Admissible holomorphic structures on the line bundle over the torus 110

ˆ

xˆsˆt⁻¹, z

χ=C

Π(ˆstˆ⁻¹)

[ˆx, z]_χ . The notationsΠ and C are transferred from Definition 2.33 as well.

On the other hand, the map Tˆ×L˜_χ −→L˜_χ (ˆs,ˆt, z

χ) 7−→

ˆ sˆt, z

χ

is smooth which makes Tˆ a Lie transformation group ofL˜_χ.

Corollary 5.15:

Let χ be a character ofπ1(G).

ThenL:= ˜G×_χC becomes a smooth, complex line bundle over the fibre bundle G→G/T in the sense of Definition 2.36.

Its fibre type is the smooth, complex line bundleL˜ = ˆT×_χC→T.

The aim of this subsection, i.e. Section 5.4, is to show thatLbecomes compatible.

Therefore, in order to fulfil the first property of Definition 3.1, we still have to show that

˜

ρrespects the holomorphic structure and acts Hermitian.

But, so far, we did not introduce a holomorphic and Hermitian structure onL˜. The following two sub-subsections are dedicated to describe these two structures onL˜ and to show thatρ˜indeed acts onL˜ as wanted, i.e. Hermitian and respecting the holomorphic structure.

5.4.2 Admissible holomorphic structures on the line bundle over the torus This sub-subsection is dedicated to describe the set of holomorphic structures on the complex line bundleL˜ = ˆT ×_χCand to show thatρ˜respects these holomorphic structures.

Recall that we saw in Section 5.3 that every the representationχ:π1(G)→C^∗ induces a holomorphic structure onL˜ = ˆT ×_χC. On the other hand, every flat complex line bundle over the torus is trivial (compare Remark 5.7).

Thus, we study the different holomorphic structures induced by those representationsχ on the trivial line bundle.

5.4 Compatible line bundles

Although, it is not necessary, for the rest of this thesis, to understand the different holomorphic structures on the trivial line bundleL˜₁:=T ×Cinduced by different charactersχ, we investigate this for the convenience of the reader.

The reader, who wants to skip this investigation, may resume reading at Lemma 5.21 on page 116.

The general theory of holomorphic line bundles over complex tori is described and proven in [7]. We are interested in the groupPic⁰(T) only, i.e. the holomorphic line bundles that are isomorphic to the trivial line bundle as smooth complex vector bundles. Therefore, we will not review the whole theory here.

However, there is one fact stated in [7] that we need here. It is a statement included in theAppell-Humbert Theorem.

Theorem (Appell-Humbert):

LetΛ be a lattice of maximal rank in Cⁿ and let further onT be the quotient space Cⁿ/Λ, then the map

Hom(Λ, U(1)) −→ Pic⁰(T)

which sends every unitary character χ: Λ =π₁(T)−→U(1) to its associated, holomorphic vector bundle Cⁿ×_χCis an isomorphism of groups.

Remark 5.16:

The theorem of Appell-Humbert implies that we obtain all possible equivalence classes of holomorphic structures on T×C that way.

In particular, it suffices to chose unitary representations χ, although π₁(T) is not finite.

Now, in order to compute the different holomorphic structures onL˜₁, we write down an explicit isomorphism of vector bundles betweenL˜₁ and L˜_χ:=Cⁿ×_χC.

Lemma 5.17:

Let χ: Λ−→U(1)be a character and let T be a torus given by T =Cⁿ/Λ for a lattice Λ⊂Cⁿ of maximal rank.

Furthermore, letωχ be in Hom_R(Cⁿ,R) such that χ(λ) =e^iωχ(λ) for every λ∈Λ.

1. We get an isomorphism of smooth vector bundles induced by ω_χ: L˜₁

ψ_ωχ

// ˜L_χ

via (tˆ , z)

!!

ψ_ωχ

//h

ˆt, e^{−iωχ(ˆt)}·zi 4 χ

yyT ˆt

depending on ωχ.

2. Every other choice ω˜_χ∈Hom_R(Cⁿ,R) with χ(λ) =e^i˜ωχ(λ) fulfils:

(ωχ−ω˜χ) (Λ)⊂2πZ. (49)

3. The isomorphism ψ_ωχ differs from ψ_ω˜χ by a map µ:T →U(1).

Proof.

1. At first, we check thatψ_ωχ is well defined:

(ˆt

, z) ^{ψωχ //}

hˆt, e^{−iωχ(ˆt)}·zi

χ

(ˆt+λ

, z) ψ_ωχ //h

ˆt+λ, e^{−iωχ(ˆt+λ)}·z i

χ.X The right vertical equality holds because

hˆt+λ, e^{−iωχ(ˆt+λ)}zi

χ=



ˆt, χ(λ)e^{−iωχ(ˆt)}e^−iωχ(λ)

| {z }

=χ(λ)⁻¹

z





χ

=h

ˆt, e^{−iωχ(ˆt)}zi

χ. ψωχ is obviously smooth, bijective andC-linear on fibres, hence, it is an isomorphism of smooth complex line bundles.

2. The mapχ is defined onΛ.

We deduce that we obtain for every λ∈Λ: χ(λ) =e^iωχ(λ)=e^i˜ωχ(λ).

Equivalently, we observe that e^{i(ωχ(λ)−˜ωχ(λ))} = 1. Now, Equation (49) follows trivially.

3. Letδω denote the differenceωχ−ω˜χ∈Hom_R(Cⁿ,R). It follows, by 2., that

δω(Λ)⊂2πZ.

We compute for tˆ∈Cⁿ and z∈C:

ψω˜χ(tˆ , z) =

hˆt, e^{−i(ωχ(t)−δω(ˆt))}·z i

χ =e^{−iδω(ˆt)}

| {z }

=:˜µ(ˆt)

hˆt, e^{−iωχ(ˆt)}·z i

χ

= ˜µ(ˆt)·ψωχ(ˆt , z)

5.4 Compatible line bundles

The next step is to compare the different holomorphic structures on the trivial line bundleL˜₁=T×Cinduced by the charactersχ.

We saw in Lemma 5.10 that the natural holomorphic structure onL˜_χ= ˆT ×_χCis given via the following diagram

Γ(T,Tˆ×_χC)

ˆ κχ

∂¯Lχ˜ //A^(0,1)(T,Tˆ×_χC)

C^∞( ˆT ,C)^χ _¯

∂

//A^(0,1)( ˆT)^χ

ˆ κ⁻¹χ

OO

for the isomorphism ˆ

κ_χ : A^(0,∗)(T,Tˆ×_χC) −→ A^(0,∗)( ˆT)^χ

where we added a hat to the symbol of the mapκ_χ in order to discern it from the map κ_χ : A^(0,∗)(G,G˜×_χC) −→ A^(0,∗)( ˜G)^χ.

We now apply the bundle isomorphismψ_ωχ in order describe the holomorphic structure onT ×Cinduced byL˜_χ.

Note therefore thatω_χ∈Hom_R(Cⁿ,R) = Hom_R(T_eT,R)can be identified canonically with a one form inA¹(T) because the co-tangent bundle of the torus is trivial.

Lemma 5.18:

The holomorphic structure ∂¯ωχ on T×Cinduced by the isomorphism of complex line bundles

ψ_ωχ : T ×C= ˜L₁ −→ L˜_χ, defined in Lemma 5.17, is given by

∂¯ωχ = ¯∂−iε

proj^(0,1)(ωχ)

where proj^(0,1) denotes the projection onto the antiholomorphic subbundle of the complexified cotangent bundle(T^∗T)⊗_RC.

In particular, the following diagram commutes for κˆ1 = ˆκχ₁≡1: Γ(T,L˜_χ) ^ψ

−1ωχ //

∂¯L˜χ

Γ(T, T×C)

ˆ

κ1 //C^∞(Cⁿ,C)^χ1

∂¯_ωχ

A^(0,1)(T,L˜_χ)

ψ⁻¹_ωχ

//A^(0,1)(T, T ×C) _ˆκ

1

//A^(0,1)(Cⁿ,C)^χ1 Now, sinceωχ∈Hom_R(Cⁿ,R)⊂A¹(T) is real valued, we obtain

∂¯ωχ = ¯∂+¹₂ε(ωχ−iJ ωχ).

Proof.

The morphismψ_ωχ is covered by a morphism ψ˜_ωχ : T˜×C −→ T˜×C

˜t, z

7−→ (˜t, e^{−iωχ(˜t)}·z), thus, the following diagram commutes:

T˜×C

which maps periodic antiholomorphic forms toχ-equivariant antiholomorphic forms both living on the universal coveringT˜=Cⁿ ofT.

It is given by multiplication with the function e^−iωχ : T˜ −→ U(1) In order to compute∂¯_ωχ, it is sufficient to walk around the outer rectangle of this diagram.

Letf be a periodic function onT˜, i.e. f ∈C^∞( ˜T ,C)^χ1.

5.4 Compatible line bundles

We obtain:

∂¯ωχf = ˇψ_ω⁻¹_χ ◦∂¯◦ψˇωχf

= ˇψ_ω⁻¹_χ ◦∂ e¯ ^−iωχ·f

= ˇψ_ω⁻¹_χ

−iproj^(0,1)(ωχ)·e^−iωχ·f +e^−iωχ·∂f¯

= ¯∂f−iproj^(0,1)(ω_χ)·f which completes the proof.

Although, it is already clear, by the Appell-Humbert theorem, that the equivalence class of holomorphic structures onL˜₁ induced by ψ_ωχ does not depend on the choice of ω_χ, we show this once more in the subsequent corollary.

Corollary 5.19:

A differentω˜χ ∈Hom_R(Cⁿ,R) for the same representation χ, gives us a holomorphic structure∂¯_ω˜χ on T ×C equivalent to ∂¯_ωχ.

Proof.

Lemma 5.17) states that two isomorphismsψ_ωχ and ψ_ω˜χ differ by a map µ:T →U(1) which is given fort˜∈T˜through µ(˜t

) =e^iδω(˜t) whereδω denotes the difference ω˜_χ−ω_χ. Consequently, we obtain:

∂¯ω˜χ−∂¯ωχ =−iε

proj^(0,1)(δω)

=ε(¯µ∂µ).¯

Thus, both holomorphic structures onL˜₁ =T ×Care equivalent by Definition 2.5.

Remark 5.20:

For a complex torusT, the groupPic⁰(T)is very well understood. Using the Appell-Humbert theorem, we see that it is isomorphic to Hom(Λ, U(1)).

In addition to that, we have the following exact sequence of groups.

0→Hom(Λ,Z)−→^2π· Hom(Λ,R)^exp(i·)−→ Hom(Λ, U(1))→1

We obtain Hom(Λ, U(1))∼= Hom(Λ,R)/Hom(Λ,Z)∼=R²ⁿ/Λ^∨ whereΛ^∨ denotes the dual lattice.

Consequently,Pic⁰(T)has the structure of a torus.

As a last statement concerning the holomorphic structure∂¯L˜χ, we show thatρ˜is compatible with this holomorphic structure.

Lemma 5.21:

Let ρ˜: ˆT −→Diff(T,L˜_χ) be the action described in Proposition 5.13.

Let furthermoreρˇdenote theρ˜-induced action on A^(0,∗)(T,L˜_χ). (compare Remark 3.3).

Then the following identity holds for every ˆs∈Tˆ: ˇ

ρ(ˆs)◦∂¯_L˜

χ = ¯∂_˜L

χ◦ρ(ˆˇs),

i.e. the action ρ˜respects the holomorphic structure ∂¯_L˜

χ. Proof.

Without loss of generality, assume thatTˆ = ˜T. We have to show that the following diagram commutes. Consequently, the following identity holds:

ˇ

ρ(˜s)(f) =κ⁻¹_χ ◦(L_˜s⁻¹)^∗◦κ_χ(f). (50) This equality extends from sectionsΓ(T,L˜_χ) to antiholomorphic forms A^(0,∗)(T,L˜_χ). Observe now that the following diagram commutes.

A^(0,q)( ˜T ,C)^χ

5.4 Compatible line bundles

The left and right trapezoid commute because of the definition of the natural holomorphic structure∂¯_L˜

χ and the upper and lower trapezoid commute because of Equation (50).

At last, the outer rectangle commutes becauseLs˜: ˜T −→T˜ is obviously biholomorphic.

Therefore, the inner rectangle commutes and the assertion is proven.

Summarising, we showed thatL˜_χ, although being isomorphic to the trivial complex line bundleL˜₁=T×C, does have an excelled, in general non-trivial, equivalence class of holomorphic structures. Furthermore, this class of holomorphic structures is respected by the actionρ˜of Tˆ.

Now, in order to show the first property of Definition 3.1, we still have to equipL˜_χ with a Hermitian metric such thatρ˜acts Hermitian onL˜_χ. This is the content of the

subsequent sub-subsection.

5.4.3 Invariant Hermitian metric

In the previous subsection, we investigated the natural holomorphic structure on the line bundleL˜_χ=Cⁿ×_χCand showed that it is respected by the actionρ.˜

In this sub-subsection, we define a natural Hermitian metrich^L˜χ on L˜_χ which is ρ˜ invariant.

We already restricted our representation to be unitary, i.e. χ:π1(G)→U(1). Thus, we inherit an induced Hermitian metric by

h^L_t^˜χ(t, zˆ 0

χ,ˆt, z1

χ) :=z0·z¯1. (51)

This metric is well defined and as the following equation shows it isρ˜-invariant.

h^L_t^˜χ

Thisρ˜-invariance is necessary as well as sufficient to induce a Hermitian metric h^Lχ on L_χ = ˜G×_χC.

To show this is the content of the next lemma.

Lemma 5.22:

Let ν be the isomorphism from Equation (48), i.e.:

ν : G˜×_ρ,˜TˆL˜_χ → L_χ= ˜G×_χC→G

We obtain an induced Hermitian metric h^Lχ on L_χ, given by:

Theρ˜-invariance of h^L˜χ now implies that this expression does not depend on the representing elementsh

˜

g,[ˆe, z_k]_χi

˜ ρ.

Up to this point, we have shown the following facts.

The bundleL= ˜G×_χCis a smooth complex line bundle over the manifoldGthat is a smooth vector bundle over the fibre bundleG→G/T. In particular, its fibre type as a bundle overG/T isL˜_χ:= ˆT ×χC which itself is a Hermitian, holomorphic line bundle.

Furthermore,L_χ is equipped with a Hermitian metric h^Lχ such that the induced Hermitian metrich^L˜χ onL˜_χ is ρ˜invariant.

Even further, we know thatL_χ has a naturally excelled holomorphic structure ∂¯_Lχ. What we show in the subsequent sub-subsection is, that the natural holomorphic

structure∂¯Lχ onL_χ induces the natural holomorphic structure ∂¯˜Lχ on L˜_χ. More general, we describe a set of holomorphic structures∂¯L on L_χ such that the induced holomorphic structure onL˜_χ is the naturally excelled holomorphic structure∂¯L˜χ.

5.4.4 Implications for the holomorphic structure on the line bundle over the

Im Dokument Equivariant holomorphic torsion for a fibre bundle (Seite 116-124)