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On positivity and semistability of vectorbundles in finite and mixed characteristics

Adrian Langer

March 24, 2015

ADDRESS:Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warszawa,Poland

Dedicated to Professor C. S. Seshadri on his 20th birthday1

Abstract

We survey results concerning behavior of positivity of linebundles andpossible vanishing theorems in positive characteristic. We also try to de-scribe variation of positivity in mixed characteristic. These problems arevery much related to behavior of strong semistability of vector bundles,which is another main topic of the paper.

Introduction

The main aim of this paper is to survey problems concerning positivity of linebundles and stability of vector bundles on schemes defined over finite fields orover finitely generated rings overZ. Note that these two topics are very muchrelated because a degree zero vector bundleE on a curve is strongly semistable ifand only if the line bundleOP(E)(1) on the projectivization ofE is nef (see, e.g.,[Mr, Proposition 7.1]).

The motivating problems are the following:

1In this rare case the number of years does not coincide with the number of birthdays.

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• What can we say about relation between nefness, semiampleness, effectivityand pseudoeffectivity for line bundles on varieties definedover finite fields?

• What vanishing theorems can hold for suitably positive linebundles in pos-itive characteristic (or overFp)?

• Is there any relation between nefness in characteristic zero and in positivecharacteristic?

• What can we say about variation in families of positivity of line bundles andsemistability of vector bundles?

The known results do not answer any of these questions. In this paper we poseand study some conjectures that try to answer all of the abovequestions. Someof these question are very arithmetic in nature and in fact they imply very strongproperties of reductions of varieties. In some simple casesthey can be recoveredusing known results or they give another point of view on wellknown conjecturesfrom arithmetic algebraic geometry.

The paper is divided in several sections describing each of these problemsand surveying known results. First we recall some notation used throughout thepaper. In Section 1 we describe positivity of line bundles onvarieties definedover finite fields. In Section 2 we survey known results on Kodaira type vanishingtheorems in positive characteristic. In Section 3 we study vanishing theorems forgeneral reductions from characteristic zero. In Section 4 we recall several knownconstructions of strictly nef line bundles in characteristic zero. This is related toKeel’s question of existence of such bundles over finite fields. In Section 5 westudy variation of positivity of line bundles in mixed characteristic. In Section6 we consider a related question concerning vector bundles.In both Sections 6and 7 we pose several conjectures that should fully explain behavior of strongsemistability in mixed characteristic.

0.1 Notation

Let X be a complete variety defined over some algebraically closedfield k.Let N1(X) (N1(X)) be the group of 1-cycles (divisors, respectively) modulo

numerical equivalence. By the Neron-Severi theoremN1(X)Q = N1(X)⊗Q andN1(X)Q = N1(X)⊗Q are finite dimensionalQ-vector spaces, dual to each otherby the intersection pairing.

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A Q-divisor D is calledpseudoeffectiveif its numerical class inN1(X)Q iscontained in the closure of the cone generated by the classesof effective divisors.

A line bundleL on X is calledsemiample, if there exists a positive integernsuch thatL⊗n is globally generated.

A line bundleL on a varietyX is calledstrictly nef if it has positive degree onevery curve inX.

A locally free sheafE on X is nef if and only if for anyk-morphismf : C →X from a smooth projective curveC/k each quotient off ∗E has a non-negativedegree. We say thatE is numerically flatif both E andE∗ are nef.

Let X be a normal projectivek-variety and letH be an ample Cartier divisoron X. Let E be a rankr torsion free sheaf onX. Then we define theslopeµH(E)of E as quotient of the degree of detE = (

∧r E)∗∗ with respect toH by the rankr.We say thatE is slope H-semistableif for every subsheafE′ ⊂ E we have

µH(E′)≤ µH(E).If k has positive characteristic then we say thatE isstrongly slope H-semistable

if all the Frobenius pull backs(FnX)

∗E of E for n≥ 0 are slopeH-semistable.Let X be an algebraick-variety. We say that avery general pointof X satisfies

some property if there exists a countable union of proper subvarieties ofX suchthat the property is satisfied for all points outside of this union.

1 Nef line bundles over finite fields

The following fact (see, e.g., [Ke1, Lemma 2.16]) is standard and it follows easilyfrom existence of the Picard scheme and the fact that an abelian variety has onlyfinitely many rational points over a given finite field.

PROPOSITION1.1. A numerically trivial line bundle on a projective scheme de-fined overFp is torsion. In particular, a nef line bundle on a projective curve overFp is semiample.

In the surface case Artin [Ar, 2-2.11] proved the following result:

THEOREM 1.2. A nef and big line bundle on a smooth projective surface definedoverFp is semiample.

In [Ke1, Theorem 0.2] Sean Keel gave the following criterionfor semiample-ness:

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THEOREM 1.3. Let L be a nef line bundle on a projective scheme X defined overa field of positive characteristic. Let L⊥ be the closure of the union of all subva-rieties Y⊂ X such that LdimY ·Y = 0, taken with the reduced scheme structure.Then L is semiample if and only if its restriction to L⊥ is semiample.

This theorem, combined with earlier ideas of Seshadri, occurred to be the mainnew ingredient in Seshadri’s new proof of Mumford’s conjecture (see [Se]).

A basic tool used in proofs of Theorems 1.2 and 1.3 is Proposition 1.1.Keel’s theorem implies Artin’s theorem, because ifX/Fp is a smooth pro-

jective surface andL is a nef and big line bundle onX thenL⊥ is at most one-dimensional and henceL|L⊥ is numerically trivial. Thus by Proposition 1.1L|L⊥

is torsion and Theorem 1.3 implies thatL is semiample.

Note that Keel’s theorem trivially fails in the characteristic zero case. As anexample one can take, e.g., any non-torsion line bundle of degree zero on a smoothprojective curve. It is more difficult to produce counterexamples to Artin’s theo-rem in the characteristic zero case but they also exist:

THEOREM 1.4. (see [Ke1, Theorem 3.0])Let C be a smooth projective curve ofgenus g≥ 2over a field of characteristic zero. Let X=C×C and let L= p∗1ωC(∆),where∆ is the diagonal and p1 is the projection of X onto the first factor. Then Lis nef and big but it is not semiample.

Note that in positive characteristic the bundleL in the above theorem is semi-ample. All these results and lack of good construction methods raised the ques-tion whether there exist any nef line bundles on varieties defined over finite fieldswhich are not semiample. In [Ke2, Section 5] Keel gives Koll´ar’s example of anef but non-semiample line bundle on a non-normal surface defined over a finitefield. The example is obtained by glueing two copies ofP1×P1 but the obtainedline bundle is not strictly nef.

Keel’s proof of non-semiampleness in Theorem 1.4 goes via showing that therestriction ofL to 2∆ is non-torsion. Interestingly, Totaro used a similar strategy toshow the following example of a nef but non-semiample line bundle on a smoothprojective surface overFp:

Example1.5. Let C be a smooth projective curve of genus 2 defined overFp.Assume that for every line bundleL of order≤ p the mapH1(C,L)→H1(C,F∗L),induced by the Frobenius morphism on C, is injective. In [To,Lemma 6.4] Totaroshowed that a general curve of genus 2 satisfies this assumption.

Then one can embeddC into P1×P1 as a curve of bidegree(2,3). In thiscase there exists twelveFp-pointsp1, ..., p12 onC such that ifX is the blow up of

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P1×P1 at these points then the line bundleL, associated to the strict transformCof C, has orderp after restricting toC but the restriction ofL⊗p to 2C is non-trivial.In this case Totaro shows the following theorem (see [To, proof of Theorem 6.1]):

THEOREM 1.6. The line bundle L is nef but it is not semiample. In fact, we haveh0(X,Ln) = 1 for every positive integer n.

Totaro used the above theorem to show the first example of nef and big linebundle on a smooth projective threefold, which is not semiample. This shows thatArtin’s theorem does not generalize to higher dimensions. These examples donot answer the following question of Keel (see [Ke2, Question 0.9]), which weprovocatively formulate as a conjecture:

CONJECTURE1.7. Let L be a strictly nef line bundle on a smooth projective sur-face X defined overFp. Then L is ample.

By the Nakai-Moishezon criterion (see [Ht2, Chapter V, Theorem 1.10]), orby Theorem 1.2, this conjecture is equivalent to non-existence of strictly nef linebundlesL onX with L2 = 0. In fact, in view of Totaro’s example, one can pose aneven stronger conjecture:

CONJECTURE1.8. Let L be a nef line bundle on a smooth projective surface Xover Fp. Then the Iitaka dimensionκ(L) of L is non-negative. Equivalently, wecan find some positive integer m such that L⊗m has a section.

If L is nef andL2 > 0 thenκ(L) = 2, so in the above conjecture we can assumethatL2= 0. We can also try to relax the nefness assumption and pose thefollowingconjecture:

CONJECTURE1.9. Let D be a pseudoeffectiveQ-divisor on a smooth projectivesurface X overFp. Then D isQ-linearly equivalent to an effectiveQ-divisor.

Conjecture 1.9 is equivalent to non-existence of a nef line bundleL with Iitakadimensionκ(L) =−∞ and the numerical Iitaka dimensionν(X) = 1. Obviously,all of the above conjectures can be also considered in higherdimensions but simi-larly to the surface case no answer seems to be known up to date. In fact, in higherdimensions Conjecture 1.9 can be generalized into two different ways: either asasking wether the cone of curves NE(X)⊂ N1(X)Q is closed or as asking wetherthe cone of effective divisors is closed.

The assertion of Conjecture 1.9 seems to be much stronger than the one ofConjecture 1.8 but in fact we have the following lemma:

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LEMMA 1.10. Conjectures 1.8 and 1.9 are equivalent.

Proof. We only need to check that Conjecture 1.8 implies Conjecture1.9.Let D be a pseudoeffectiveQ-divisor. Then there exists a decomposition (so

calledZariski decomposition) D = P+N, whereP is a nefQ-divisor andN is a(negative) effectiveQ-divisorN such thatP ·N = 0. By our assumption we knowthat some positive multiple ofP, and therefore also ofD, has a section.

2 Killing cohomology by finite morphisms

If L is an ample line bundle on a smooth varietyX defined over a field of charac-teristic zero then Kodaira’s vanishing theorem says thatH i(X,L −1) vanishes fori < dimX. Kawamata–Viehweg vanishing theorem says that the same vanishingholds ifL is only nef and big. However, Raynaud in [Ra] constructed an exampleshowing that already Kodaira’s vanishing theorem fails in positive characteristic.In this section we do not try to recover Kodaira’s vanishing theorem adding addi-tional assumptions on the base variety as was done by Deligneand Illusie in [DI].Instead try to kill cohomology on all varieties but using finite morphisms:

THEOREM 2.1. Let X be a proper variety over a field of positive characteristicand letL be a semiample line bundle on X.

1. For any i> 0 there exists a finite surjective morphismπ : Y → X such thatthe induced map Hi(X,L )→ H i(Y,π∗L ) is zero.

2. If L is big then for any i< dimX there exists a finite surjective morphismπ :Y →X such that the induced map Hi(X,L−1)→ H i(Y,π∗L −1) is zero.

This theorem was proven by Hochster and Huneke [HH, Theorem 1.2] in caseL is a tensor power of a very ample line bundle (see also [Sm, Theorem 2.1] andits errratum for the case whenL is a tensor power of an ample line bundle), andby Bhatt [Bh, Propositions 7.2 and 7.3] in general. Note thatin caseX is Cohen–Macaulay andL is a tensor power of an ample line bundle, then the only non-trivial case is whenL = OX. In the remaining cases, it is sufficient to use Serre’svanishing theorem (see [Ht2, Chapter III, Theorem 5.2]) andSerre’s duality (see[Ht2, Chapter III, Corollary 7.7]) in the dual case.

One can ask wether Theorem 2.1 works under weaker assumptions on L ,possibly after restricting the base field to the algebraic closure of a finite field(this is the most interesting case, as it is the only case thatarises when reducing

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from characteristic zero). By Proposition 1.1, Theorem 2.1.1 holds for nef linebundles on curves overFp but it fails for nef line bundles on smooth projectivesurfaces overFp. More precisely, one can prove that in Example 1.5 we have thefollowing non-vanishing theorem (see [La2, Theorem 3.1]):

THEOREM 2.2. Let M= L−p−1 or M = Lp−1. Then for any completeFp-surfaceY and any generically finite surjective morphismπ : Y → X the induced mapH1(X,M)→ H1(Y,π∗M) is non-zero.

Similarly, Theorem 1.2 implies that Theorem 2.1.1 holds fornef and big linebundles on smooth projective surfaces overFp but one can show that it fails fornef and big line bundles on smooth projective threefolds over Fp (see [La2, Propo-sition 4.1]).

In analogy to the Kawamata–Viehweg vanishing theorem, it ismore naturalto generalize Theorem 2.1.2 to nef and big line bundles on smooth projectivevarieties. In fact, in low dimensions one can show an even stronger theorem:

THEOREM 2.3. Let L be a nef and big line bundle on a normal projective varietyover field of positive characteristic. Fix an integer0 ≤ i < min(dimX,2). Thenfor sufficiently large m the map

H i(X,L−1)→ H i(X,L−pm)

induced by the m-th Frobenius pull back is zero.

Unfortunately, the vanishing holds for trivial reasons because under the aboveassumptions one hasH i(X,L−n) = 0 for n ≫ 0 (see [Fu, Theorem 10]; see also[La1, Theorem 2.22 and Corollary 2.27] for effective versions of this theorem).

The only known examples of nef and big line bundleL on a smooth projectivevariety X of dimension> 2 such thatH2(X,L−n) 6= 0 for all n ≫ 0 were con-structed by Fujita (see [Fu, pp. 526–527]). He used Raynaud’s counterexampleto Kodaira’s vanishing theorem in positive characteristic(see [Ra]). By construc-tion, in Fujita’s example the map induced by them-th Frobenius pull back onH2(X,L−1) vanishes for allm≫ 0. This leaves open the following question:

QUESTION 2.4. Let L be a nef and big line bundle on a smooth projective varietydefined over an algebraically closed field of positive characteristic. Fix an inte-ger 0 ≤ i < dimX. Is the map Hi(X,L−1) → H i(X,L−pm

) induced by the m-thFrobenius pull back zero for m≫ 0?

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Note that [La2, Example 5.4] shows that the answer to this question is negativeif one allows singular varieties. But for smooth varieties an answer to the abovequestion is not known even ifL is semiample and big.

One can also try to weaken conditions onL in Theorem 2.3 still hoping thatwe can kill cohomology using the Frobenius morphism. This works in some casesas shown by the following theorem proven in [La2, Theorem 6.1]:

THEOREM 2.5. Let X be a smooth projective surface defined over an algebraicclosure of some finite field. Let L be a nef line bundle on X such thatκ(X,L)=−∞(i.e., no power of L has any sections). Then for large n the mapH1(X,L−1) →H1(X,(Fn

X)∗L−1) induced by the n-th Frobenius morphism Fn

X is zero.

Note that if in Example 1.5 we takeM = Lp+1 then we get a nef line bundlewith M2 = 0 on a smooth projective surface overFp such thatH1(X,M−1) →H1(X,M−pn

) induced by then-th Frobenius pull back is always non-zero (seeTheorem 2.2).

The above theorem is consistent with Conjecture 1.8 saying that there does notexist a nef line bundleL on a smooth projective surface defined overFp such thatκ(L) =−∞ (cf. Corollary 3.4).

An interesting point in proof of Theorem 2.5 is that we use thehigher rank caseof Proposition 1.1, which follows from boundedness of the family of semistablevector bundles with trivial Chern classes.

As a corollary to Theorem 2.5 we get the following theorem analogous toTheorem 2.3:

COROLLARY 2.6. Let X be a smooth projective variety of dimension d≥ 2 definedover an algebraic closure of some finite field. Let L be a strictly nef line bundle onX. Then for large n the map H1(X,L−1)→ H1(X,(Fn

X)∗L−1) induced by the n-th

Frobenius morphism FnX is zero.

3 Vanishing theorems in mixed characteristic

Let R be a domain which containsZ and which, as a ring, is finitely generatedoverZ. Let X be a projectiveR-scheme and letL be an invertible sheaf ofOX -modules. LetXs denote the fibre overs∈ S and letLs be the restriction (i.e.,pull-back) ofL to Xs.

Let R⊂K be an algebraic closure of the field of quotients ofR. By assumptionK is of characteristic zero, so we can think ofX → S= SpecRas a model of thegeneric geometric fibreXK with polarizationLK.

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The following theorem (see [Sm, 3.5]), conjectured by Huneke and K. Smithin [HS, 3.9], was proven (in more general setting of rationalsingularities) by N.Hara in [Ha, Theorem 4.7] and later by V. Mehta and V. Srinivasin [MSr, Theorem1.1].

THEOREM 3.1. Let us assume thatXK is smooth andLK is ample. Then thereexists a non-empty Zariski open subset U⊂ S such that for every closed points∈U the natural map

H i(Xs,L−1s )→ H i(Xs,F

∗L

−1s ),

induced by the Frobenius morphism on the fiberXs, is injective for all i≥ 0.

Note that fori <dimXK Kodaira’s vanishing theorem says thatH i(XK,L−1K )=

0 so by semicontinuity of cohomology (see [Ht2, III, Theorem12.8]) we haveH i(Xs,L

−1s ) = 0 for s from some open subset ofS. So the above theorem is non-

trivial only in casei = dimX. On the other hand, one can ask if similar theoremshold in other cases when we do not have vanishing of cohomology at the genericfibre. Here is one such example in the surface case:

PROPOSITION3.2. Let us assume thatXK is a smooth surface andLK is a linebundle withκ(LK) =−∞. Assume also that there exists an ample line bundleAK

on XK such that c1LK · c1AK > 0. Then there exists a non-empty Zariski opensubset U⊂ S such that for every closed point s∈U and every positive integer nthe natural map

H1(Xs,L−1s )→ H1(Xs,(F

n)∗L −1s ),

induced by composition of n absolute Frobenius morphisms onthe fiberXs, isinjective.

Proof. Let B1Xs

be the sheaf of exact 1-forms. By definition we have an exactsequence

0→ OXs → F∗OXs → F∗B1Xs

→ 0.

Therefore to check that

H1(Xs,L−1s )→ H1(Xs,F∗OXs⊗L

−1s ) = H1(Xs,F

∗L

−1s )

is injective, it is sufficient to prove thatH0(Xs,F∗B1Xs

⊗L −1s ) = 0. ButF∗B1

Xsis

a subsheaf ofF∗Ω1Xs

, so by the projection formula we have

H0(F∗B1Xs

⊗L−1s )⊂ H0(F∗ΩXs⊗L

−1s ) = H0(ΩXs⊗F∗

L−1s ).

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So it is sufficient to show that there exists an open subsetU ⊂Ssuch that for everyclosed points∈ U the sheafΩXs ⊗F∗L −1

s has no sections. Similarly, to checkthat

H1(Xs,(Fn−1)∗L −1

s )→ H1(Xs,(Fn)∗L −1

s )

is injective it is sufficient to prove thatΩXs⊗ (Fn)∗L −1s has no sections.

We can find a Zariski open subsetV ⊂ Sand a line bundleA extendingAK.Since ampleness is an open property, shrinkingV if necessary, we can assume thatA on XV →V is relatively ample. Existence of the relative Harder-Narasimhanfiltration of ΩXV/V (see [HL, Theorem 2.3.2]) implies that further shrinkingV wecan assume that for all closed pointss∈V we have

µmax,As(ΩXs) = µmax,H(ΩXK).

Sincec1Ls·c1As= c1LK ·c1AK > 0, we see that if the characteristicp at a closedpoint s∈ V is larger thanµmax,A (ΩXk

)/(c1LK · c1AK), then for every positiveintegern

µmax,As(ΩXs⊗ (Fn)∗L −1s ) = µmax,AK(ΩXK)− pn(c1LK ·c1AK)< 0.

But existence of sections ofΩXs ⊗ (Fn)∗L −1s would contradict this inequality.

LEMMA 3.3. Let C be aQ-divisor on a smooth projective surface X. If C2 ≥ 0and CP> 0 for some nef divisor P then C is pseudoeffective.

Proof. If CA< 0 for some ample divisorA then taking appropriate combinationH = aA+ bP for somea,b > 0 we haveCH = 0. SinceH is ample andC isnumerically non-trivial, the Hodge index theorem (see [Ht2, Chapter V, Theorem1.9]) givesC2 < 0.

COROLLARY 3.4. Let L be a pseudoeffective line bundle on a smooth projectivesurface defined over a field of characteristic zero. Let us assume that L2 ≥ 0 andH1(X,L−1) is non-zero. Then for almost all primes p the reduction of L modulop has a non-negative Iitaka dimension.

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Proof. If L is pseudoeffective then andL2 ≥ 0 then by Lemma 3.3 almost allreductions ofL are pseudoeffective. LetL = P+N be the Zariski decomposition(see proof of Lemma 1.10). IfL is not nef thenP2 = L2−N2 > 0 (sinceN isnon-zero we haveN2 < 0 as follows fromPN= 0 by the Hodge index theorem).HenceP is big, which implies thatL is also big. The same argument shows thatif we take a reduction ofL which is pseudoeffective but not nef then it is big. Sowe can assume that a reduction ofL is nef. In this case the assertion follows fromProposition 3.2 and Theorem 2.5.

Remarks3.5. 1. In the above corollary, instead of assuming thatH1(X,L−1)is non-zero it is sufficient to assume that there exist a smooth projective sur-faceY and a generically finite morphismπ : Y → X such thatH1(Y,π∗L−1)is non-zero.

2. Corollary 3.4 implies that if a line bundleL is strictly nef with non-vanishingH1(X,L−m) for some positive integerm, then its reduction to positive char-acteristic is almost never strictly nef. This happens, e.g., in Mumford’s ex-ample (see Example 4.1). In fact, in this case Biswas and Subramanian (see[BS, Theorem 1.1]) proved that strictly nef line bundles on ruled surfacesoverFp are always ample.

4 Examples of strictly nef line bundles

Note that ifL is a strictly nef line bundle on a proper varietyX and f : Y → Xis a finite morphism thenf ∗L is also strictly nef. This gives a lot of examplesof strictly nef line bundles once we have constructed some such bundles. In thissection we review known constructions of strictly nef line bundles on smoothprojective surfaces that do not come from this construction.

Example4.1. The most famous example of a strictly nef line bundle is due toMumford (see [Ht1, I, Example 10.6]). Namely, letC be a smooth complex pro-jective curve of genus≥ 2. Then onC there exists a rank 2 stable vector bundleE with trivial determinant and such that all symmetric powersSnE are also stable.Let π : X = P(E)→C be the projectivization ofE and letL=OP(E)(1). ThenL isa strictly nef line bundle onX with L2 = 0. Note that in this exampleH1(X,L−2)is non-zero. More precisely, let us not that the relative Euler exact sequence

0→ ΩX/C → π∗E⊗L−1 → OX → 0

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is non split, as it is non-split after restricting to the fibers ofπ . After tensoring thissequence byL and using detE⊗OC we get the sequence

0→ L−1 → π∗E → L → 0,

which gives a non-zero element in Ext1(L,L−1) = H1(X,L−2).For generalization of Mumford’s example to higher dimensions see S. Sub-

ramanian’s paper [Su]. For uncountable fields of positive characteristic a similarexample was considered by V. Mehta and S. Subramanian [MSu].The next ex-ample shows existence of strictly nef line bundles even overcountable fields ofpositive characteristic, provided they have sufficiently large transcendental degreeover its prime field.

Example4.2. Consider the projective planeP2 over some fieldk and let us taker = s2, wheres> 3, k-rational pointsp1, ..., pr ∈ P2(k). Let p : X → P2 be theblow up at these points and let us takeL = p∗OP2(s)⊗O(−E), whereE is theexceptional divisor ofp. Clearly, we haveL2 = 0. If all the chosen points lie ona geometrically irreducible degrees curveC ⊂ P2 defined overk thenL is nef.This follows from the fact that the strict transformC gives an element of the linearsystem|L| and hence for every irreducible curveD ⊂Y we haveD ·L = D ·C≥ 0with equality if and only ifD = C. This is also the main idea behind Totaro’sconstruction of a nef non-semiample line bundle, except that to obtain an examplewhereC has genus 2 he blow upsP1×P1 instead ofP2. Obviously, the bundleLobtained in this way is not strictly nef asL ·C = 0. However, Nagata proved thefollowing theorem:

THEOREM 4.3. Assume that the points p1, ..., pr are very general. Then L isstrictly nef.

Proof. Let D be any reduced curve on the blow upX and letC ∈ |OP2(d)| be itsimage. Letm1, ...,mr be the multiplicities ofC at the pointsP1, ...,Pr, respectively.ThenLD = sd−∑r

i=1mi . But by [Na, Chapter 3, Proposition 1] we havesd−∑r

i=1mi > 0.

Unfortunately, this theorem does not say anything for varieties defined overFp.

Note that a similar construction can be used also in different cases: we canblow up some pointsp1, ..., pr (wherer can be arbitrary) on a smooth projectivesurfaceX and take the pull back of an ample line bundle onX twisted by a suitable

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negative combination of exceptional divisors, arranging this so that the obtainedline bundle has self intersection 0. If the numberr of points is sufficiently largeand the points are in a very general position then the obtained line bundle shouldbe strictly nef. This type of construction was used, e.g., in[LR, Example 3.3] butit seems that the proof of strict nefness of the obtained divisor is incorrect.

Example4.4. Let F be a real quadratic field and letD be a totally indefinite quater-nion F-algebra. Let us recall that a quaternion algebra overF is anF-algebraD = F +Fi +F j +Fi j given byi2 = a, j2 = b andi j = − ji , wherea,b∈ F aresome non-zero elements.D is totally indefinite, if for both embeddingsF →R wehaveR⊗F D ≃ M2(R). In this case we get two inequivalent real representationsρi : D → M2(R), i = 1,2. On the algebraD we can introduce anorm N: D → Fby

N(x0+x1i +x1 j +x2i j ) = x20−ax2

1−bx22+abx23

for xi ∈ F. Let G be the group of elements of norm 1 in a fixed maximal orderRin D and letG= G/〈±1〉. Let H be the complex upper half plane. The groupGacts on the productH×H by

λ (z1,z2) = (ρ1(λ )z1,ρ2(λ )z2).

In caseD is a division algebra, the quotient surfaceX =H×H/G is compact. Letus also assume thatX is smooth (all these assumptions are satisfied in some cases).Let p1, p2 : X =H×H→H be the two projections. ThenΩ1

X≃ p∗1Ω1

H⊕ p∗2Ω1H as

G-linearized bundles. So by descent we haveΩ1X ≃ L⊕M for some line bundles

L andM. Then we have the following lemma:

LEMMA 4.5. ([SB1, Lemma 3])The line bundles L and M are strictly nef withL2 = M2 = 0.

Proof. Let C be a reduced and irreducible curve inX and letC be an irreduciblecomponent of its pre-image inX. The line bundleL|C is represented by a formwhose pull-back toC is the pull-back of a positive form fromH. ThereforeCL = degL|C > 0. This shows thatL is strictly nef and in particularL2 ≥ 0.If L2 > 0 thenL is ample by the Nakai–Moishezon criterion (see [Ha, V, Theorem1.10]). But by Bogomolov’s vanishing theoremΩ1

X does not contain any amplesubbundles. ThereforeL2 = 0. The same proof works also forM.

[SB1] contains a more general example of the same type but we will need thisparticular case later on (see Example 5.6).

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5 Variation of positivity of line bundles

It is known that ampleness is an open condition in families (not necessarily flat).More precisely, letSbe an irreducible noetherian scheme and letπ : X → Sbe aproper morphism. LetL be a line bundle onX .

THEOREM 5.1. (see [Gr, III, Theorem 4.7.1])If Ls0 is ample onXs0 for somepoint s0 ∈ S thenLs is ample for a general point of S, i.e., there exists an openneighborhood U⊂ S of s0 such thatLs is ample onXs for all s∈U.

COROLLARY 5.2. If Ls0 is nef onXs0 for some geometric point s0 ∈ S thenLs isnef for a very general point of S, i.e., there exist countablymany open and densesubsets Um ⊂ S such thatLs is nef for every geometric point s∈

⋂

Um.

Proof. Using Chow’s lemma we can reduce to the case whereπ is projective.Let OX (1) be aπ-ample line bundle onX . By Theorem 5.1 we know thatfor every positive integerm the setUm of points for which(L ⊗m⊗OX (1))s isample is open and dense inS. It is easy to see that these sets satisfy the requiredassertion.

Note that we can assume that the sequenceUmm∈N is descending, i.e.,Um+1⊂Um for all m and one can ask if such a sequence must stabilize. In general,this istoo much to hope for but

⋂

Um contains the generic geometric point ofSso we canask if it contains any closed points. This is interesting only if S has only count-ably many points as only then the set of closed geometric points s∈ S for whichLs is nef can be empty. Indeed, this can really happen as shown bythe followingexample due to Monsky [Mo1], Brenner [Br2] and Trivedi [Tr]:

Example5.3. Let us start with recalling the following result of Monsky [Mo1,Theorem]:

THEOREM 5.4. Let Rt = Kt [x,y,z]/(Pt), where Kt is an algebraic closure ofF2(t)and set

Pt = z4+xyz2+x3z+y3z+ tx2y2.

Then the Hilbert-Kunz multiplicity of Rt is equal to3+4−m(t), where

m(t) =

degree ofλ overF2, if t = λ 2+λ is algebraic overF2,∞, if t is transcendental overF2.

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Now let k= F2 and let us setS= A1k with coordinatet andP2

k with homoge-neous coordinates[x : y : z]. LetY ⊂ P2×k Sbe given by

z4+xyz2+x3z+y3z+ tx2y2 = 0

and letE = p∗1ΩP2, wherep1 : Y → P2 is the canonical projection. Consider theprojectionp2 : Y → S. ThenEs is not strongly semistable for every closed points∈ S (even on the singular fiber over 0∈ S) but Eη is strongly semistable forthe generic pointη ∈ S. This follows from Monsky’s theorem and the computa-tion of the Hilbert-Kunz multiplicity ofRt in terms of strong Harder–Narasimhanfiltration of bundlesEs for s∈ S due to Brenner [Br2, Theorem 1] and Trivedi[Tr, Theorem 5.3]. This computation implies thatEs is strongly semistable fors : SpecKt → S if and only if the Hilbert-Kunz multiplicity ofRt is equal to 3.

LetX be the projectivization ofF = p∗1(S2ΩP2(1)) overY. LetL =OP(F )(1)

and letπ : X → Sbe the composition of the projectionsX →Y andp2 : Y → S.ThenLη is nef for a generic geometric pointη ∈ S but Ls is not nef for everyclosed geometric points∈ S.

One can also show a similar example in equal characteristic 3(see [Mo2]).

Note that in the above exampleS was defined over an algebraic closure of afinite field. It seems to be unknown if similar examples can occur for S definedover a countable field of positive characteristic containing transcendental elementsover its prime field, or even in caseS is defined overQ. One might expect thatthe strange behavior of variation of nefness in positive equal characteristic cannotoccur in mixed characteristic:2

CONJECTURE 5.5. Let R be a finitely generated integral domain overZ, con-taining Z. Let π : X → S= SpecR be a smooth proper morphism. LetL bean invertible sheaf ofOX -modules and assume that the restriction ofL to thegeneric geometric fibre ofπ is nef. Then the set T of closed points s∈ S such thatLs is semiample is dense in S.

Totaro’s Example 1.5 comes from characteristic zero by reduction modulop.The above conjecture suggests that such examples are ratherrare and almost allreductions of a fixed nef line bundle are semiample.

Conjecture 5.5 generalizes [Mi, Problem 5.4] which considers the same ques-tion in caseX is a projectivization of a rank 2 vector bundle over a curve (in this

2Recently the author constructed a counterexample to this conjecture (see [La3]). But theconjecture can still be true under appropriate assumptions, e.g., if we require that the rank of theNeron-Severi group stays the same on the fibers ofπ .

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case ifs∈ S is a closed point then nefness ofLs implies its semiampleness by theLange–Stuhler theorem; see Proposition 7.1).

Note that one can show examples in which the setT is not open in the setof closed points ofS. The first such examples come from an unpublished work[EST] of Ekedahl, Shepherd–Barron and Taylor:

Example5.6. ConsiderX from Example 4.4. The line subbundleL−1 ⊂ TX ≃L−1⊕M−1 defines a foliation. If we reduceX modulo some prime of character-

istic p then thep-curvature mapL⊗(−p)p = F∗(L−1

p )→ TXp/L−1p = M−1

p , given bytaking thep-th power of a derivation, isOXp-linear. If p is inert inF then this map

is non-zero (see [EST, p. 23]). In this case we get a section ofL⊗pp ⊗M−1

p and,

similarly, we get a section ofM⊗pp ⊗L−1

p . Note thatLp is not nef (and hence it isnot semiample). Otherwise, we would have−LM = Lp(pLp−Mp) ≥ 0, whereasLM > 0. SinceLp is pseudoeffective andL2

p = 0, existence of the Zariski decom-position ofLp implies thatLp is big (see proof of Corollary 3.4). Let us recall thatby Chebotarev’s density theorem the number of rational primes p which remaininert in F is infinite (of Dirichlet density 1/2). So in this case we have a strictlynef line bundleL for which infinitely many reductions are not semiample.

In fact, it is not clear how to prove that in the remaining cases the reduction ofL is semiample (possibly apart from finitely many primes).

Other examples of a similar type were obtained by Brenner [Br1] in caseX isa projectivization of a rank 2 vector bundle over a curve (note that these examplesdid not solve Miyaoka’s problem [Mi, Problem 5.4]).

6 Variation of semistability of vector bundles

Let X be a smooth complex projective variety and letOX(1) be an ample linebundle onX. Let E be a slope semistable (with respect toOX(1)) locally freeOX-module.

We are interested in behavior ofE when taking reduction modulop. Moreprecisely, all of the above data can be described by a finite number of equa-tions. Therefore there exist a subringR⊂ C, finitely generated as an algebraoverZ, and a triple(X ,OX (1),E ) consisting of a smooth projectiveR-schemeπ : X → S= SpecR, anR-ample line bundleOX (1) and a familyE of locallyfree slope semistable sheaves on the fibers ofπ , such that on the fiber over thegeneric geometric point SpecC → S we recover the triple(X,OX(1),E). Note

16

that we have implicitly used openness of slope semistability in flat families ofsheaves.

Let us recall that for every maximal idealm⊂ R the residue fieldk= R/m isfinite of characteristicp> 0. Now we would like to relate various properties ofEto the behavior of its reductions modulop. We pose a series of conjectures thatshould completely describe the behavior of strong semistability in mixed charac-teristic. The first conjecture is motivated by [SB2], where it was proven in therank 2 case:

CONJECTURE6.1. Let Σnss be the set of closed points s∈ S such thatEs is notstrongly slope semistable. IfΣnss is infinite 3 then End E is a numerically flatvector bundle. Moreover, End E is notetale trivializable.

LEMMA 6.2. If Σnss is infinite then End E is notetale trivializable. In particular,Conjecture 6.1 is true in the curve case.

Proof. If End E is etale trivializable thenEndE is etale trivializable overXU forsome open subsetU ⊂ S. In particular,EndEs, is strongly semistable fors∈ U .We claim thatEs is also strongly semistable. IfEs is not strongly semistable thenthere exists somen such that thenth Frobenius pull back ofEs is destabilized bysome subsheafE′. But then

µ(E′⊗ (Fn)∗E ∗s ) = µ(E′)+µ((Fn)∗E ∗

s )> µ((Fn)∗Es)+µ((Fn)∗E ∗s ) = 0

and henceE′⊗(Fn)∗E ∗s destabilizes(Fn)∗(EndEs), a contradiction. This implies

that Σnss is contained in the set of closed points ofS−U , and thereforeΣnss isfinite.

If X is a curve then for every semistableE the bundleEnd E is semistable ofdegree 0, so it is numerically flat and the conjecture followsfrom the first part ofthe lemma.

This shows that Conjecture 6.1 is of interest only in the surface case and theonly non-trivial part of the conjecture is thatEnd E is numerically flat. Indeed,the higher dimensional case can be easily reduced to the surface case by means ofrestriction theorems. More precisely, ifX has dimensiond greater than 2 andE is

3This assumption is tentative and works well only in the number field case. In general, it shouldprobably be modified so that the setΣnss is dense in S. (Unfortunately, in the published versionthis footnote was misplaced.)

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a vector bundle for whichΣnssis infinite then the restriction ofE to a general com-plete intersection surfaceY ⊂ X is semistable and it satisfies the assumptions ofthe conjecture. So if we know the conjecture forE|Y thenEnd E|Y is a numericallyflat vector bundle. But thenEnd E is also numerically flat because it is semistablewith respect to some ample polarizationH such thatc1(E)Hd−1= c2(E)Hd−2 = 0(cf. [Si, Theorem 2]).

7 Arithmetic of numerically flat vector bundles

Conjecture 6.1 implies that to study strong semistability of reductions of a com-plex vector bundle, it is sufficient to study reductions of numerically flat vectorbundles. The following subsection recalls a special role ofsuch vector bundlesand their relation to representations of the fundamental group.

7.1 Flat bundles

Let X be a smooth complex projective variety. Giving a representation of thetopological fundamental groupπ1(X,x) on a complex vector spaceVx is equivalentto giving a complex local systemV (a sheaf of complex vector spaces locallyisomorphic to the constant sheafCn, n∈ N). Given a local system we can recoverthe corresponding representation as themonodromy representation.

GivenV we can construct a holomorphic vector bundleOX ⊗CV with (holo-morphic) integrable connection∇ such that∇( f v) = d f · v, where f is a localsection ofOX andv is a local section ofV. On the other hand, given a holomor-phic vector bundleE with integrable connection∇ we can recover a local systemV as a sheaf of local sectionsv of E for which ∇(v) = 0. This constructions pro-vide functors giving an equivalence of categories of complex local systems andholomorphic vector bundles with integrable connection.

In [Si, Corollary 3.10] Simpson proved that these categories are equivalent tothe category of (Higgs) semistable Higgs bundles(E,θ) with vanishing (rational)Chern classes. This category contains the category of semistable vector bundleswith vanishing Chern classes. If a representation ofπ1(X,x) is an extension ofunitary representations, then the corresponding Higgs bundle is an extension ofstable vector bundles and the equivalence preserves the holomorphic structure.In particular, every semistable vector bundle with vanishing Chern classes has aholomorphic flat structure which is an extension of unitary flat bundles. Finally,

18

let us recall that a vector bundle is semistable with vanishing Chern classes if andonly if it is numerically flat.

We also need to recall a few basic results about etale trivializable bundles.

7.2 Etale trivializable bundles

Let X be a smooth projective variety over an algebraically closedfield k. A rankr locally free sheafE on X is calledetale trivializableif there exists a finite etalecoveringπ : Y → X such thatπ∗E ≃ O r

Y. Over finite fields etale trivializablebundles are characterized as Frobenius periodic bundles:

PROPOSITION7.1. (see [LS])Assume that k= Fp and let F: X →X be the Frobe-nius morphism. A locally free sheaf E isetale trivializable if and only if thereexists an isomorphism(Fn

X)∗E ≃ E for some positive integer n.

It is easy to see that every etale trivializable bundle is numerically flat. Sowe can try to characterize such bundles fork = C in terms of their monodromyrepresentation. If we have a representationρ : π1(X,x)→ GLr(C) whose imageG is a finite group then by Weyl’s trickG is a unitary subgroup of GLr(C). Sinceevery complex representation of a finite group is a direct sumof irreducible repre-sentations, the corresponding Higgs bundle(E,θ) is a direct sum of stable vectorbundles. Passing to the etale covering defined by the quotient π1(X,x)→ G wesee that each direct summand is etale trivializable and theHiggs fieldθ = 0.

On the other hand, if a bundle is etale trivializable then itis etale trivializableby a finite Galois covering and hence the corresponding monodromy representa-tion has finite image.

7.3 Etale trivializability of reductions of numerically flat bu n-dles

We keep the notation from Section 6 but now we restrict to the case whereE is anumerically flat vector bundle.

CONJECTURE7.2. The setΣet of closed points s∈ S such thatEs is etale trivial-izable, is infinite.

The following example shows that this conjecture is interesting even for verysimple semistable vector bundles:

19

Example7.3. Let X be a smooth complex projective variety withh1(X,OX) > 0.Let us consider vector bundleE corresponding to the extension

0→ OX → E → H1(OX)⊗OX → 0

defined by the identity idH1(OX)∈End(H1(OX))=Ext1(H1(OX)⊗OX,OX). This

is clearly a numerically flat vector bundle.For every finite etale morphismπ : Y → X the mapπ∗ : H1(OX)→ H1(OY) is

injective as it can be split by the trace map. LetEY be the extension correspondingto idH1(OY)

∈ End(H1(OY)) = Ext1(H1(OY)⊗OY,OY) and consider the commu-tative diagram

0 // OY // π∗E //

H1(OX)⊗OY //

π∗⊗idOY

0

0 // OY // EY // H1(OY)⊗OY // 0.

If π∗E is trivial then it injects intoEY and henceEY has at leastrkE = h1(OX)+1linearly independent global sections. But by the definitionof EY the connectingmapH0(Y,H1(OY)⊗OY)→H1(Y,OY) is an isomorphism and henceh0(EY) = 1.ThereforeE is not etale trivializable.

Let X → Sbe a model ofX as in the beginning of Section 6.

LEMMA 7.4. There exists a non-empty open subset U⊂ S such that the reductionEs of E for a closed point s∈ U is etale trivializable if and only if the Frobeniusmorphism F= FXs acts on H1(OXs) bijectively.

Proof. If F∗ acts onV = H1(OXs) bijectively then the diagram

0 // OXs// F∗Es //

V ⊗OXs//

F∗⊗idOXs

0

0 // OXs// Es // V ⊗OXs

// 0

shows thatF∗Es ≃ Es and henceEs is etale trivializable by the Lange–Stuhlertheorem (see Proposition 7.1).

Now assume thatEs is etale trivializable. Let us consider the unique decompo-sitionV =Vs⊕Vn such that the Frobenius morphismF∗ acts onVs as an automor-phism and it is nilpotent onVn. Let G be the bundle obtained as the extension of

20

Vn⊗OXs by OXs defined by the canonical inclusion(Vn → V) ∈ Hom(Vn,V) =Ext1(Vn⊗OXs,OXs). Then we have the diagram

0 // OXs// G //

Vn⊗OXs//

_

0

0 // OXs// Es // V ⊗OXs

// 0,

which shows thatG → Es. By the definition ofG there exists somem0 such that(Fm0)∗G is trivial. Let r = dimVn. This shows that for everym≥ m0 we have

h0((Fm)∗Es)≥ h0((Fm)∗G) = r +1.

By the Lange–Stuhler theorem we know that for somem≥m0 we have(Fm)∗Es≃Es and henceh0(Es) ≥ r +1. By the definition ofE we know that the connectingmapδ : H0(H1(OX)⊗OXs)→ H1(OX) is an isomorphism and henceh0(E) = 1.Using semicontinuity of cohomology, we see that there exists an open subsetU ⊂Ssuch thath0(Es) = 1 for everys∈U . This implies that for any closeds∈U wehaver = 0 andV =Vs.

Therefore Conjecture 7.2 for vector bundleE is equivalent to the assertionthat there are infinitely many closed pointss∈ S for which the Frobenius acts onH1(OXs) bijectively. In the curve case this is equivalent to saying that there areinfinitely many places of ordinary reduction. This is known in case of genusg≤ 2but it is still an open problem in general.

Remark7.5. Note that if the reduction ofEs is etale trivializable byπ : Y → Xs

then the degree ofπ is divisible by the characteristicp of the residue fieldk(s).Indeed, if the characteristicp does not divide the degree ofπ then 1

degπ TrXs/Y :π∗OY → OXs splits the injectionOXs → π∗OY. Then the same argument as in thecharacteristic zero case gives a contradiction.

7.4 Analogue of the Grothendieck-Katzp-curvature conjecture

In this subsection we try to relate etale trivializabilityof reductions of a vectorbundle to finiteness of the image of its monodromy representation. Before formu-lating the corresponding conjecture we provide its original motivation: the globalcase of the Grothendieck–Katz conjecture.

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Let X be a smooth variety defined over a field of characteristicp > 0 and let∇ : E → ΩX ⊗E be an integrablek-connection on a locally freeOX-moduleE. Incharacteristicp, the p-th powerDp of a derivationD is again a derivation so wecan consider∇(Dp)−∇(D)p. When this is zero for all local derivationsD thenwe say that∇ has zerop-curvature. IfFg : X → X(1) is the geometric Frobeniusmorphism then(E,∇) is equivalent to giving a locally freeOX(1)-moduleG. ThesheafG can be recovered from(E,∇) as a sheaf of local sectionsv of E for which∇(v) = 0. On the other hand, givingG we can construct a canonical connection onE = F∗

g G by differentiating along the fibers ofFg, i.e., we set∇( f ⊗g) = d f ⊗g.

CONJECTURE7.6. (Grothendieck–Katz, see [Ka])Let (E,∇) be a holomorphicvector bundle with an integrable connection on a complex manifold X. Then(E,∇) has a finite monodromy group if and only if almost all its reductions topositive characteristic have vanishing p-curvature.

Note that ifX projective then(E,∇)with finite monodromy group correspondsvia Simpson’s correspondence described in Subsection 7.1 to an etale trivializablebundle (with zero Higgs field). So we can try to describe representations of thefundamental group with finite image on the Higgs bundle side in the followingway:

CONJECTURE7.7. In the notation of Section 6 assume that E is notetale trivializ-able. Then the setΣnet of closed points s∈ S such thatEs is notetale trivializable,is infinite.

In case of bundles described in Example 7.3, the conjecture can be refor-mulated as saying that for a given smooth complex projectivevariety X withh1(X,OX) > 0, there are infinitely many pointss∈ S for which the nilpotent partof the Frobenius action onH1(OXs) is non-trivial. In particular, ifX is a complexelliptic curve then this is equivalent to saying that there are infinitely many primesfor which the reduction ofX is supersingular. In case of elliptic curves definedoverQ (and also in some other cases) this is a celebrated Elkies’ result [El].

Example7.8. Let A be an abelian variety over a number fieldK and letL be aline bundle on some modelA → S= SpecRof A for a finitely generated subringR⊂ K. Note that by Theorem 7.1 a line bundleL on a smooth projective vari-ety overFp is etale trivializable if and only if there exists somen ∈ N such that(Fn)∗L ≃ L. Therefore Conjecture 7.7 predicts that in the above case iffor almostall closed pointss∈ S there existsns ∈ N such that(Fns)∗Ls ≃ Ls thenLK isetale trivializable onA.

22

In this case a slightly weaker result is known. Namely, assume that there existssomen∈N such that for almost all closed pointss∈ Swe have(Fn)∗Ls≃Ls (sons in the above reformulation is independent ofs). ThenLK is etale trivializableonA. This is just a dual version of [Pi, Theorem 5.3] and it implies that Conjecture7.7 reduces to existence of a uniform bound on allns.

Note that ifLK is etale trivializable then there exists a positive integer msuchthatL m

s ≃OXs for all s from some non-empty open subsetU ⊂S. Since for every(rational) primep not dividingm the numberpm! −1 is divisible bymwe see that(Fm!)∗Ls ≃ Ls for all closed pointss from some smaller non-empty open subsetV ⊂U . This provides us with the converse to Pink’s theorem.

Using the same methods as in proof of [An, Theoreme 7.2.2] and [EL, Theo-rem 5.1] one can show that an analogue of Conjecture 7.7 holdsin case of equalcharacteristic zero:

THEOREM 7.9. Let f : X → S be a smooth projective morphism of varieties de-fined over an algebraically closed field k of characteristic0. Let η be the genericgeometric point of S and letE be a locally free sheaf onX . Let us assume thatthere exists a dense subset U⊂ S(k) such that for every s in U the bundleEs isetale trivializable. Then we have the following:

1) There exists a finite Galoisetale coveringπ : Y → Xη such thatπ∗Eη is adirect sum of line bundles.

2) If U is open in S(k) thenEη is etale trivializable.

Note that, similarly as in other cases, an analogue of this theorem is false forfamilies defined over an algebraic closure of a finite filed:Example7.10. In [EL, Corollary 4.3] the authors used Laszlo’s example [Ls, Sec-tion 3]) to construct a locally free sheafE on X = X ×k S→ S, whereX is asmooth projective curve,S is a smooth curve, both defined overk = F2 and suchthat for every closed points∈ S the bundleEs is etale trivializable butEη is notetale trivializable for the generic geometric pointη of S.

The above example can occur only because the monodromy groups ofEs haveorders divisible by the characteristic ofk(s). For positive results in other cases see[EL, Theorem 5.1].

Acknowledgements.The author would like to thank D. Rossler and H. Esnault for useful conver-

sations related to Section 7. Author’s work was partially supported by PolishNational Science Centre (NCN) contract number 2012/07/B/ST1/03343.

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