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Completions of mapping class groups

arXiv:alg-geom/9207001v1 23 Jul 1992Completions of mapping class groupsand the cycle C −C−July, 1992Richard M. Hain1Department of MathematicsDuke UniversityDurham, NC 277061IntroductionThe classical Malcev (or unipotent completion) of an abstract group π is aprounipotent group P defined over Q together with a homomorphism φ : π →P.It is characterized by the property that if ψ : π →U is a homomorphism of πinto a prounipotent group, then there is a unique homomorphism Ψ : P →U ofprounipotent groups such that ψ = Ψφ.πφ→PցψyΨUWhen H1(π; Q) = 0, the unipotent completion is trivial, and it is therefore auseless tool for studying mapping class groups. Deligne has suggested a notion ofrelative Malcev completion: Suppose Γ is an abstract group and that ρ : Γ →Sis a homomorphism of Γ into a semisimple linear algebraic group defined over Q.Suppose that ρ has Zariski dense image.

The completion of Γ relative to ρ is aproalgebraic group G over Q, which is an extension of S by a prounipotent groupU, and a homomorphism ˜ρ : Γ →G which lifts ρ. When S is the trivial group, itreduces to the unipotent completion.

The relative completion is characterizedby a universal mapping property which generalizes the one in the unipotent case(see (2.3)).Denote the mapping class group associated to a surface of genus g with rboundary components and n ordered marked distinct points by Γng,r. The map-ping class group Γng,r has a natural representation ρ : Γng,r →Spg(Z) obtainedfrom the action of Γng,r on the first homology group of the underlying compactRiemann surface.

Its kernel is, by definition, the Torelli group T ng,r. One cantherefore form the completion of Γng,r relative to ρ.

It is a proalgebraic groupGng,r which is an extension1 →Ung,r →Gng,r →Spg →11Supported in part by grants from the National Science Foundation.1

of Spg by a prounipotent group. The homomorphism˜ρ : Γng,r →Gng,rinduces a homomorphism T ng,r →Ung,r.

This, in turn, induces a homomorphismT ng,r →Ung,r from the unipotent completion of the Torelli group into Ung,r. Ourmain result is:Theorem.When g ≥3, the natural homomorphism T ng,r →Ung,r is surjectivewith nontrivial kernel which is contained in the center of T ng,r and which isisomorphic to Q whenever g ≥8.We prove this by relating the central extension above to the line bundle overthe moduli space of genus three curves associated to the archimedean heightof the algebraic cycle C −C−in the jacobian of a curve C of genus 3.2 Thistheorem is related to, and complements, the work of Morita [21].

The constant8 in the theorem can surely be improved, possibly to 3.3One reason for introducing relative completions of fundamental groups ofvarieties, and of mapping class groups in particular, is that their coordinaterings are, under suitable conditions, direct limits of variations of mixed Hodgestructure over the variety. This result and some of its applications to the actionof the mapping class group of a surface S on the lower central series of π1(S, ∗)will be presented elsewhere.Part of a general theory of relative completions is worked out in Section4.Many of the results of that section were worked out independently andcontemporaneously by Eduard Looijenga.

I would like to thank him for hiscorrespondence. I would also like to thank P. Deligne for his correspondence,and the Mathematics Department of the University of Utrecht for its hospitalityand support during a visit in May, 1992 when this paper was written.Conventions: The group Spg(R) will denote the group of automorphisms ofa free R module of dimension 2g which preserve a unimodular skew symmetricbilinear form.

In short, elements of Spg(R) are 2g × 2g matrices.2Relative completionFix a field F of characteristic zero. Suppose that π is a abstract group andthat ρ : π →S is a representation of π into a linear algebraic group S definedover F. Assume that the image of ρ is Zariski dense.

In this section we definethe completion of π relative to ρ. When S is the trivial group, this reduces2When the genus g of C is ≥3, one can relate this central extension to the height pairingbetween the cycles C(a) −C(a)−and C(b) −C(b)−in Jac C, where a + b = g −1 and C(r)denotes the rth symmetric power of C. We chose not to do this in order to keep the Hodgetheory straightforward.3The optimal constant is the smallest integer d such that H2(Spg(Z), A) vanishes for allrational representations A of Spg(Q) whenever g ≥d.2

to the Malcev completion (a.k.a. unipotent completion) which is defined, forexample, in [24], [3] and [25].

The idea of relative completion is due to Deligne[4] and is a refinement of the idea of the “algebraic hull” of a group introducedby Hochschild and Mostow [14, p. 1140].To construct the relative completion of π with respect to ρ, consider allcommutative diagrams of the form1→U→E→S→1˜ρ ↑ր ρπwhere E is a linear algebraic group over F, U a unipotent subgroup of E, andwhere ˜ρ is a lift of ρ to E whose image is Zariski dense. All morphisms in thetop row are algebraic group homomorphisms.

One can define morphisms of suchdiagrams in the obvious way.Proposition 2.1 The set of such diagrams forms an inverse system.Proof. If1→Uα→Eα→S→11→Uβ→Eβ→S→1are two extensions of S by a unipotent algebraic group, then one can form thefibered productE→Eα↓↓Eβ→S.The natural homomorphism E →S is surjective with kernel the unipotent groupUα × Uβ.Now suppose that ρα : π →Eα and ρβ : π →Eβ are lifts of ρ : π →Sto Eα and Eβ, respectively, both with Zariski dense image.They induce ahomomorphism ραβ : π →E which lifts both ρα and ρβ.

Denote the Zariskiclosure of the image of ραβ in E by Eαβ. Then Eαβ is a linear algebraic groupand the kernel of the natural homomorphism Eαβ →S is unipotent as it is asubgroup of Uα × Uβ.

The natural map Eαβ →S is surjective as the image ofρ is Zariski dense in S.Definition 2.2 The completion PF of π (over F) relative to ρ : π →S isdefined to be the proalgebraic groupPF = lim←E,where the inverse limit is taken over all commutative diagrams1→U→E→S→1x˜ρրρπ3

whose top row is an extension of S by a unipotent group in the category of linearalgebraic groups over F, and where ˜ρ has dense image. The homomorphisms˜ρ : π →E induce a canonical homomorphism π →PF .Often we will simply say that π →PF is the S-completion of π.Thecoordinate ring of PF is the direct limit of the coordinate rings of the groupsE.

It will be denoted O(PF ). It is a commutative Hopf algebra with antipode.There is a natural surjection PF →S whose kernel is a prounipotent group.When S is the trivial group, we obtain the classical Malcev completion.

TheS-Malcev completion is characterized by the following easily verified universalmapping property.Proposition 2.3 Suppose that E is a linear proalgebraic group defined over F,and that E →S is a homomorphism of proalgebraic groups with prounipotentkernel. If ϕ : π →E is a group homomorphism, then there is a unique homo-morphism τ : PF →E of pro-algebraic groups over F such that the diagramPFˆρ րցπyτSϕ ցրEcommutes.Suppose that G is a (pro)algebraic group over the field F. Suppose that k isa field extension of F. We shall denote G, viewed as an algebraic group over kby extension of scalars, by G(k).

The following assertion follows directly fromthe universal mapping property.Corollary 2.4 If k is a field extension of the field F, then there is a naturalhomomorphism Pk →PF (k).We will show in the next two sections that, with some extra hypotheses, thishomomorphism is always an isomorphism.3A construction of the Malcev completionThere is an explicit algebraic construction of the Malcev completion which isdue to Quillen [24]. We will use it to show that the Malcev completion of agroup over F is isomorphic to the F points of its Malcev completion over Q.Denote the group algebra of a group π over a commutative ring R by Rπ.The augmentation is the homomorphism ǫ : Rπ →R defined by taking eachγ ∈π to 1.

The kernel of the augmentation is called the augmentation idealand will be denoted by JR, or simply J when there is no chance of confusion.4

With the coproduct ∆: Rπ →Rπ ⊗Rπ defined by ∆(γ) = γ ⊗γ, for all γ ∈π,Rπ has the structure of a cocommutative Hopf algebra.The powers of the augmentation ideal define a topology on Rπ that is calledthe J-adic topology. The J-adic completion of the group ring is the R moduleRπb := lim←Rπ/Jl.The completion of J will be denoted by bJ.

Since the coproduct is continuous,it induces a coproduct∆: Rπb →Rπb⊗Rπ,where b⊗denotes the completed tensor product. This gives Rπb the structureof a complete Hopf algebra.The proof of the following proposition is straightforward.Proposition 3.1 If π is a group and R a ring, then the function π →JR/J2Rdefined by taking γ ∈π to the coset of γ −1, induces an R-module isomorphismH1(π, R) ≈JR/J2R.Now let R be a field F of characteristic zero.

The logarithm and exponentialmaps are mutually inverse homeomorphismslog : 1 + bJF →bJF and exp : bJF →1 + bJF .The set of primitive elements p of Fπb is defined byp = {X ∈bJF : ∆(X) = X ⊗1 + 1 ⊗X}.With the bracket [X, Y ] = XY −Y X, it has the structure of a Lie algebra. Thetopology of Fπb induces a topology on p, giving it the structure of a completetopological Lie algebra.The set of group-like elements P of Fπb is defined byP = {X ∈Fπˆ : ∆(X) = X ⊗X and ǫ(X) = 1}.It is a subgroup of the group of units of Fπb.

The logarithm and exponentialmaps restrict to mutually inverse homeomorphismslog : P →p and exp : p →P.The filtration of Fπb by the powers of J induces filtrations of P and p: Setpl = p ∩bJl and Pl = P ∩(1 + bJl).These satisfyp = p1 ⊇p2 ⊇p3 · · ·5

andP = P1 ⊇P2 ⊇P3 · · · .These filtrations define topologies on p and P. Both are separated and complete.For each l setpl = p/pl+1 and Pl = P/Pl+1.It follows easily from (3.1) that if H1(π, Q) is finite dimensional (e.g. π is finitelygenerated), then each Pl is a linear algebraic group.Since the logarithm and exponential maps induce isomorphisms between1 + bJ/ bJl+1 and bJ/ bJl+1, and sincepl ⊆bJ/ bJl+1 and Pl ⊆1 + bJ/ bJl+1,it follows that the logarithm and exponential maps induce polynomial bijectionsbetween pl and Pl.

Consequently, when H1(π, Q) is finite dimensional, each ofthe groups Pl is a unipotent algebraic group over F with Lie algebra pl. Itfollows that if H1(π, F) is finite dimensional, then P is a prounipotent groupover F.The composition of the canonical inclusion of π into Fπ followed by thecompletion map Fπ →Fπb yields a canonical map π →Fπb.

Since the imageof this map is contained in P, there is a canonical homomorphism π →P.The composition of the natural homomorphism π →P with the quotient mapP →Pl yields a canonical homomorphism π →Pl.Proposition 3.2 If H1(π, F) is finite dimensional, then each of the homomor-phisms π →Pl has Zariski dense image.Proof. Denote the Zariski closure of the image of π in Pl by Zl.

Each Zl is analgebraic subgroup of Pl. Since the compositeH1(π; F) →H1(Zl) →H1(Pl)is an isomorphism, it follows that the second map is surjective.Since Pl isunipotent, this implies that the inclusion Zl ֒→Pl is surjective.Theorem 3.3 If H1(π, F) is finite dimensional, then the natural map π →Pis the Malcev completion of π over F.Proof.

By the universal mapping property (2.3), there is a canonical homomor-phism from the Malcev completion UF of π to P. It follows from (3.2) that thishomomorphism is surjective.We now establish injectivity. Suppose that U is a unipotent group definedover F. This means that U can be represented as a subgroup of the group ofupper triangular unipotent matrices in GLn(F) for some n. A representationρ : π →U induces a ring homomorphism˜ρ : Fπ →gln(F).6

Since the representation is unipotent, it follows that ˜ρ(J) is contained in theset of nilpotent upper triangular matrices. This implies that the kernel of ˜ρcontains Jn.

Consequently, ρ induces a homomorphism¯ρ : Fπ/Jn →gln(F).Since the image of J is contained in the nilpotent upper triangular matrices, theimage of the subgroup Pn−1 of 1 + J/Jn is contained in the group of unipotentupper triangular matrices. Because the image of π is Zariski dense in Pn−1, itfollows that the image of Pn−1 is contained in U.

That is, there is a homomor-phism Pn−1 →U of linear algebraic groups over F such that the diagramπ→Pn−1ցρ↓Ucommutes. It follows that UF →P is injective.Corollary 3.4 If PF is the Malcev completion of the group π over F, then thenatural homomorphismPF →PQ(F)is an isomorphism.Proof.

This follows as PF is the set of group-like elements of Fπb, while PQ(F)is the set of group-like elements of Qπb⊗F ≈Fπb.4Basic theory of relative completionIn this section we establish several basic properties of relative completion. Webegin by recalling some basic facts about group extensions.

Once again, F willdenote a fixed field of characteristic zero. All algebraic groups will be linear.Suppose that L is an abstract group and that A is an L module.

The groupof congruence classes of extensions0 →A →G →L →1,where the natural action of L on A is the given action, is naturally isomorphicto H2(L; A). The identity is the semidirect product L ⋉A ([20, Theorem 4.1,p.

112]). If H1(L; A) vanishes, then any 2 splittings s0, s1 : L →L ⋉A areconjugate via an element of A ([20, Prop.

2.1,p. 106]).

That is, there existsa ∈A such that s1 = as0a−1.If S is a connected semisimple algebraic group over F, and if A is a rationalrepresentation of S, then every extension0 →A →G →S →17

in the category of algebraic groups over F splits. Moreover, any 2 splittingss0, s1 : S →G are conjugate by an element of A [15, p. 185].These results extend to the case when the kernel is unipotent.

For this weneed the following construction. ( 4.1 ) Construction.

Suppose that1 →U →G →L →1(1)is an extension of an abstract group L by a group U. Suppose that Z is a centralsubgroup of U and that the extension1 →U/Z →G/Z →L →1(2)is split.

Let s : L →G/Z be a splitting. Pulling back the extension1 →Z →G →G/Z →1along s, one obtains an extension1 →Z →Es →L →1.

(3)This determines, and is determined by, a class ζs in H2(L; Z) which dependsonly on s up to inner automorphisms by elements of G. The extension (3) splitsif and only if ζs = 0.Proposition 4.2 If ζs = 0, then the extension (1) splits. Moreover, if any 2splittings of (2) are conjugate, and if H1(L; Z) = 0, then any 2 splittings of (1)are conjugate.Proof.

If ζs vanishes, there is a splitting σ : L →Es. By composing σ with theinclusion of Es ֒→G, one obtains a splitting of the extension (1).Suppose now that H1(L; Z) vanishes, and that any 2 splittings of L →G/Zare conjugate.

If s0, s1 : L →G are 2 splittings of (1), then their reductionss0, s1 : L →G/Z mod Z are conjugate. We may therefore assume that s0 ands1 agree mod Z.

The images of s0 and s1 are then contained in the subgroupEs of G determined by the section s0 = s1 : L →G/Z. Since H1(L; Z) = 0,there is an element z of Z which conjugates s0 into s1.A similar argument, combined with the facts about extensions of algebraicgroups at the beginning of this section, can be used to prove the following result.Proposition 4.3 Suppose that S is a connected semisimple algebraic groupover F. If1 →U →G →S →1is an extension in the category of algebraic groups over F, and if U is unipotent,then the extension splits, and any two splittings S →G are conjugate.8

By choosing compatible splittings and then taking inverse limits, we obtainthe analogous result for proalgebraic groups.Proposition 4.4 Suppose that S is a connected semisimple algebraic groupover F. If1 →U →G →S →1is an extension in the category of proalgebraic groups over F, and if U isprounipotent, then the extension splits, and any two splittings S →G areconjugate.Suppose that Γ is a abstract group, S an algebraic group over F, and thatρ : Γ →S a representation with Zariski dense image. Denote the image of ρ byL and its kernel by T .

Thus we have an extension1 →T →Γ →L →1.Let Γ →GF be the completion over F of Γ relative to ρ and UF its prounipotentradical. There is a commutative diagram1→T→Γ→L→1↓↓↓1→UF→GF→S→1Denote the unipotent (i.e., the Malcev) completion of T over F by TF .

Theuniversal mapping property of TF gives a homomorphism Φ : TF →UF ofprounipotent groups whose composition with the natural map T →TF is thecanonical map T →UF . Denote the kernel of Φ by KF .Proposition 4.5 Suppose that H1(T ; F) is a finite dimensional.

If the actionof L on H1(T ; F) extends to a rational action of S, then the kernel KF of Φ iscentral in TF .Proof. First, Γ acts on the completion FT b of the group algebra of T byconjugation.

This action preserves the filtration by the powers of bJ, so it actson the associated graded algebraGr•JFTb =∞Mm=0bJm/ bJm+1.If H1(T ; F) is finite dimensional, each truncation FT/Jl of FTb is finite dimen-sional. This implies that each of the groups Aut FT/Jl is an algebraic group.Since Aut Gr•JFT b is generated by J/J2 = H1(T ; F), it follows that whenH1(T ; F) is finite dimensional, Aut FTb, the group of augmentation preservingalgebra automorphisms of FT b, is a proalgebraic group which is an extensionof a subgroup of Aut H1(T ; F) by a prounipotent group.9

If the action of Γ on H1(T ; F) factors through a rational representationS →Aut H1(T ; F) of S, then we can form a proalgebraic group extension1 →J−1Aut FTb →E →S →1of S by the prounipotent radical of Aut FTb by pulling back the extension1 →J−1Aut FTb →Aut FTb →Aut H1(T ; F)along S →Aut H1(T ; F). The representation Γ →Aut FTb lifts to a represen-tation Γ →E whose composition with the projection E →S is ρ : Γ →S.

Thisinduces a homomorphism GF →E. Since the compositeTF →GF →E →Aut FTbis the action of TF ⊆FT b on FT b by inner automorphisms, it follows thatthe kernel of this map is the center Z(TF ) of TF .

It follows that the kernel ofTF →UF is a subgroup of Z(T ).The following result and its proof were communicated to me by P. Deligne[5].Proposition 4.6 Suppose that S is semisimple and that the natural action ofL on H1(T ; F) extends to a rational representation of S. If H1(L; A) = 0 forall rational representations A of S, then Φ is surjective.Proof. The homomorphism TF →UF is surjective if and only if the inducedmap H1(T ; F) →H1(UF ) is surjective.

Let A be the cokernel of this map. Thisis a rational representation of S as both H1(T ; F) and H1(UF ) are.

By pushingout the extension1 →UF →GF →S →1along the map UF →H1(U) →A, we obtain an extension1 →A →G →S →1of algebraic groups and a homomorphism Γ →G which lifts ρ and has Zariskidense image in G. Let1 →A →GL →L →1be the restriction of this extension to L. Since A is the cokernel of the mapTF →H1(U), the image of Γ in G is Γ/T = L. So ρ induces a homomorphism˜ρ : L →G which has Zariski dense image.The image of ˜ρ lies in GL andinduces a splitting σ : L →GL of the projection GL →L.Since G is analgebraic group, there is a splitting s : S →G of the projection in the categoryof algebraic groups. This restricts to a splitting s′ : L →G.

Since H1(L; A)vanishes, there exists a ∈A which conjugates s′ into σ. Thus the image of σ is10

contained in the algebraic subgroup as(S)a−1 of G. Since the image of σ in Gis Zariski dense, it follows that A must be trivial.A direct consequence of this result is a criterion for the map ρ : Γ →Sitself to be the ρ completion. This criterion is satisfied by arithmetic groups insemisimple groups where each factor has Q rank ≥2 [23].Corollary 4.7 Suppose that ρ : Γ →S is a homomorphism of an abstractgroup into a semisimple algebraic group.

If H1(L; A) vanishes for all rationalrepresentations of S, and if H1(T ; F) is zero (e.g., ρ injective), then the relativecompletion of Γ with respect to ρ is ρ : Γ →S.Next we consider the problem of imbedding an extension of L by a unipotentgroup U in an extension of S by U.Proposition 4.8 Suppose that1 →U →G →L →1is a split extension of abstract groups, where U is a unipotent group over Fand where the action of L on H1(U) extends to a rational representation ofS. If H1(L; A) vanishes for all rational representations of S, then there is anextension1 →U →˜G →S →1of algebraic groups such that the original extension is the restriction of this oneto L.Proof.

Since the first extension splits, we can write it as a semi direct productG = L ⋉U.Denote the Lie algebra of U by u. The group of Lie algebra automorphisms ofu is an algebraic group over F. It can be expressed as an extension1 →J−1Aut u →Aut u →Aut H1(u)where the kernel consists of those automorphisms which act trivially on H1(u),and therefore on the graded quotients of the lower central series of u.It isa unipotent group.

We can pull this extension back along the representationS →Aut H1(u) to obtain an extension1 →J−1Aut u →˜A →S →1The representation L →Aut u lifts to a homomorphism L →˜A. This induces ahomomorphism of the S-completion of L into ˜A.

By (4.7), the completion of Lis simply the inclusion L ֒→S. That is, the representation L →˜A extends to an11

algebraic group homomorphism S →˜A. This implies that the representation ofL on u extends to a rational representation S →Aut u.

We can therefore formthe semi direct product S ⋉U, which is an algebraic group. The homomorphismL ⋉U →S ⋉U exists because of the compatibility of the actions of L and S onU.Combining this with (4.2), we obtain:Corollary 4.9 Suppose that U is a prounipotent group over F with H1(U)finite dimensional, and suppose that Z is a central subgroup of U. Supposethat1 →U →G →L →1(4)is an extension of abstract groups where the action of L on H1(U) extends to arational representation of S. Suppose that1 →U/Z →G →S →1is an extension of proalgebraic groups which gives the extension1 →U/Z →G/Z →L →1when restricted to L. If the class in H2(L; Z) given by (4.1) vanishes, then thereexists an extension1 →U →˜G →S →1of proalgebraic groups whose restriction to L is the extension (4).By pushing out the extension1 →T →Γ →L →1along the homomorphism T →TF , we obtain a “fattening” ˆΓ of Γ.

Let GLbe the inverse image of L in GF .Using the universal mapping property ofpushouts, one can show easily that the natural homomorphism Γ →GF induceshomomorphism ˆΓ →GL.These groups fit into a commutative diagram ofextensions:1→T→Γ→L→1↓↓∥1→TF→ˆΓ→L→1↓↓∥1→UF→GL→L→1∥↓↓1→UF→GF→S→1Next we introduce conditions we need to impose on our extension for theremainder of the section.12

( 4.10 ) We will now assume that the extension Γ of L by T satisfies the followingconditions. First, H1(T ; F) is finite dimensional and the action of L on it extendsto a rational representation of S.Next, we assume that H1(L; A) vanishesfor all rational representations of S. Finally, we add the new condition thatH2(L; A) vanishes for all nontrivial irreducible rational representations of S.These conditions are satisfied by arithmetic groups in semisimple groups whereeach factor has real rank at least 8 [1, 2].The following result is an immediate consequence of (4.9).Proposition 4.11 If the conditions (4.10) are satisfied, then KF = ker Φ iscontained in the center of the thickening ˆΓ of Γ.Applying the construction (4.1) to the thickening ˆΓ of Γ and a splitting ofGL →L, we obtain an extension0 →KF →G →L →1(5)which is unique up to isomorphism.

It follows from (4.11) that this is a centralextension. Since H1(L; F) vanishes, there is a universal central extension withkernel an F vector space.

It is the extension0 →H2(L; F) →˜L →L →1with cocycle the identity mapnH2(L; F)id→H2(L; F)o∈Hom(H2(L; F), H2(L; F)) ≈H2(L; H2(L; F)).The central extension (5) is classified by a linear map ψF : H2(L; F) →KF .Because all splittings s : L →GL are conjugate (4.2), the class of this extensionis independent of the choice of the splitting.Since KF is an abelian unipotent group, KF (k) = KF ⊗k for all fields k whichcontain F. The homomorphism ψF satisfies the following naturality property:Proposition 4.12 If k is an extension field of F, then the diagramH2(L; k)→Kk||↓H2(L; F) ⊗k→KF (k)commutes.The next result bounds the size of KF .Proposition 4.13 If the conditions (4.10) hold, then the natural map ψF :H2(L; F) →KF is surjective.13

Proof. As above, we shall denote by G the central extension of L by KF .

LetA be the cokernel of ψF and E the cokernel of ψF : H2(L; F) →G. Then E isa central extension of L by A.

Because the composite H2(L; F) →KF →A istrivial, this extension is split. From (4.2) it follows that the extension1 →TF /im ψF →ˆΓ/im ψF →L →1is split.

By (4.8), this implies that there is a proalgebraic group E which is asemidirect product of S by TF /im ψF into which ˆΓ/im ψF injects. The map ofΓ to E induces a map GF →E.

Since the kernel of the map TF →E is im ψF ,it follows that KF is contained in im ψF .Corollary 4.14 If the conditions (4.10) hold, then the natural map Gk →GF (k) associated to a field extension k : F is an isomorphism.Proof. By (3.4), the natural map TF →TF (k) is an isomorphism.

Since TK →UK is surjective for all fields K, the natural map Uk →UF (k) is also surjective.Consequently, the natural map Kk →KF (k) is injective. Since KF is abelianunipotent, KF (k) = KF ⊗k.

But it follows from (4.13) that Kk →KF (k) issurjective, and therefore an isomorphism.5The Johnson homomorphismThis section is a brief review of the construction of Johnson’s homomorphism[19, 17].There are two equivalent ways, both due to Johnson, to define ahomomorphismTg →Λ3H1(C; Z)where C is a compact Riemann surface of genus g.Choose a base point x of C. The first construction uses the action of T 1g onπ1(C, x). Denote the lower central series of π1(C, x) byπ1(C, x) = π1 ⊇π2 ⊇π3 ⊃· · ·The first graded quotient π1/π2 is H1(C; Z).

The second is naturally isomorphicto Λ2H1(C; Z)/⟨q⟩, where q : Λ2H1(C; Z) →H2(C; Z) ≈Z is the cup product.If γj, j = 1, . .

. , 2g, are generators of π1(C, x), then the residue class of thecommutator γjγkγ−1jγ−1kmodulo π3 is the element cj ∧ck of Λ2H1(C; Z)/⟨q⟩,where cj denotes the homology class of γj.

The form q is just the equivalenceclass of the standard relation in π1(C, x).If φ : (C, x) →(C, x) is a diffeomorphism which represents an element of T 1g ,then φ acts trivially on the homology of C. It therefore acts as the identity oneach graded quotient of the lower central series of π1(C, x). Define a functionπ →π by taking γ to φ(γ)γ−1.

Since φ acts trivially on H1(C), it follows thatthis map takes πl into πl+1. In particular, it induces a well defined function˜τ(φ) : H1(C; Z) →Λ2H1(C; Z)/⟨q⟩14

between the first two graded quotients of π, which is easily seen to be linear.Using Poincar´e duality, ˜τ(φ) can be regarded as an element ofH1(C; Z) ⊗Λ2H1(C; Z)/⟨q⟩.The map φ 7→˜τ(φ) induces a group homomorphismT 1g →H1(C; Z) ⊗Λ2H1(C; Z)/⟨q⟩.and therefore a homomorphismˆτ : H1(T 1g ) →H1(C; Z) ⊗Λ2H1(C; Z)/⟨q⟩.There is a natural inclusionΛ3H1(C; Z) →H1(C; Z) ⊗Λ2H1(C; Z)/⟨q⟩.defined byx ∧y ∧z 7→x ⊗(y ∧z) + y ⊗(z ∧x) + z ⊗(x ∧y).Johnson has proved that the image of ˆτ is contained in the image of this map,so that ˆτ induces a homomorphismτ 1g : H1(T 1g ) →Λ3H1(C; Z).It is not difficult to check that this homomorphism is Spg(Z) equivariant.This story can be extended to Tg as follows. There is a natural extension1 →π1(C, x) →T 1g →Tg →1.Applying H1, we obtain the diagramH1(C; Z)→H1(T 1g )→H1(Tg)→0yτ 1gΛ3H1(C; Z)whose top row is exact.Identify the lower group with H3(Jac C; Z).SinceJac C is a group with torsion free homology, the group multiplication induces aproduct on its homology which is called the Pontrjagin product.

The compositeof τ1g with the map from H1(C) takes a class in H1(Jac C) to its Pontrjaginproduct with q = [C] ∈H2(Jac C; Z). It follows that τ 1g induces a mapτg : H1(Tg) →Λ3H1(C; Z)/ (q ∧H1(C; Z)) .The following fundamental theorem is due to Dennis Johnson.15

Theorem 5.1 When g ≥3, the homomorphisms τ 1g and τg are isomorphismsmodulo 2 torsion.The second way to construct a map T 1g →Λ3H1(C; Z) is as follows. Supposethat φ represents an element of T 1g .

Let Mφ →S1 be the bundle over the circleconstructed by identifying the point (z, 1) of C × [0, 1] with the point (φ(z), 0).Since φ fixes the basepoint x, the map t →(x, t) induces a section of Mφ →S1.Similarly, one can construct the bundle of jacobians; this is trivial as φ actstrivially on H1(C). One can imbed Mφ in this bundle of jacobians using thissection of basepoints.

Let p be the projection of the bundle of jacobians ontoone of its fibers. Then one has the 3 cycle p∗Mφ in Jac C.Proposition 5.2 [19] When g ≥3, the homology classp∗[Mφ] ∈H3(Jac C; Z) ≈Λ3H1(C; Z)is τ1g (φ).This can be proved, for example, by checking that both maps agree on whatJohnson calls “bounding pair” maps.

These maps generate the Torelli groupwhen g ≥3 [16].6The cycle C −C−In this section we relate the algebraic cycle C−C−to Johnson’s homomorphism.Let C be a compact Riemann surface. Denote its jacobian by Jac C. This isdefined to be Pic0C, the group of divisors of degree zero on C modulo principaldivisors.

Each divisor D of degree zero may be written as the boundary of atopological 1-chain: D = ∂γ. Taking D to the functional ω 7→Rγ ω on the spaceΩ(C) of holomorphic 1-forms yields a well defined mapPic0C →Ω(C)∗/H1(C; Z).

(6)This is an isomorphism by Abel’s Theorem.For each x ∈C, we have an Abel-Jacobi mapνx : C →Jac Cwhich is defined by νx(y) = y −x. This map is an imbedding if the genus gof C is ≥1.

Denote its image by Cx. This is an algebraic 1-cycle in Jac C.Denote the involution D 7→−D of Jac C by i, and the image of Cx underthis involution by C−x .

Since i induces −id on H1(Jac C; Z), and since i∗isa ring homomorphism, we see that i∗acts as (−1)k on Hk(Jac C). It followsthat Cx and C−x are homologically equivalent.

Set Zx = Cx −C−x . This is ahomologically trivial 1-cycle.16

Griffiths has a construction which associates to a homologically trivial an-alytic cycle in a compact K¨ahler manifold a point in a complex torus.Hisconstruction is a generalization the construction of the map (6). We review thisconstruction briefly.

Suppose that X is a compact K¨ahler manifold, and that Zis an analytic k-cycle in X which is homologous to 0. Write Z = ∂Γ, where Γis a topological 2k + 1 chain in X. DefineF pHm(X) =Ms≥pHs,m−s(X).Each class in F pHm(X) can be represented by a closed form where each term ofa local expression in terms of local holomorphic coordinates (z1, .

. .

, zn) has atleast p dzjs. Integrating such representatives of classes in F k+1H2k+1(X) overΓ gives a well defined functionalZΓ: F k+1H2k+1(X) →C.The choice of Γ is unique up to a topological 2k + 1 cycle.

So Z determines apoint of the complex torusJk(X) := F k+1H2k+1(X)∗/H2k+1(X; Z).This group is called the kth intermediate jacobian of X.In our case, the cycle Zx determines a point ζx(C) in the intermediate jaco-bianJ1(Jac C) = F 2H3(Jac C)∗/H3(Jac C; Z).The homology class of Cx is easily seen to be independent of x. Taking thePontrjagin product with [C] defines an injective mapH1(Jac C; Z) ֒→H3(Jac C; Z).The dual H3(Jac C) →H1(C) is a morphism of Hodge structures of type(−1, −1) — that is, Hs,t gets mapped into Hs−1,t−1.

It follows that this mapinduces an imbeddingΦ : Jac C ֒→J1(Jac (C))of complex tori. We shall denote the cokernel of this map by JQ1(Jac C).

It istrivial when g < 3.The following result is not difficult; a proof may be found in [22].Proposition 6.1 If x, y ∈C, thenζx(C) −ζy(C) = 2Φ(x −y).In particular, the image of ζx(C) in JQ1(Jac C) is independent of x.17

Denote the common image of the ζx(C) in JQ1(Jac C) by ζ(C).We now suppose that the genus g of C is ≥3. Fix a level l so that themoduli space Mng (l) of curves of genus g and n marked points with a level lstructure is smooth.

(Any l ≥3 will do for the time being.) Denote the spaceof principally polarized abelian varieties of dimension g with a level l structureby Ag(l).One can easily construct bundles Jm →Ag(l) of complex tori whose fiberover the abelian variety A is Jm(A).

The pullback of J0 along the period mapMg(l) →Ag(l) is the bundle of jacobians associated to the universal curve.The imbedding J0 ֒→J1 defined over Mg(l) extends to all of Ag(l) as thehomology class [C] ∈H2(Jac C; Z) extends to a class q ∈H2(A; Z) for everyabelian variety A. Denote the bundle of quotient tori by Q →Ag(l).Denote the level l congruence subgroup of Spg(Z) by L(l). Since Ag(l) isan Eilenberg-Mac Lane space K(L(l), 1), and since the fiber of J1 →Ag(l) is aK(H3(Jac C; Z), 1), it follows that J1 is also an Eilenberg-Mac Lane space whosefundamental group is an extension of L(l) by H3(Jac C; Z).

Since this bundlehas a section, viz., the zero section, this extension splits. The action of L(l) onH3(Jac C; Z) is the restriction of the third exterior power of the fundamentalrepresentation of Spg.

There is a similar story with J1 replaced by Q.We shall denote the quotient Λ3H1(C; Z)/ ([C] · H1(C; Z)) by QΛ3H1(C).This is the fundamental group of JQ1(Jac C).Proposition 6.2 The spaces J1 and Q are Eilenberg-Mac Lane spaces withfundamental groupsπ1(J1, 0C) ≈L(l) ⋉Λ3H1(C; Z)andπ1(Q, 0C) ≈L(l) ⋉QΛ3H1(C),respectively. Here 0C denotes the identity element in the fiber over Jac C.The normal functions (C, x) 7→ζx(C) and C 7→ζ(C) give lifts of the periodmap:J1րζ1g↓M1g(l)→Ag(l)Qրζg↓Mg(l)→Ag(l)These induce maps of fundamental groupsζ1g ∗: Γ1g(l) →L(l) ⋉Λ3H1(C)andζg∗: Γg(l) →L(l) ⋉QΛ3H1(C)Since these maps commute with the canonical projections to L(l), these induceL(l) equivariant mapsζ1g : H1(T 1g ) →Λ3H1(C; Z),ζ1g : H1(Tg) →QΛ3H1(C; Z)18

The following result follows easily from (5.2). The factor of 2 arises as bothC and C−each contribute a copy of the Johnson homomorphism.Proposition 6.3 The map ζng is twice Johnson’s map τng for n = 0, 1.It is natural to try to give a “motivic description” of the Johnson homo-morphism rather than of twice it.

Looijenga (unpublished) has done this byconstructing a normal function which compares the cycle Cx to a fixed topolog-ical (but not algebraic) cycle in Jac C which is homologous to Cx. At the costof being more abstract, we give another description which does not make use ofLooijenga’s topological cycle.We will only consider the pointed case, the unpointed case being similar.For each pointed curve (C, x), the cycle Cx determines a point cx in the Delignecohomology group H2g−2D(Jac (C), Z(g −1)).

This group is an extension of theHodge classes Hg−1,g−1Z(Jac C) by the intermediate jacobian J1(Jac C):0 →J1(Jac C) →H2g−2D(Jac (C), Z(g −1)) →Hg−1,g−1Z(Jac C) →0One can consider the bundle over Ag(l) whose fiber over the abelian variety Ais H2g−2D(A, Z(g −1)). The subbundle whose fiber over A is the J1(A) coset ofthe class of the polarization q ∈Hg−1,g−1Z(A) is a principal J1 torsor over Ag(l).Denote it by Z →Ag(l).

The cycle gives a liftZրc↓M1g(l)→Ag(l)of the period map. The total space Z is an Eilenberg Mac Lane space whosefundamental group is an extension of L(l) by Λ3H1(C; Z).

The map induced byc on fundamental groups induces the Johnson homomorphism τ 1g . This followsdirectly from (5.2).7Completion of mapping class groupsDenote the completion of the mapping class group Γng,r with respect to thecanonical representation Γng,r →Spg by Gng,r and its prounipotent radical byUng,r.

Denote the Malcev completion of the Torelli group T ng,r by T ng,r and thekernel of the natural homomorphism T ng,r →Ung,r by Kng,r. These groups are alldefined over Q by (4.14).For all g ≥3, and all arithmetic subgroups L of Spg(Z), H1(L; A) vanishesfor all rational representations A of Spg.

By the results of Borel [1, 2], thehypotheses (4.10) are satisfied by all arithmetic subgroups L of Spg(Z) andH2(L; Q) is one dimensional when g ≥8. This, combined with (4.13) yields thefollowing result.19

Proposition 7.1 For all g ≥3 and all n, r ≥0, the natural map T ng,r →Ung,ris surjective. When g ≥8, the kernel Kng,r is either trivial or isomorphic to Q.Our main result is:Theorem 7.2 For all g ≥3 and all r, n ≥0, the kernel Kng,r is non-trivial, sothat Kng,r ≈Q when g ≥8.Let λ1, .

. .

, λg be a fundamental set of weights of Spg.For a dominantintegral weight λ, denote the irreducible representation with highest weight λby V (λ).In this section we will reduce the proof of Theorem 7.2 to the proof of the thecase g = 3 and r = n = 0. The following assertion follows easily by inductionon n and r from Johnson’s result.Proposition 7.3 If g ≥3, then for all n, r ≥0, there is an Spg equivariantisomorphismH1(T ng,r; Q) ≈V (λ3) ⊕V (λ1)n+r.The important fact for us is that the multiplicity of λ3 in H1(T ng,r; Q) isalways 1.

Denote the second graded quotient of the lower central series of T ng,rby Vng,r. The commutator induces a linear surjectionΛ2H1(T ng,r; Q) →Vng,rwhich is Spg equivariant.

By Schur’s lemma, there is a unique copy of the trivialrepresentation in Λ2V (λ3). Let βng,r : Q →V be the compositeQ ֒→Λ2V (λ3) →Λ2H1(T ng,r; Q) →Vng,r,where the first map is the inclusion of the trivial representation.

Consider thefollowing assertions:Ang,r:The map βng,r is injective.Proposition 7.4 If h ≥g ≥3, s ≥r ≥0 and m ≥n ≥, then Ang,r impliesAmh,s. Furthermore Ag,1 implies Ag.

In particular, A3 implies Ang,r for all g ≥3and n, r ≥0.Proof. It is easy to see that βng,r is the composite of βmg,s with the canonicalquotient map Vmg,s →Vng,r whenever m ≥n and s ≥r.

So Ang,r implies Amg,s.Moreover, when r ≥1, the composition of βng,r with the map Vng,r →Vng+1,rinduced by any one of the natural maps Γng,r →Γng+1,r is βng+1,r. So, in thiscase, Ang,r implies Ang+1,r.20

To see that Ag,1 implies Ag, consider the group extension1 →π1(T ∗1 C,⃗v) →Tg,1 →Tg →1where T ∗1 C denotes the unit cotangent bundle of C. It is not difficult to showthat Vg,1 is the direct sum of Vg and the second graded quotient of the lowercentral series of π1(T ∗1 C,⃗v), from which the assertion follows.The assertions Ang,r can be proved using Harer’s computations of H2(Γng,r; Q)[9, 10]. However, we will prove A3 in the course establishing the rest of Theorem7.2 directly, without appeal to Harer’s computation.Denote the fattening (see discussion following (4.2)) of the mapping classgroup Γng,r by ˆΓng,r.

This is an extension1 →T ng,r →ˆΓng,r →Spg(Z) →1.Dividing out by the commutator subgroup of T ng,r, we obtain an extension0 →H1(T ng,r; Q) →Eng,r →Spg(Z) →1. (7)This extension is split.

This can be seen using a straightforward generalizationof (6.3).Now suppose Ang,r holds. Then there is a quotient Gng,r of T ng,r which is anextension of H1(T ng,r; Q) by Q.

The cocycle of the extension being a non-zeromultiple of the polarizationθ ∈Λ2V (λ3) ⊆H2(T ng,r; Q).Dividing ˆΓng,r by the kernel of T ng,r →Gng,r, we obtain an extension1 →Gng,r →Eng,r →Spg(Z) →1. (8)Since the extension (7) splits, we can apply the construction (4.1) to obtain anextension0 →Q →Hng,r →Spg(Z) →1or equivalently, a class eng,r ∈H2(Spg(Z); Q).

By (4.2), the non-triviality ofthis class is the obstruction to splitting the extension (8), which, by (4.8) isthe obstruction to imbedding it in an algebraic group extension of Spg by Gng,r.This proves the following statement.Proposition 7.5 If Ang,r holds and if eng,r is non-zero, then Kng,r is non-trivial.To reduce the proof of Theorem (7.2) to the genus 3 case, we need to relatethe classes eng,r.Proposition 7.6 For fixed g ≥3, the classes eng,r are all equal. The image ofeg+1,1 under the natural map H2(Spg+1(Z); Q) →H2(Spg(Z); Q) is eg,1.Proof.

Both statements follow from the naturality of the construction.Combining (7.4) and (7.6), we have:Proposition 7.7 If A3 holds and e3 is non trivial, then Theorem 7.2 is true.21

8Proof of Theorem 7.2We prove Theorem 7.2 by proving (7.7). In this section we assume that thereader is familiar with mixed Hodge theory.

We will use the notation and con-ventions of [8, §§2–3]. The moduli space Ag can be thought of as the modulispace of principally polarized Hodge structures of weight −1, level 1 and dimen-sion 2g; the abelian variety A ∈Ag corresponds to the Hodge structure H1(A)and its natural polarization.

We can construct bundles over Ag by consideringmoduli spaces of various mixed Hodge structures derived from such a Hodgestructure of weight −1. To guarantee that we have a smooth moduli space, wefix a level l so that Ag(l) is smooth.For a Hodge structure H ∈Ag, where g ≥3, with principal polarization q,we define QH to be the Hodge structureΛ3H/ (q ∧H)⊗Z(−1)which is of weight −1.

Denote the dual Hodge structureHom(QH, Z(1)) ≈ker∧q : Λ2g−3H →Λ2g−1H⊗Z(2 −g)by PH.The set of all mixed Hodge structures with weight graded quotients Z andQH is naturally isomorphic to the complex torusJ(QH) := QHC/F 0QHC + QHZ.If H = H1(Jac C), then J(QH) is the torus JQ1(Jac C) defined in §6. As we letH vary over Ag(l), we obtain the bundle Q →Ag(l) of intermediate jacobiansconstructed in §6.The set of all mixed Hodge structures with weight graded quotients QH andZ(1) is naturally isomorphic to the complex torusJ(PH) := PHC/F 0PHC + PHZ.This torus is the dual of J(QH).

Performing this construction for each H inAg(l), we obtain a bundle P →Ag(l) of complex tori.We now construct a line bundle over the fibered productP ×Ag(l) Q →Ag(l).It is the biextension line bundle. For a Hodge structure H ∈Ag, let B(H)be the set of mixed Hodge structure with weight graded quotients canonicallyisomorphic to Z, QH, and Z(1).

SetGZ =1QHZZ(1)01PHZ001G =1QHCC01PHC00122

F 0GZ =1F 0QH001F 0PH001.There is a natural isomorphismB(H) = GZ\G/F 0G.The natural projectionB(H) →J(QH) × J(PH),which takes V ∈B(H) to (V/W−2, W−1V ), is a principal C∗bundle. Doingthis construction over Ag(l), we obtain a C∗bundleB →Q ×Ag(l) PDenote the corresponding line bundle by L →Q ×Ag(l) P.Since the fiber of B →Ag(l) over H is the nilmanifold B(H), which is anEilenberg-Mac Lane space, it follows that B is also an Eilenberg-Mac Lane spacewhose fundamental group of B is an extension1 →GZ →π1(B, ∗) →L(l) →1.

(9)Taking H ∈Ag(l) to the split biextension Z ⊕H ⊕Z(1) defines a section of thebundle B →Ag. It follows that the extension (9) is split.Proposition 8.1 π1(B, ∗) ≈L(l) ⋉GZ.We now restrict ourselves to genus 3 and consider the problem of liftingthe period map M3(l) →A3(l) to B.The idea is that such a lifting willbe the period map of a variation of mixed Hodge structure.

To this end wedefine certain algebraic cycles which are just more canonical versions of thecycle C −C−considered in §6.Suppose that C is a curve of genus 3 and that α is a theta characteristic ofC — i.e., α is a square root of the canonical divisor κC. Denote the algebraiccycle corresponding to the canonical inclusion C ֒→Pic1C by C. Denote theinvolution x 7→α −x of Pic1C by iα.

For D ∈Pic0C, denote the translationmap x 7→x + D byτ D : Pic1C →Pic1C.For D ∈Pic0C, define CD = τ D∗C and Zα,D = CD −iα∗CD. Each Zα,D ishomologous to zero.

Set Zα = Zα,0.Let Θα be the theta divisor{x + y −α : x, y ∈C} ⊆Pic0Cand ∆be the difference divisor{x −y : x, y ∈C} ⊆Pic0C.The following fact is easily verified.23

Proposition 8.2 The cycles Zα and Zα,D have disjoint supports if and only ifD ̸∈Θα ∪∆.Now choose a point δ ∈Pic0C of order 2 such that β := α + δ is an eventheta characteristic. (i.e., h0(C, β) = 0 or 2.

)Proposition 8.3 The cycles Zα and Zα,δ have disjoint supports except whenC is hyperelliptic and either δ is the difference between two distinct Weierstrasspoints or α + δ is the hyperelliptic series.Proof. By (8.2), Zα and Zα,δ intersect if and only if δ ∈∆or δ ∈ϑα.

In thefirst case, there exist x, y ∈C such that x −y = δ ̸= 0. So 2x −2y = 0, whichimplies that C is hyperelliptic and that x and y are distinct Weierstrass points.In the second case, there are x, y ∈C such thatx + y = α + δ,which implies that α + δ is an effective theta characteristic.

Since α + δ is evenby assumption, h0(C, α + δ) = 2, which implies that C is hyperelliptic and thatα + δ is the hyperelliptic series.Next we use these cycles to construct various variations of mixed Hodgestructure whose period maps give lifts of the period map M3(l) →A3(l) toQ ×A3(l) P, and to B generically.For each D ∈Pic0C, one has the extension of mixed Hodge structure0 →H3(Pic1C; Z(−1)) →H3(Pic1C, Zα,D; Z(−1)) →Z →0where the generator 1 of Z is the image of any relative class [Γ] with ∂Γ = Zα,D.Pushing this extension out along the projectionH3(Pic1C) →QH3(Pic1C)one obtains an extension0 →QH3(Pic1C; Z(−1)) →E →Z →0. (10)This extension determines the same point in J(QH3(Pic1C)) ≈Q1(Jac C) asthe cycle Cx −C−x , and is independent of the choice of D.Dually, one can consider the extension0 →Z(1) →H3(Pic1C −Zα,D; Z(−1)) →H3(Pic1C; Z(−1)) →0(11)which comes from the Gysin sequence.

Here the canonical generator of Z(1)is the boundary of any 4 ball which is transverse to Zα,D and has intersectionnumber 1 with it.24

Proposition 8.4 If D is a point of order 2 in Jac C, then there is a morphismof Hodge structuresH1(C, Z) →H3(Pic1C −Zα,D; Z(−1))whose composition with the natural mapH3(Pic1C −Zα,D; Z(−1)) →H3(Pic1C; Z(−1))is the map× [C] given by Pontrjagin product with [C].Proof. The exact sequence of Hodge structures0 →H1(C; Z)×[C]−→H3(Pic1C; Z(−1)) →QH3(Pic1C; Z(−1)) →0induces an exact sequence of Ext groups0 →Ext1(QH3(Pic1C, Z(−1)), Z(1)) →Ext1(H3(Pic1C, Z(−1)), Z(1)) →Ext1(H1(C; Z), Z(1)) →0.This sequence may be identified naturally with the sequence0 →JPH3(Pic1C; Z(−1)) →JH3(Pic1C; Z(−1))φ→Jac C →0,where φ is the map induced by the morphism of Hodge structures H3(Jac C; Z) →H1(C; Z) defined byx × y × z 7→(x · y)z + (y · z)x + (z · x)y.Here x, y, z are elements of H1(C; Z), × denotes the Pontrjagin product, and( · ) denotes the intersection pairing.

To prove the assertion, it suffices to showthat the image in Jac C of the class of the extension (11) vanishes. Lefschetzduality gives an isomorphism of (11) with the the extension0 →H3(Pic1C; Z(−1)) →H3(Pic1C, Zα,D; Z(−1)) →Z →0It follows directly from [22, (6.7)] that the image of this extension under themap φ is KC −2(α + D) = 0.Taking D = 0 and dividing out by this copy of H1(C; Z), we obtain anextension0 →Z(1) →F →QH3(Pic1C; Z(−1)) →0.

(12)Let M3(l, α, δ) be the moduli space of genus 3 curves with a level l structure,a distinguished even theta characteristic α, and a distinguished point δ ∈Pic0Cof order 2 such that α + δ is also an even theta characteristic.Denote theuniversal jacobian over M3(l, α, δ) by J →M3(l, α, δ). The period maps for25

the extensions (10), (12), respectively, define maps J →Q and J →P. Theseinduce a map φ into their fibered product over A3(l) such that the diagramJφ−→Q ×A3(l) P↓↓M3(l, α, δ)→A3(l)commutes.

Pulling back the C∗bundle B →Q ×A3(l) P along φ gives a C∗bundle L∗→J . Denote the corresponding line bundle by L.Denote the relative difference divisor in J by D and the relative theta divisorassociated to α by ϑαLemma 8.5 The Chern class of this line bundle is the divisor J is 2D −4ϑα.Proof.

We construct a meromorphic section of L. A point of J is a curve C anda point D of Jac C. According to (8.2), the cycles Zα and Zα,D have disjointsupports when D ̸∈∆∪Θα. In this case we can consider the mixed Hodgestructure H3(Pic1C −Zα, Zα,D; Z(−1)).4 Dividing this biextension out by theimage of the compositeH1(C, Z) →H3(Pic1C −Zα; Z(−1)) →H3(Pic1C −Zα, Zα,D; Z(−1))of the map of (11) with the natural inclusion produces a biextension bC,α withweight graded quotients canonically isomorphic toZ,QH3(Jac C; Z(−1)),Z(1);the generator 1 of Z corresponding to any Γ with ∂Γ = Zα,D, and the canonicalgenerator 2πi of Z(1) being the class of the boundary of any small 4 ball whichis transverse to Zα and having intersection number 1 with it.

This defines a lift˜φ : J −(D ∪ϑα) →Bof the map φ : J →Q ×A3(l) P.It therefore defines a nowhere vanishingholomorphic section s of L →J on the complement of D ∪ϑα.It followsfrom [8, (3.4.3)] that s extends to a meromorphic section of L on all of J .Consequently, the Chern class of this bundle is supported on the divisor D ∪ϑα.Since the divisors D and ϑα are irreducible, the Chern class can be computedby restricting to a general enough fiber.With the aid of (8.4), and the formula [8, (3.2.11)], one can easily show thatthe height of the biextension bC,α equals that ofH3(Pic1C −Zα, Zα,D; Z(−1)).4This is everywhere a local system. One has to replace it with another group when C ishyperelliptic and either α or α + δ is the hyperelliptic series.

For details, see the footnote onpage 887 of [8].26

It follows from the main theorem of [8] that the divisor of s restricted to Jac Cis 2D −4Θα for all C. The result follows.The point δ of order 2 is a section of the bundle J →M3(l, α, δ). A liftζ : M3(l, α, δ) →Q ×A3(l) P of the period map can be defined by composingδ with φ.

The pullback of the line bundle L along δ equals the pullback ofthe biextension line bundle to M3. It follows that the Chern class of this linebundle is 2δ∗(D −2ϑα).Proposition 8.6 If α and α + δ are even theta characteristics, then the push-forward of the divisor δ∗(D −2ϑα) in M3(l, α, δ) to M3(l) is 28.35 times thehyperelliptic locus.

Consequently, the line bundleδ∗L ∈Pic M3(l, α, δ) ⊗Qis non-trivial.In the proof of this result, we will need the following fact.Lemma 8.7 The section δ : M3(l, α, δ) →J is transverse to the divisors Dand ϑα.Proof. We first prove that δ intersects ϑα transversally.

We view M3(l, α, δ)as a subvariety of J via the section δ. Let (C, α, δ) be a point in ϑα.

Then,by (8.3), C is hyperelliptic, and α + δ is the hyperelliptic series. So h0(α) = 0and h0(α + δ) = 2.

Let Z0 be the period matrix of C with respect to somesymplectic basis of H1(C; Z), and θα(u, Z) the theta function which defines ϑα.The point δ of order 2 may be viewed as a function δ(Z). Set δ0 = δ(Z0).

Sinceδ0 ∈Θα, θα(δ0, Z0) = 0. We have to show that there exist a, b such that∂∂Zabθα(δ(Z), Z)|Z0 ̸= 0.By Riemann’s Theorem [6, p. 348], the multiplicity of δ on Θα is h0(α + δ) = 2.That is,∂θα∂ua(δ0, Z0) = 0(13)for all indices a, but there exist indices a, b such that∂2θα∂ua∂ub(δ0, Z0) ̸= 0.Substituting (13) into the chain rule, we have∂θα∂Zab(δ(Z), Z)|Z0 =gXj=1∂θα∂uj(δ0, Z0) ∂uj∂Zab(δ0, Z0)+ ∂θα∂Zab(δ0, Z0) = ∂θα∂Zab(δ0, Z0).27

Plugging this into the heat equation, we obtain the desired result:∂θα∂Zab(δ(Z), Z)|Z0 = ∂θα∂Zab(δ0, Z0) = 2πi(1 + δab) ∂2θα∂ua∂ub(δ0, Z0) ̸= 0.To prove that δ is transverse to D, we use an argument suggested to usby Nick Katz. Consider a family of curves C →Spec C[ǫ]/(ǫ2) over the dualnumbers.

There is a relative difference divisor ∆→Spec C[ǫ]/(ǫ2) contained inthe Picard scheme Pic0C →Spec C[ǫ]/(ǫ2). Suppose that we have a point oforder 2δ : Spec C[ǫ]/(ǫ2) →Pic0C,defined over the dual numbers which lies in ∆.

To prove transversality, it sufficesto show that C is hyperelliptic over the dual numbers. But this is immediate asδ gives a 2:1 map C →P1 defined over the dual numbers.Proof of (8.6).

Denote the hyperelliptic series of a hyperelliptic curve by H.It follows from (8.5) and (8.7) thatδ∗D = H∆andδ∗ϑα = H0whereH∆=(C, α, δ) : C is hyperelliptic , δ ∈∆and h0(C, α + δ) is evenandH0 = {(C, α, δ) : C is hyperelliptic , α ̸= H, α + δ = H} .If C is hyperelliptic and if δ = x −y ∈∆, then x and y are distinct Weierstrasspoints, andH + δ = 2y + x −y = x + ywhich is an odd theta characteristic. It follows thatH∆=(C, α, δ) : C is hyperelliptic , α ̸= H, δ ∈∆and h0(C, α + δ) even.Apart from the hyperelliptic series, every even theta characteristic on ahyperelliptic curve C is of the form−H + p1 + p2 + p3 + p4 = −H + q1 + q2 + q3 + q4where p1, .

. .

, p4, q1, . .

. , q4 are the Weierstrass points.

If α = −H+p1+p2+p3+p4 and α + x −y is an even theta characteristic, then it is not difficult to showthat x −y = qi −pj for some i, j. So, for each even theta characteristic α ̸= H,there are 16 points δ ∈∆such that α + δ is also an even theta characteristic.Since there are 35 even theta characteristics α ̸= H, it follows that H∆hasdegree 16.35 over the hyperelliptic locus H of M3(l).

Since there is only onepoint δ of order 2 such that α + δ = H, H0 has degree 35 over H.28

Putting this together we see that the pushforward of δ∗L to M3 isπ∗(2H∆−4H0) = (2.16.35 −4.35)H = 28.35H.We are now ready to prove (7.7). Denote the subgroup of Γ3 which cor-responds to M3(l, α, δ) by Γ3(l, α, δ), and its intersection with T3 by T3(α, δ).Proof of (7.7).

Let N be the line bundle over A3(l) which is the determinant ofR1f∗O, where f : J →A3(l) is the universal abelian variety. The restriction ofthis to M3(l) is H/9 [12, p. 134].

We shall also denote its pullback to Q×A3(l)Pby N. It follows from (8.6) that the line bundle L ⊗N ⊗(−9.28.35) pulls back tothe trivial line bundle over M3(l, α, δ). There is therefore a lift of the periodmapM3(l, α, δ) →L ⊗N ⊗(−9.28.35)∗to the C∗bundle associated to L ⊗N ⊗(−9.28.35).

This induces a group homo-morphismΓ3(l, α, δ) →π1(L ⊗N ⊗(−9.28.35)∗, ∗).This last group is an extension1 →GZ →π1(L ⊗N ⊗(−9.28.35)∗, ∗) →L(l) →1.It follows from (6.3) and the fact that T3(α, δ) has finite index in T3 that theimage of the mapH1(T3(α, δ); Q) →H1(GZ; Q) = QH3(Jac C; Q) ⊕PH3(Jac C; Q) ≈V (λ3)2is the diagonal copy of V (λ3). The restriction of the extension1 →Q →GQ →V (λ3)2 →1to the the diagonal is the extension given by the polarization of V (λ3).

It followsthat the homomorphismQ ֒→Λ2H1(T3; Q) →Qgiven by evaluating the bracket on the polarization is an isomorphism, as claimed.Second, the element of H2(L(l); Q) which corresponds to the extension0 →Q →H →L(l) →1constructed from the Torelli group is just the Chern class of the pullback ofthe line bundle L ⊗N ⊗(−9.28.35) pulled back to A3(l) along the zero sectionof Q ×A3(l) P →A3(l).This is just −9.28.35c1(N), which is nonzero inH2(A3(l); Q). Consequently, the extension is non-trivial as claimed.29

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