Relative fundamental groups and rational points

Relative fundamental groups and rational points

CHRISTOPHERLAZDA(*)

ABSTRACT- In this paper we define a relative rigid fundamental group, which associates to a section of a smooth and proper morphism f :X!S in characteristic p, a Hopf algebra in the ind-category of overconvergent F-isocrystals onS. We prove a base change property, which says that the fibres of this object are the Hopf algebras of the rigid fundamental groups of the fibres off. We explain how to use this theory to define period maps as Kim does for varieties over number fields, and show in certain cases that the targets of these maps can be interpreted as varieties.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 11G35; 14G05, 14F35.

KEYWORDS. Unipotent fundamental group, non-abelian Jacobian.

Contents

Introduction . . . 2

1. Relative de Rham fundamental groups . . . 5

1.1 ± The relative fundamental group and its pro-nilpotent Lie algebra . . . 6

1.2 ± An algebraic proof of Theorem 1.6 . . . 9

2. Path torsors, non-abelian crystals and period maps . . . 16

2.1 ± Torsors in Tannakian categories . . . 16

2.2 ± Path torsors under relative fundamental groups . . . 21

3. Crystalline fundamental groups of smooth families in charp . . . 22

3.1 ± Base change . . . 24

3.2 ± Frobenius structures . . . 33

3.3 ± Cohomology and period maps . . . 34

(*) Indirizzo dell'A.: Department of Mathematics, Room 6.13 Huxley Building, 180 Queen's Gate, Imperial College, London, SW7 2AZUK.

The author was supported by an EPSRC Doctoral Training Grant during the writing of this paper.

E-mail: [email protected]

Introduction

Let K be a number field and letC=K be a smooth, projective curve of genus g>1, with JacobianJ. Then a famous theorem of Faltings states that the setC K… † ofK-rational points onCis finite. The groupJ K… †is finitely generated, and under the assumption that its rank is strictly less than g, Chabauty in [14] was able to prove this theorem using elementary methods as follows. Letvbe a place ofK, of good reduction forC, and denote byC_v;J_vthe base change toK_v. Then Chabauty defines a homomorphism

log :J K… † !_v H⁰J_v;V¹_Jv=Kv …1†

and shows that there exists a non zero linear functional on H⁰(Jv;V¹_Jv=Kv) which vanishes on the image ofJ K… †. He then proves that pulling this back toJ K… †v gives an analytic function onJ K… †, which is not identically zero onv C K… †, and whichv

vanishes onJ K… †. HenceC K… † C K… † \_v J K… †must be finite as it is contained in the zero set of a non-zero analytic function onC K… †._v

In [29], Kim describes what he calls a `non-abelian lift' of this method. Fix a pointp2C K… †. By considering the Tannakian category of integrable connections onCv, one can define a `de Rham fundamental group'U^dRˆp^dR₁ …Cv;p†, which is a pro-unipotent group scheme over K_v, as well as, for any other x2C K… †, path_v torsors P^dR… † ˆx p^dR₁ …C_v;x;p† which are right torsors under U^dR. These group schemes and torsors come with extra structure, namely that of a Hodge filtration and, by comparison with the crystalline fundamental group of the reduction ofCv, a Frobenius action. He then shows that such torsors are classified byU^dR=F⁰, and hence one can define `period maps'

jn:C K… † !v U_n^dR=F⁰ …2†

whereU_n^dRis the nth level nilpotent quotient ofU^dR. Ifnˆ2 then jn is just the composition of the above log map with the inclusionC K… † !_v J K… †. By analysing_v the image of this map, he is able to prove finiteness ofC K… †under certain condi- tions, namely if the dimension ofU_n^dR=F⁰is greater than the dimension of the target of a global period map defined using the category of lisse eÂtale sheaves on C.

Moreover, when nˆ2, this condition on dimensions is essentially Chabauty's condition that rank_ZJ K… †<genus… †C (modulo the Tate-Shafarevich conjecture).

Our interest lies in trying to develop a function field analogue of these ideas.

The analogy between function fields in one variable over finite fields and number fields has been a fruitful one throughout modern number theory, and indeed the analogue of Mordell's conjecture was first proven for function fields by Grauert. In this paper we discuss the problem of defining a good analogue of the global period map. This is defined in [29] using the Tannakian category of lisseQ_psheaves onX, and this approach will not work in the function field setting. Neitherp-adic nor`- adic eÂtale cohomology will give satisfactory answers, the first because, for example,

the resulting fundamental group will be moduli dependent, i.e. will not be locally constant in families (see for example [34]), and the second because the`-adic to- pology on the resulting target spaces for period maps will not be compatible with thep-adic topology on the source varieties. Instead we will work with the category of overconvergentF-isocrystals.

LetKbe a finite extension ofFp… †, and lett kbe the field of constants ofK, i.e.

the algebraic closure ofFp insideK. LetSbe the unique smooth projective, geo- metrically irreducible curve overkwhose function field isK. IfC=K is a smooth, projective, geometrically integral curve then one can choose a regular model forC.

This is a regular, proper surface X=k, equipped with a flat, proper morphism f :X!Swhose generic fibre is C=K. Let SS be the smooth locus of f, and denote byf also the pullbackf :X!S. The idea is to construct, for any sectionp of f, a `non-abelian isocrystal' onSwhose fibre at any closed points`is' the rigid fundamental groupp^rig₁ …X_s;p_s†. The idea behind how to construct such an object is very simple.

Suppose thatf :X!Sis a Serre fibration of topological spaces, with connected base and fibres. If p is a section, then for any s2S the homomorphism p₁…X;p s… †† !p₁…S;s†is surjective, andp₁…S;s†acts on the kernel via conjugation.

This corresponds to a locally constant sheaf of groups onS, and the fibre over any point s2S is just the fundamental group of the fibreX_s. This approach makes sense for any fundamental group defined algebraically as the Tannaka dual of a category of `locally constant' coefficients. So iff :X!Sis a morphism of smooth varieties with section p, then f:p<sup>C</sup><sub>1</sub><sup>X</sup>…X;x† !p<sup>C</sup><sub>1</sub><sup>S</sup>…S;s†is surjective, andp<sup>C</sup><sub>1</sub><sup>S</sup>…S;s† acts on the kernel. Here C<sub>… †</sub> is any appropriate category of coefficients, for ex- ample vector bundles with integrable connection, unipotent isocrystals etc., and e:g:p<sup>C</sup><sub>1</sub><sup>X</sup>…X;x†is the Tannaka dual of this category with respect to the fibre functor x. This gives the kernel offthe structure of an `affine group scheme overCS', and it makes sense to ask what the fibre is over any closed point s2S. The main theorem of the first section is the following.

THEOREM. Suppose that f :X!S is a smooth morphism of smooth varieties over an algebraicallyclosed field k of characteristic zero. Assume that both S and the fibres of f are connected, and that X is the complement of a relative normal crossings divisor in a smooth and proper S-scheme X. Let C_S be the categoryof vector bundles with a regular integrable connection on S, and letC_Xbe the category of vector bundles with a regular integrable connection on X which are iterated extensions of those of the form fE, with E 2 CS. Then the fibre of the corre- sponding affine group scheme overC_Sat s2S is the de Rham fundamental group p^dR₁ …X_s;p_s†of the fibre.

In Section 2 we discuss path torsors in the relative setting. We show in parti- cular that for any other sectionqoffone can define an affine schemep^dR₁ …X=S;q;p† over C_S which is a right torsor under the relative de Rham fundamental group

p^dR₁ …X=S;p†. The upshot of this is that we obtain

jn:X S… † !H¹ S;p^dR₁ …X=S;p†_n …3†

which are a coarse characteristic zero function field analogue of Kim's global period maps. Of course, if we were really interested in the characteristic zero picture, we would want to define Hodge structures on these objects, and thus obtain finer period maps. However, our main interest lies in the positive characteristic case, and so we don't pursue these questions.

In Section 3 we define the relative rigid fundamental group in positive char- acteristic, mimicking the definition in characteristic zero. Instead of the category of vector bundles with regular integrable connections, we consider the category of overconvergent F-isocrystals (throughout Section 3 we will be over a finite field, and Frobenius will always mean the linearFrobenius). We then proceed to use Caro's theory of cohomological operations for arithmetic D-modules in order to prove the analogue of the above theorem in positive characteristic.

The upshot of this is that for a smooth and proper map f :X!S with geo- metrically connected fibres and smooth, geometrically connected base over a finite fieldk, and a sectionpoff, we can define an affine group schemep^rig₁ (X=S;p) over the category of overconvergent F-isocrystals on S, which we call the relative fundamental group atp. The fibre of this over any points2Sis just the unipotent rigid fundamental group of the fibreXs off over s. As in the zero characteristic case, the general Tannakian formalism gives us path torsorsp^rig₁ (X=S;p;q) for any otherq2X(S), and hence we can define a period map

X(S)!H¹_F;rig(S;p^rig₁ (X=S;p)) …4†

where the RHS is a classifying set ofF-torsors underp^rig₁ (X=S;p), as well as finite level versions given by pushing out along the quotient map p^rig₁ (X=S;p)! p^rig₁ (X=S;p)_n.

Finally, we study the targets of these period maps, and show that after replacing H¹_F;rig(S;p^rig₁ (X=S;p)), the set classifyingF-torsors, by the Frobenius invariant part of the set classifying torsors without F-structure, H¹_rig(S;p^rig₁ (X=S;p))^fˆid, then under very restrictive hypotheses on the morphism f :X!S, we obtain the structure of an algebraic variety. The argument here is just a translation of the original argument of Kim into our context, and what for us are restrictive hy- potheses are automatically satisfied in his case.

We are still a long way away from getting a version of Kim's methods to work for function fields. There is still the question of how to define the analogue of the local period maps, and also to show that the domains of the period maps have the structure of varieties. Even then, it is very unclear what the correct analogue of the local integration theory will be in positive characteristic. There is still a very large amount of work to be done if such a project is to be com- pleted.

1. Relative de Rham fundamental groups

Letf :X!Sbe a smooth morphism of smooth complex varieties, and suppose that f admits a good compactification, that is, there existsX smooth and proper overS, an open immersionX,!XoverS, such thatDˆXnXis a relative normal crossings divisor inX. Letp2X S… †be a section. For every closed points2Swith fibreXs, one can consider the topological fundamental groupGs:ˆp1 X_s^an;p s… †

, and assvaries, these fit together to give a locally constant sheafp₁…X=S;p†onS^an. Let

U^…LieG_s†:ˆlimC[G_s]=aⁿ …5†

denote the completed enveloping algebra of the Malcev Lie algebra ofGs, where aC[G_s] is the augmentation ideal. According to Proposition 4.2 of [25], assvaries, these fit together to give a pro-local system on S^an, i.e. a pro-object U^^top_p in the category of locally constant sheaves of finite dimensionalC-vector spaces onS^an. (Their theorem is a lot stronger than this, but this is all we need for now). According to TheÂoreÁme 5.9 in Chapter II of [18], the pro-vector bundle with integrable con- nection U^^top_{p C}O_S^an has a canonical algebraic structure. Thus given a smooth morphismf :X!Sas above, with sectionp, one can construct a pro-vector bundle with connection U^_p onS, whose fibre at any closed point s2S is the completed enveloping algebra of the Malcev Lie algebra ofp1 X_s^an;p s… †

. Denoting byg_s the Malcev Lie algebra of p1 X_s^an;p s… †

,U^… † ˆg_s (U^p)_s can be constructed algebraically, asg_s is equal to Liep^dR₁ …Xs;ps†, the Lie algebra of the Tannaka dual of the category of unipotent vector bundles with integrable con- nection on X_s. This suggests the question of whether or not there is an algebraic construction ofU^p?

We will not directly answer this question - instead we will construct the Lie algebra associated toU^p- this is a pro-systemL^pof Lie algebras with connection onS. The way we will do so is very simple, and is closely related to ideas used in [35]

to study relatively unipotent mixed motivic sheaves.

DEFINITION1.1. To save ourselves saying the same thing over and over again, we make the following definition. A `good' morphism is a smooth morphism f :X!Sof smooth varieties over a fieldk, with geometrically connected fibres and base, such that X is the complement of a relative normal crossings divisor in a smooth, proper S-scheme X. Throughout this section we will assume that the ground fieldkis algebraically closed of characteristic 0.

We will assume that the reader is familiar with Tannakian categories, a good introductory reference is [31]. IfT is a Tannakian category over a fieldk, andvis a fibre functor on T, in the sense of §1.9 of [20], we will denote the group scheme representing tensor automorphisms of v by G(T;v). We will also use the rudi- ments of algebraic geometry in Tannakian categories, as explained in §5 of [19] - in particular we will talk about affine (group) schemes over Tannakian categories. We

will denote the fundamental group of a Tannakian category by p(T), this is an affine group scheme over T which satisfies v(p(T))ˆG(T;v) for every fibre functorv(see for example 6.1 of [19]). IfT is a Tannakian category overk, andk⁰=k is a finite extension, then we will denote the category ofk⁰-modules inT by either T kk⁰, orTk⁰.

We will also assume familiarity with the theory of integrable connections and regular holonomic D-modules onk-varieties, and will generally refer to [18] and [23] for details. We say that a regular integrable connection onXis unipotent if it is a successive extension of the trivial connection, and these form a Tannakian sub- category NIC(X)IC(X) of the Tannakian category of regular integrable con- nections.

DEFINITION1.2. For X=ksmooth and connected, the algebraic and de Rham fundamental groups ofXat a closed pointx2Xare defined by

p^alg₁ …X;x†:ˆx…p…IC… †X †† ˆG…IC… †;X x† …6†

p^dR₁ …X;x†:ˆx…p…NIC… †X †† ˆG…NIC… †;X x†:

…7†

REMARK 1.3. It follows from the Riemann-Hilbert correspondence that if kˆC, then these affine group schemes are the pro-algebraic and pro-unipotent completions ofp1…X^an;x†respectively.

Iff :X!Yis a morphism of smoothk-varieties, then we can form the pullback of vector bundles with integrable connection onY, which preserves regularity and is the usual pull-back on the underlyingO_Y-module. This induces a homomorphism f:p^#₁(X;x)!p^#₁(Y;f(x)) for#ˆdR;alg.

1.1 ±The relative fundamental group and its pro-nilpotent Lie algebra

Letf :X!Sbe a `good' morphism. A regular integrable connectionEonXis said to be relatively unipotent if there exists a filtration by horizontal sub-bundles, whose graded objects are all in the essential image off:IC… † !S IC… †. We willX denote the full subcategory of relatively unipotent objects in IC… †X byN_fIC… †,X which is a Tannakian subcategory. Suppose thatp2X S… †is a section off. We have functors of Tannakian categories

N_fIC… †X ^p!

f IC… †S …8†

and hence, after choosing a points2S(k), homomorphisms G N_fIC… †;X p s… † !^f

p G…IC… †;S s† …9†

between their Tannaka duals. LetKsdenote the kernel off. Then the splittingp

induces an action of p^alg₁ …S;s† ˆG…IC… †;S s† on K_s via conjugation. This corre- sponds to an affine group scheme over IC… †.S

LEMMA1.4. This affine group scheme overIC(S)is independent of s.

PROOF. Thanks to [19], §6.4,f;pabove come from homomorphisms p p N_fIC… †X !^f

p p…IC… †S† …10†

of affine group schemes over IC… †. If we letS K denote the kernel of f, then Ksˆs… †, and this induces the action ofK p^alg₁ (S;s) onKs. p DEFINITION1.5. The relative de Rham fundamental groupp^dR₁ …X=S;p†ofX=S atpis defined to be the affine group schemeKover IC(S).

Letis:Xs!Xdenote the inclusion of the fibre overs. Then there is a canonical functor i_s :N_fIC… † ! NX IC… †. This induces a homomorphismXs p^dR₁ …Xs;ps† ! G N_fIC… †;X p_s

which is easily seen to factor through the fibre p^dR₁ …X=S;p†_s:ˆ

s(K)ˆK_s ofp^dR₁ …X=S;p†overs.

THEOREM1.6. Suppose that kˆC. Thenf:p^dR₁ …Xs;ps† !p^dR₁ …X=S;p†_s is an isomorphism.

PROOF. The pointsgives us fibre functorsp_s onNIC(X_s),p s… † onN_fIC… †X andson IC… †. WriteS

K ˆG NIC(Xs);p_s

; G ˆG N_fIC… †;X p s… †

; H ˆG…IC… †;S s† …11†

and also let

Kˆp₁ X_s^an;p s… †

; Gˆp₁…X^an;p s… ††; Hˆp₁…S^an;s† …12†

be the topological fundamental groups ofX_s;X;Srespectively. ThenK ˆK^un, the pro-unipotent completion ofK, and H ˆH^alg, the pro-algebraic completion ofH.

We need to show that the sequence of affine group schemes 1! K ! G ! H !1

…13†

is exact, and we will use the equivalences of categories

IC… †!X Rep_C…p₁…X^an;p s… †††; IC… †!S Rep_C…p₁…S^an;s†† …14†

IC… †!X_s Rep_C p₁ X_s^an;p s… † : …15†

By Proposition 1.3 in Chapter I of [35], ker…G ! H†is pro-unipotent. Hence ac- cording to Proposition 1.4 ofloc. cit., in order to show thatfis an isomorphism, we must show the following.

IfE2 N_fIC… †X is such thati_s… †E is trivial, thenEf… †F for someF in IC… †.S

Let E2 N_fIC(X), and let F₀i_s(E) denote the largest trivial sub-object.

Then there existsE0Esuch thatF0ˆi_s(E0).

There is a pro-action ofGonU^…LieK†such that the corresponding action of LieGextends the left multiplication by LieK.

The first is straightforward. Sincef is topologically a fibration with sectionp, we have a split exact sequence

1!K!G!H!1 …16†

and a representationVofGsuch thatKacts trivially. We must show thatVis the pullback of anH-representation - this is obvious! The second is no harder, we must show that ifVis aG-representation, thenV^Kis a sub-G-module ofV. But sinceK is normal in G, this is clear. For the third, note that U^…LieK† ˆU^…LieK† ˆ limC[K]=aⁿ, whereais the augmentation ideal ofC[K]. LetHact onC[K]=aⁿby conjugation and K by left multiplication. I claim that C[K]=aⁿ is finite dimen- sional, and unipotent as a K-representation.

Indeed, there are extensions ofK-representations

0!aⁿ=a^n‡1!C[K]=a^n‡1!C[K]=aⁿ!0 …17†

and hence, since the action ofKonaⁿ=a^n‡1is trivial, it follows by induction that each C[K]=aⁿis unipotent. There are also surjections

a=a²_n

!!aⁿ=a^n‡1 …18†

for eachn, and hence by induction, to show finite dimensionality it suffices to show thata=a²is finite dimensional. Buta=a²K^ab_ZCis finite dimensional, asKis finitely generated.

Now, sinceC[K]=aⁿis unipotent as aK-representation, it is relatively unipotent as aGˆK^jH-representation, henceC[K]=aⁿ is naturally an object in Rep_C… †.G Thus there is a pro-action of G on U^…LieK†, and the action extends left multi-

plication by LieKas required. p

REMARK 1.7. The co-ordinate algebra of p^dR₁ (X=S;p) is an ind-object in the category of regular integrable connections onS. Hence we may viewp^dR₁ (X=S;p) as an affine group scheme over S in the usual sense, together with a regular in- tegrable connection on the associatedO_S-Hopf algebra.

If g:T!S is any morphism of smooth varieties over k, then there is a homomorphism of fundamental groups

p^dR₁ …XT=T;pT† !p^dR₁ …X=S;p† ST:ˆg(p^dR₁ …X=S;p†) …19†

which corresponds to a horizontal morphism O_p^dR

1 …X=S;p†_OSO_T! O_p^dR

1 …XT=T;pT†: …20†

PROPOSITION1.8. If kˆCthen this is an isomorphism.

PROOF. We know by the previous theorem that this induces an isomorphism on fibres over any pointt2T(C). Hence by rigidity, it is an isomorphism. p WriteGˆp^dR₁ …X=S;p†and letGndenote the quotient ofGby thenth term in its lower central series. LetAn denote the Hopf algebra ofGn, andInAn the aug- mentation ideal. L_n:ˆ Hom_OS I_n=I²_n;O_S

is the Lie algebra of G_n. This is a co- herent, nilpotent Lie algebra with connection, i.e the bracket [;]:L_nL_n!L_nis horizontal. There are natural morphisms Ln‡1!Ln, which form a pro-system of nilpotent Lie algebras with connectionL^p, whose universal enveloping algebra is the objectU^pconsidered in the introduction to this section.

1.2 ±An algebraic proof of Theorem1.6

Although we have a candidate for the relative fundamental group of a `good' morphismf :X!Sat a sectionp, so far we have only proved it is a good candidate when the ground field is the complex numbers. One might hope to be able to reduce to the casekˆCvia base change and finiteness arguments, but this approach will not work in a straightforward manner. Also, such an argument will not easily adapt to the case of positive characteristic, as in general one will not be able to lift a smooth proper family, even locally on the base. In this section, we will give a more algebraic proof, that will more easily adapt to positive characteristic. Recall that we have an affine group schemep^dR₁ …X=S;p†over IC(S), and a comparison morphism

f:p^dR₁ …X_s;p_s† !p^dR₁ …X=S;p†_s …21†

for any points2S. We will prove that this map is an isomorphism.

It follows from Proposition 1.4 in Chapter I of [35] and Appendix A of [22] that we need to prove the following:

(Injectivity) Every E2 NIC… †X_s is a quotient of i_s… †F for some F2 NfIC… †.X

(Surjectivity I) Suppose thatE2 N_fIC… †X is such thati_s… †E is trivial. Then there existsF2IC(S) such thatEf… †.F

(Surjectivity II) Let E2 NfIC(X), and let F0i_s(E) denote the largest trivial sub-object. Then there existsE0Esuch thatF0ˆi_s(E0).

To do so, we will need to use the language of algebraicD-modules. We define the functor

f^dR:NfIC(X)!IC(S)

byf^dR(E)ˆ H ^d(f‡E) wheref‡ is the usual push-forward for regular holonomic complexes of D-modules, d is the relative dimension of f :X!S, and we are considering a regular integrable connection onXas aD_X-module in the usual way.

LEMMA1.9. The functor f^dR lands in the categoryof regular integrable con- nections, and is a right adjoint to f.

PROOF. The content of the first claim is in the coherence of direct images in de Rham cohomology, using the comparison result 1.4 of [21], and the fact that a regular holonomicDX-module is a vector bundle iff it is coherent as anOX-module.

To see this coherence, we first use adjointness off‡andf^‡, together with the facts thatf^‡O_S ˆ O_X[ d] andf‡O_X is concentrated in degrees d, to get ca- nonical adjunction morphism f^dR(O_X)! O_S of regular holonomicD_X-modules. I claim that this is an isomorphism. Indeed, by 1.4 of [21] we have f^dR(O_X) R⁰f(V_X=S), and since f admits a good compactification f :X!S the latter is isomorphic toR⁰f(V_X=S(logD)) whereDˆXnX. Sincefis proper with connected fibres, we know thatf(O_X) O_S, and sinceR⁰f(V_X=S(logD)) is a sub-O_S-module off(O_X), the claim follows.

Hence in particular f^dR(O_X) is coherent, and via the projection formula, so is f^dR(fF) for anyF2IC(S). Hence using exact sequences in cohomology and in- duction on unipotence degree,f^dREis coherent wheneverEis relatively unipotent.

To prove to the second claim, we just use thatf^‡is adjoint tof‡,f^‡ˆf[ d]

on the subcategory of regular integrable connections, and f_‡Eis concentrated in degrees dwheneverEis a regular integrable connection. p REMARK1.10. Although the proposition is stated in [21] forkˆC, the same proof works for any algebraically closed field of characteristic zero.

Thus we get a canonical morphism e_E:ff^dRE!Ewhich is the counit of the adjunction betweenf andf^dR.

EXAMPLE1.11. Suppose thatSˆSpec… †. Thenk

f^dREˆH⁰_dR…X;E† ˆHom_{NIC… †X} …O_X;E† …22†

and the adjunction becomes the identification

Hom_{NIC… †X}…VkOX;E† ˆHomVeck V;Hom_{NIC… †X} …OX;E† : …23†

Sincef^dRtakes objects inN_fIC… †X to objects in IC… †, it commutes with baseS change and there is an isomorphism of functors

H⁰_dR…Xs; † i_s sf^dR:NfIC… † !X Veck: …24†

To see this, first note that by using the five lemma, the projection formula and

induction on the relative unipotence degree, it suffices to prove thatf^dR(OX) and R¹_dRf(O_X) commute with base change, where we have written R¹_dRf(O_X)ˆ H ^d‡1(f‡O_X). Again, using 1.4 of [21] this boils down to proving thatRⁱf(V_X=S) commute with base change foriˆ0;1. Since we know that they are coherent (and hence locally free), and thatf :X!Sadmits a good compactificationf :X!S, this follows from Corollary 8.6 of [27] together with the standard isomorphism Rⁱf(V_X=S)Rⁱf(V_X=S(logD)) whereDˆXnX.

PROPOSITION1.12. Suppose that i_sE is trivial. Then the couniteE:ff^dRE!E is an isomorphism.

PROOF. Pulling backe_Ebyi_s, and using base change, we get a morphism O_XskH⁰_dR Xs;i_sE

!i_sE …25†

which by the explicit description of 1.11 is seen to be an isomorphism (as i_sEis trivial). Hence by rigidity,eEmust be an isomorphism. p PROPOSITION 1.13. Let E2 N_fIC(X), and let F0i_s(E) denote the largest trivial sub-object. Then there exists E₀E such that F₀ˆi_s(E₀).

PROOF. Let Fˆi_s(E), then we have H⁰_dR(Xs;F)ˆHomIC(Xs)(OXs;F). I first claim that the natural map OXs_kH⁰_dR…Xs;F† !F is injective and identifies F0

with O_XskH⁰_dR…X_s;F†. But since the category NIC(X_s) is Tannakian, this is equivalent to claiming that for an affine group scheme G over k, and V a re- presentation ofG, the natural map ofGrepresentationsH⁰(G;V)!Vis injective, and identifiesH⁰(G;V) with the largest trivial sub-G-module ofV. This is obvious.

To complete the proof, setE0ˆff^dR(E), then by the base change results we know thati_s(E₀)F₀, and that the natural mapE₀!Erestricts to the inclusion F₀!Fon the fibreX_s.

COROLLARY1.14. The mapp^dR₁ …Xs;ps† !p^dR₁ …X=S;p†_sis a surjection.

We now turn to the proof of injectivity of the comparison map, borrowing heavily from ideas used in Section 2.1 of [24]. We define objectsU_nofNIC(X_s), the category of unipotent integrable connections onX_s inductively as follows.U₁will just beOXs, andUn‡1 will be the extension ofUnbyOXskH¹_dR Xs;U_n^__{_}

corre- sponding to the identity under the isomorphisms

Ext_{IC… †Xs} Un;OXs_kH¹_dR Xs;U_n^__{_}

H¹_dRXs;U^__nkH¹_dR Xs;U^__{n_} …26†

H¹_dR Xs;U^__n

kH¹_dR Xs;U_n^__{_}

End_k H¹_dR X_s;U^__n :

If we look at the long exact sequence in de Rham cohomology associated to the short exact sequence 0!U_n^_!U^__n‡1!H¹_dR X_s;U_n^_

_kO_Xs !0 we get 0!H⁰_dR X_s;U^__n

!H⁰_dR X_s;U_n‡1^_

!H¹_dR X_s;U_n^_ …27†

!^d H¹_dR X_s;U_n^_

!H¹_dR X_s;U_n‡1^_ : LEMMA1.15. The connecting homomorphismdis the identity.

PROOF. By dualising, the extension

0!U_n^_!U_n‡1^_ ! O_XskH¹_dR X_s;U^__n

!0 …28†

corresponds to the identity under the isomorphism Ext_{IC… †Xs} OXskH¹_dR Xs;U_n^_

;U_n^_

Endk H¹_dR Xs;U_n^_ : …29†

Now the lemma follows from the fact that for an extension 0!E! F! OXskV!0 of a trivial bundle byE, the class of the extension under the isomorphism

Ext_{IC… †Xs} …O_XskV;E† V^_H¹_dR…X_s;E† Hom_k V;H¹_dR…X_s;E† …30†

is just the connecting homomorphism for the long exact sequence 0!H⁰_dR…X_s;E† !H⁰_dR…X_s;F† !V !H¹_dR…X_s;E†:

…31† p

In particular, any extension of U_n by a trivial bundle V_kO_Xs is split after pulling back to Un‡1, and H⁰_dR Xs;U^__n‡1

H⁰_dR Xs;U^__n

. It then follows by in- duction thatH⁰_dR Xs;U_n^_

H⁰_dR…Xs;OXs† kfor alln.

DEFINITION1.16. We define the unipotent class of an objectE2 NIC…X_s†in- ductively as follows. IfEis trivial, then we sayEhas unipotent class 1. If there exists an extension

0!V_kO_Xs !E!E⁰!0 …32†

withE⁰of unipotent classm 1, then we say thatEhas unipotent classm.

Now let xˆp(s),u₁ˆ12… †U_{1 x}k, and choose a compatible system of ele- mentsun2… †Un _xmapping tou1.

PROPOSITION1.17. Let F2 NIC(X_s)be an object of unipotent classm. Then for all nm and anyf 2F_x there exists a morphism a:U_n !F such that a_x… † ˆu_n f .

PROOF. We copy the proof of Proposition 2.1.6 of [24] and use induction onm.

The case mˆ1 is straightforward. For the inductive step, letF be of unipotent

classm, and choose an exact sequence

0!E!^c E!^f G!0 …33†

withEtrivial andGof unipotent class<m. By induction there exists a morphism b:Un 1!Gsuch thatf_x… † ˆf b_x…un 1†. Pulling back the extension (33) first by the morphismband then by the natural surjectionUn!Un 1gives an extension of Un byE, which must split, as observed above.

…34†

Letg:Un!Fdenote the induced morphism, thenf_x…g_x… †un f† ˆ0. Hence there exists somee2Exsuch thatc_x… † ˆe g_x… †un f. Again by induction we can choose g⁰:Un!E with g⁰_x… † ˆun e. Finally let aˆg cg⁰, it is easily seen that

a_x(u_n)ˆf. p

COROLLARY 1.18. EveryE in NIC(Xs) is a quotient of U^N_m for some m;N2N.

PROOF. Suppose thatEis of unipotent classm. Lete1;. . .;eNbe a basis for Ex. Then there is a morphisma:U_m^N!Ewith everyeiin the image of the induced map on fibres. Thusaxis surjective, and hence so isa. p We now try to inductively define relatively nilpotent integrable connectionsW_n. onXwhich restrict to theUn on fibres. Define higher direct images in de Rham cohomology byRⁱ_dRf(E)ˆ H^{i d}(f‡E), and begin the induction withW1 ˆ OX. As part of the induction we will assume that R⁰_dRf W_n^_

R⁰_dRf…O_X† ˆ O_S, that R¹_dRf(W^_) andR¹_dRf(W) are both coherent, i.e. regular integrable connections, and that there exists a morphism pW_n^_! O_S such that the composite OSfW_n^_pffW_n^_!pW_n^_! OSis an isomorphism. We will defineWn‡1to be an extension ofWnby the sheaffR¹_dRf W_n^__{_}

, and thus consider the extension group

Ext_{IC… †X} Wn;fR¹_dRf W_n^__{_}

H¹_dRX;W_n^_OX fR¹_dRf W_n^__{_} : …35†

The Leray spectral sequence, together with the induction hypothesis and the projection formula, gives us the 5-term exact sequence

0!H¹_dRS;R¹_dRf W_n^__{_}

!Ext_{IC… †X} W_n;fR¹_dRf W_n^__{_} …36†

!End_{IC… †S}R¹_dRf W_n^_

!H²_dRS;R¹_dRf W_n^__{_}

!H²_dR(X;W_n^__OX (R¹_dRfW_n^_)^_):

Now, the projectionpW_n^_! OSinduces a map Hⁱ_dRS;pW_n^_OX (R¹_dRfW_n^_)^_

!Hⁱ_dRS;R¹_dRf W_n^__{_} …37†

such that the composite (dotted) arrow

…38†

is as isomorphism, since it can be identified with the map induced by the composite arrowO_SfW_n^_pffW_n^_!pW_n^_! O_S. Hence both the maps

H¹_dRS;R¹_dRf W_n^__{_}

!Ext_{IC… †X} W_n;fR¹_dRf W_n^__{_} …39†

H²_dRS;R¹_dRf W_n^__{_}

!H²_dR(X;W_n^__OX(R¹_dRfW_n^_)^_)

appearing in the 5-term exact sequence split. So there is a commutative diagram

…40†

where the horizontal arrows are just restrictions to fibres, and the left hand vertical arrow is surjective. The identity morphism in End_k H¹_dR X_s;U_n^_

clearly lifts to End_{IC… †S} R¹_dRf(W_n^_)

, and there exists a unique element of the extension group Ext_{IC… †X} Wn;fR¹_dRf(W_n^_)^_

lifting the identity in End_{IC… †S} R¹_dRf(W_n^_) , and which maps to zero under the above splitting

Ext_{IC… †X} W_n;fR¹_dRf W_n^__{_}

!H¹_dRS;R¹_dRf W_n^__{_} : …41†

LetWn‡1be the corresponding extension.

PROPOSITION 1.19. Everyobject of NIC(X_s) is a quotient of i_sE for some E2 N_fIC… †.X

PROOF. To finish the induction step, we must show that R⁰_dRf W_n‡1^_

R⁰_dRf W_n^_

; …42†

thatR¹_dRf(W_n‡1^_ ) andR¹_dRf(W_n‡1) are coherent, and that there exists a morphism pW_n‡1^_ ! O_Sas in the induction hypothesis. For the first claim, if we look at the long exact sequence of relative de Rham cohomology

0!R⁰_dRf W_n^_

!R⁰_dRf W_n‡1^_

!. . . …43†

we simply note that the given map restricts to an isomorphism on the fibre overs, and is hence an isomorphism. For the second, we simply use the long exact sequence in cohomology and the inductive hypothesis forR¹_dRf…W_n†andR¹_dRf W_n^_

. For the third, note that it follows from the construction ofW_n‡1that the exact sequence

0!pW_n^_!pW_n‡1^_ !(R¹_dRfW_n^_)^_!0 …44†

splits when we push out via the map pW_n^_! O_S. This splitting induces a map pW_n‡1^_ ! O_Ssuch that the diagram

…45†

commutes. Now the fact that the diagram

…46†

commutes implies that the composite along the top row is an isomorphism, finishing

the proof. p

COROLLARY 1.20. Let f :X!S be a `good' morphism over an algebraically closed field of characteristic zero, and p a section of f . Then the natural `base change' mapp^dR₁ (Xs;ps)!p^dR₁ (X=S;p)_s is an isomorphism.

REMARK1.21. It is possible to define a relative fundamental group whenkis not necessarily algebraically closed (but still of characteristic 0) using identical methods. One can then show that the corresponding `base change' question can be deduced from what we have proved in the algebraically closed case. Since this argument is rather fiddly, and not necessary in the context of this paper, we have omitted it.

2. Path torsors, non-abelian crystals and period maps

IfT is a Tannakian category over an arbitrary fieldk, andv_iare fibre functors onT, iˆ1;2, with values in the category of quasi-coherent sheaves on some k- schemeS, then the functor of isomorphismsv1!v2is representable by an affine S-scheme, which is a (G(T;v1);G(T;v2))-bitorsor. This allows us to define path torsors under the algebraic and de Rham fundamental groups. In this section, we show how to do this in the relative case.

2.1 ±Torsors in Tannakian categories

Let C be a Tannakian category over a field k. A Tannakian C-category is a Tannakian category Dtogether with an exact,k-linear tensor functort:C ! D.

We say it is neutral overCif there exists an exact, faithfulk-linear tensor functor v:D ! Csuch thatvtid. Such functors will be called fibre functors. If such a functor v is fixed, we say D is neutralised. Thanks to §6.4 of [19], we have a homomorphism

t :p… † !D t…p… †C † …47†

of affine group schemes overD. Hence applyingvgives us a homomorphism v… †t :v p… … †D† !p… †C

…48†

of affine group schemes overC. We defineG…D;v†:ˆkerv… †.t

For an affine group scheme G over C, let O_G be its Hopf algebra, a re- presentation of Gis then defined to be anO_G-comodule. That is an objectV 2 C together with a mapd:V ! O_GV satisfying the usual axioms.

DEFINITION2.1. A torsor underGis a non-empty affine scheme Sp(O_P) overC, together with a O_G-comodule structure on OP, such that the induced map O_P O_P! O_P O_Gis an isomorphism.

EXAMPLE 2.2. Suppose that C ˆRep_k… †, for some affine group schemeH H overk. Then an affine group schemeGoverC`is' just an affine group schemeG<sub>0</sub> over ktogether with an action of H. A representation of G `is' then just anH- equivariant representation ofG0, or in other words, a representation of the semi- direct productG₀^jH.

Representations have another interpretation. Suppose that V is anO_G-como- dule, and letRbe aC-algebra. A pointg2G R… †is then a morphismO_G!RofC- algebras, and hence for any suchgwe get a morphism

V !VR …49†

which extends linearly to a morphism

VR!VR:

…50†

This is an isomorphism, with inverse given by the map induced byg ¹. Hence we get anR-linear action ofG R… †onVR, for allC-algebrasR. The same proof as in the absolute case (Proposition 2.2 of [31]) shows that a representation ofG(defined in terms of comodules) is equivalent to anR-linear action ofG R… †onVR, for allR.

For G an affine group scheme overC, let Rep_C… †G denote its category of re- presentations, which is a Tannakian category overk. There are canonical functors

C !_v^t Rep_C… †G …51†

given by `trivial representation' and `forget the representation'. This makes Rep_C… †G neutral over C. There is a natural homomorphism G!v(p(Rep_C(G))) which comes from the fact that by definition,Gacts onv(V) for allV2Rep_C(G).

Since this action is trivial on everything of the formt(W),W2 C, again by definition, this homomorphism factors to give a homomorphism

G!G(Rep_C(G);v):

…52†

Conversely, ifDis neutral overC, with fibre functorv, then the action ofv(p(D)) on v(V), for allV 2 D, induces an action ofG(D;v) onv(V), and hence a functor

D !Rep_C(G(D;v)):

…53†

PROPOSITION2.3. In the above situation, the homomorphism G!G(Rep_C(G);v)

…54†

is an isomorphism, and the functor

D !Rep_C(G(D;v)) …55†

is an equivalence of categories.

PROOF. Suppose first that C Rep_k(H) is neutral. In the first case, we can identifyGwith an affine group schemeG₀overktogether with an action ofH, and the category Rep_C(G) with the category of representations of the semi-direct pro- duct G₀^jH. The functor v:Rep_C(G)! C can be identified with the forgetful functor fromG0^jH-representations toH-representations, and the morphism

v(p(Rep_C(G)))!p(C) …56†

with the natural map

G0^jH!H …57†

of affine group schemes withH-action. Thus the kernel of this map is identified with

G0together with its givenH-action. In other words, G!G(Rep_C(G);v) …58†

is an isomorphism.

In the second case, D is also neutral, and corresponds to representations of some affine group scheme G. The functors t;v give a surjection G!Hand a splitting H!G which induces an action of H on G₀:ˆker(G!H) such that GG₀^jH. Then G(D;v) is identified with G₀ together with its H-action, and D !Rep_C(G(D;v)) with the natural functor fromGˆG0^jH-representations to H-equivariantG0-representations. It is thus an equivalence.

IfCis not neutral, then we can choose a fibre functor with values in somek- schemeS, apply TheÂoreÁme 1.12 of [20] and replace the affine group schemeHby a certain groupoid acting on aS(for more details see §3.3). The argument is then

formally identical. p

REMARK2.4. Our definition of the fundamental groupp^dR₁ …X=S;p†is then just G N_fIC… †;X p

, as an affine group scheme over IC… †.S

In order to define torsors of isomorphisms in the relative setting, we must first recall Deligne's construction in the absolute case, which uses the notion of a coend.

So suppose that we have categoriesXandS, and a functorF:X X^op! S. The coend ofFis the universal pair… †z;s wheresis an object ofSandz:F!sis a bi- natural transformation. Here s is the constant functor ats2Ob… †, and by bi-S natural we mean that it is natural in both variables. If such an object exists, we will denote it by

Z^X

F x;… x†:

…59†

If S is cocomplete then the coend always exists and is given concretely by the formula (see Chapter IX, Section 6 of [30])

Z^X

F x;… x† ˆcolim a

f:x!y2Mor… †X

F x;… y† !! a

x2Ob… †X

F x;… x† 0

@

1 A: …60†

Suppose thatCis a Tannakian category, and letv1;v2:C !QCoh… †S be two fibre functors onC. In [20], Deligne defines

L_S…v₁;v₂† ˆ Z^C

v₁… † V v₂… †V ^_ …61†

to be the coend of the bifunctor

v₁v^_₂ :C C^op!QCoh… †;S …62†

and in §6 of loc. cit., uses the tensor structure of Cto define a multiplication on L_S…v₁;v₂†which makes it into a quasi-coherentO_S-algebra. He then proves that Spec…L_S…v₁;v₂††represents the functor of isomorphisms fromv₁tov₂.

Now letCbe a Tannakian category, letDbe neutral overC, and suppose that v1;v2:D ! Care two fibre functors fromDtoC. Define the coend

LC…v1;v2†:ˆ

Z^D

v1… † V v2… †V ^_2Ind… †:C …63†

Ifh:C !QCoh… †S is a fibre functor, then hcommutes with colimits, and hence h…L_C…v₁;v₂†† ˆL_S…hv₁;hv₂†: this is a quasi-coherent O_S-algebra, functorial inh.

Since algebraic structures in Tannakian categories, such as commutative algebras, Hopf algebras, and so on, can be constructed `functorially in fibre functors', (see for example §5.11 of [19]), it follows that there is a unique way of defining aC-algebra structure on L_C…v₁;v₂† lifting the O_S-algebra structure on each h…L_C…v₁;v₂††.

Moreover, since h…Sp…L_C…v₁;v₂††† is a …hv₁…p… †D†;hv₂…p… †D††-bitorsor, functo- rially inh, the affine scheme

PC…v1;v2†:ˆSp…LC…v1;v2†† …64†

is a…v₁…p… †D†;v₂…p… †D††-bitorsor overC.

What we actually want, however, is a …GC…D;v2†;GC…D;v2††-bitorsor. We get this as follows. Suppose thatV 2 D, then by the definition ofLC…v1;v2†we get a morphism

v₁… † V v₂… †V ^_!L_C…v₁;v₂† …65†

which corresponds to a morphism

v₁… † !V v₂… † V L_C…v₁;v₂†:

…66†

Thus a morphism LC…v1;v2† !R for some C-algebra R induces an R-linear morphism

v₁… † V R!v₂… † V R …67†

which is in fact an isomorphism, since it is so after applying any fibre functor.

DEFINITION2.5. DefineP_triv…v1;v2†to be the sub-functor ofPC…v1;v2†which takesR to the set of all morphismsL_C…v₁;v₂† !Rsuch that for everyV in the essential image oft:C ! D, the induced automorphism ofRv₁(V)ˆRv₂(V) is the identity.

PROPOSITION2.6. The functor P_triv…v₁;v₂†is representable byan affine scheme overC, and is a G… _C…D;v₁†;G_C…D;v₂††-bitorsor in the categoryof affine schemes overC.