IfFpis not contained inF, thenF′is given by elementary modification ofF alongF ∩Fp. In particular,rk(F) = rk(F′)anddeg(F) = deg(F′) + 1.Ebeing semistable implies:
µ(F′) =deg(F)−1
rk(F) ≤deg(E) rk(E) − 1
rk(F)< deg(E)−1
rk(E) =µ(E′).
IfFpis contained inFthen the sheaf inclusionE′→Erestricts to an isomor- phism betweenF′andF. In this case (this is where we needEto be of degree 1):
deg(F′)rk(E′) = deg(F)rk(E)<deg(E)rk(F) = 1·rk(F) = rk(F′).
This can only be the case ifdeg(F′)rk(E′)≤0 = deg(E′)rk(F′).
Definition 5.8. Letq0 : P → M(n,∆0)be the map that takes a class of pairs [(E,F)]to the elementary modification ofEin the direction ofF.
Remark5.9. Notice thatq1is aCPn−1-fibration. For later use we note the follow- ing explicit isomorphism:
GivenE ∈ M(n,∆1), the fibreq1−1(E)is canonically isomorphic toP(Ep∗).
Forω∈Ep∗\ {0}, the isomorphism takes[ω]∈P(Ep∗)to[(E,Ker(ω))]∈q1−1(E).
Remark5.10. In fact,P is isomorphic to a moduli spaceM(χ, a, d)of parabolic bundles of rankn, sequence of multiplicitiesχ= (1,(n−1)), weightsa= (0,0) and degreed. (See chapter 3 in [27].) Hence the methods of section 2.5 could be used to show thatq0andq1are in fact morphisms.
Definition 5.11. There is a natural action ofJ(n)onP. The action ofα∈ J(n) maps a class[(E,F)]into[(E⊗Lα,F ⊗Lα)]. By usual abuse of notation, this map will be denoted simply byα.
Lemma 5.12. With the above definition, bothq0andq1becomeJ(n)-equivariant.
Proof. It is obvious thatq1becomes equivariant.
As for q0, Let [(E,F)] ∈ P, and let E′ = q0([(E,F)]), such that Γ(E′) = Ker(λ), withλas in definition 5.1. Notice that elementary modification ofE⊗Lα
in the direction ofF ⊗Lαis equal toKer(˜λ), whereλ˜: Γ(E⊗Lα)→Cpis given by
λ˜:s∈ΓU(E⊗Lα)7→
0 , p /∈U
[s(p)]∈(E⊗Lα)p/(F ⊗Lα)∼=C , p∈U Now,Ker(˜λ) = Ker(λ)⊗Γ(Lα) = Γ(E′⊗Lα). This means thatE′⊗Lα∼= q0([(E⊗Lα,F ⊗Lα)]).
Suppose thatE ∈M(n,∆1)is fixed by a primitive element,α∈J(n). By the above,αacts on the fibreq−11 (E)∼=P(Ep∗). We will need an explicit description of this action.
Recall that the construction of∆α1, back in section 3.6 involved picking a pointpα ∈ πα−1(p)⊆ (Lα)p. Also recall thatπα∗ : ϑ−1α (∆1)→ |M(n,∆1)|αis a surjective morphism.
Lemma 5.13. SupposeE=πα∗(L). Letψ:E→E⊗Lαbe the explicit isomorphism constructed in lemma 3.17.
DefineA:Ep→Epby the following commutative diagram:
Ep A
//
ψp
$$J
JJ JJ JJ JJ
J Ep
Ep⊗(Lα)p φtpαttttttt::
tt
(5.3)
– whereφpαis the map:(x⊗z·pα)7→z·x.
We then have the following commutative diagram:
P(E∗p)
∼=
P(AT)
//P(E∗p)
∼=
q−1(E) α //q−1(E)
–WhereAT is the dual map toA, and the vertical isomorphisms are the one from remark 5.9.
Proof. Let ω ∈ Ep∗. Sending[ω] through the diagram, to the right, and then down, yields:
[(E,Ker(ω◦A))].
Sending it down and then to the right yields:
[(E⊗Lα,Ker(ω)⊗(Lα)p)].
But by (5.3),ψ:E→E⊗Lαtakesx∈Ker(ω◦A)toAx⊗pαwhich lies in Ker(ω)⊗(Lα)p. This shows that the two classes are the same.
Again letαbe a primitive element inJ(n)and denote by|P|αthe subvariety of P fixed by α. For every L ∈ ϑ−1α (∆1)the projection induces a canonical isomorphism:
(M
ζ∈µn
ζ∗L)pα ∼=πα∗(L)p
and thereby specifies an(n−1)dimensional subspace of πα∗(L)p, namely the one corresponding to(L
ζ6=1ζ∗(L))pα.
Proposition 5.14. Letα∈J(n)be primitive. There is a bijection: jpα :ϑ−1α (∆1)→
|P|αdefined byjpα(L) = [(πα∗(L),(L
ζ6=1ζ∗(L))pα)], making the following diagram commutative:
|P|α
q1
%%K
KK KK KK KK
K ϑ−1α (∆1)
πα∗
xxppppppppppp
jpα
oo
|M(n,∆1)|α
Proof. Let L ∈ ϑ−1α (∆1) and denote πα∗(L) by E and (L
ζ6=1ζ∗(L))pα byF.
From lemma 3.17 we have an isomorphism ψ : E ∼= E ⊗Lα given by de- scent ofψ, where˜ ψ˜is induced by the matrixB= diag(1, ζn, . . . , ζnn−1)mapping Ln−1
j=0 ζnj∗(L)to itself. SinceEis stable and therefore simple, any other such iso- morphism is given by a non-zero complex scalar timesψ. Thus,jpα(L)is fixed by the action ofαprecisely ifψtakesF toF ⊗Lα. Clearly, this is always the case.
Ifjpα(L) =jpα(L′)forL, L′ ∈ϑ−1α (∆1), lemma 3.13 implies thatL′∼=ζ′∗(L) for someζ′∈µn. But then,(L
ζ6=1ζ∗(L′))pα ∼= (L
ζ6=ζ′ζ∗(L))pα. Since this must be mapped to(L
ζ6=1ζ∗(L))pα by an automorphism ofE (again, those are all constant), we get thatζ′= 1. This shows thatjpαis injective.
As for surjectivity, let[(E,F)] ∈ |P|α. SinceE ∈ |M(n,∆d)|α, pick L ∈ ϑ−1α (∆1)withπα∗(L) =E. The fact thatαfixes[(E,F)]implies thatFis induced by an(n−1)dimensional subspace of(Ln−1
j=0 ζnj∗(L))pα, which is invariant under B. These are all given by(L
ζ6=ζ′ζ∗(L))pα)for someζ′ ∈µn. Hence,[(E′,F)] = jpα(ζ′∗(L)).
Remark5.15. Again, with a little more theory on parabolic bundles, the results of section 2.5 could be applied to show thatjpα is an isomorphism of varieties.
However, we will only need the fact that it is a bijection.
The mapjpαfits nicely with the other half of the Heche diagram:
Proposition 5.16. For anyL∈ϑ−1α (∆1), we have:
q0(jpα(L)) =πα∗(L⊗[pα]−1).
Proof. LetL ∈ ϑ−1α (∆1). LetE = πα∗(L)andF ⊆ Ep the subspace such that jpα(L) = [(E,F)]. LetE′ =q0(jpα(L)).
Choose an isomorphism: Lpα ∼= C. Using the canonical isomorphism be- tweenEpand(L
ζ∈µnζ∗L)pα, this gives an isomorphism:
Ep/F ∼= (M
ζ∈µn
ζ∗(L))pα/(M
ζ6=1
ζ∗(L))pα ∼=Lpα∼=C
and hence a mapλ: Γ(E)→Cp.
LetE = Γ(E),E′= Γ(E′) = Ker(λ), and letE˜= Γ(π∗α(E))∼= Γ(L
ζ∈µnζ∗(L)).
We have a commutative diagram:
0 //E˜′ //E˜ ˜λ //S //0
0 //E′ //E?OO λ //
C?OOp //
0 –whereS = L
ζ∈µnζ∗(Cpα), the vertical maps are inclusions of invariant sec- tions, and˜λis given by evaluation of sections in ζ∗(L)in ζ−1(pα)(composed with the chosen isomorphismLpα ∼=C). The kernelE˜′ ofλ˜is simply the sheaf of sections inL
ζ∈µnζ∗(L⊗[pα]−1).
The diagram shows thatE′ is the invariant part ofE˜′, and therefore, E′ is isomorphic to the descent ofL
ζ∈µnζ∗(L⊗[pα]−1), i.e. toπα∗(L⊗[pα]−1).
Lifting the action
This chapter introduces certain groups ofliftsof the action ofJ(n)onM(n,∆d) to the canonical ample generator ofPic(M(n,∆d)).
The groups constitute central extensions of J(n), and they turn out to be defined independently of the complex structure on Σ. In chapter 7 we shall further see how the lifts induce actions on the Verlinde bundles over Teichm ¨uller space of the underlying surface ofΣ, and thereby become represented on the Verlinde vector spaces in the construction of topological quantum field theories from gauge theory and conformal field theory.
During the investigation of the groups we gradually strengthen the assump- tions, finally giving a complete presentation of the groups of lifts in the special case wherenis an odd prime. (Buta priorin≥2is arbitrary.)
By ongoing abuse of notation, we will make no distinction between elements α∈J(n)and the automorphisms ofM(n,∆d)given by the action ofα.
6.1 Definition of the lifts
A main result of [8] is that the Picard group ofM(n, d)is generated by pull- backs fromJ(Σ)(under the determinant map) along with a canonical, ample1 line bundle,Ln,d. Furthermore, each subvarietyM(n,∆d)have Picard group isomorphic toZwith the restriction ofLn,das its generator. This restriction is of course still ample and will also be denoted byLn,d.
1For a brief introduction to ample line bundles, see [35] p.143-156.
91
Definition 6.1. By aliftofαtoLn,d, we shall mean an invertible bundle mapρ fromLn,dto itself, inducingαon the base.
Lemma 6.2. For eachα∈J(n), there exist lifts ofα.
Proof. Pull back byαinduces an endomorphismα∗ofPic(M(n,∆d))∼=Zwith (α∗)n=1. Henceα∗(Ln,d) =L⊗(±1)n,d . (The option ”−1” being relevant only when nis even, of course.) But sinceLn,dis ample, so isα∗(Ln,d)and by the Kodaira vanishing theorem,L−1n,dis not ample. Thus,α∗(Ln,d)∼=Ln,d.
By choosing an isomorphismLn,d ∼=α∗(Ln,d)and composing this with the canonical (invertible) bundle mapα∗(Ln,d)→ Ln,d(inducingαon the base), we get a lift ofα.
Definition 6.3. LetG(J(n),Ln,d)denote the group consisting of all possible lifts of elements inJ(n).
Lemma 6.4. G(J(n),Ln,d)is a central extension:
{1} →C∗→ G(J(n),Ln,d)→J(n)→ {0} (6.1) Consequently, a unique liftρof a givenα ∈ J(n)may be specified by demanding thatρact by multiplication with a certain scalar on the fibre ofLn,d over a pointx∈ M(n,∆d)fixed byα.
The specified lift then acts by multiplication with the same scalar in every fibre above the connected component of|M(n,∆d)|αcontainingx.
Proof. A lift of0∈J(n)is simply an algebraic function onM(n,∆d)and hence, M(n,∆d)being complete, a non-zero constant. This shows that the sequence (6.1) is a central extension.
Therefore, two different lifts of an elementα ∈J(n)differ only by a scalar.
And since a liftρofαmultiplies by a non-zero scalar in a fibre ofLn,d over a point in|M(n,∆d)|α, fixing this scalar determinesρuniquely.
The final claim is true becauseρnis a lift of the identity, and hence constant.
Therefore, the action on fibres above fixed points can vary only byn’th roots of unity. By continuity, it must be constant on connected components.
Definition 6.5. Ifnis odd, for each elementα∈J(n), defineρα,dto be the lift of αtoLn,d, which acts as the identity on the fibre over each point in|M(n,∆d)|1α. Denote byE(n, d)the subgroup of G(J(n),Ln,d)generated by {ρα,d | α ∈ J(n)}.
Definition 6.6. Ifnis even, for each element a ∈ J(2n), defineρa,d to be the lift ofα= 2atoLn,d, which acts as the identity on the fibre over each point in
|M(n,∆d)|1a.
Denote by E(n, d) the subgroup ofG(J(n),Ln,d) generated by{ρa,d | a ∈ J(2n)}.
Remark6.7. Notice that we suppress the dependence onnofρα,dandρa,din the notation. This should not cause confusion, sincenwill always be fixed when discussing the lifts.
Whenever appropriate, we will also suppress the dependence ond, writing:
ρα,d=ραandρa,d =ρa.