closure K̄ of K and let η̄ be the corresponding generic geometric point of S. By contravariance, a relative stratified sheaf {E_{i},σ_{i}} on X/S restricts to a relative stratified sheaf {E_{i},σ_{i}}|_{X_{s}} in fibers X_{s} for s a point of S. We are interested in the relation between {E_{i},σ_{i}}|_{Xη̄} and {E_{i},σ_{i}}|_{X_{s}} for closed points s ∈ |S|. More precisely, we want to understand under which assumptions the finiteness of ⟨{E_{i},σ_{i}}|_{X_{s}}⟩ for all closed points s ∈ |S| implies the finiteness of ⟨{E_{i},σ_{i}}|_{Xη̄}⟩. Recall that finiteness of 𝔼 ⊂ Strat(X_{s}) means that all objects of ⟨𝔼⟩ are subquotients in Strat(X_{s}) of direct sums of a single object, which is equivalent to saying that after the choice of a rational point, the monodromy group of 𝔼 is finite ([DM, Proposition 2.20 (a)]).

Let X be a smooth variety defined over 𝔽_{q} with q = p^{r}. For all n ∈ ℕ \ {0}, one has the commutative diagram

\xymatrix{X \ar[dr] \ar[rr]^{(F_{X}^{r})^{n}=F_{X}^{rn}} \ar[r]_{F_{X/𝔽q}^{rn}} & X^{(rn)} \ar[d] \ar[r]_{W_{X/𝔽q}^{rn}} & X \ar[d] \\  & Spec𝔽_{q} \ar[r]_{F_{𝔽q}^{rn}=id} & Spec𝔽_{q}}  (1)

which allows us to identify X^{(rn)} with X (as an 𝔽_{q}-scheme). Let S be an 𝔽_{q} connected scheme, with field of constants k, i.e. k is the normal closure of 𝔽_{q} in H^{0}(S,O_{X}). We define X_{S}:= X ×_{𝔽_{q}} S.

Proposition 1.3. Let E be a vector bundle on X_{S}. Assume that there exists a positive integer n such that we have an isomorphism

τ : ((F^{r} ×_{𝔽_{q}} id_{S})^{n})^{∗}E ≃ E.  (2)

Then E has a natural stratification 𝔼_{τ} = {E_{i},σ_{i}}, E_{0} = E relative to S.

Proof. We define

E_{rn} = (W_{X/𝔽_{q}}^{rn}×_{𝔽_{q}} id_{S})^{∗}E.  (3)

Then we use the factorization

\xymatrix{X \ar@{-->}[drrrr] \ar[r]^{F_{X/𝔽q}} & X^{(1)} \ar[drrr] \ar[r]^{F_{X^{(1)}/𝔽q}} & ··· \ar[r] & X^{(rn−1)} \ar[dr] \ar[r]^{F_{X^{(rn−1)} /𝔽q}} & X^{(rn)} \ar[d] \\  &  &  &  & Spec𝔽_{q}}  (4)

of F_{X/𝔽_{q}}^{rn} and we define

E_{nr−1} = (F_{X^{(rn−1)} /𝔽_{q}}×_{𝔽_{q}} id_{S})^{∗}E_{rn},...,E_{1} = (F_{X^{(1)}/𝔽_{q}}×_{𝔽_{q}} id_{S})^{∗}E_{2}  (5)

with identity isomorphisms σ_{nr−1} ,...,σ_{1}. Then we use the isomorphism τ to define

σ_{0} : E ≃ (F_{X/𝔽_{q}}×_{𝔽_{q}} id_{S})^{∗}E_{1}.  (6)

Assume we constructed the bundles E_{i} on X^{(i)} for all i ≤ arn for some integer a ≥ 1. We now replace the diagram (1) by the diagram

\xymatrix{X^{(arn)} \ar[dr] \ar[rr]^{(F_{X^{(arn)}}^{r})^{n}} \ar[r]_{F_{X^{(arn)}/𝔽q}^{rn}} & X^{((a+1)rn)} \ar[d] \ar[r]_{W_{X^{(arn)}/𝔽q}^{rn}} & X^{(arn)} \ar[d] \\  & Spec𝔽_{q} \ar[r]_{F_{𝔽q}^{rn}=1} & Spec𝔽_{q}}  (7)

We then define

E_{(a+1)rn} = (W_{X^{(arn)}/𝔽_{q}}^{rn}×_{𝔽_{q}} id_{S})^{∗}E_{arn}  (8)

(which is equal to E under identification of X^{(arn)} with X). Then we use the factorization

\xymatrix{X^{(arn)} \ar@{-->}[drrrr] \ar[r]^{F_{X^{(arn)}/𝔽q}} & X^{(arn+1)} \ar@{-->}[drrr] \ar[r]^{F_{X^{(arn+1)} /𝔽q}} & ··· \ar[r] & X^{((a+1)rn−1)} \ar[dr] \ar[r]^{F_{X^{((a+1)rn−1)} /𝔽q}} & X^{((a+1)rn)} \ar[d] \\  &  &  &  & Spec𝔽_{q}}  (9)

of F_{X^{(arn)}/𝔽_{q}}^{rn} to define

E_{(a+1)rn−1} = (F_{X^{((a+1)rn−1)} /𝔽_{q}}×_{𝔽_{q}} id_{S})^{∗}E_{(a+1)rn} ,...,

E_{arn+1} = (F_{X^{(arn+1)} /𝔽_{q}}×_{𝔽_{q}} id_{S})^{∗}E_{arn+2} (10) with identity isomorphisms σ_{(a+1)nr−1} ,...,σ_{arn+1} . Then we again use τ to define

σ_{arn}: E_{arn} ≃ (F_{X^{(arn)}/𝔽_{q}}^{rn})^{∗}E_{arn+1} .  (11)

The above construction and [Gi, Proposition 1.7] imply

Proposition 1.4. Assume in addition to (2) that X is proper and 𝔽_{q} ⊂ k ⊂ 𝔽̄_{q}. Fix a rational point x ∈ X_{S}(k). Then for any closed point s ∈ |S|, the Tannaka group scheme π(𝔼_{τ_{s}},ω_{x⊗_{k}k(s)}) of 𝔼_{τ_{s}} := 𝔼_{τ}|_{X_{s}} over the residue field k(s) of s is finite.

Proof. The bundle E is base changed of a bundle E^{0} defined over X ×_{𝔽_{q}} S_{0} for some form S_{0} of S defined over a finite extension 𝔽_{q^{a}} of 𝔽_{q} such that x is base change of an 𝔽_{q^{a}}-rational point x_{0} of X ×_{𝔽_{q}} S_{0}. We can also assume that τ comes by base change from τ_{0} : ((F^{r} ×_{𝔽_{q}} id_{S_{0}})^{n})^{∗}E^{0} ≃ E^{0}. Proposition 1.3 yields then a relative stratification 𝔼_{τ_{0}}^{0} = (E_{i}^{0},σ_{i}^{0}) of E^{0} defined over 𝔽_{q^{a}}, with E_{i} = E_{i}^{0} ⊗_{𝔽_{q}^{a}} k. A closed point s of S = S_{0}⊗_{𝔽_{q}^{a}} k is a base change of some closed point s_{0} of S_{0} of degree b say over 𝔽_{q^{a}}. By Corollary 1.2 we just have to show that π(𝔼_{(τ_{0})_{s_{0}}},ω_{x_{0}⊗_{𝔽_{q^{a}}}k(s_{0})}) is finite. So we assume that k = 𝔽_{q^{a}}, S = S_{0}, s = s_{0}. The underling bundles of 𝔼_{τ} and 𝔼_{τ^{m}} are by construction all isomorphic for m = ab. Thus by [Gi, Proposition 1.7], 𝔼_{τ} ≃ 𝔼_{τ^{m}} in Strat(X/k). But this implies that F_{X×_{𝔽_{q^{a}}}s}^{mn}(𝔼_{τ_{s}}) ≅ 𝔼_{τ_{s}}. Thus E is algebraically

trivializable on the Lang torsor h: Y → X ×_{𝔽_{q^{a}}} 𝔽_{q^{m}} and the bundles E_{i} are trivializable on Y ×_{X×_{𝔽_{q^{a}}}𝔽_{q^{m}}} X^{(i)} = Y^{(i)}/𝔽_{q^{m}}. Thus the stratified bundle h^{∗}𝔼_{τ} on Y relative to 𝔽_{q^{m}} is trivial. We apply Lemma 1.1 to finish the proof.

2 Étale trivializable bundles

Let X be a smooth projective variety over an algebraically closed field k. Let F_{X}: X → X be the absolute Frobenius morphism. A locally free sheaf on X is called étale trivializable if there exists a finite étale covering of X on which E becomes trivial. Note that if E is étale trivializable then it is numerically flat (see Definition 3.2 and the subsequent discussion). In particular, stability and semistability for such bundles are independent of a polarization (and Gieseker and slope stability and semistability are equivalent). More precisely, such E is stable if and only if it does not contain any locally free subsheaves of smaller rank and degree 0 (with respect to some or equivalently to any polarization).

Proposition 2.1. (see [LSt]) If there exists a positive integer n such that (F_{X}^{n})^{∗}E ≃ E then E is étale trivializable. Moreover, if k = 𝔽̄_{p} then E is étale trivializable if and only if there exists a positive integer n and an isomorphism (F_{X}^{n})^{∗}E ≃ E.

Proposition 2.2. (see [BD]) If there exists a finite degree d étale Galois covering f : Y → X such that f^{∗}E is trivial and E is stable, then one has an isomorphism α : (F_{X}^{d})^{∗}E ≃ E.

As a corollary we see that a line bundle on X/k is étale trivializable if and only if it is torsion of order prime to p. One implication follows from the above proposition. The other one follows from the fact that (F_{X}^{d})^{∗}L ≃ L is equivalent to L^{⊗(p^{d}−1)} ≃ O_{X} and for any integer n prime to p we can find d such that p^{d}− 1 is divisible by n. We recall that if E is any vector bundle on X such that there is a d ∈ ℕ \ {0} and an isomorphism α : (F_{X}^{d})^{∗}(E) ≅ E, then E carries an absolute stratified structure 𝔼_{α}, i.e. a stratified structure relative to 𝔽_{p} by the procedure of Proposition 1.3. On the other hand, any stratified stratified structure {E_{i},σ_{i}} relative to 𝔽_{p} induces in an obvious way a stratified structure relative to k: the absolute Frobenius F_{X}^{n} : X → X factors through W_{X/k}^{n}: X^{(n)}→ X, so {(W_{X/k}^{n})^{∗}E_{n},(W_{X/k}^{n})^{∗}σ_{n}} is the relative stratified structure, denoted by 𝔼_{α/k}. Proposition 2.2 together with Lemma 1.1 2) show

Corollary 2.3. Under the assumptions of Proposition 2.2, we can take d = length_{k}k[π(⟨𝔼_{α/k}⟩,ω_{x})].

Let us also recall that there exist examples of étale trivializable bundles such that (F_{X}^{n})^{∗}E ≄ E for every positive integer n (see Laszlo’s example in [BD]).

Proposition 2.4. (Deligne; see [Ls, 3.2]) Let X be an 𝔽_{p^{n}}-scheme. If G is a connected linear algebraic group defined over a finite field 𝔽_{p^{n}} then the embedding G(𝔽_{p^{n}}) ↪ G induces an equivalence of categories between the category of G(𝔽_{p^{n}})- torsors on X and G-torsors P over X with an isomorphism (F_{X}^{n})^{∗}P ≃ P.

In particular, if G is a connected reductive algebraic group defined over an algebraically closed field k and P is a principal G-bundle on X/k such that there exists an isomorphism (F_{X}^{n})^{∗}P ≃ P for some natural number n > 0, then there exists a Galois étale cover f : Y → X with Galois group G(𝔽_{p^{n}}) such that f^{∗}P is trivial. Indeed, every reductive group has a ℤ-form so we can use the above proposition.

3 Preliminaries on relative moduli spaces of sheaves

Let S be a scheme of finite type over a universally Japanese ring R. Let f : X → S be a projective morphism of R-schemes of finite type with geometrically connected fibers and let O_{X}(1) be an f-very ample line bundle. A family of pure Gieseker semistable sheaves on the fibres of X_{T} = X ×_{S}T → T is a T -flat coherent O_{X_{T}}-module E such that for every geometric point t of T the restriction of E to the fibre X_{t} is pure (i.e., all its associated points have the same dimension) and Gieseker semistable (which is semistability with respect to the growth of the Hilbert polynomial of subsheaves defined by O_{X}(1) (see [HL, 1.2]). We introduce an equivalence relation ∼ on such families in the following way. E ∼ E′if and only if there exist filtrations 0 = E_{0} ⊂ E_{1} ⊂ ... ⊂ E_{m}= E and 0 = E′_{0} ⊂ E′_{1} ⊂ ... ⊂ E′_{m}= E′by coherent O_{X_{T}}-modules such that ⊕_{i=0}^{m} E_{i}/E_{i−1} is a family of pure Gieseker semistable sheaves on the fibres of X_{T} and there exists an invertible sheaf L on T such that ⊕_{i=1}^{m} E′_{i}/E′_{i−1} ≃ (⊕_{i=1}^{m} E_{i}/E_{i−1}) ⊗_{O_{T}}L. Let us define the moduli functor

M_{P}(X/S) : (Sch/S)^{o} → Sets

from the category of locally noetherian schemes over S to the category of sets by

M_{P}(X/S)(T) = \{ \begin{matrix}∼ equivalence classes of families of pure Gieseker \\ semistable sheaves on the fibres ofT ×_{S}X → T, \\ which have Hilbert polynomial P. \end{matrix} \} .

Then we have the following theorem (see [La1, Theorem 0.2]).

Theorem 3.1. Let us fix a polynomial P. Then there exists a projective S-scheme M_{P}(X/S) of finite type over S and a natural transformation of functors

θ : M_{P}(X/S) → Hom_{S}(·,M_{P}(X/S)),

which uniformly corepresents the functor M_{P}(X/S). For every geometric point s ∈ S the induced map θ(s) is a bijection. Moreover, there is an open scheme M_{X/S}^{s}(P) ⊂ M_{P}(X/S) that universally corepresents the subfunctor of families of geometrically Gieseker stable sheaves.

Let us recall that M_{P}(X/S) uniformly corepresents M_{P}(X/S) means that for every flat base change T → S the fiber product M_{P}(X/S)×_{S}T corepresents the fiber product functor Hom_{S}(·,T)×_{Hom_{S}(·,S)}M_{P}(X/S). For the notion of corepresentability, we refer to [HL, Definition 2.2.1]. In general, for every S-scheme T we have a well defined morphism M_{P}(X/S)×_{S}T → M_{P}(X_{T}/T) which for a geometric point T = Speck(s) → S is bijection on points. The moduli space M_{P}(X/S) in general depends on the choice of polarization O_{X}(1).

Definition 3.2. Let k be a field and let Y be a projective k-variety. A coherent O_{Y}-module E is called numerically flat, if it is locally free and both E and its dual E^{∗} = H om(E,O_{Y}) are numerically effective on Y ⊗k̄, where k̄is an algebraic closure of k.

Assume that Y is smooth. Then a numerically flat sheaf is strongly slope semistable of degree 0 with respect to any polarization (see [La2, Proposition 5.1]). But such a sheaf has a filtration with quotients which are numerically flat and slope stable (see [La2, Theorem 4.1]). Let us recall that a slope stable sheaf is Gieseker stable and any extension of Gieseker semistable sheaves with the same Hilbert polynomial is Gieseker semistable. Thus a numerically flat sheaf is Gieseker semistable with respect to any polarization. Let P be the Hilbert polynomial of the trivial sheaf of rank r. In case S is a spectrum of a field we write M_{X}(r) to denote the subscheme of the moduli space M_{P}(X/k) corresponding to locally free sheaves. For a smooth projective morphism X → S we also define the moduli subscheme M(X/S,r) → S of the relative moduli space M_{P}(X/S) as a union of connected components which contains points corresponding to numerically flat sheaves of rank r. Note that in positive characteristic numerical flatness is not an open condition. More precisely, on a smooth projective variety Y with an ample divisor H, a locally free sheaf with numerically trivial Chern classes, that is with Chern classes c_{i} in the Chow group of codimension i cycles intersecting trivially H^{dim(Y)−i} for all i ≥ 1, is numerically flat if and only if it is strongly slope semistable (see [La2, Proposition 5.1]). By definition for every family E of pure Gieseker semistable sheaves on the fibres of X_{T} we have a well defined morphism φ_{E} = θ([E]) : T → M_{P}(X/S), which we call a classifying morphism.

Proposition 3.3. Let X be a smooth projective variety defined over an algebraically closed field k of positive characteristic. Let S be a k-variety and let E be a rank r locally free sheaf on X ×_{k}S such that for every s ∈ S(k) the restriction E_{s} is Gieseker semistable with numerically trivial Chern classes. Assume that the classifying morphism φ_{E} : S → M_{X}(r) is constant and for a dense subset S′⊂ S(k) the bundle E_{s} is étale trivializable for s ∈ S′. Then Eη̄ is étale trivializable.

Proof. If E_{s} is stable for some k-point s ∈ S then there exists an open neighbourhood U of φ_{E}(s), a finite étale morphism U′→ U and a locally free sheaf U on X ×_{k}U′ such that the pull backs of E and U to X ×_{k}(φ_{E}^{−1} (U) ×_{U} U′) are isomorphic (this is called existence of a universal bundle on the moduli space in the étale topology). But φ_{E}(S) is a point, so this proves that there exists a vector bundle on X such that E is its pull back by the projection X ×_{k}S → X. In this case the assertion is obvious. Now let us assume that E_{s} is not stable for all s ∈ S(k). If 0 = E_{0}^{s} ⊂ E_{1}^{s} ⊂ ... ⊂ E_{m}^{s}= E_{s} is a Jordan–Hölder filtration (in the category of slope semistable torsion free sheaves), then by assumption the isomorphism classes of semi-simplifications ⊕_{i=1}^{m} E_{i}^{s}/E_{i−1}^{s} do not depend on s ∈ S(k). Let (r_{1},...,r_{m}) denote the sequence of ranks of the components E_{i}^{s}/E_{i−1}^{s} for some s ∈ S(k). Since there is only finitely many such sequences (they differ only by permutation), we choose some permutation that appears for a dense subset S′′⊂ S′. Now let us consider the scheme of relative flags f : Flag(E/S;P_{1},...,P_{m}) → S, where P_{i} is the Hilbert polynomial of O_{X}^{r_{i}}. By our assumption the image of f contains S′′. Therefore by Chevalley’s theorem it contains an open subscheme U of S. Let us recall that the scheme of relative flags Flag(E|_{X×_{k}U}/U;P_{1},...,P_{m}) → U is projective. In particular, using Bertini’s theorem (k is algebraically closed) we can find a generically finite morphism W → U factoring through this flag scheme. Let us consider pull back of the universal filtration 0 = F_{0} ⊂ F_{1} ⊂ ... ⊂ F_{m} = E_{W} to X ×_{k}W. Note that the quotients F^{i} = F_{i}/F_{i−1} are W -flat and by shrinking W we can assume that they are families of Gieseker stable locally free sheaves (since by assumption F_{s}^{i} is Gieseker stable and locally free for some points s ∈ W (k) ∩ S′). This and the first part of the proof implies that Eη̄ has a filtration by subbundles such that the associated graded sheaf isétale trivializable. By Lemma 5.2this implies that Eη̄ isétale trivializable.

4 Laszlo’s example

Let us describe Laszlo’s example of a line in the moduli space of bundles on a curve fixed by the second Verschiebung morphism (see [Ls, Section 3]). Let us consider a smooth projective genus 2 curve X over 𝔽_{2} with affine equation

y^{2}+ x(x + 1)y = x^{5}+ x^{2}+ x.

In this case the moduli space M_{X}(2,O_{X}) of rank 2 vector bundles on X with trivial determinant is an 𝔽_{2}-scheme isomorphic to ℙ^{3}. The pull back of bundles by the relative Frobenius morphism defines the Verschiebung map

V : M_{X^{(1)}}(2,O_{X^{(1)}}) ≃ ℙ^{3} −−→ M_{X}(2,O_{X}) ≃ ℙ^{3}

which in appropriate coordinates can be described as

[a: b: c: d] → [a^{2}+ b^{2}+ c^{2}+ d^{2} : ab + cd : ac + bd : ad + bc].

The restriction of V to the line Δ ≃ ℙ^{1} given by b = c = d is an involution and it can be described as [a: b] → [a + b: b]. Using a universal bundle on the moduli space (which exists locally in the étale topology around points corresponding to stable bundles) and taking a finite covering S → Δ we obtain the following theorem:

Theorem 4.1. ([Ls, Corollary 3.2]) There exist a smooth quasi-projective curve S defined over some finite extension of 𝔽_{2} and a locally free sheaf E of rank 2 on X × S such that (F^{2} × id_{S})^{∗}E ≃ E, detE ≃ O_{X×S} and the classifying morphism φ_{E} : S → M_{X}(2,O_{X}) is not constant. Moreover, one can choose S so that E_{s} is stable for every closed point s in S.

Now note that the map (F_{X})^{∗}: M_{X}(2,O_{X}) −−→ M_{X}(2,O_{X}) defined by pulling back bundles by the absolute Frobenius morphism can be described on Δ as [a: b] → [a^{2}+ b^{2} : b^{2}]. In particular, the map (F_{X}^{2n})^{∗}|_{Δ} is described as [a: b] → [a^{2n},b^{2n}]. It follows that if a stable bundle E corresponds to a modular point of Δ(𝔽_{2}^{n})\Δ(𝔽_{2}^{n−1} ) (or, equivalently, E is defined over 𝔽_{2^{n}}) then (F_{X}^{2n})^{∗}E ≃ E and (F_{X}^{m})^{∗}E ≄ E for 0 < m < 2n. This implies that for k = 𝔽̄_{2} and for every s ∈ S(k), the bundle E_{s} which is the restriction to X ×_{𝔽_{2}} s of the bundle E from Theorem 4.1, is étale trivializable.

Let X,S be varieties defined over an algebraically closed field k of positive characteristic. Assume that X is projective. Let us set K = k(S). Let η̄ be a generic geometric point of S.

Proposition 4.2. Let E be a bundle on X_{S} = X ×_{k}S → S which is numerically flat on the closed fibres of X_{S} = X ×_{k}S → S. Assume that for some s ∈ S the bundle E_{s} is stable and the classifying morphism φ_{E} : S → M_{X}(r) is not constant. Then Eη̄ = E|_{Xη̄} is not étale trivializable.

Proof. Assume that there exists a finite étale cover π′: Y′→ Xη̄ such that (π′)^{∗}Eη̄ ≃ O_{Y′}^{r}. As k is algebraically closed, one has the base change π_{1}(X) ⟶^{≅} π_{1}(XK̄) for the étale fundamental group ([SGA1, Exp. X, Cor.1.8]), so there exists a finite étale cover π : Y → X such that π′= π ⊗ K̄. Hence there exists a finite morphism T → U over some open subset U of S, such that π_{T}^{∗}(E_{T}) is trivial where π_{T} = π ×_{k}id_{T}: Y ×_{k}T → X ×_{k}T and E_{T} =pull back by X ×_{k}T → X ×_{k}U of E|_{X×_{k}U}. So for any k-rational point t ∈ T, one has π^{∗}E_{t} ⊂ O_{Y}^{r}, where r is the rank of E. Hence E_{t} ⊂ π_{∗}π^{∗}E_{t} ⊂ π_{∗}O_{Y}^{r}, i.e., all the bundles E_{t} lie in one fixed bundle π_{∗}O_{Y}^{r}.

Since π is étale, the diagram

\xymatrix{Y \ar[d]^{π} \ar[r]^{F_{Y}} & Y \ar[d]^{π} \\ X \ar[r]^{F_{X}} & X}

is cartesian (see, e.g., [SGA5, Exp. XIV, §1, Prop. 2]). Since X is smooth, F_{X} is flat. By flat base change we have isomorphisms F_{X}^{∗}(π_{∗}O_{Y}) ≃ π_{∗}(F_{Y}^{∗}O_{Y}) ≃ π_{∗}O_{Y}. In particular, this implies that π_{∗}O_{Y} is strongly semistable of degree 0. Therefore if E_{t} is stable then it appears as one of the factors in a Jordan–Hölder filtration of π_{∗}O_{Y}. Since the direct sum of factors in a Jordan–Hölder filtration of a semistable sheaf does not depend on the choice of the filtration, there are only finitely many possibilities for the isomorphism classes of stable sheaves E_{t} for t ∈ T (k). It follows that in U ⊂ S there is an infinite sequence of k-rational points s_{i} with the property that E_{s_{i}} is stable (since stability is an open property) and E_{s_{i}} ≅ E_{s_{i+1}} . This contradicts our assumption that the classifying morphism φ_{E} is not constant.

Corollary 4.3. There exist smooth curves X and S defined over an algebraic closure k of 𝔽_{2} such that X is projective and there exists a locally free sheaf E on X ×_{k}S → S such that for every s ∈ S(k), the bundle E_{s} is étale trivializable but Eη̄ is not étale trivializable. Moreover, on E there exists a structure of a relatively stratified sheaf 𝔼 such that for every s ∈ S(k), the bundle 𝔼_{s} has finite monodromy but the monodromy group of 𝔼η̄ is infinite.

The second part of the corollary follows from Proposition 1.3. The above corollary should be compared to the following fact:

Proposition 4.4. Let X be a projective variety defined over an algebraically closed field k of positive characteristic. Let S be a k-variety and let E be a rank r locally free sheaf on X ×_{k}S. Assume that there exists a positive integer n such that for every s ∈ S(k) we have (F_{X}^{n})^{∗}E_{s} ≃ E_{s}, where F_{X} denotes the absolute Frobenius morphism. Then the classifying morphism φ_{E} : S → M_{X}(r) is constant and Eη̄ is étale trivializable.

Proof. By Proposition 2.1, if (F_{X}^{n})^{∗}E_{s} ≃ E_{s} then there exists a finite étale Galois cover π_{s} : Y_{s} → X with Galois group G = GL_{r}(𝔽_{p^{n}}) such that π_{s}^{∗}E_{s} is trivial (in this case it is essentially due to Lange and Stuhler; see [LSt]). This implies that E_{s} ⊂ (π_{s})_{∗}π_{s}^{∗}E_{s} ≃ ((π_{s})_{∗}O_{Y})^{⊕r} and hence gr_{JH}E_{s} ⊂ (gr_{JH}(π_{s})_{∗}O_{Y})^{⊕r} . Since X is proper, the étale fundamental group of X is topologically finitely generated and hence there exists only finitely many finite étale coverings of X of fixed degree (up to an isomorphism). This theorem is known as the Lang–Serre theorem (see [LS, Théorème 4]). Let S be the set of all Galois coverings of X with Galois group G. Then for every closed k-point s of S the semi-simplification of E_{s} is contained in (gr_{JH}α_{∗}O_{Y})^{⊕r} for some α ∈ S. Therefore there are only finitely many possibilities for images of k-points s in M_{X}(r). Since S is connected, it follows that φ_{E}: S → M_{X}(r) is constant.

The remaining part of the proposition follows from Proposition 3.3.

Note that by Proposition 4.2 together with Corollary 2.3, the monodromy groups of E_{s} in Theorem 4.1 for s ∈ S(k) are not uniformly bounded. In fact, only if k is an algebraic closure of a finite field do we know that the monodromy groups of E_{s} are finite because then E_{s} can be defined over some finite subfield of k and the isomorphism (F^{2})^{∗}E_{s} ≃ E_{s} implies that for some n we have (F_{X}^{n})^{∗}E_{s} ≃ E_{s} (see the paragraph following Theorem 4.1). Moreover, the above proposition shows that in Theorem 4.1, we cannot hope to replace F with the absolute Frobenius morphism F_{X}.

5 Analogue of the Grothendieck-Katz conjecture in positive equicharacteristic

As Corollary 4.3 shows, the positive equicharacteristic version of the Grothendieck– Katz conjecture which requests a relatively stratified bundle to have finite monodromy group on the geometric generic fiber once it does on all closed fibers, does not hold in general. But one can still hope that it holds for a family of bundles coming from representations of the prime-to-p quotient of the étale fundamental group. In this section we follow André’s approach [An, Théorème 7.2.2] in the equicharacteristic zero case to show that this is indeed the case.

Let k be an algebraically closed field of positive characteristic p. Let f : X → S be a smooth projective morphism of k-varieties (in particular, integral k-schemes). Let η be the generic point of S. In particular, Xη̄ is smooth (see [SGA1, Defn 1.1]).

Theorem 5.1. Let E be a locally free sheaf of rank r on X. Let us assume that there exists a dense subset U ⊂ S(k) such that for every s in U, there is a finite Galois étale covering π_{s} : Y_{s} → X_{s} of Galois group of order prime-to-p such that π_{s}^{∗}(E_{s}) is trivial.

1) Then there exists a finite Galois étale covering πη̄ : Yη̄ → Xη̄ of order prime-to-p such that π∗ ; η̄Eη̄ is a direct sum of line bundles.

2) If k is not algebraic over its prime field and U is open in S, then Eη̄ is étale trivializable on a finite étale cover Zη̄ → Xη̄ which factors as a Kummer (thus finite abelian of order prime to p) cover Zη̄ → Yη̄ and a Galois cover Yη̄ → Xη̄ of order prime to p.

Proof. Without loss of generality, shrinking S if necessary, we may assume that S is smooth. Moreover, by passing to a finite cover of S and replacing U by its inverse image, we can assume that f has a section σ : S → X. By assumption for every s ∈ U there exists a finite étale Galois covering π_{s} : Y_{s} → X_{s} with Galois group Γ_{s} of order prime-to-p and such that π_{s}^{∗}E_{s} is trivial. To these data one can associate a representation ρ_{s} : π_{1}^{p′}(X_{s},σ(s)) → Γ_{s} ⊂ GL_{r}(k) of the prime-to-p quotient of the étale fundamental group.

By the Brauer–Feit version of Jordan’s theorem (see [BF, Theorem]) there exist a constant j(r) such that Γ_{s} contains an abelian normal subgroup A_{s} of index ≤ j(r) (here we use assumption that the p-Sylow subgroup of Γ_{s} is trivial). For a k-point s of S we have a homomorphism of specialization

α_{s} : π_{1}(Xη̄,σ(η̄)) ↠ π_{1}(X_{s},σ(s)),

which induces an isomorphism of the prime-to-p quotients of the étale fundamental groups. So for every s ∈ U we can define the composite morphism

ρ̃_{s} : π_{1}^{p′}(Xη̄,σ(η̄)) ⟶^{α_{s}} π_{1}^{p′}(X_{s},σ(s)) ⟶^{ρ_{s}} Γ_{s} ↠ Γ_{s}/A_{s}.

Let K be the kernel of the canonical homomorphism π_{∗} : π_{1}(X,σ(η̄))−→ π_{1}(S,η̄), let K^{p′} be its maximal pro-p′-quotient. Then by [SGA1, Exp. XIII, Proposition 4.3 and Exemples 4.4], one has K^{p′} = π_{1}^{p′}(Xη̄,σ(η̄)), the maximal pro-p′-quotient of π_{1}(Xη̄,σ(η̄)), and one has a short exact sequence

{1} → π_{1}^{p′}(Xη̄,σ(η̄))−→ π′_{1}(X,σ(η̄))−→^{π_{∗}} π_{1}(S,η̄) → {1},

where π′_{1}(X,σ(η̄)) is defined as the push-out of π_{1}(X,σ(η̄)) by K → K^{p′}. Since Xη̄ is proper, π_{1}(Xη̄,σ(η̄)) is topologically finitely generated. Therefore π_{1}^{p′}(Xη̄,σ(η̄)) is also topologically finitely generated and hence it contains only finitely many subgroups of indices ≤ j(r). Let G be the intersection of all such subgroups in π_{1}^{p′}(Xη̄,σ(η̄)). It is a normal subgroup of finite index. Since ker(ρ̃_{s}) is a normal subgroup of index ≤ j(r) in π_{1}^{p′}(Xη̄,σ(η̄)) we have

G ⊂ ⋂_{s∈U} ker(ρ̃_{s}).

Now let us consider the commutative diagram

\xymatrix{ & π_{1}(Xη̄,σ(η̄)) \ar[r] \ar[d] & π_{1}(X,σ(η̄)) \ar[r] \ar[d] & π_{1}(S,η̄) \ar[r] \ar[d] & \{1\} \\ \{1\} \ar[r] & π_{1}^{p′}(Xη̄,σ(η̄)) \ar[r] & π1^{′}(X,σ(η̄)) \ar[r] & π_{1}(S,η̄) \ar[r] & \{1\}}

Then G · σ_{∗}(π_{1}(S,η̄)) ⊂ π′_{1}(X,σ(η̄)) is a subgroup of finite index. It is open by the Nikolov–Segal theorem [NS, Theorem 1.1]. So the pre-image H of this subgroup under the quotient homomorphism π_{1}(X,σ(η̄)) → π1^{′}(X,σ(η̄)) defines a finite étale covering h: X′→ X. Let us take s ∈ S(k). Since the composition

H ⊂ π_{1}(X,σ(η̄)) → π_{1}(X,σ(s)) → π_{1}(S,s) is surjective, the geometric fibres of X′→ S are connected. Let us choose a k-point in X′lying over σ(s). By abuse of notation we call it σ′(s). Similarly, let us choose a geometric point σ′(η̄) of X ′ ; η̄ lying over σ(η̄). Then for any s ∈ U we have the following commutative diagram:

\xymatrix{π_{1}^{p′}(X ′ ; η̄,σ′(η̄)) \ar[rr]^{0} \ar[r]^{h_{∗}} \ar[d]^{≃} & π_{1}^{p′}(Xη̄,σ(η̄)) \ar[r] \ar[d]^{α_{s} ≃} & π_{1}^{p′}(Xη̄,σ(η̄))/G \ar[d] \\ π_{1}^{p′}(X′_{s},σ′(s)) \ar[r]^{h_{∗}} & π_{1}^{p′}(X_{s},σ(s)) \ar[r] & Γ_{s}/A_{s}}

This diagram shows that π_{1}^{p′}(X′_{s},σ′(s)) → Γ_{s} factors through A_{s} and hence E′_{s}= (h^{∗}E)_{s} is trivialized by a finiteétale Galois covering π′_{s}: Y′_{s}→ X′_{s}with an abelian Galois group of order prime to p, which is a subgroup of A_{s}. Since

E′_{s}⊂ (π′_{s})_{∗}(π′_{s})^{∗}E′_{s}≃ ((π′_{s})_{∗}O_{Y′_{s}})^{⊕r} ,

and (π′_{s})_{∗}O_{Y′_{s}}is a direct sum of torsion line bundles of orders prime to p, it follows that for every s ∈ U the bundle E′_{s}is also a direct sum of torsion line bundles of order prime to p. We consider the union M(X′/S,r) of the components of M_{P}(X′/S) containing moduli points of numerically flat bundles, as defined in Section 3. Let us consider the Smorphism ψ : M(X′/S,1)^{×_{S}r} → M(X′/S) given by ([L_{1}],...,[L_{r}]) → [⊕L_{i}] (in fact we give it by this formula on the level of functors; existence of the morphism follows from the fact that moduli schemes corepresent these functors). The bundle E′gives us a section τ : S → M(X′/S,r), and by the above for every k-rational point s of U, the point τ(s) is contained in the image of ψ. Therefore τ(S) is contained in the image of ψ as ψ is projective (thus proper). Let us consider the fibre product

\xymatrix{M(X′/S,1)^{×_{S}r} ×_{M(X′/S,r)}S \ar[r] \ar[d] & S \ar[d]^{τ} \\ M(X′/S,1)^{×_{S}r} \ar[r] & M(X′/S,r)}

Let us recall that in positive characteristic the canonical map M(X′×_{S} S′/S′,r) → M(X′/S,r)×_{S}S′need not be an isomorphism (although it is an isomorphism for r = 1). Anyway we can find an étale morphism S′→ S over some non-empty open subset of S, such that there exists a map υ : S′→ M(X′×_{S} S′/S′,1)^{×_{S′}r} which composed with M(X′×_{S}S′/S′,1)^{×_{S′}r} → M(X′×_{S}S′/S′,r) → M(X′/S,r) gives the composition of S′→ S with τ. This shows that the pull back E′′of E′to X′×_{S}S′has a filtration whose quotients are line bundles which are of degree 0 on the fibres of X′×_{S}S′→ S′. Now let us note the following lemma:

Lemma 5.2. Let f : X → S be a projective morphism of k-varieties. Let 0 → G_{1} → G → G_{2} → 0 be a sequence of locally free sheaves on X. Assume that there exists a dense subset U ⊂ S(k) such that for each s ∈ U this sequence splits after restricting to X_{s}. Then it splits on the fibre X_{η} over the generic point η of S.

Proof. By shrinking S if necessary, we may assume that S is affine and the relative cohomology sheaf R^{1}p_{∗}H om(G_{2},G_{1}) is locally free. The above short exact sequence defines a class λ ∈ Ext^{1}(G_{2},G_{1}) ≃ H^{0}(S,R^{1}f_{∗}H om(G_{2},G_{1})), such that λ(s) = 0 for every k-rational point s of U. It follows that λ = 0 and hence the sequence is split over the generic point of S.

Now let us note that on a smooth projective variety every short exact sequence of the form 0 → G_{1} → G → G_{2} → 0 in which G is a direct sum of line bundles of degree 0 and G_{2} is a line bundle of degree 0 splits. So the filtration of E′′restricted to the closed fibers splits. Therefore the above lemma and easy induction show that E′′_{η′} is a direct sum of line bundles, where η′is the generic point of S′. This shows the first part of the theorem.

To prove the second part of the theorem, we may assume that U = S. Let us take a line bundle L on X such that for every k-rational point s the line bundle L_{s} is étale trivializable. We need to prove that there exists a positive integer n prime to p and such that L_{η}^{⊗n} ≃ O_{X_{η}}.

We thank the referee for showing us the following lemma.

Lemma 5.3. Let g: A → S be an abelian scheme and let σ be a section of g such that for all s ∈ S(k), σ(s) is torsion of order prime to p. Then σ is torsion of order prime to p.

Proof. We may assume that S is normal and affine. Let us choose a subfield k′⊂ k that is finitely generated and transcendental over 𝔽_{p} and such that A → S and σ come by base change Speck → Speck′from an abelian scheme g′: A′→ S′and a section σ′ defined over k′. Let m > 1 be prime to p and let Γ be the subgroup A′(S′)∩[m]^{−1} (ℤ.σ′) of A′(S′). Then Γ is a finitely generated group. Note that assumptions of Néron’s specialization theorem [L, Chapter 9, Theorem 6.2] are satisfied and therefore there exists a Hilbert set Σ of points s′∈ S′ for which the specialization map A′(S′) → A′_{s′}(k(s′)) is injective on Γ. Since the Hilbert subset Σ ⊂ S′contains infinitely many closed points (see [L, Chapter 9, Theorems 5.1, 5.2 and 4.2]), there is a closed point s ∈ S the image of which in S′lies in Σ. The specialization of ℤ.σ at s is injective and hence σ is torsion of order dividing the order of σ(s), which is prime to p.

Let us first assume that X → S is of relative dimension 1. By passing to a finite cover of S we can assume that f has a section. The relative Picard scheme A = Pic^{0}(X/S) → S is smooth. Using the above lemma to the section corresponding to the line bundle L we see that there exists some positive integer n prime to p and a line bundle M on S such that L^{⊗n} ≃ f^{∗}M. In particular, L_{η}^{⊗n} ≃ O_{X_{η}}.

Now we use induction on the relative dimension of f : X → S to prove the theorem in the general case. Note that our assumptions imply that Lη̄ is numerically flat and therefore the family {L⊗n ; η̄ }_{n∈ℤ} is bounded. Thus for any sufficiently ample divisor H on Xη̄ we have H^{1}(Xη̄,L⊗n ; η̄ (−H)) = 0 for all integers n. We consider such an H which is defined over η. Using Bertini’s theorem we can find a very ample divisor Y ⊂ X in the linear system |H| such that f|_{Y} : Y → S is smooth (possibly after shrinking S) and such that for every positive integer n we have H^{1}(X_{η},L^{⊗n} (−Y)|_{Y_{η}}) = 0. Indeed, shrinking S and using semicontinuity of cohomology, we may assume that H is defined over S, that the function dimH^{0}(X_{s},O_{X_{s}}(H)) is constant and S is affine. Let us choose a k-rational point s in S. Then by Grauert’s theorem (see [Ha, Chapter III, Corollary 12.9]) the restriction map

H^{0}(X,O_{X}(H)) → H^{0}(X_{s},O_{X_{s}}(H))

is surjective. By Bertini’s theorem in the linear system |O_{X_{s}}(H)| there exists a smooth divisor. By the above we can lift it to a divisor Y ⊂ X, which after shrinking S is the required divisor. Applying our induction assumption to L|_{Y}on Y → X we see that there exists a positive integer n prime to p such that (L|_{Y})_{η}^{⊗n} ≃ O_{Y_{η}}. Using the short exact sequence

0 → L_{η}^{⊗n} (−Y_{η}) → L_{η}^{⊗n} → (L_{Y}^{⊗n})_{η} → 0

we see that the map

H^{0}(X_{η},L_{η}^{⊗n}) → H^{0}(Y_{η},(L_{Y}^{⊗n})_{η})

is surjective. In particular, L_{η}^{⊗n} has a section and hence it is trivial.

Remarks 5.4. 1. Laszlo’s example shows that the first part of the theorem is false if one does not assume that orders of the monodromy groups of E_{s} are prime to p (in this example Eη̄ is a stable rank 2 vector bundle). Note that in this example, E has even the richer structure of a relatively stratified bundle (see Proposition 1.3).

2. Let E be a supersingular elliptic curve defined over k = 𝔽̄_{p}. Let M be a line bundle of degree 0 and of infinite order on E\overline{𝔽_{p}(t)}. Then one can find a smooth curve S defined over k such that there exists a line bundle L on X = S ×_{k}E → S such that Lη̄ ≃ M. In this example the line bundle L_{s} is torsion for every krational point s of S as it is defined over a finite field. Since E is a supersingular elliptic curve, there are no torsion line bundles of order divisible by p. So in this case all line bundles L_{s} for s ∈ S(k) are étale trivializable (and the monodromy group has order prime to p). This shows that the second part of Theorem 5.1 is no longer true if k is an algebraic closure of a finite field.

Let us keep the notation from the beginning of the section, i.e., k is an algebraically closed field of positive characteristic p and f : X → S is a smooth projective morphism of k-varieties (in particular connected) with geometrically connected fibers. For simplicity, we also assume that f has a section σ : S → X.

Lemma 5.5. Let E be a locally free sheaf on X. If there exists a point s_{0} ∈ S(k) such that E_{s_{0}} is numerically flat then Eη̄ is also numerically flat. In particular, if there exists a point s_{0} ∈ S(k) such that there is a finite covering π_{s_{0}} : Y_{s_{0}} → X_{s_{0}} such that π_{s_{0}}^{∗}(E_{s_{0}}) is trivial, then Eη̄ is also numerically flat.

Proof. Let us fix a relatively ample line bundle. If E_{s_{0}} is numerically flat then it is strongly semistable with numerically trivial Chern classes (see [La2, Proposition 5.1]). Since E is S-flat, the restriction of E to any fiber has numerically trivial Chern classes (as intersection numbers remain constant on fibres). Now note that for any n the sheaf (F_{X_{s_{0}}/k}^{n})^{∗}E_{s_{0}} is slope semistable. Since slope semistability is an open property, it follows that (F_{X_{η} /K}^{n})^{∗}E_{η} is also slope semistable. By [HL, Corollary 1.3.8] it follows that (F_{Xη̄/K̄}^{n})^{∗}Eη̄ is also slope semistable. Thus Eη̄ is strongly semistable with vanishing Chern classes and hence it is numerically flat by [La2, Proposition 5.1].

Let us recall that numerically flat sheaves on a proper k-variety Y form a Tannakian category. A rational point y ∈ Y (k) neutralizes it. Thus we can define S-fundamental group scheme of Y at the point y (see [La2, Definition 6.1]). For a numerically flat sheaf E on Y, we consider the Tannaka k-group π_{S}(⟨E⟩,y) := Aut^{⊗} (⟨E⟩,y) ⊂ GL(E_{y}), where now ⟨E⟩ is the full tensor subcategory of numerically flat bundles spanned by E. We call it the S-monodromy group scheme. Using this language we can reformulate Theorem 5.1 in the following way (for simplicity we reformulate only the second part of the theorem).

Theorem 5.6. Let E be an S-flat family of numerically flat sheaves on the fibres of X → S. Let us assume that k is not algebraic over its prime field and there exists a non-empty open subset U ⊂ S(k) such that for every s in U, the S-monodromy group scheme π_{S}(⟨E_{s}⟩,σ(s)) is finite étale of order prime-to-p. Then π_{S}(⟨Eη̄⟩,σ(η̄)) is also finite étale.

6 Verschiebung divisible points on abelian varieties: on the theorem by M. Raynaud

Let K be an arbitrary field of positive characteristic p and let A be an abelian variety defined over K. The multiplication by p^{n} map [p^{n}] : A → A factors through the relative Frobenius morphism F_{A/K}^{n}: A → A^{(n)} and hence defines the Verschiebung morphism V^{n}: A^{(n)} → A such that V^{n}F_{A/K}^{n}= [p^{n}].

Definition 6.1. A K-point P of A is said to be V-divisible if for every positive integer n there exists a K-point P_{n} in A^{(n)} such that V^{n}(P_{n}) = P.

Let T be an integral noetherian separated scheme of dimension 1 with field of rational functions K. Let us recall that a smooth, separated group scheme of finite type A → T is called a Néron model of A if the general fiber of A → T is isomorphic to A and for every smooth morphism X → T, a morphism X_{K} → A_{K} extends (then uniquely) to a T -morphism X → A.

Assume that the base field K is the function field of a normal projective variety S defined over a field k of positive characteristic p. We say that A has a good reduction at a codimension 1 point s ∈ S if the Néron model of A over SpecO_{S,s} is an abelian scheme (the usual definition is slightly different as it assumes that the identity component of the special fibre of the Néron model is an abelian variety; it is equivalent to the above one by [BLR, 7.4, Theorem 5]). We say that A has potential good reduction at a codimension 1 point s ∈ S if there exists a finite Galois extension K′of K such that if S′is the normalization of S in K′then A_{K′} has good reduction at every codimension 1 point s′∈ S′lying over s. We say that A has (potential) good reduction if it has (potential) good reduction at every codimension 1 point of S. Assume that A has good reduction at every codimension 1 point of S. Then there exists a big open subset U ⊂ S (i.e., the codimension of the complement of U in S is ≥ 2) and an abelian U-scheme A → U. Note that the group A(K) of K-points of A is isomorphic via the restriction map to the group of rational sections U −−→ A of A → U defined over some big open subset of U. The section corresponding to P ∈ A(K) will be denoted by P̃: U −−→ A . Let c ∈ PicA be a class of a line bundle L. By the theorem of the cube c, satisfies the following equality:

m_{123}^{∗}c − m_{12}^{∗}c − m_{13}^{∗}c − m_{23}^{∗}c + m_{1}^{∗}c + m_{2}^{∗}c + m_{3}^{∗}c = 0,

where m_{I} for I ⊂ {1,2,3} is the map A ×_{K}A ×_{K}A → A defined by addition over the factors in I. (In particular, m_{i} is the i-th projection). Combining [MB, Chapter III, 3.1] (relying on [MB, Chapter II, Proposition 1.2.1]), the line bundle L ∈ Pic(A) extends uniquely (at least if we fix a rigidification) to a line bundle L̃ over A_{V} such that the class c̃= [L̃] ∈ Pic(A_{V}) is cubical, i.e., satisfies the relation

m̃_{123}^{∗}c̃−m̃_{12}^{∗}c̃−m̃_{13}^{∗}c̃−m̃_{23}^{∗}c̃+m̃_{1}^{∗}c̃+m̃_{2}^{∗}c̃+m̃_{3}^{∗}c̃= 0,

where V ⊂ U is a big open subset and where m̃_{I} for I ⊂ {1,2,3} is the map A ×_{S} A ×_{S}A → A defined by addition over the factors in I. Now let us choose an ample line bundle H on S. Then the map ĥ_{c}: A(K) → ℤ given by

ĥ_{c}(P) = deg_{H}(P̃− ˜0) ^{∗}c̃

is well defined as P̃ is defined on a big open subset of S and P̃^{∗}L̃ extends to a rank 1 reflexive sheaf on S. This map is the canonical (Néron–Tate) height of A corresponding to c (see [MB, Chapter III, Section 3]).

The following theorem was suggested to the authors by M. Raynaud (in the good reduction case over a curve S, and with a somewhat different proof).

Theorem 6.2. Assume that A has potential good reduction. If P ∈ A(K) is V-divisible and c is symmetric then ĥ_{c}(P) = 0.

Proof. Let us first assume that A has good reduction. By assumption there exists a K-point P_{n} of A^{(n)} such that V^{n}(P_{n}) = P. Since A → U is an abelian scheme, so is A^{(n)} → U, thus P_{n} is the restriction to SpecK of P̃_{n}∈ A^{(n)}(U).

Let us factor the absolute Frobenius morphism F_{A}^{n} into the composition of the relative Frobenius morphism F_{A/K}^{n}: A → A^{(n)} and W_{n} : A^{(n)} → A. Let us set c_{n} = W_{n}^{∗}c. Its

cubical extension c̃_{n} ∈ Pic(A_{V_{n}}^{(n)}), for some big open V_{n} ⊂ U, together with H allows one to define ĥ_{c_{n}}(P_{n}) by the corresponding formula. Since (F_{A}^{n})^{∗}c = p^{n}c, we have (F_{A/K}^{n})^{∗}c_{n} = p^{n}c. On the other hand, since c is symmetric, we have [p^{n}]^{∗}c = p^{2n}c and hence (F_{A/K}^{n})^{∗}((V^{n})^{∗}c) = p^{2n}c. Therefore

(F_{A/K}^{n})^{∗}((V^{n})^{∗}c − p^{n}c_{n}) = 0.

Since F_{A/K}^{n}is an isogeny this implies that the class d = (V^{n})^{∗}c − p^{n}c_{n} is torsion. By additivity and functoriality of the canonical height (see [Se, Theorem, p. 35]) we have

ĥ_{c}(P) = ĥ_{(V^{n})^{∗}c}(P_{n}) = ĥ_{p^{n}c_{n}}(P_{n}) + ĥ_{d}(P_{n}) = p^{n}· ĥ_{c_{n}}(P_{n})

(note that additivity implies that ĥ_{md}= mĥ_{d}, so since md = 0 for some m, we get ĥ_{d} = 0). Therefore if ĥ_{c}(P) ≠ 0 then |ĥ_{c}(P)| ≥ p^{n} and we get a contradiction if n is sufficiently large. Now let us consider the general case. Since there exist only finitely many codimension 1 points s ∈ S at which A has bad reduction, one can find a finite Galois extension K′ of K such that if S′ is the normalization of S in K′ then A_{K′} has good reduction at every codimension 1 point s′∈ S′. On the other hand, if P ∈ A(K) is V -divisible on A, P ⊗ K′∈ A(K′) is V divisible on A_{K′}. Then by the above we have ĥ_{π^{∗}c}(P′) = 0 and functoriality of the canonical height implies that ĥ_{c}(P) = 0.

Remark 6.3. It is an interesting problem whether Theorem 6.2 holds for an arbitrary abelian variety A/K. Its proof shows that one can use the semiabelian reduction theorem to reduce the general statement to the case when A has semiabelian reduction (see [BLR, 7.4, Theorem 1]).

Now assume that S is geometrically connected. Then the extension k ⊂ K is regular (i.e., K/k is separable and k is algebraically closed in K). Let (B,τ) be the K/ktrace of the abelian K-variety A, where B is an abelian k-variety and τ : B_{K} → A is a homomorphism of abelian K-varieties (it exists by [Co, Theorem 6.2]). Let us recall that by definition (B,τ) is a final object in the category of pairs consisting of an abelian k-variety and a K-map from the scalar K-extension of this variety to A. Since the extension k ⊂ K is regular, the kernel K-group scheme of τ is connected (with connected dual) ([Co, Theorem 6.12]). Therefore τ is injective on K-points and in particular we can treat B(k) as a subgroup of A(K).

Corollary 6.4. Assume that A has potential good reduction. If P ∈ A(K) is Vdivisible then [P] ∈ (A(K)/B(k))_{tors}. In particular, if k is algebraically closed then P ∈ B(k) + A(K)_{tors}⊂ A(K).

Proof. We can choose the class c ∈ Pic(A) so that it is ample and symmetric. Then the first part of the corollary follows from Theorem 6.2 and [Co, Theorem 9.15] (which is true for regular extensions K/k).

To prove the second part take positive integer m such that mP = Q ∈ B(k). Since k is algebraically closed, the set B(k) is divisible and there exists Q′∈ B(k) such that mQ′= Q. Then P = Q′+ (P − Q′), where m(P − Q′) = 0.

Let us assume that the field k is algebraically closed. It is an interesting question whether a V -divisible K-point P of A can be written as a sum of Q+R, where Q ∈ B(k) and R ∈ A(K)_{tors}is torsion of order prime-to-p. By the Lang–Néron theorem ([Co, Theorem 2.1]), the groups A^{(i)}(K)/B^{(i)}(k) are finitely generated. It follows that the groups G_{i} = (A^{(i)}(K)/B^{(i)}(k))_{tors}are finite. Note that the homomorphism B(k) → B^{(i)}(k) induced by F_{B/k}^{i} is a bijection. One

has a factorization F_{A/K}^{i}: A(K^{1/p^{i}}) → A^{(i)}(K) → A^{(i)}(K^{1/p^{i}}), inducing a bijection

A(K^{1/p^{i}}) → A^{(i)}(K). Thus in particular,

F_{i} : A(K)/B(k) → A^{(i)}(K)/B^{(i)}(k)

is injective. Moreover, the Verschiebung morphism induces the homomorphisms

V_{i} : A^{(i)}(K)/B^{(i)}(k) → A(K)/B(k)

such that V_{i}F_{i} = p^{i} and F_{i}V_{i} = p^{i}. This shows that prime-to-p torsion subgroups of groups G_{i} are isomorphic and in particular have the same order m. Now let us assume that orders of the p-primary torsion subgroups of the abelian groups G_{i} are uniformly bounded by some p^{e}. Then for all i ≥ e

F_{i}(m[P]) = F_{i}(V_{i}(m[P_{i}])) = p^{i}m[P_{i}] = 0.

This implies that m[P] = 0, so mP ∈ B(k). Now B(k) is a divisible group so there exists some Q′∈ B(k) such that mP = mQ. Then R = P −Q ∈ A(K) is torsion of order prime to p. So we conclude

Lemma 6.5. If the order of the G_{i} is bounded as i goes to infinity, under the assumption the Theorem 6.2, there exists a positive integer m, prime to p and such that m · P_{i} ∈ B(k) for every integer i.

Note that the above assumption on G_{i} is satisfied, e.g., if A is an elliptic curve over the function field K of a smooth curve over k = k̄. If A is isotrivial then the assertion is clear. If A is not isotrivial then the j-invariant of A is transcendental over k. In this case A(K^{perf})_{tors} is finite (see [Le]) so orders of the groups G_{i} = A^{(i)}(K)_{tors} are uniformly bounded.

7 Stratified bundles

In this section we use the height estimate of the previous section and the fact that torsion stratified line bundles on a perfect field have order prime to p (apply Proposition 2.2 together with Lemma 1.1).

Let k be an algebraically closed field of positive characteristic p. Let f : X → S be a smooth projective morphism of k-varieties with geometrically connected fibres. Assume that S is projective, which surely is a very strong assumption. Indeed, if k ≠ 𝔽_{p} , and in the statement of Theorem 7.1, S′is open, then one obtains the stronger Theorem 7.2. For simplicity, let us also assume that f has a section σ : S → X. Consider the torsion component Pic^{τ}(X/S) → S of identity of Pic(X/S) → S. Let φ_{n}: Pic(X/S) → Pic(X/S) be the multiplication by n map. Then there exists an open subgroup scheme Pic^{τ}(X/S) of Pic(X/S) such that every geometric point s of S the fibre of Pic^{τ}(X/S) over s is the union

⋃_{n>0} φ_{n}^{−1} (Pic^{0}(X_{s})),

where Pic^{0}(X_{s}) is the connected component of the identity of Pic(X_{s}/s). It is well known that Pic^{τ}(X/S) → S is also a closed subgroup scheme of Pic(X/S). Moreover, the morphism Pic^{τ}(X/S) → S is projective and the formation of Pic^{τ}(X/S) → S commutes with a base change of S (see, e.g., [Kl, Theorem 6.16 and Exercise 6.18]). We assume that Pic^{0}(X_{s}) is reduced for every point s ∈ S.

Theorem 7.1. Let 𝕃 = {L_{i},σ_{i}} be a relatively stratified line bundle on X/S. Assume that there exists a dense subset S′⊂ S(k) such that for every s ∈ S′the stratified bundle 𝕃_{s} = 𝕃|_{X_{s}} has finite monodromy. Then 𝕃η̄ has finite monodromy.

Proof. Replacing 𝕃 by a power 𝕃^{⊗N}, where N is sufficiently large, we may assume that 𝕃_{s} ∈ Pic^{0}(X_{s}) for all closed points s in S (see [Kl, Corollary 6.17]). By assumption π̂: Â= Pic^{0}(X/S) → S is an abelian scheme. Let us consider the dual abelian scheme A → S. We have a well defined Albanese morphism g : (X,σ) → (A ,e) (see [FGA, Exposé VI, Théorème 3.3]). Moreover, the map g^{∗} : Pic^{0}(A /S) → Â= Pic^{0}(X/S) is an isomorphism of S-schemes. Let us set Â= Â_{η}. Let P_{i} be the K-point of Â^{(i)} corresponding to (L_{i})_{η}. Note that the K-point P_{0} ∈ Âis V -divisible. Indeed, by the definition of a relative stratification we have V^{n}(P_{n}) = P_{0} for all integers n. Similarly, we see that all the points P_{i} ∈ Â^{(i)}(K) are V -divisible. By Corollary 6.4 it follows that P_{i} ∈ B̂^{(i)}(k) + Â^{(i)}(K)_{tors}, where (B̂/k,τ̂: B̂_{K}→ Â) is the K/k-trace of Â(note that (B̂^{(i)}/k,τ̂^{(i)}) is the K/k-trace of Â^{(i)}). So for every i ≥ 0 we can write P_{i} = Q_{i}+ R_{i} for some Q_{i} ∈ B̂^{(i)}(k) and R_{i} ∈ Â^{(i)}(K)_{tors}. Now we transpose the above by duality. Let A be the dual abelian K-variety of Â and B the dual abelian k-variety of B̂. We have the K/k-images τ^{(i)} : A_{η}^{(i)} → B_{K}^{(i)} and an S-morphism τ : A → B ×_{k}S (possibly after shrinking S). By abuse of notation we can treat L_{i} as line bundles on A because g^{∗} : Pic^{0}(A /S) → Pic^{0}(X/S) is an isomorphism. Let M_{i} be the line bundle on B^{(i)} corresponding to Q_{i} and let π_{i} : B^{(i)}×_{k} S → B^{(i)} denote the projection. Let us fix a non-negative integer i and take a positive integer n_{i} such that n_{i}R_{i} = 0. Then the line bundle L_{i}^{⊗n_{i}} ⊗ τ^{∗}π_{i}^{∗}M_{i}^{⊗−n_{i}} has degree 0 on every fiber of A → S. Thus it is trivial after restriction to A_{η}. Hence after shrinking S we can assume that L_{i}^{⊗n_{i}} ≃ τ^{∗}π_{i}^{∗}M_{i}^{⊗n_{i}} . Let us fix a point s ∈ S(k) and consider the morphism

π′_{i}= (τ^{(i)}π_{i})_{A_{s}^{(i)}} : A_{s}^{(i)} → B^{(i)}.

Note that τ^{(i)} has connected fibres and hence (π′_{i})_{∗}O_{A_{s}^{(i)}} = O_{B^{(i)}}. By assumption there exists a positive integer a_{s}, such that for every i the order of the line bundle (L_{i})_{A_{s}} divides a_{s}. The important point is that a_{s} is prime to p. Therefore (π′_{i})^{∗}M_{i}^{⊗a_{s}n_{i}} ≃ O_{A_{s}} and by the projection formula

M_{i}^{⊗a_{s}n_{i}} ≃ (π′_{i})_{∗}(π′_{i})^{∗}M_{i}^{⊗a_{s}n_{i}} ≃ (π′_{i})_{∗}O_{A_{s}} ≃ O_{B}.

This implies that M_{i} is a torsion line bundle and hence Q_{i} ∈ Â^{(i)}(K)_{tors}. Therefore

P_{i} = Q_{i}+ R_{i} ∈ Â^{(i)}(K)_{tors}.

Let us recall that the set of p-torsion points of Â(K) is finite. Assuming it is not empty, we can therefore find a non-empty open subset U ⊂ S such that for every s ∈ U(k) and every p-torsion point T ∈ Â(K) the section T̃ is defined on U and the point T̃(s) is non-zero. Let us write the order of P_{i} as m_{i}p^{e_{i}}, where m_{i} is not divisible by p. If e_{0} ≥ 1 then the point m_{0}p^{e_{0}−1} P_{0} is p-torsion in Â(K). If we take s ∈ S′∩U(k), then a_{s}m_{0}p^{e_{0}−1} P̃_{0}(s) = [L_{0}^{⊗a_{s}m_{0}} ]_{s} = 0, a contradiction. It follows that m_{0}P_{0} = 0. Similarly, the order of all P_{i} is prime to p. As already mentioned in the last section, the homomorphism Â(K^{1/p^{i}}) → Â^{(i)}(K) induced by F_{A/K}^{i}is a bijection. So we have an induced injection

F_{i} : Â(K) → Â^{(i)}(K).

On the other hand, the Verschiebung morphism induces homomorphisms

V_{i} : Â^{(i)}(K) → Â(K)

such that V_{i}F_{i}(P) = p^{i}P and F_{i}V_{i}(Q) = p^{i}Q for all P ∈ Â(K) and Q ∈ Â^{(i)}(K). Hence

p^{i}m_{0}P_{i} = F_{i}V_{i}(m_{0}P_{i}) = F_{i}(m_{0}P_{0}) = 0

and since the order of P_{i} is prime to p we have m_{0}P_{i} = 0 for all i ≥ 0. Therefore (L_{i})⊗m_{0} ; η̄ ≃ O_{Xη̄} for all i and the stratified line bundle 𝕃η̄ has finite monodromy.

Now we fix the following notation: k is an algebraically closed field of positive characteristic p and f : X → S is a smooth projective morphism of k-varieties with geometrically connected fibres.

Theorem 7.2. Let 𝔼 = {E_{i},σ_{i}} be a relatively stratified bundle on X/S. Assume that there exists a dense subset U ⊂ S(k) such that for every s ∈ U the stratified bundle 𝔼_{s} = 𝔼|_{X_{s}} has finite monodromy of order prime to p.

1) Then there exists a finite Galois étale covering πη̄ : Yη̄ → Xη̄ of order prime-to-p such that π∗ ; η̄𝔼η̄ is a direct sum of stratified line bundles.

2) If k ≠ 𝔽̄_{p} and U is open in S(k), then the monodromy group of 𝔼η̄ is finite, and 𝔼η̄ trivializes on a finite étale cover Zη̄ → Xη̄ which factors as a Kummer (thus finite abelian of order prime to p) cover Zη̄ → Yη̄ and a Galois cover Yη̄ → Xη̄ of order prime to p.

Proof. We prove 1). Let us first remark that the schemes X (i) ; η̄, i ≥ 0, are all isomorphic (as schemes, not as k-schemes). Therefore the relative Frobenius induces an isomorphism on fundamental groups. By the first part of Theorem 5.1we know that there exists a finite Galois étale covering π_{i} : Y_{η̄,i} → X (i) ; η̄ of degree prime to p such that π_{i}^{∗}(E_{i}) is a direct sum of line bundles ⊕_{1}^{r}L_{ij}. Note that from the proof of Theorem 5.1 the degree of π_{i} depends only on π_{1}^{p′}(X (i) ; η̄,σ^{(i)}(η̄)) and the Brauer-Feit constant j(r), and therefore it can be bounded independently of i. Using the Lang–Serre theorem (see [LS, Théorème 4]) we can therefore assume that Y_{η̄,i} = Y (i) ; η̄, where Yη̄ = Y_{η̄,0}. Now we know that

⊕_{j=1}^{r} L_{ij} ≃ (F_{Y (i) ; η̄/η̄}^{i})^{∗}( ⊕_{j′=1}^{r} L_{i+1,j′} ).

By the Krull-Schmidt theorem, the set of isomorphism classes of line bundles {L_{ij}}_{j} is the same as the set of isomorphism classes of lines bundles which come by pull-back {(F_{Y (i) ; η̄/η̄}^{i})^{∗}(L_{i+1,j′} )}_{j′}. So we can reorder the indices j′so that

(F_{Y (i) ; η̄/η̄}^{i})^{∗}(L_{i+1,j} ) ≅ L_{i,j}.

This finishes the proof of 1). To prove 2), we do the proof 1) replacing Yη̄ → Xη̄ by Zη̄ → Xη̄ of Theorem 5.1 2). This finishes the proof of 2).

Remarks 7.3. 1) Case 2) of Theorem 7.2 applied to a line bundle extends Theorem 7.1, where S was assumed to be projective, Pic^{0}(X_{s}) reduced for all s ∈ S closed, S′⊂ S(k) dense, to the case when S is not necessarily projective and S′⊂ S(k) is open and dense, but we have to assume that k is not algebraic over its prime field.

2) If Yη̄ has a good projective model satisfying assumptions of Theorem 7.1 then it follows that 𝔼η̄ has finite monodromy.

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Hélène Esnault Freie Universität Berlin FB Mathematik und Informatik Arnimallee 3 14195 Berlin, Germany Helene.Esnault@fu-berlin.de Adrian Langer University of Warsaw Institute of Mathematics ul. Banacha 2 02-097 Warszawa, Poland alan@mimuw.edu.pl