The Variation of the Monodromy Group in Families

of Stratified Bundles in Positive Characteristic

Giulia Battiston

Received: June 03, 2014

Communicated by Takeshi Saito

Abstract. In this article we study smooth families of stratified bundles in positive characteristic and the variation of their monodromy group. Our aim is, in particular, to strengthen the weak form of the positive equicharacteristic p-curvature conjecture stated and proved by Esnault and Langer in [9]. The main result is that if the ground field is uncountable then the strong form holds. In the case where the ground field is countable we provide positive and negative answers to possible generalizations.

2010 Mathematics Subject Classification: 2010 MSC Primary: 14D05, 14H30, 13N10

1. Introduction

Let (E, ∇) be a vector bundle endowed with a flat connection on a smooth complex variety X. Then, there exists a smooth scheme S over (some open subscheme of) Spec ℤ such that (E, ∇) = (E_{S}, ∇_{S}) ⊗_{S} ℂ and X = X_{S} ⊗_{S} ℂ with X_{S} smooth over S and (E_{S}, ∇_{S}) flat connection on X_{S} relative to S. The p-curvature conjecture of Grothendieck and Katz predicts (see [2, Conj. 3.3.3]) that if for all closed points s of a dense open subscheme S̃ ⊂ S we have that E_{S} ×_{S} s is spanned by its horizontal sections, then (E, ∇) must be trivialized by an étale finite cover of X. An analogue problem can be studied in equicharacteristic zero, and in fact it reduces the p-curvature conjecture to the number field case. Y. André in [2, Prop. 7.1.1] and E. Hrushovski in [14, 116] stated and proved the following equicharacteristic zero version of the p-curvature conjecture: let X → S be a smooth morphism of varieties over a field K of characteristic zero; let (E, ∇) be a flat connection on X relative to S such that, for every closed point s in a dense open S̃ ⊂ S, the flat connection (E, ∇)×_{S} s is trivialized by a finite étale cover. Then, there exists a finite étale cover of the generic geometric fiber over η̄ that trivializes (E, ∇) ×_{S} η̄, where η̄ is a geometric generic point of S.

The theorem of André and Hrushovsky translates naturally in positive characteristic, providing a positive equicharacteristic analogue to the p-curvature conjecture. Here, the role of relative flat connections is played by relative stratified bundles. A stratified bundle on X relative to S is a vector bundle of finite rank with an action of the ring of differential operators 𝒟_{X/S} on X relative to S. In [9, Cor. 4.3, Rmk. 5.4.1] H. Esnault and A. Langer proved, using an example of Y. Laszlo (see [18]), that there exists X → S a projective smooth morphism of varieties over 𝔽̄_{2} and a stratified bundle over X relative to S which is trivialized by a finite étale cover on every closed fiber but not on the geometric generic one. In particular, this provides a counterexample to the positive equicharacteristic version of André and Hrushovsky’s theorem. Nevertheless, they were able to prove what they call a weak form of the theorem (see [9, Thm. 7.2]): let X → S be a projective smooth morphism and let 𝔼 = (E, ∇) be a stratified bundle on X relative to S such that, for all closed points of a dense open S̃ ⊂ S, the stratified bundle 𝔼 ×_{S} s is trivialized by a finite étale cover of order prime to p. Then, if K ≠ 𝔽̄_{p}, there exists a finite étale cover of order prime to p of the generic geometric fiber that trivializes 𝔼 ×_{S} η̄. In case K = 𝔽̄_{p}, there exists a finite étale cover of order prime to p such that the pullback of 𝔼 ×_{S} η̄ is a direct sum of stratified line bundles.

In this article we answer two natural questions aiming at generalizing this last theorem: the first one is whether we can relax the assumption of coprimality to p of the order of the trivializing covers of the 𝔼 ×_{S} s, while keeping the assumption that X is proper over S. The second one is if we can drop this last assumption as well. Our main result, in particular, is that if K is uncountable then the positive equicharacteristic version of André and Hrushovsky’s theorem holds. Let assume from now on that K has positive characteristic p. Bearing in mind the counterexample of Esnault and Langer ([9, Cor. 4.3]) we cannot hope in general to completely eliminate the assumption of coprimality to p of the order of the trivializing covers of the 𝔼 ×_{S} s. Still, we can answer positively the first question proving that it suffices to impose to the power of p dividing the order of such trivializing covers to be bounded:

Theorem 1 (See Theorem 4.3). Let K be an algebraically closed field, X → S a smooth proper morphism of K-varieties and 𝔼 = (E, ∇) a stratified bundle on X relative to S. Assume that for every closed point s in a dense open S̃ ⊂ S the stratified bundle 𝔼_{s} = 𝔼 ×_{S} s is trivialized by a finite étale cover whose order is not divisible by p^{N} for some fixed N ≥ 0. Then, if K ≠ 𝔽̄_{p}, there exists a finite étale cover of the generic geometric fiber that trivializes 𝔼η̄ = 𝔼 ×_{S} η̄. In case K = 𝔽̄_{p}, there exists a finite étale cover such that the pullback of 𝔼η̄ is the direct sum of stratified line bundles.

The Variation of the Monodromy Group in Families … 67

The second question has a more involved answer. The assumption on X being proper over S is more delicate to eliminate; the order of the trivializing covers does not play any role while the cardinality of K becomes the main obstruction:

Counterexample (See Proposition 5.1). If K is a countable field, then there exists a stratified bundle on 𝔸_{K}^{2} relative to 𝔸_{K}^{1} which is trivial on every closed fiber but is not trivialized by any finite étale cover on the generic geometric fiber.

On the other hand the main result of this article is that in case K is uncountable the strong version of the theorem holds, namely:

Theorem 2 (See Theorem 6.1). Let K be an uncountable algebraically closed field, X → S a smooth morphism of K-varieties and 𝔼 = (E, ∇) a stratified bundle on X relative to S such that, for every closed point s in a dense open S̃ ⊂ S, the stratified bundle 𝔼_{s} = 𝔼 ×_{S} s is trivialized by a finite étale cover. Then, there exists a finite étale cover of the generic geometric fiber that trivializes 𝔼η̄ = 𝔼 ×_{S} η̄.

In the case where K is countable and X is not proper over S there is still room for improvement, using the theory of regular singular stratified bundles (introduced in [11]). Roughly speaking, a stratified bundle is regular singular if it has only mild (that is logarithmic) singularities along the divisor at infinity. In characteristic zero there is a parallel notion of regular singular flat connections, and one of the first steps in the proof of André’s theorem is to show that if a relative flat connection (E, ∇) on X over S is regular singular on the fiber over all closed points of a dense subset of S then it is regular singular on the generic fiber (see [2, Lemma 8.1.1]). In positive characteristic this is no longer true, as our counterexample shows. The converse still holds (see the proof of Lemma 7.4): if X admits a good compactification over S and 𝔼 = (E, ∇) is a stratified bundle on X relative to S such that 𝔼η̄ is regular singular then for every closed point s of some dense open S̃ ⊂ S the stratified bundle 𝔼_{s} is regular singular as well. Moreover, assuming 𝔼η̄ to be regular singular we obtain the same results than in the proper case:

Theorem 3 (See Theorem 7.7). Let K be an algebraically closed field of any cardinality and X → S a smooth morphism of K-varieties. Let 𝔼 = (E, ∇) be a stratified bundle on X relative to S such that, for every closed point s in a dense open S̃ ⊂ S, the stratified bundle 𝔼_{s} = 𝔼 ×_{S} s is trivialized by a finite étale cover whose order is not divisible by p^{N} for some fixed N ≥ 0. Assume moreover that 𝔼η̄ = 𝔼 ×_{S} η̄ is regular singular. Then, if K ≠ 𝔽̄_{p}, there exists a finite étale cover of the generic geometric fiber that trivializes 𝔼η̄. In case K = 𝔽̄_{p}, there exists a finite étale cover such that the pullback of 𝔼η̄ is the direct sum of stratified line bundles.

The proofs of these generalizations are of two different kinds. The ones of Theorem 1 and Theorem 3 rely on a reduction to Esnault and Langer’s result ([9, Thm. 7.2]). Theorem 3 is reduced to Theorem 1 using the theory of exponents for regular singular stratified bundles ([11],[16]) and an adaptation of Kawamata coverings ([15, Thm. 17]) to positive characteristic (see Theorem 7.6). Theorem 1 is then proved by reduction to [9, Thm. 7.2] constructing a suitable finite étale cover of X that kills the p-powers in the orders of the trivializing covers on the closed fibers. The proof of Theorem 2 is of another flavour: it relies on the invariance of the Tannakian monodromy group under algebraically closed extension of fields for finite stratified bundles (Lemma 3.4), and on the easy but fundamental fact that a (relative) stratified bundle is defined by countably many data.

Acknowledgments. The results contained in this article are part of my PhD work under the supervision of Hélène Esnault. I would like to thank her for introducing me to the subject, and for her great patience and support. Moreover, I would like to thank Lars Kindler for many useful and pleasant discussions, in particular regarding Lemma 3.2.

Notation. If S is an integral scheme k(S) will denote its field of fractions, η its generic point and η̄ a geometric generic point given by the choice of an algebraically closure \overline{k(S)} of k(S). If K is a field a variety over K is a separated integral scheme of finite type over K.

2. The category of stratified bundles

Throughout the whole article K will denote an algebraically closed field of positive characteristic p and u : X → S a smooth morphism of varieties over K, of relative dimension d. Let 𝒟_{X/S} be the quasi-coherent 𝒪_{X}-module of relative differential operators as defined in [12, §16]; recall that if U is an open subscheme of X admitting global coordinates x_{1}, …, x_{d} relative to S, then for every k ∈ ℕ there are 𝒪_{S}-linear maps ∂_{x_{i}}^{(k)} : 𝒪_{U} → 𝒪_{U} given by

∂_{x_{i}}^{(k)}(x_{j}^{h}) = δ_{ij}\binom{h}{k}(x_{j}^{h−k})

where δ_{ij} is the Kronecker delta. These maps are differential operators of order k and generate locally the ring of differential operators:

𝒟_{X/S|_{U}} = 𝒪_{U}[∂_{x_{i}}^{(k)} | i ∈ {1, …, d}, k ∈ ℕ_{>0}].

For higher differential operators we have an extension of the Leibniz rule, namely if f, g ∈ 𝒪_{U} then

(1)  ∂_{x_{i}}^{(k)}(fg) = ∑_{\begin{matrix}a+b=k \\ a,b≥0 \end{matrix}} ∂_{x_{i}}^{(a)}(f)∂_{x_{i}}^{(b)}(g).

Definition 2.1. A stratified bundle 𝔼 (relative to S) is a 𝒪_{X}-locally free module E of finite rank r endowed with a 𝒟_{X/S}-action extending the 𝒪_{X}-module structure via the inclusion 𝒪_{X} ⊂ 𝒟_{X/S}. A morphism of stratified bundles is a morphism of 𝒟_{X/S}-modules. We denote by Strat(X/S) the category of stratified bundles on X relative to S; if S = Spec K we use the notation Strat(X/K) for Strat(X/ Spec K). The structure sheaf 𝒪_{X} together with the natural 𝒟_{X/S}- action is denoted by 𝕀_{X/S}; if S = Spec K we use the notation 𝕀_{X/K} for 𝕀_{X/ Spec K}. A stratified bundle is trivial if it is isomorphic to 𝕀_{X/S}^{⊕r} for some r ∈ ℕ.

If h : Y → X is a morphism of smooth S-varieties then the pullback along h induces a functor h^{∗} : Strat(X/S) → Strat(Y/S) and if h is finite and étale then the pushforward along h induces a functor h_{∗} : Strat(Y/S) → Strat(X/S). For 𝔼, 𝔽 ∈ Strat(X/S) we can construct the dual 𝔼^{∨}, the tensor product 𝔼 ⊗ 𝔽 and the direct sum 𝔼 ⊕ 𝔽, all of which are objects of Strat(X/S).

3. The monodromy group

If X is a smooth connected K-variety, Strat(X/K) is an abelian tensor rigid K-linear category and the choice of a rational point x ∈ X(K) defines a fiber functor to the category of finite dimensional K-vector spaces by:

ω_{x} : Strat(X/S) → Vecf_{K}
𝔼 ↦ E_{x}

where E is the vector bundle underlying 𝔼 ([21, §VI.1]). Hence (Strat(X/S), ω_{x}) is a neutral Tannakian category and by Tannakian duality ([7, Thm. 2.11]) there exists an affine group scheme π_{1}^{Strat}(X, x) ≐ π(Strat(X/S), ω_{x}) = \underline{Aut}_{K}^{⊗}(ω_{x}) over K such that Strat(X/K) is equivalent via ω_{x} to the category of finite dimensional representations of π_{1}^{Strat}(X, x) over K. For every 𝔼 ∈ Strat(X/K) we denote by ⟨𝔼⟩_{⊗} ⊂ Strat(X/K) the full Tannakian subcategory spanned by 𝔼 with fiber functor ω_{x} defined as above. The affine group scheme π(𝔼, x) ≐ π(⟨𝔼⟩_{⊗}, ω_{x}) is called the monodromy group of 𝔼. If U ⊂ X is an open dense subscheme of X then by [16, Lemma 2.5(a)] the restriction functor ρ_{U} : ⟨𝔼⟩_{⊗} → ⟨𝔼_{|U}⟩_{⊗} is an equivalence; hence, in particular, the monodromy group of 𝔼 is invariant under restriction to dense open subschemes. Moreover, as K is algebraically closed, the monodromy group does not depend on the choice of x, up to non-unique isomorphism (this can be deduced from [7, Thm. 3.2]). We will hence sometimes use the notation π(𝔼) instead of π(𝔼, x).

Definition 3.1. We say that 𝔼 ∈ Strat(X/K) is finite if its monodromy group is finite. By what we have just remarked, this property is independent of the choice of x. We say that 𝔼 is isotrivial if it is étale trivializable; that is, there exists h : Y → X finite étale cover such that h^{∗}𝔼 is trivial in Strat(Y/K).

These two properties are equivalent:

Lemma 3.2. For a stratified bundle 𝔼 ∈ Strat(X/K) the following are equivalent:

i) 𝔼 is isotrivial; ii) 𝔼 is finite.

Moreover, if 𝔼 is finite, then there exists an étale π(𝔼, x)-torsor h_{𝔼,x} : Y_{𝔼,x} → X, called the Picard–Vessiot torsor of 𝔼 such that, for any 𝔼′ ∈ Strat(X/K), the pullback h_{𝔼,x}^{∗}𝔼′ is trivial if and only if 𝔼′ ∈ ⟨𝔼⟩_{⊗}.

Finally, for a finite étale cover h : Y → X such that h^{∗}𝔼 is trivial, the following conditions are equivalent:

i) h : Y → X is the Picard–Vessiot torsor for 𝔼; ii) every finite étale cover trivializing 𝔼 factors (non-uniquely) through

h : Y → X; iii) h : Y → X is Galois and ⟨𝔼⟩_{⊗} = ⟨h_{∗}𝕀_{Y/K}⟩_{⊗}; iv) h : Y → X is Galois of Galois group π(𝔼, x)(K).

Proof. The first part of the lemma is [9, Lemma 1.1]. As for the second part, first notice that point (b) and (f) of [16, Prop. 2.15], together with [16, Cor. 2.16] imply that if h : Y → X is a finite étale cover trivializing 𝔼 then ⟨𝔼⟩_{⊗} ⊂ ⟨h_{∗}𝕀_{Y/K}⟩_{⊗} and that ⟨𝔼⟩_{⊗} = ⟨h_{𝔼,x∗}𝕀_{Y_{𝔼,x}/K}⟩_{⊗}. Moreover, if h : Y → X is Galois of Galois group G, then π(h_{∗}𝕀_{Y/K}, x) is the finite constant group G and if h̃ : Ỹ → X is an étale cover factoring through h then ⟨h_{∗}𝕀_{Y/K}⟩_{⊗} ⊂ ⟨h̃_{∗}𝕀_{Ỹ /K}⟩_{⊗}. We are now ready to prove the rest of the lemma.

(i)⇒(ii) Because ⟨𝔼⟩_{⊗} = ⟨h_{𝔼,x∗}𝕀_{Y_{𝔼,x}/K}⟩_{⊗}, a cover h̃ : Ỹ → X trivializes 𝔼 if and only if it trivializes h_{𝔼,x∗}𝕀_{Y_{𝔼,x}/K}. Let Z = Ỹ ×_{X} Y_{𝔼,x}, and let p_{1} and p_{2} be the projections on the first and second factor. Then by flat base change (notice that the flat base change morphism is compatible with the 𝒟_{Ỹ /K}-action) there is an isomorphism of 𝒟_{Ỹ /K}- modules h̃^{∗}h_{𝔼,x∗}𝕀_{Y_{𝔼,x}/K} ≃ p_{1∗}𝕀_{Z/K}; hence, the latter is also a trivial stratified bundle. This, together with [16, Cor. 2.17], implies that p_{1} : Z → Ỹ is a trivial covering. In particular, it admits a section s; hence, h̃ = s ∘ p_{2} ∘ h_{𝔼,x} and h̃ factors through h.

(ii)⇒(iii) Because h trivializes 𝔼, we have the inclusion ⟨𝔼⟩_{⊗} ⊂ ⟨h_{∗}𝕀_{Y/K}⟩_{⊗}. On the other side, by assumption, h_{𝔼,x} : Y_{𝔼,x} → X factors through h : Y → X; hence, ⟨h_{∗}𝕀_{Y/K}⟩_{⊗} ⊂ ⟨h_{𝔼,x∗}𝕀_{Y_{𝔼,x}/K}⟩_{⊗} = ⟨𝔼⟩_{⊗}.

(iii)⇒(iv) As ⟨𝔼⟩_{⊗} = ⟨h_{∗}𝕀_{Y/K}⟩_{⊗}, then we have the equality π(𝔼, x) = π(h_{∗}𝕀_{Y/K}, x) and as h : Y → X is Galois, then its Galois group is π(h_{∗}𝕀_{Y/K})(K) = π(𝔼, x)(K).

(iv)⇒ (i) By what we already proved there must be a factorization f : Y → Y_{𝔼,x} such that h = h_{𝔼,x} ∘ f. Hence, if G is the Galois group of h : Y → X then h_{𝔼} : Y_{𝔼} → X corresponds to a normal subgroup H of G. But by assumption G = π(𝔼, x)(K) = H; hence, h = h_{𝔼}. □

Corollary 3.3. If 𝔼 ∈ Strat(X/K) is finite then the set of finite étale covers of X trivializing 𝔼 has a minimal element which is Galois of Galois group π(𝔼, x)(K).

By [8, Cor. 12] for every 𝔼 ∈ Strat(X/K) the group scheme π(𝔼, x) is smooth (which is equivalent to being reduced). In particular, as K is algebraically closed, if 𝔼 is finite we are allowed to identify the abstract group π(𝔼, x)(K) and the algebraic group π(𝔼, x). Given a finite stratified bundle 𝔼 ∈ Strat(X/K) it is straightforward to see that for every L ⊃ K algebraically closed field extension 𝔼_{L} = 𝔼 ⊗_{K} L ∈ Strat(X_{L}/L) is finite as well. In fact, the following stronger statement holds:

Lemma 3.4. Let 𝔼 ∈ Strat(X/K) and let L ⊃ K be an algebraically closed field extension such that 𝔼_{L} is finite. Then for every L′ ⊃ K algebraically closed such that trdeg_{K} L′ ≥ trdeg_{K} L (or, if both are infinite, such that there exists an immersion L ↪ L′ which is the identity on K) we have that 𝔼_{L′} is finite. Moreover for any x ∈ X(K)

π(𝔼_{L}, x)(L) ≃ π(𝔼_{L′}, x)(L′),

where we consider x ∈ X_{L}(L) via K ⊂ L and similarly for L′.

Proof. Let L and L′ as in the hypothesis, then we can construct an immersion L ↪ L′ which is the identity on K, just by sending any transcendence basis of L to a algebraically independent set in L′ over K and using the fact that L′ is algebraically closed to see that this extends to an immersion L ↪ L′. Hence, we have reduced the problem to proving that if 𝔼 is finite and L ⊃ K is an algebraically closed field extension then 𝔼_{L} is finite and has the same monodromy group of 𝔼 as abstract groups. In order to do so we need first to establish a result on Galois covers:

Claim. Let h : Y → X be a Galois cover of Galois group G and let h_{L} : Y_{L} → X_{L} the extension of scalars of h : Y → X to L, then h_{L} is a Galois cover of Galois group G.

Proof. Certainly h_{L} : Y_{L} → X_{L} is a finite étale morphism as these properties are stable under base change. We are left to check that (i) Y_{L} is connected, (ii) Aut(Y_{L}/X_{L}) acts transitively on the fiber over some geometric point of X_{L} and finally (iii) Aut(Y_{L}/X_{L}) ≃ Aut(Y/X).

i) As K is algebraically closed (hence, in particular, separably closed) Y is connected if and only if Y_{L} is connected for any field extension L ⊃ K. In particular, Y_{L} is connected.

ii) Let x ∈ X_{L}(L) be any closed (in particular, geometric) point of X_{L}, then the composition x̄ : Spec(L) → X_{L} → X is a geometric point for X. As h : Y → X is Galois, Aut(Y/X) acts transitively on Yx̄ = Y ×_{X} x̄ = Y_{L} ×_{X_{L}} x = Y_{L,x}. Now, the action of Aut(Y/X) on Y_{L,x} factors through Aut(Y_{L}/X_{L}) via the inclusion Aut(Y/X) ⊂ Aut(Y_{L}/X_{L}) defined by ϕ ↦ ϕ_{L}. Hence, the action of Aut(Y_{L}/X_{L}) on Y_{L,x} is transitive as well; therefore, h_{L} : Y_{L} → X_{L} is Galois.

iii) As Yx̄ = Y_{L,x} and both h and h_{L} are Galois, it follows that the order of their Galois group is the same, as it is the cardinality of the respective geometric fibers over x̄ and over x. Moreover we have a natural inclusion Aut(Y/X) ⊂ Aut(Y_{L}/X_{L}) so as they have the same cardinality they must be equal; hence, Aut(Y_{L}/X_{L}) = G. □

Until the end of the proof let us denote by h_{𝔼,x} : Y → X the Picard–Vessiot torsor of 𝔼 (see Lemma 3.2), then h_{𝔼,x}⊗_{K}L : Y_{L} → X_{L} is a Galois cover trivializing 𝔼_{L} which is then finite, by Lemma 3.2. Recall that ⟨(h_{𝔼,x})_{∗}𝕀_{Y/K}⟩_{⊗} = ⟨𝔼⟩_{⊗}. But then in particular, ⟨(h_{𝔼,x})_{∗}𝕀_{Y/K} ⊗_{K} L⟩_{⊗} = ⟨𝔼_{L}⟩_{⊗} and as (h_{𝔼,x})_{∗}𝕀_{Y/K} ⊗_{K} L = (h_{𝔼,x} ⊗_{K} L)_{∗}𝕀_{Y_{L}/L}, it follows that ⟨(h_{𝔼,x} ⊗_{K} L)_{∗}Y_{L}⟩_{⊗} = ⟨𝔼_{L}⟩_{⊗}. Hence, by Lemma 3.2, we have that h_{𝔼,x} ⊗_{K} L : Y_{L} → X_{L} is the minimal trivializing cover for 𝔼_{L}. Now, the Galois group of h_{𝔼,x} is the same as that of h_{𝔼,x} ⊗_{K} L by the previous claim; hence, again by Lemma 3.2, we have that π(𝔼_{L}, x)(L) = π(𝔼, x)(K). □

Finite stratified bundle have an additional propriety that will turn out to be very useful to prove that some stratified bundle cannot be isotrivial:

Lemma 3.5. Let 𝔼 ∈ Strat(X/K) be a finite stratified bundle. Then there exists a subfield K′ ⊂ K of finite type over 𝔽_{p} over which X and 𝔼 are defined; that is, there exists X′ smooth variety over K′ and 𝔼′ ∈ Strat(X′/K′) such that X = X′ ×_{Spec K′} Spec K and 𝔼 = 𝔼′ ⊗_{K′} K.

Proof. Let h_{𝔼,x} : Y_{𝔼,x} → X be the Picard–Vessiot torsor of 𝔼 (see Lemma 3.2), and let ℍ = (h_{𝔼,x})_{∗}𝕀_{Y_{𝔼,x}/K}, then (see Lemma 3.2) 𝔼 ∈ ⟨ℍ⟩_{⊗}. Certainly there exists K′′ of finite type over 𝔽_{p} on which h_{𝔼,x} : Y_{𝔼,x} → X is defined; hence, ℍ is also defined over K′′. Notice that 𝔼 is a subquotient of ℙ where ℙ ∈ ℤ[ℍ, ℍ^{∨}] (see e.g. [16, def. 2.4]); that is, 𝔼 ≃ ℙ̃/ℙ̄ with ℙ̄ ⊂ ℙ̃ ⊂ ℙ. It is clear that ℙ̃ and ℙ̄ are defined over some extension K′ of finite type of K′′ (thus over 𝔽_{p}). Therefore, so does 𝔼 ≃ ℙ̃/ℙ̄. □

In particular, we have the following:

Corollary 3.6. Let 𝔼 ∈ Strat(X/K) with K algebraically closed, such that for some algebraically closed field extension L ⊃ K the stratified bundle 𝔼_{L} is finite. Assume that K has infinite transcendence degree over 𝔽_{p}, then 𝔼 is finite as well.

Proof. As 𝔼_{L} is finite then by Lemma 3.5 there exists K′ ⊂ L algebraically closed of finite transcendence degree over 𝔽_{p} over which 𝔼_{L} and is defined. But then 𝔼 is defined over K′ as well, hence we can assume that K′ ⊂ K. Let h_{𝔼_{L}} : Y → X_{L} be the Picard–Vessiot torsor of 𝔼_{L}, then it is defined over some algebraically closed field K′′ of finite transcendence degree over K′. In particular 𝔼_{K′′} is finite and as ∞ = trdeg_{K′} K ≥ trdeg_{K′} K′′ by Lemma 3.2 𝔼 = 𝔼_{K′} ⊗_{K′} K is finite. □

4. Families of finite stratified bundles

We can view a relative stratified bundle 𝔼 ∈ Strat(X/S) as a family of stratified bundles parametrized by the points of S. In particular, for every s ∈ S(K), let 𝔼_{s} ∈ Strat(X_{s}/k(s)) denote the restriction of 𝔼 on X_{s} and 𝔼η̄ ∈ Strat(Xη̄/k(η̄)) its restriction on the geometric generic fiber given by a choice of an algebraic closure \overline{k(S)} of k(S). It is natural then to ask how the property of being isotrivial behaves in families: the main question we want to study is whether it is true that if 𝔼_{s} is finite for every s ∈ S(K) then so is 𝔼_{η}. André proved in [2, Prop. 7.1.1] that the analogous result in characteristic zero holds. In positive characteristic, following an idea of Laszlo, in [9, Cor. 4.3, Rmk. 5.4.1] the authors proved that there exists X → S a projective smooth morphism of varieties over 𝔽̄_{2} and a stratified bundle on X relative to S which is finite on every closed fiber but not on the geometric generic one. Nevertheless, assuming X to be projective over S and imposing a coprimality to p condition on the order of the monodromy group on the closed fibers, they proved the following:

Theorem 4.1. [9, Thm. 7.2] Let X → S be a smooth projective morphism of Kvarieties with geometrically connected fibers and let 𝔼 ∈ Strat(X/S). Assume that there exists a dense subset S̃ ⊂ S(K) such that, for every s ∈ S̃, the stratified bundle 𝔼_{s} has finite monodromy of order prime to p. Then

i) there exists fη̄ : Yη̄ → Xη̄ a finite étale cover of order prime to p such that f^{∗}𝔼η̄ decomposes as direct sum of stratified line bundles;

ii) if K ≠ 𝔽̄_{p} then 𝔼η̄ is trivialized by a finite étale cover of order prime to p.

Note that the cover of order prime to p in the second point of the theorem factors through the Picard–Vessiot torsor of 𝔼η̄ by its minimality (see Lemma 3.2). In particular, this implies that the order of the monodromy group of 𝔼η̄ is prime to p. This article is devoted to determine how the assumptions of X being projective over S and of the order of the monodromy groups to be prime to p can or cannot be relaxed in order to get similar results. A first strengthening of the theorem comes rather directly from the ideas in the proof of Theorem 4.1. In order to prove it we need first to establish the following

Lemma 4.2. Let h : X → S be a proper flat separable morphism of connected varieties with geometrically connected fibers over an algebraically closed field K and suppose it has a section σ : S → X. Let S̃ ⊂ S(K) be any subset of the closed points of S, let N ∈ ℕ and let us fix for every s ∈ S̃ a finite étale cover g_{s} : Z_{s} → X_{s} of degree less than N. Then there exists an open subscheme U ⊂ S and a finite étale cover f : W → X ×_{S} U dominating all the g_{s} for s ∈ S̃ ∩ U; that is, for every s ∈ S̃ ∩ U the finite étale cover f_{s} : W_{s} → X_{s} factors through g_{s} : Z_{s} → X_{s}.

Proof. The proof of this lemma is a generalization of the construction that one can find in the beginning of the proof of [9, Thm. 5.1]. First notice that if the order of the g_{s} : Z_{s} → X_{s} is bounded by N then the order of their Galois closures is bounded by N!, hence we can assume all the g_{s} : Z_{s} → X_{s} to be Galois. Moreover if S′ is connected and S′ → S is étale and generically finite then there is a non-trivial open U over which S′ ×_{S} U → U is finite and étale. As X → S is smooth its image is open; hence, by shrinking S, we can assume that X → S is surjective. Let S′ → S be finite étale, then so is X′ = X ×_{S} S′ → X. Let s ∈ S̃ and s′ ∈ S′(K) lying over s. Assume that we have found f′ : W → X′ such that f′_{s} : W_{s} → X′_{s} factors through g_{s′} : Z_{s′} = Z_{s} ×_{k(s)} k(s′) → X′_{s′} then the composition f : W → X′ → X is a finite étale cover of X and f_{s} : W_{s} → X_{s} factors through g_{s}. As h : X → S has geometrically connected fibers so does h′ : X′ → S′ as if s′ ∈ S lies over s ∈ S then X′_{s′} = X_{s} ⊗_{k(s)} k(s′). Therefore, h′ : X′ → S′ is proper, flat, separable and has geometrically connected fibers. So if S′ is connected the morphism h′ : X′ → S′ together with the section σ′ : S′ → X′ induced by σ : S → X satisfy the assumptions of the theorem.

For any s ∈ S̃ let G_{s} ⊂ π_{1}^{ét}(X_{s}, σ(s)) be the open normal subgroup corresponding via Galois duality to the cover g_{s} : Z_{s} → X_{s}. Let η̄ be a generic geometric point of S given by the choice of an algebraic closure \overline{k(S)} of k(S). The fibers of X → S are geometrically connected and the morphism is proper, flat and separable; hence, the specialization map π_{1}^{ét}(Xη̄, σ(η̄)) ↠ π_{1}^{ét}(X_{s}, σ(s)) is surjective. Composing it with the quotient of π_{1}^{ét}(X_{s}, σ(s)) by G_{s} we get

ρ_{s} : π_{1}^{ét}(Xη̄, σ(η̄)) ↠ π_{1}^{ét}(X_{s}, σ(s)) ↠ π_{1}^{ét}(X_{s}, σ(s))/G_{s}.

Notice that the index of ker(ρ_{s}) in π_{1}^{ét}(Xη̄, σ(η̄)) is bounded by N. Let τ : S′ → S be any finite étale cover and let s′ ∈ S lying over s. As K is algebraically closed, then k(s′) ≃ k(s) and hence X′_{s′} ≃ X_{s}. In particular, the natural morphism π_{1}^{ét}(X′_{s′}, σ′(s′)) → π_{1}^{ét}(X_{s}, σ(s)) is an isomorphism. Let G_{s′} ⊂ π_{1}^{ét}(X′_{s′}, σ′(s′)) be the open subgroup corresponding to g_{s′} : Z_{s′} → X′_{s′}, that is, the preimage of G_{s} under this isomorphism and let us denote by

ρ_{s′} : π_{1}^{ét}(Xη̄, σ(η̄)) ↠ π_{1}^{ét}(X′_{s′}, σ′(s′)) ↠ π_{1}^{ét}(X′_{s′}, σ′(s′))/G_{s′},

then ker(ρ_{s′}) = ker(ρ_{s}) ⊂ π_{1}^{ét}(Xη̄, σ(η̄)). As Xη̄ is a projective \overline{k(S)}-variety, then π_{1}^{ét}(Xη̄, σ(η̄)) is topologically finitely generated and hence has finitely many subgroups of index less than N, which are all opens by Nikolov–Segal theorem ([19, Thm 1.1]), the intersection of which we denote by G: It is a normal open subgroup and it has finite index. Moreover

G ⊂ ⋂_{s∈S̃} ker(ρ_{s}) = ⋂_{s′∈τ^{−1}(S̃)} ker(ρ_{s′}).

At this point, we need an additional step before concluding similarly than in the proof of [9, Thm. 5.1]. By Galois duality G corresponds to a finite étale cover Zη̄ → Xη̄. Let k(S)^{sep} be the separable closure of k(S) in \overline{k(S)}. The base change functor from the category of finite étale covers over X ⊗_{S} k(S)^{sep} to the one of finite étale covers over Xη̄ is an equivalence. Hence, Zη̄ is defined over some separable extension of k(S). In particular, there exists an étale generically finite cover S′ → S such that Zη̄ descends to a finite étale cover of X′ = X ×_{S} S′.

Let η̄′ be the geometric generic point of S′ given by k(S) ⊂ k(S′) ⊂ \overline{k(S)}. Then X′_{η̄′} = Xη̄ and σ(η̄) = σ′(η̄′). Hence, the following diagram commutes:

[figure: π_{1}^{ét}(X′ ; η̄, σ′(η̄′)); π_{1}^{ét}(X′, σ′(η̄′)); π_{1}^{ét}(Xη̄, σ(η̄)); π_{1}^{ét}(X, σ(η̄)).]

Let K′ be the kernel of π_{1}^{ét}(Xη̄, σ(η̄)) → π_{1}^{ét}(X′, σ′(η̄′)). As X → S is projective we have the following exact sequence:

\xymatrix{ & π_{1}^{ét}(Xη̄, σ(η̄)) \ar[d]^{q} \ar[dr]^{α} \\ \{1\} \ar[r] & π_{1}^{ét}(Xη̄, σ(η̄))/K′ \ar[r]^{i} & π_{1}^{ét}(X′, σ′(η̄′)) \ar[r] & π_{1}^{ét}(S′, η̄′) \ar[r] \ar[l]_{σ′_{∗}} & \{1\}.}

By [1, V Cor 6.7] we have the inclusion G ⊃ K′. Moreover if we denote by Π_{K′} = π_{1}^{ét}(Xη̄, σ(η̄))/K′, then the section σ′_{∗} induces a splitting

π_{1}^{ét}(X′, σ′(η̄′)) ≃ Π_{K′} ⋊ σ′_{∗}(π_{1}^{ét}(S′, η̄′))

as abstract groups. It is also a splitting of topological groups (see for example [4, §2.10 Prop. 28] and following discussion). In particular, the topology on π_{1}^{ét}(X′, σ′(η̄′)) is the product topology. Note that \overline{G} = q(G) is invariant by the action of σ′_{∗}(π_{1}^{ét}(S′, η̄′)); hence, we can define

H = \overline{G} ⋊ σ′_{∗}(π_{1}^{ét}(S′, η̄′)).

By definition α^{−1}(H) = G, and H has finite index in π_{1}^{ét}(X′, σ′(η̄′)). It is also open: π_{1}^{ét}(X′, σ′(η̄′)) is endowed with the product topology, and \overline{G} is open because G = q^{−1}(\overline{G}) is open as well. Hence, H corresponds to a finite étale cover W → X′ and as the composition H ⊂ π_{1}^{ét}(X′, σ′(η̄′)) ↠ π_{1}^{ét}(S′, η̄′) is surjective, then W has geometrically connected fibers over S′. In particular, the specialization map is again surjective. Let z ∈ W be a closed point lying over s′ and let ζ̄ be a geometric generic point lying over σ(η̄), then we have the following commutative diagram

\xymatrix{π_{1}^{ét}(Wη̄, ζ̄) \ar[rr]^{0} \ar[r] \ar@{->>}[d] & π_{1}^{ét}(Xη̄, σ(η̄)) \ar[r] \ar@{->>}[d] & π_{1}^{ét}(Xη̄, σ(η̄))/G \ar[d] \\ π_{1}^{ét}(W_{s}, z) \ar[r] & π_{1}^{ét}(X′_{s′}, σ′(s′)) \ar[r] & π_{1}^{ét}(X′_{s′}, σ′(s′))/G_{s′}.}

Using the surjectivity of the specialization map on W it follows that the composition of the morphisms on the second line is zero as well; hence, if G̃_{s′} ⊂ π_{1}^{ét}(X′_{s′}, σ′(s′)) is the open normal subgroup corresponding to f_{s′} : W_{s′} → X′_{s′} then G̃_{s} ⊂ G_{s′}. Therefore, f_{s′} factors through g_{s′}. □

We can summarize the previous lemma by saying that with the assumptions of the theorem, up to shrinking S every family of finite étale covers of the closed fibers with bounded order can be dominated by a finite étale cover of X (notice that the existence of the section σ : X → S is not essential for the proof).

Theorem 4.3. Let X → S be a smooth proper morphism of K-varieties with geometrically connected fibers and 𝔼 ∈ Strat(X/S) of rank r. Assume that there exists a dense subset S̃ ⊂ S(K) such that, for every s ∈ S̃, the stratified bundle 𝔼_{s} has finite monodromy and that the highest power of p dividing |π(𝔼_{s})| is bounded over S̃. Then

i) there exists fη̄ : Yη̄ → Xη̄ a finite étale cover such that f^{∗}𝔼η̄ decomposes as direct sum of stratified line bundles;

ii) if K ≠ 𝔽̄_{p} then 𝔼η̄ is finite.

Proof. We will reduce this theorem to Theorem 4.1. By the invariance of the monodromy group it suffices to prove the theorem for f^{∗}𝔼 where f : Y → X is a morphism of smooth S-varieties which is generically finite étale. By Chow’s lemma and using de Jong alterations ([6]) there exists f : Y → X projective and generically finite étale, hence we can assume X to be projective. Up to taking an étale open of S we can assume that there exists a section σ : S → X. For any s ∈ S̃ let Γ_{s} = π(𝔼, σ(s)) and h_{s} : Y_{s} → X_{s} the Picard–Vessiot torsor of 𝔼_{s} (see Lemma 3.2). Let G_{s} ⊂ π_{1}^{ét}(X_{s}, σ(s)) be the normal open subgroup corresponding via Galois duality to the cover h_{s}. By Tannakian duality 𝔼_{s} corresponds to the image of an r-dimensional representation of π_{1}^{Strat}(X_{s}, σ(s)) ([7, Prop. 2.21]) and as 𝔼_{s} is finite by [8, Prop. 13] this representation factors through the étale fundamental group, considered as a constant group scheme

π_{1}^{Strat}(X_{s}, σ(s)) ↠ π_{1}^{ét}(X_{s}, σ(s)) ↠ π_{1}^{ét}(X_{s}, σ(s))/G_{s} = Γ_{s} ⊂ GL_{r}(K)

where r is the rank of 𝔼. By Brauer–Feit generalization of Jordan’s theorem [5, Theorem], as the orders of the Sylow-p-subgroups of every G_{s} are bounded by p^{N}, there exists an integer M = f(r, N) and, for every s ∈ S̃, a normal abelian subgroup A_{s} such that |Γ_{s} : A_{s}| < M. This gives for every s ∈ S(K) a Galois cover g_{s} : Z_{s} → X_{s} of order bounded by M and a factorization

Y_{s} → Z_{s} → X_{s}

where Y_{s} → Z_{s} is Galois of Galois group A_{s}. Therefore, by Lemma 4.2, up to shrinking S there exists a cover g′ : Z′ → X such that g′_{s} : Z′_{s} → X_{s} factors through g_{s} : Z_{s} → X_{s}. In particular, if 𝔼′ is the pullback of 𝔼 via g′ then π(𝔼′_{s}) is abelian for every s. Up to taking an étale open of S the section σ : S → X extends to a section σ′ : S → Z′. Let Γ′_{s} = π(𝔼′, σ′(s)), as we just noticed for every s ∈ S̃ we have that Γ′_{s} is abelian; hence, we can write it as the direct product of its p part with its prime to p part:

Γ′_{s} = Γ_{s}^{p} × Γ_{s}^{p′}

and Γ_{s}^{p′} corresponds to a Galois cover over Z′_{s} whose index is by assumption bounded by p^{N} for some N ∈ ℕ. Applying Lemma 4.2 and up to shrinking S we get a Galois cover g′′ : Z′′ → Z′ dominating all such covers. Let 𝔼′′ be the pullback of 𝔼′ along g′′, then π(𝔼′′_{s}) is (abelian) of order prime to p for every s ∈ S̃. Therefore, we have reduced the problem to Theorem 4.1. □

5. A counterexample over countable fields

Our next aim is to drop the assumption of X being projective over S. However, before getting to the positive results, let us present a counterexample to understand what we can reasonably expect to hold without this assumption. Assume for the rest of this section K to be an algebraically closed countable field. Let X = 𝔸_{K}^{2}, S = 𝔸_{K}^{1} and let X → S be given by K[y] → K[x, y]. The main result of this section is the following:

Proposition 5.1. There exists 𝔼 ∈ Strat(X/S) such that 𝔼_{s} is trivial for every point s ∈ S(K) but 𝔼η̄ is not isotrivial.

The rest of the section will be spent constructing such a stratified bundle and proving it satisfies the proposition. As x is a global coordinate of X relative to S, it is clear that

𝒟_{X/S} = 𝒪_{X}[∂_{x}^{(k)} | k ∈ ℕ_{>0}].

Moreover any vector bundle is free over X. If E is a vector bundle on X, then a 𝒟_{X/S}-module structure on E is a 𝒪_{S}-linear morphism

ϕ : 𝒟_{X/S} → End_{𝒪_{S}} (E)

extending the 𝒪_{X}-module structure on E; hence, the image of 𝒪_{X} ⊂ 𝒟_{X/S} under ϕ is always fixed. Therefore, to determine the action of the whole 𝒟_{X/S} it is enough to consider the image of the generators ∂_{x}^{(k)} under ϕ. Let e_{1}, …, e_{r} be a basis for E, and let A_{k} = (a_{ij}^{k}) be given by ∂_{x}^{(k)}(e_{i}) = ∑ a_{ij}^{k}e_{j}. Then the A_{k}, for k ∈ ℕ_{>0}, determine the 𝒟_{X/S}-action: If s = ∑_{i=1}^{r} f_{i} · e_{i} is a section of E, with f_{i} ∈ 𝒪_{X}, then using (1) we have

(2)  ∂_{x_{i}}^{(k)}(s) = ∑_{i=1}^{r} ∑_{\begin{matrix}a+b=k \\ a,b≥0 \end{matrix}} ∂_{x}^{(a)}(f_{i})A_{b} · e_{i}.

Note that if e′_{1}, …, e′_{r} is an other basis of E and U = (u_{ij}) ∈ H^{0}(X, GL_{r}) is given by e′_{i} = ∑ u_{ij}e_{j} then by (1) it follows that in this new basis the matrices A′_{k} = (a_{ij}^{′k}) describing the action of ∂_{x}^{(k)} are given by

(3)  A′_{k} = [ ∑_{\begin{matrix}a+b=k \\ a,b≥0 \end{matrix}} ∂_{x}^{(a)}(U)A_{b}]U^{−1}.

In order to construct our example, let us fix a bijection n ↦ a_{n} between the natural numbers and K = S(K). Let 𝔼 ∈ Strat(X/S) be the rank-two relative stratified bundle E = 𝒪_{X} · e_{1} ⊕ 𝒪_{x} · e_{2} with 𝒟_{X/S}-action given by ∂_{x}^{(k)}(e_{1}) = 0 and

(4)  ∂_{x}^{(k)}(e_{2}) = \begin{cases}∏_{i=0}^{h}(y − a_{i}) · e_{1} & if k = p^{h}, \\ 0 & else. \end{cases}

In order to prove Proposition 5.1 we need to show that this actually defines an action of 𝒟_{X/S} over E and that 𝔼 satisfies the two properties of the proposition, namely that it is trivial on every closed fiber and not isotrivial on the geometric generic fiber.

Lemma 5.2. The formulae in (4) define a 𝒟_{X/S}-module structure on E.

Proof. As we fixed the action of the generators of 𝒟_{X/S}, for it to extend to a 𝒟_{X/S}-action we only need to check that the relations of the generators in the ring of differential operators are satisfied by their images in End_{𝒪_{X}} (E). By [3, Cor. 2.5] the only relations are

[∂_{x}^{(l)}, ∂_{x}^{(k)}] = 0

∂_{x}^{(k)} ∘ ∂_{x}^{(l)} = \binom{k + l}{k}∂_{x}^{(k+l)}

[∂_{x}^{(k)}, x] = ∂_{x}^{(k−1)}.

Let us begin with the second relation: for k, l > 0

∂_{x}^{(k)} ∘ ∂_{x}^{(l)}(e_{1}) = 0

∂_{x}^{(k)} ∘ ∂_{x}^{(l)}(e_{2}) = \{ \begin{matrix}∂_{x}^{(k)}(∏_{i=0}^{h}(y − a_{i}) · e_{1}) = 0 & if l = p^{h} \\ 0 & else \end{matrix}

Hence, we just need to verify that if k + l = p^{h} then \binom{k+l}{k} = 0 but this holds by Lucas’s theorem and the first relation follows immediately. Moreover by (2) we have

∂_{x}^{(k)} · x(e_{i}) = ∂_{x}^{(k)}(x · e_{i}) = ∑_{\begin{matrix}a+b=k \\ a,b≥0 \end{matrix}} ∂_{x}^{(a)}(x)∂_{x}^{(b)}(e_{i}) = x∂_{x}^{(k)}(e_{i}) + ∂_{x}^{(k−1)}(e_{i});

hence, the third relation trivially holds. □

In order to prove that 𝔼_{s} is trivial for every closed fiber, let us fix n ∈ ℕ and let s = a_{n} ∈ S(K), that is, X_{s} = {y = a_{n}} ⊂ X. Let us consider the basis change on 𝒪_{X_{s}} · e_{1} ⊕ 𝒪_{X_{s}} · e_{2} = E_{s} given by e′_{1} = e_{1} and

e′_{2} = e_{2} − [(y − a_{0})x + (y − a_{0})(y − a_{1})x^{p} + ⋯ + [ ∏_{i=0}^{n−1}(y − a_{i})]x^{p^{n−1}}] · e_{1}

then by (3) in this new basis the action of 𝒟_{X_{s}/k(s)} is given by ∂_{x}^{(k)}(e′_{1}) = ∂_{x}^{(k)}(e′_{2}) = 0 hence is the trivial action. We are now left to prove that 𝔼η̄ is not isotrivial:

Lemma 5.3. Let 𝔼 ∈ Strat(X/S) be the stratified bundle defined by (4), then 𝔼η̄ is not isotrivial.

Proof. In order to prove that 𝔼η̄ is not isotrivial it suffices by Lemma 3.5 to show that it cannot be defined over any K′ of finite type over 𝔽_{p}. Remark that 𝔼η̄ is defined over 𝔸1 ; η̄ which descends to 𝔸_{K′}^{1} for any K′ ⊂ K. By way of contradiction assume there exists such a K′ and let 𝔼′ be the descent of 𝔼η̄ over 𝔸_{K′}^{1}. This means that there is a basis e′_{1}, e′_{2} of 𝔼η̄ such that the matrices A′_{k} in this new basis take values in K′[x]. Let U be the basis change matrix between e_{i} and e′_{i}, then U is defined over some K′′ of finite type over K′, hence over 𝔽_{p}, so by (3) we have that ∏_{i=0}^{h}(y −a_{i}) ∈ K′′[x]. In particular, if we denote by A = 𝔽_{p}[∏_{i=0}^{h}(y − a_{i}) | h ∈ ℕ], our assumption implies that A ⊂ K′′[x]. To see that this leads to a contradiction it suffices to show that K ⊈ K′′(x) where K is the quotient field of A. Note that K ⊂ K; therefore, it is enough to prove that for every K′ of finite type over 𝔽_{p} we have that K ⊈ K′(x) and as 𝔽̄_{p} ⊂ K it is sufficient to show 𝔽̄_{p} ⊈ K′(x), which follows from the following:

Claim. Let 𝔽_{q} be a finite field with q = p^{n} for some n ∈ ℕ and let K ⊃ 𝔽_{q} an algebraic extension such that [K : 𝔽_{q}] = +∞. Then for every m ∈ ℕ and every ε_{1}, …, ε_{m} non-algebraic over 𝔽_{q} we have that

K ⊈ 𝔽_{q}(ε_{1}, …, ε_{m}).

Proof. By induction on m, the case m = 0 being evident. Let m = 1, and γ ∈ K − 𝔽_{q} and let μ_{γ}(t) its minimal polynomial over 𝔽_{q}. By way of contradiction assume γ ∈ 𝔽_{q}(ε_{1}); then γ = f(ε_{1})/g(ε_{1}) and

g(ε_{1})^{deg μ_{γ}} · μ_{γ}(\frac{f(ε_{1})}{g(ε_{1})}) = 0

gives an algebraic dependence of ε_{1} over 𝔽_{q} which is a contradiction with our assumption that ε_{1} is not algebraic over 𝔽_{q}. Let now m ≥ 1, by induction step we know that for every n ∈ ℕ, q = p^{n}, then no infinite algebraic extension of 𝔽_{q} is contained in 𝔽_{q}(ε_{1}, …, ε_{m−1}); hence, there exists an r such that 𝔽_{q^{r}} = 𝔽_{q}(ε_{1}, …, ε_{m−1}) ∩ K. Then

𝔽_{q}(ε_{1}, …, ε_{m−1})(ε_{m}) ∩ K ⊂ 𝔽_{q^{r}}(ε_{m}) ≠ K

by the m = 1 step applied to q = p^{rn}. In particular, K ⊈ 𝔽_{q}(ε_{1}, …, ε_{m}). □

Note that if K′ is of finite type over 𝔽_{p} then K′ can be always be written as 𝔽_{q}(ε_{1}, …, ε_{m}) for some q = p^{n} and ε_{i} non-algebraic over 𝔽_{q}; hence, 𝔽̄_{p} ⊈ K′(x). Therefore, 𝔼 cannot be defined over any K′ of finite type over 𝔽_{p} and by Lemma 3.5 it cannot be finite. □

Remark 5.4. Let us observe that if K is uncountable the same construction provides an example of a relative stratified bundle 𝔼 ∈ Strat(𝔸_{K}^{2}/𝔸_{K}^{1}) and a dense subset S̃ ⊂ 𝔸_{K}^{1}(K) such that 𝔼_{s} is trivial for every s ∈ S̃ but 𝔼η̄ is not isotrivial. Therefore, the density condition on S̃ of Theorem 4.1 will not be sufficient for our purposes, in parallel with the similar problem that one encounters in the equicharacteristic zero case (see [2, Rmk. 7.2.3]).

6. The main theorem

From the example in previous section it appears that in the case where X is not projective over S the situation is significantly different from the one in Theorem 4.1. In the latter one big obstruction for the theorem to hold is related to p dividing the order of the monodromy group on the closed fibers. In the counterexample of Section 5 these are trivial and the obstruction seems more related to the cardinality of K. As noticed in Section 3, the monodromy group does not depend (up to a non-unique isomorphism) on the choice of x ∈ X. Therefore, in this section we will denote the monodromy group of a stratified bundle 𝔼 simply by π(𝔼). In order to phrase the statement of the main theorem let us introduce the following notation: we will denote by (X, S; 𝔼) (and call it a triple over K) any triple consisting of X → S smooth morphism of K-varieties with geometrically connected fibers and 𝔼 ∈ Strat(X/S). We denote furthermore by K′ = K′(X, S; 𝔼) a (minimal) algebraically closed subfield of K such that (X, S; 𝔼) is defined over K′, and by (X′, S′; 𝔼′) the descent of the triple (X, S; 𝔼) to K′. Then the following result holds

Theorem 6.1. Let (X, S; 𝔼) be a triple over K, let K′ = K′(X, S; 𝔼) ⊂ K and (X′, S′; 𝔼′) the descended triple to K′. Let k(S′) be the function field of S′. Let us assume:

∃ i : k(S′) ↪ K extending K′ ⊂ K. (∗)

Assume that there exists a dense open S̃ ⊂ S such that 𝔼_{s} is finite for every s ∈ S̃(K), then so is 𝔼η̄. Moreover there exists s ∈ S̃(K) such that π(𝔼_{s})(K) ≃ π(𝔼η̄)(\overline{k(S)}), in particular

i) |π(𝔼_{s})| is bounded over S̃(K); ii) if p ∤ |π(𝔼_{s})| for every s ∈ S̃(K) then p ∤ |π(𝔼η̄)|; iii) any group property holding for π(𝔼_{s}) for every s ∈ S̃(K) holds for

π(𝔼η̄).

Proof. Up to shrinking S, we can assume S̃ = S. Let Δ : S′ → S′ ×_{Spec K′} S′ be the diagonal morphism and i : k(S′) ↪ K be an immersion as in (∗). Then the base change along

\xymatrix{Spec K \ar[r]^{i} & Spec k(S′) \ar[r] & Spec S′}

induces (s : Spec K → S) ∈ S(K) such that X′ ⊗_{k(S′)} K ≃ X_{s} and

i^{∗}𝔼′ = 𝔼_{s},

where we are considering i : k(S′) ↪ K as a geometric generic point of S′. In particular, π(i^{∗}𝔼′) = π(𝔼_{s}). Therefore, i^{∗}𝔼′ is finite; hence, by Lemma 3.4, 𝔼η̄ is finite as well, moreover their monodromy group are isomorphic as constant groups. □

Corollary 6.2. If K is uncountable then the assumption (∗), hence the theorem, always holds.

Proof. It suffices to show that if K is uncountable, then for any triple (X, S; 𝔼) over K there exists K′ = K′(X, S; 𝔼) and an inclusion k(S′) ⊂ K extending K′ ⊂ K. But it is easy to check that a triple (X, S; 𝔼) is defined by countably many data; hence, we can choose K′ such that it has countable transcendence degree over 𝔽_{p}. As K is uncountable, it has infinite transcendence degree over K′; hence, there always exists k(S′) ⊂ K as in (∗). □

Remark 6.3. Notice that if the smooth morphism X → S does not have geometrically connected fibers then we lose the notion of monodromy group on the closed and geometric generic fibers: if X is a K-variety which is not connected and 𝕀_{X/K} is the trivial stratified bundle on X then End(𝕀_{X/K}) ≠ K, hence Strat(X/K) is not a Tannakian category. Nevertheless, if we do not assume X → S to have geometrically connected fibers, the same proof shows that if 𝔼_{s} is finite when restricted to every connected component of X_{s}, then the same holds for 𝔼η̄ on every connected component of Xη̄.

7. Regular singularity and a refinement of the theorem

Regardless of the example in Section 5, there is a way to broaden Theorem 4.1 in the case where K is countable, making the additional assumption that the stratified bundle is regular singular on the geometric generic fiber. Let X be a smooth variety over K and let (X, \overline{X}) be a good partial compactification of X; that is: \overline{X} is a smooth variety over K such that X ⊂ \overline{X} is an open subscheme and D = \overline{X}\X is a strict normal crossing divisor. Let 𝒟_{\overline{X}/K}(log D) ⊂ 𝒟_{\overline{X}/K} the subalgebra generated by the differential operators that locally fix all powers of the ideal of definition of D. If U ⊂ \overline{X} admits global coordinates x_{1}, …, x_{d} and D is smooth and given by {x_{1} = 0} then

𝒟_{\overline{X}/K}(log D)_{|U} = 𝒪_{U}[x_{1}^{k}∂_{x_{1}}^{(k)}, ∂_{x_{i}}^{(k)} | i ∈ \{2, …, d\}, k ∈ ℕ_{>0}].

Definition 7.1. A stratified bundle 𝔼 ∈ Strat(X/K) is called (X, \overline{X})-regular singular if it extends to a locally free 𝒪\overline{X}-coherent 𝒟_{\overline{X}/k}(log D)-module \overline{𝔼} on \overline{X}. It is regular singular if it is (X, \overline{X})-regular singular for every partial good compactification (X, \overline{X}).

Remark 7.2. There is a parallel notion of regular singularities in characteristic zero. Despite the fact that isotrivial implies regular singular over the complex numbers, this is not longer true in positive characteristic, due to the existence of wild coverings (for a more precise statement, see [16, Thm. 1.1]).

For a (X, \overline{X})-regular singular stratified bundle 𝔼 we have a theory of exponents (see [11, §3]) of 𝔼 along D: it is a finite subset Exp_{D}(𝔼) ⊂ ℤ_{p}/ℤ given by the following:

Proposition 7.3. [11, Lemma 3.8],[16, Prop. 4.12] Let \overline{X} = Spec A be a smooth variety over K = K̄ with global coordinates x_{1}, …, x_{d} and let D be the smooth divisor defined by \{x_{1} = 0\}. Let 𝔼 ∈ Strat(X/K) a (X, \overline{X})-regular singular stratified bundle and \overline{𝔼} a locally free 𝒟_{\overline{X}/K}(log D)-module extending 𝔼 . Then there exists a decomposition of \overline{𝔼}_{|D} = ⨁ K_{α} with α ∈ ℤ_{p} such that x_{1}^{k}∂_{x_{1}}^{(k)} acts on K_{α} by multiplication by \binom{α}{k}. The image in ℤ_{p}/ℤ of the α ∈ ℤ_{p} such that K_{α} ≠ 0 are called the exponents of 𝔼 along D and do not depend on the choice of \overline{𝔼}.

If D is not smooth Exp_{D}(𝔼) is defined to be the union of the exponents along all the irreducible components of D. By [16, Cor. 5.4] 𝔼 extends to a stratified bundle \overline{𝔼} on \overline{X} if and only if its exponents are zero. In particular, [16, Prop. 4.11] implies that if 𝔼 is finite then its exponents are torsion. Moreover:

Lemma 7.4. Let 𝔼 be a 𝒟_{X/S}-module such that 𝔼_{s} is finite for every s ∈ S̃ a dense subset of S(K). If 𝔼η̄ is regular singular then the exponents of 𝔼η̄ with respect to any partial good compactification of Xη̄ are torsion.

Proof. Let us fix (Xη̄, \overline{X}η̄) a partial good compactification and let Dη̄ = \overline{X}η̄\Xη̄. As the exponents can be checked locally, we can shrink \overline{X}η̄ around the generic point of one of the irreducible components of Dη̄ at a time. Moreover in order to prove the lemma we are allowed to take a generically finite étale open S′ of S and substitute X by X ×_{S} S′ (and S̃ by its preimage) as the geometric generic fiber is the same. Finally for every s ∈ S(K) we have that X′_{s} is either empty or a finite union of copies of X_{s}; hence, we will still denote by s any point s′ ∈ S′ lying over it. Hence, without loss of generality, we can assume that we are in the following situation: the partial good compactification (Xη̄, \overline{X}η̄) is the restriction of a relative good partial compactification (X, \overline{X}) defined on the whole S, \overline{X} is the spectrum of a ring A, with global relative coordinates x_{1}, …, x_{d} over S and finally D = \overline{X}\X is defined by \{x_{1} = 0\}. Moreover we can assume that 𝔼 is globally free and that on the geometric generic fiber 𝔼η̄ extends to a globally free 𝒟_{X̄η̄/\overline{k(S})}(log Dη̄)-module \overline{𝔼}η̄.

Let s ∈ S̃ be any point such that X_{s} ∩ D ≠ ∅, and let us consider the globally free 𝒪_{X}-module Ē = 𝒪\overline{X} ē_{1} ⊕ ⋯ ⊕ 𝒪\overline{X} ē_{r}. Then the ē_{i} induce a basis on the restriction of \overline{E} to the closed fiber over s (as well as to the geometric generic one) and to the boundary divisor (as well as to its complement) as in the following commutative diagram:

\xymatrix{E_{s} = ⨁_{i=1}^{r} 𝒪_{X_{s}}e_{i}^{s} & E = ⨁_{i=1}^{r} 𝒪_{X}e_{i} \ar[l]_{⊗k(s)} \ar[r]^{⊗\overline{k(S)}} & Eη̄ = ⨁_{i=1}^{r} 𝒪_{Xη̄}ε_{i} \\ \overline{E}_{s} = ⨁_{i=1}^{r} 𝒪_{\overline{X}_{s}}ē_{i}^{s} \ar[u]^{|X_{s}} \ar[d]_{|_{Ds}} & \overline{E} = ⨁_{i=1}^{r} 𝒪\overline{X} ē_{i} \ar[l]_{⊗k(s)} \ar[r]^{⊗\overline{k(S)}} \ar[u]^{|_{X}} \ar[d]_{|_{D}} & \overline{E}η̄ = ⨁_{i=1}^{r} 𝒪_{\overline{X}η̄} ε̄_{i} \ar[u]^{|_{Xη̄}} \ar[d]_{|_{Dη̄}} \\ \overline{E}_{|D_{s}} = ⨁_{i=1}^{r} 𝒪_{D_{s}}ẽ_{i}^{s} & \overline{E}_{|D} = ⨁_{i=1}^{r} 𝒪_{D}ẽ_{i} \ar[l]_{⊗k(s)} \ar[r]^{⊗\overline{k(S)}} & \overline{E}_{|Dη̄} = ⨁_{i=1}^{r} 𝒪_{Dη̄} ε̃_{i}.}

Consider the first line of the diagram: on the first (respectively second and third) column there is an action of 𝒟_{X_{s}/k(s)} (respectively 𝒟_{X/S} and 𝒟_{Xη̄/\overline{k(S)}}), compatible with each other. On the last column this action extends to a logarithmic action on \overline{E}η̄ that we want to extend compatibly to \overline{E}. Similarly as in Section 5, let A_{i,k} be the matrices describing the action of ∂_{x_{i}}^{(k)} ∈ 𝒟_{X/S} in the basis e_{i}, then the same ones describe the action of ∂_{x_{i}}^{(k)} ∈ 𝒟_{Xη̄/\overline{k(S)}} in the basis ε_{i}. By regular singularity of 𝔼η̄ this action extends to a 𝒟_{\overline{X}η̄/\overline{k(S)}}(log Dη̄)-action. Therefore, there is a second basis ε′_{1}, …, ε′_{d} on the geometric generic fiber such that in the new basis the matrices A′_{i,k} have no poles in x_{1} for i ≠ 1 and logarithmic poles for i = 1. Let U ∈ H^{0}(Xη̄, GL_{r}) the basis change matrix from ε_{i} to ε′_{i}. Taking a generically finite étale open of S we can assume that U is defined on the whole S; hence, the A′_{i,k} are defined over the whole S as well and this defines an action of

𝒟_{X/S}(log D) ≐ 𝒪_{X}[x_{1}^{k}∂_{x_{1}}^{(k)}, ∂_{x_{i}}^{(k)} | i ∈ {2, …, d}, k ∈ ℕ_{>0}]

on \overline{E}, compatible with the logarithmic action on the fibers over η̄. In particular, this induces a 𝒟_{\overline{X}_{s}}(log D_{s})-action on \overline{E}_{s}; hence, 𝔼_{s} is (X_{s}, \overline{X}_{s})-regular singular (notice that if S′ is an étale open of S then for s ∈ S(K) the fiber X′_{s} of X′ = X ×_{S} S′ is either empty or the disjoint union of finitely many copies of X_{s}). We want now to compare Exp_{Dη̄}(𝔼η̄) and Exp_{D_{s}}(𝔼_{s}). By Proposition 7.3 we have that \overline{𝔼}_{η̄|_{Dη̄}} = ⊕F_{α}; hence, there exists ε̃_{i} a basis of \overline{E}_{|Dη̄} such that the matrices B̃_{k} defining the action of x_{1}^{k}∂_{(k),x_{1}} are diagonal with values \binom{α}{k} ∈ 𝔽_{p}. Let ε̄_{i} be a lift of ε̃_{i}, then up to taking an étale generically finite open of S we can assume that ε̄_{i} is a restriction of a basis ē_{i} of \overline{E} over \overline{X}. In particular, the decomposition extends as well and \overline{𝔼}_{|D} = ⊕F_{α} induces a decomposition on \overline{𝔼}_{s|_{Ds}}. This decomposition must coincide with the one given by Proposition 7.3; hence, the exponents must be the same of the ones of 𝔼η̄. As 𝔼_{s} is isotrivial, its exponents are torsion; hence, so must be the ones of 𝔼η̄. □

Remark 7.5. While the previous proof shows that if 𝔼η̄ is regular singular so are the 𝔼_{s} for every s ∈ S(K), the example in Section 5, together with Theorem 7.7, shows that the converse does not hold in general (however, one can prove it is the case when K is uncountable). On the contrary, in characteristic zero it is always true that if a relative flat connection is regular with respect to some smooth good compactification on the fibers over a dense set of points of S, then it is regular on the geometric generic fiber, as proven in [2, Lemma 8.1.1].

Before stating and proving the main theorem of this section we need to prove the existence of Kawamata coverings in positive characteristic. Analogously to the original construction in characteristic zero ([15, Thm. 17]) we have the following

Theorem 7.6. Let X be a projective smooth variety of dimension d over an algebraically closed field K of characteristic p and let D be a simple normal crossing divisor on X. Let m ∈ ℕ prime to p, then there exist a projective smooth variety Y and a finite surjective mapping f : Y → X such that (f^{∗}D)_{red} is a simple normal crossing divisor on Y and if f^{∗}D = ∑ m_{i}D̃_{i} is the decomposition in irreducible components with D̃_{i} ≠ D̃_{j} for i ≠ j then m | m_{i} for all i and m_{i} are all prime to p.

Proof. The proof follows the one of the original theorem ([15, Thm. 17], see also [10, Lemma 3.17]). One does the construction one irreducible component D′ of D at a time, choosing an ample line bundle M on X and N ≫ 0 such that NM − D′ is very ample. The only additional care that needs to be taken, is to choose N so that m | N and (N, p) = 1, which is possible as m is prime to p (this will be enough to prove that Y is smooth in the very same way by [13, Lemma 1.8.6]). One needs moreover to use [17, Cor. 12] instead of the classical smoothness theorem for general members of a very ample linear system. □

We can now state and prove the following

Theorem 7.7. Let X → S be a smooth morphism of K-varieties with geometrical connected fibers and let 𝔼 ∈ Strat(X/S). Assume that there exists a dense subset S̃ ⊂ S(K) such that, for every s ∈ S̃, the stratified bundle 𝔼_{s} has finite monodromy and that the highest power of p dividing |π(𝔼_{s})| is bounded over S̃. Assume moreover that 𝔼η̄ is regular singular, then

i) there exists fη̄ : Yη̄ → Xη̄ a finite étale cover such that f^{∗}𝔼η̄ decomposes as direct sum of stratified line bundles;

ii) if K ≠ 𝔽̄_{p} then 𝔼η̄ is finite.

Proof. Let U ⊂ X be a dense open, then by invariance of the monodromy group it is enough to show the theorem for 𝔼_{|U} moreover it is enough to prove finiteness for its pullback along any finite étale cover. Therefore, we can always work up to generically finite étale covers. Using [6] we can find an alteration generically finite étale f : X′ → X such that X′ admits a good projective compactification relative to S. By [16, Prop. 4.4] the pullback of a regular singular stratified bundle is again regular singular. Hence, without loss of generality, we can assume that X admits a good projective compactification \overline{X} relative to S. We will denote by D = \overline{X}\X the divisor at infinity. Let Exp_{D}(𝔼η̄) ⊂ ℤ_{p}/ℤ be the finite set of exponents of Eη̄ along Dη̄ (as defined in Lemma 7.3). As 𝔼η̄ is regular singular then by Lemma 7.4 the exponents of 𝔼η̄ are torsion; let m ∈ ℕ an integer prime to p killing the torsion of Exp_{D}(𝔼η̄) and let f : \overline{Y} η̄ → \overline{X}η̄ be the Kawamata covering constructed in Theorem 7.6: it ramifies on a simple normal crossing divisor D̃η̄ containing the divisor at infinity Dη̄ = \overline{X}η̄ − Xη̄ and it is Kummer on \overline{X}η̄ − D̃η̄. As m divides the ramification order along Dη̄ by [16, Prop. 4.11] the exponents of the pullback of 𝔼η̄ along (f^{∗}Dη̄)_{red} are zero; hence, it extends to the whole Yη̄. Up to taking an étale open of S and using a similar argument as in the proof of Lemma 7.4 we can assume that this extension is defined on the whole S. Therefore, we have reduced the problem to Theorem 4.3. □

8. Finite vector bundles

The notion of isotriviality has as well relevance in the category of vector bundles over a proper smooth K-variety, even though it is not equivalent to the notion of finiteness (see [20, Lemma 3.1] and following definition) for vector bundles, at least in positive characteristic. In the same spirit of Theorem 4.1, Esnault and Langer proved in the same paper the following:

Theorem 8.1. [9, Thm. 5.1] Let X → S be a smooth projective morphism of K-varieties with geometrically connected fibers and let E be a locally free sheaf over X. Assume that there exists a dense subset S̃ ⊂ S(K) such that, for every s ∈ S̃, there is a finite étale Galois cover h_{s} : Y_{s} → X_{s} of order prime to p such that h_{s}^{∗}(E_{s}) is trivial. Then

i) there exists fη̄ : Yη̄ → Xη̄ a finite étale cover of order prime to p such that f^{∗}Eη̄ decomposes as direct sum of stratified line bundles; ii) if K ≠ 𝔽̄_{p} then Eη̄ is trivialized by a finite étale cover of order prime

to p.

Then, a reasoning similar to the proof of Theorem 4.3 proves the following:

Theorem 8.2. Let X → S be a smooth projective morphism of K-varieties with geometrically connected fibers and let E be a locally free sheaf over X. Assume that there exists a dense subset S̃ ⊂ S(K) such that, for every s ∈ S̃, there is a finite étale Galois cover h_{s} : Y_{s} → X_{s} such that h_{s}^{∗}(E_{s}) is trivial and that the highest power of p dividing the order of such covers is bounded over S̃. Then

i) there exists fη̄ : Yη̄ → Xη̄ a finite étale cover such that f^{∗}Eη̄ decomposes as direct sum of stratified line bundles;

ii) if K ≠ 𝔽̄_{p} then Eη̄ is trivialized by a finite étale cover.

Proof. We will reduce this theorem to Theorem 8.1. By taking an étale open of S we can assume there exists a section σ : S → X. Let r be the rank of E and fix s a closed point in S. As X_{s} is a smooth k(s)-variety, then (see [9, Definition 3.2] and following discussion) every étale trivializable vector bundle is Nori semistable. In particular, the Galois cover h_{s} : Y_{s} → X_{s} corresponds to a representation of rank r of the Nori fundamental group scheme π_{1}^{N}(X_{s}, σ_{s}) (for the definition of the Nori group scheme see [20]) that factors through the étale fundamental group:

π_{1}^{N}(X_{s}, σ(s)) ↠ π_{1}^{ét}(X_{s}, σ(s)) ↠ Γ_{s} ⊂ GL_{r}(K),

where Γ_{s} is the Galois group of h_{s} : Y_{s} → X_{s}. The rest of the proof follows exactly as in Theorem 4.3. □

If the morphism X → S is not projective but only smooth we get a similar result to Corollary 6.2:

Theorem 8.3. Let K be an algebraically closed field of positive characteristic with infinite transcendental degree over 𝔽_{p}. Let X → S be a smooth morphism of varieties over K and E a vector bundle over X. Assume that there exists a dense open S̃ ⊂ S such that E_{s} is isotrivial for every s ∈ S̃(K), then so is Eη̄.

Proof. There exists K′ a subfield of K of finite type over 𝔽_{p} such that X → S and E descend to X′ → S′ and E′. Moreover as K has infinite transcendence degree over 𝔽_{p} there exists an immersion k(S′) ↪ K over K′ and a point s ∈ S(K), like in the proof of Theorem 6.1, such that the morphism i : Spec K → Spec k(S′) given by k(S′) ⊂ K is a geometric generic point of S′, and on X′ ⊗_{k(S′)} K ≃ X_{s}

i^{∗}E′ = E_{s}.

Note that there exists an immersion ι : K ↪ \overline{k(S)} (which is not the natural one given by the fact that S is a K-variety) that is the identity on k(S′), hence via ι we have that Xη̄ ≃ X′ ⊗_{k(S′)} \overline{k(S)} ≃ X_{s} ⊗_{K} \overline{k(S)}. In particular, if we continue to consider K as a subfield of \overline{k(S)} via the immersion ι, then h_{s} ⊗_{K} \overline{k(S)} : Y_{s} ×_{Spec K} Spec \overline{k(S)} → Xη̄ trivializes Eη̄. □

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Giulia Battiston

gbattiston@mi.fu-berlin.de 88

Documenta Mathematica 20 (2015)