Bijection of certain section spaces

This subsection is dedicated to the proof of Proposition 3.14, below.

It is about a vector space isomorphism between theλ-Eigensections of the vertical Laplace operatorV and the antiholomorphic forms on M with coefficients in the associated bundleW^(λ;∗).

Recall that we described the bundleW^(λ;∗) in Remark 3.3.

We denote the operator^[s,t]_V to be the vertical Laplacian acting on the space A^(0,s)_H (E)∧A^(0,t)_V (E,L).

Proposition 3.14:

Let (E, π_E,(M, g_M),(F, g_F), T^HE) be a holomorphic fibre bundle and letL→E be a compatible holomorphic line bundle of fibre type L˜ →F.

Furthermore, letK denote a Lie group, πP :P →M a K principle fibre bundle and ρ: K→ Aut(F) as well as ρ˜: K → Diff(F,L)˜

group homomorphisms such that

E =P ×_ρF and L=P×_ρ˜L.˜

Let λbe an Eigenvalue of andW^(λ;∗) be the vector bundleP × Ker( −λ)

3.3 Bijection of certain section spaces

Furthermore, let the space of λ-Eigenforms of V be denoted by Eig_λ

^[s,t]_V

:= Ker(V −λ)∩A^(0,s)_H (E)∧A^(0,t)_V (E,L).

Then there is an isomorphism ψ: Γ(M,W^(λ;∗)) −→˜ Eig_λ

^[0,∗]_V

⊂A^(0,∗)_V (E,L), This isomorphism can be extended to an isomorphism

ψ: A^(0,s)(M,W^(λ;t)) −→˜ Eig_λ

^[s,t]_V

⊂A^(0,s)_H (E)∧A^(0,t)_V (E,L).

Definition 3.15:

We call the isomorphismψ:A^(0,s) M,W^(λ;t)

−→Eig˜ _λ ^[s,t]_V

from the proposition above, i.e. Proposition 3.14,ψ-morphism.

We divide the proof of Proposition 3.14 into smaller pieces.

At first as a direct consequence of Lemma 3.16 (in Corollary 3.17), we obtain an

isomorphism between the vertical antiholomorphic forms,A^(0,∗)_V (E,L), and a subspace of C^∞

P ×F,Λ^∗ T^(0,1)F∗

⊗L˜ .

Later on in Corollary 3.20 following from Lemma 3.18, we see that the latter space can be identified withC^∞(P,A^(0,∗)(F,L))˜ ^ρˇ.

In Lemma 3.22, we check the compatibility of these identifications above with the vertical Laplace operator.

This finally leads to an isomorphism between Eig_λ(^0,t_V ) = Ker (V −λ)∩

A^(0,t)_V (E,L) and

C^∞

P,Eig_λ

^(0,t)_L˜ ρˇ

.

The latter one is by standard arguments for associated bundles in one to one correspondence to the space of sectionsΓ

M, P×_ρˇEig_λ

^(0,t)_L˜ . Lemma 3.16:

Let K be a Lie group and letP →M be a K principle fibre bundle over a compact manifold M.

Furthermore, letE be fibre bundle over M, associated to P via a group homomorphism ρ:K →Diff(F, F), whose fibre type F is a compact manifold, i.e.

E=P×_ρF.

Additionally, letQ →˜ F be a smooth vector bundle andρ˜:K →Diff(F,Q)˜ be a

continuous group homomorphism covering ρ, makingK a Lie transformation group ofQ˜. Denote the induced vector bundle over E by:

Q:=P×_ρ˜Q.˜

At last, let C^∞(P ×F,Q)˜ ^i,ii denote the set of smooth maps

˜

s: P×F −→ Q˜ with the properties:

i) ˜s(p, f)∈Q˜_f and

ii) ˜s

p·k⁻¹, ρ(k) (f)

= ˜ρ(k)

˜ s(p, f)

. Then there is an isomorphism e of vector spaces

e : Γ(E,Q) −→ C^∞(P×F,Q)˜ ^i,ii

s 7−→ s.˜

Proof.

Lets∈Γ(E,Q)be a smooth section and let e= [p, k]_ρ be inE. Thens(e) lies inQ_e. Therefore, it has the forms(e) = [p,s(p, f˜ )]_ρ˜.

The desired bijection is now given by:

e : Γ(E,Q) −→ C^∞(P×F,Q)˜ ^i,ii

s 7−→ ˜s

Note that for e to be well defined, s˜has to fulfil the second propertyii).

Furthermore,sis a section from E toQ, in particular,πQ◦s= id_E. This is corresponds on thes˜side to propertyi).

Conversely, any maps˜∈C^∞(P ×F,Q)˜ ^i,ii induces a maps:E→ Q that fulfils πQ◦s= id_E.

Summarising, we obtain

πQ◦s ←→ i)πQ˜ ◦s˜= proj₂ s([p, f]_ρ) =s(

p·k⁻¹, ρ(k)(f)

ρ) ←→ ii) ˜s(p·k⁻¹, ρ(k)(f)) = ˜ρ(k) (˜s(p, f)). What remains to be shown is that the smoothness ofsinduces the smoothness ofs˜and vice verse.

In order to do that, we will borrow some notations from Lemma 2.39. In particular, we denote the local trivialisations the same way.

3.3 Bijection of certain section spaces

We deduce:

s:E −→ Q isC^∞ ⇔ χ_ij◦s_|

Wij : W_ij −→ W_ij ×C^m is C^∞ ∀i, j

⇔ proj₂◦χ_ij◦s_|

Wij : W_ij −→ C^m is C^∞ ∀i, j On the other handWij is diffeomorphic toUi×Vj through qi×idVj and

proj₂◦χij◦s|_Wij ◦h

qi×id_Vj i

ρ= proj₂◦ψj◦s˜◦(qi×id_Vj).

Therefore,

sisC^∞ ⇔ proj₂◦ψ_j◦˜s◦

q_i×id_Vj

: U_i×V_j → C^m is C^∞∀i, j

⇔ s˜◦ ϕ⁻¹_i (·, e)×id_F

: U_i×F → Q˜ is C^∞∀i This fact implies that ifs˜:P×F →Q˜ is smooth, then so is its corresponding maps. For the opposite direction, suppose now thatsis a smooth map. The considerations above showed thatssmooth directly implies that the maps˜◦ ϕ⁻¹_i (·, e)×idF

is smooth as well.

In order to show thats˜itself is C^∞, we have to check that, locally for eachU_i,

˜

s◦ ϕ⁻¹_i ×idF

: Ui×K×F −→ Q,˜ i.e. the map(x, k, f)7−→˜s(qi(x)·k, f), is smooth.

But this is true because of propertyii) which implies

˜

s(q_i(x)·k, f) = ˜ρ k⁻¹

◦˜s ϕ⁻¹_i (x, e), ρ(k⁻¹)(f)

and this is smooth in(x, k, f) as composition of smooth maps.

For our compatible line bundleL=P×_ρ˜L˜ →E over a holomorphic fibre bundle, we can apply Lemma 3.16 not only on sectionsΓ(E,L)but on the vertical antiholomorphic formsA^(0,q)_V (E,L) with coefficients in Las well since for

ρ^∗ : K−→ Diff(F,Λ^qT^(0,1)F), given byρ^∗(k) :=ρ(k⁻¹)^∗, we obtain:

Λ^q

T^V,(0,1)E∗

=P ×_ρ^∗Λ^q

T^(0,1)F∗

. And consequently we get forρ^∗⊗ρ˜:

Λ^q

T^V,(0,1)E∗

⊗L=P ×_ρ^∗_⊗˜ρΛ^q

T^(0,1)F∗

⊗L.˜

Corollary 3.17:

For a holomorphic fibre bundle(E, πE,(M, gM),(F, gF), T^HE) and a compatible line bundle L, i.e.

L=P×_ρ˜L˜ −→ E =P×_ρF, we get an isomorphism of vector spaces

e : A^(0,q)_V (E,L) −→ C^∞(P ×F,Λ^q T^(0,1)F∗

⊗L)˜ ^i,ii

s 7−→ s˜

The next step is to find a bijection ofC^∞(P ×F,Λ^q T^(0,1)F∗

⊗L)˜ ^i,ii on the one hand and a subspace ofC^∞(P,A^(0,q)(F,L))˜ on the other hand.

Therefore, let C^∞

P ×F,Λ^q

T^(0,1)F∗

⊗L˜i

be the subspace of smooth mapss˜from P×F to Λ^q T^(0,1)F^∗

⊗L˜ that fulfil:

i) ˜s(p, f)∈ Λ^q

T^(0,1)F∗

⊗L˜

f. Obviously, we have the following inclusion:

C^∞

P ×F,Λ^q

T^(0,1)F∗

⊗L˜i,ii

⊂C^∞

P×F,Λ^q

T^(0,1)F∗

⊗L˜i

. Now, we show thatC^∞(P ×F,Λ^q T^(0,1)F^∗

⊗L)˜ ⁱ andC^∞(P,A^(0,q)(F,L))˜ are isomorphic.

The map describing the desired isomorphism is actually a very basic one.

If we forget any structure, letA, B, C be sets and let Map(A, B) denote the maps fromA toB.

The spaceMap(A×B, C) is isomorphic to the spaceMap(A,Map(B, C))in a natural way.The identification is given by

b : Map(A×B, C) −→ Map(A,Map(B, C)), wheref(a) :ˆ b7→f(a, b) and inverselyf(a, b) :=

fˆ(a) (b).

Unfortunately, if we return to the category of smooth manifolds, the translation of properties like continuity or smoothness via b are much more intricate.

The subsequent lemma shows that in the case of ^∞ ˜ ⁱ everything behaves just

3.3 Bijection of certain section spaces

Beforehand, we state the following theorem stated by Trèves in [30, Ch.40, Thm. 40.1].

We present it, tailored to the situation at hand.

Theorem (Trèves Thm 40.1):

Let A⊂Rⁿ andB ⊂R^m be open sets and letC =R^k. Then the map b : Map(A×B, C)−→Map(A,Map(B, C)),

introduced above, induces an isomorphism of topological vector spaces between C^∞(A×B,R) and C^∞(A, C^∞(B,R))by restriction.

Here, the topology on C^∞(X,R) is given by the Fréchet topology.

We give a brief summary of the Fréchet space structure onΓ(F,Q) in Appendix B.

Lemma 3.18:

Let B, F be smooth manifolds and let πQ˜ : ˜Q →F be a smooth vector bundle.

Let furthermoreC^∞(B×F,Q)˜ ⁱ denote the following topological vector space:

C^∞(B×F,Q)˜ ⁱ:=

n

g:B×F →Q |˜ πQ˜ ◦g= proj₂ o

.

Then there is an isomorphism b of the topological vector spaces C^∞(B×F,Q)˜ ⁱ and C^∞(B,Γ(F,Q))˜ .

b : C^∞(B×F,Q)˜ ⁱ −→^1:1 C^∞(B,Γ(F,Q))˜ Proof.

The bundleQ →˜ F is locally trivial. We denote the local trivialisation maps for Q →˜ F by(ψj, Vj), i.e..

ψ_j : Q |˜ _V

j −→ V_j ×C^m Now, forg∈C^∞(B×F,Q)˜ ⁱ the map

ˆ

g: B −→ Γ(F,Q)˜

is smooth if and only if the mapsgˆ_j defined by ˆ

g_j : B −→ Γ(V_j,Q |˜ _V

j) b 7−→ ˆg(b)|_V

j

are smooth for allj∈J wheregˆ_j(b) denotes the restriction ofg(b) to the local trivialisation base setVj ⊂F ofQ˜ (compare Corollary B.5).

In particular,gˆis smooth if and only if the maps ψj◦ˆgj : B −→ C^∞(Vj, Vj×C^m)

b 7−→ ψ_j◦gˆ_j(b) are smooth for anyj∈J.

Now, for everyb∈B and everyj∈J, the map ψ_j◦ˆg_j(b) : V_j −→ V_j×C^m

has got the formψ_j◦ˆg_j(b) = id_Vj׈h_j(b)for some ˆh_j(b)∈C^∞(V_j,C^m).

Looking at the Fréchet structure, we see thatgˆ:B →Γ(F,Q)˜ is smooth if and only if ˆh_j : B −→ C^∞(V_j,C^m)

is smooth for everyj∈J (compare Corollary B.6).

Reading this statement through charts, we can apply Trèves Theorem 40.1, stated above.

Hence,gˆis smooth if and only if

h_j : B×V_j −→ C^m

(b, f) 7−→

hˆ_j(b)

(f) = proj₂◦ψ_j◦(ˆg(b)) (f) is smooth for anyj∈J.

On the other handh_j is obviously smooth if and only if proj₂×hj : B×Vj −→ Vj ×C^m

(b, f) 7−→

f, hj(b, f) is smooth.

Now, by applyingψ⁻¹_j , we see that this is equivalent to ψ⁻¹_j ◦(proj₂×h_j) : B×V_j −→ Q˜

being smooth for anyj ∈J, since ψis a diffeomorphism.

But, if we make use of Propertyi), we obtain

ψ_j⁻¹◦(proj₂×hj) (b, f) =ψ⁻¹_j f,proj₂◦ψj◦(ˆg(b)) (f)

| {z }

=g(b,f)

!

=i) g(p, f).

Consequently, we obtain:

ψ⁻¹_j ◦(proj₂×h_j) : B×V_j −→ Q˜ (b, f) 7−→ g(b, f),

3.3 Bijection of certain section spaces

We obtain a direct consequence of the Lemma 3.18 by choosingB =P. Corollary 3.19:

In the setting from Lemma 3.16, there is an isomorphism b between the topological vector spaces C^∞(P ×F,Q)˜ ⁱ and C^∞(P,Γ(F,Q)).˜

b : C^∞(P×F,Q)˜ ⁱ −→^1:1 C^∞(P,Γ(F,Q))˜

This corollary on the other hand has as a direct consequence the subsequent corollary.

Corollary 3.20:

Here, we marked the equalities coming from the isomorphism b with an ⁰!⁰. It follows thatˆg lies inC^∞(P,Γ(F,Q))˜ ^ρˇ.

For the other direction, letgˆbe inC^∞(P,Γ(F,Q))˜ ^ρˇ. We now have to show thatg fulfils the property ii). We compute forp∈P,k∈K and f ∈f: which is what we wanted to show.

Corollary 3.21:

For a holomorphic fibre bundle(E, πE,(M, gM),(F, gF), T^HE) and a compatible line bundle L→E. We have an isomorphism

b◦e= b

e : A^(0,∗)_V (E,L) −→ C^∞(P,A^(0,∗)(F,L))˜ ^ρˇ. One question has been unanswered so far.

What happens under the correspondence b

e above with theλ-Eigenspaces of V? To answer this question is the next step.

In the definition of a compatible line bundle we fixed our holomorphic structure onL in such a way that the vertical LaplacianV corresponds to the Laplacian ˜L onL˜ →F. This property has the indispensable consequence that the morphism b

e from above, identifyingA^(0,∗)_V (E,L)withC^∞(P,A^(0,∗)(F,L)˜ ^ρ˜) restricts to a morphism:

A^(0,∗)_V (E,L)⊃Eig_λ(^0,∗_V )↔C^∞(P,Eig_λ(F))^ρˇ⊂C^∞(P,A^(0,∗)(F,L))˜ ^ρˇ This is the content of the following lemma.

Lemma 3.22:

Let α be a form in A^(0,q)_V (E,L) and let αˆ˜ ∈C^∞(P,A^(0,q)(F,L))˜ ^ρˇ be its image under b e. Then we get for everyp∈P:

[] Vα

(p) =L˜

α(p)ˆ˜ .

Proof.

For the local trivialisation maps, we stick to the notations we have evolved so far, nonetheless we will repeat them here to make this proof easier to understand.

We have an open cover{U_i}_i∈I of M and local sectionsqi:Ui→P which induce local trivialisations:

• forP →M,

ϕ_i: P |_Ui −→ U_i×K (qi(x)·k) 7−→ (x, k),

• forE →M,

φ_i: E|_Ui −→ U_i×F [q_i(x), f]_ρ 7−→ (x, f),

3.3 Bijection of certain section spaces

They generalise to local trivialisations φ˜_i: Λ^∗ T^V,(0,1)E^∗ whereφi,x denotes the induced map

φ_i,x: π⁻¹_E (x) −→ F and φ˜_i,x likewise.

Keeping these maps in mind, we can start the actual proof.

For anα∈A^(0,q)_V (E,L), we obtain by construction that αˆ˜ evaluated atq_i(x) is given by α(qˆ˜ i(x))

(f) = ˜φi,x◦α(qi(x), f) = ˜φi,x◦α◦φ⁻¹_i,x(f).

LetL_i,x denote the pullback bundle φ⁻¹_i,x∗

L. A small computation shows that

ˆ˜

and the natural vector bundle homomorphism identifyingL˜ and L_i,x as an equivalence of Hermitian, holomorphic line bundles overF

Φ_i,x: L_i,x −→ L˜

Now, sinceL→E is a compatible line bundle, Property 3 of Definition 3.1 implies

An analogous result holds for∂¯_V^∗ because of Lemma 2.29.

Therefore the assertion is proven.

Finally, we are able complete the proof of Proposition 3.14.

Therefore, recall the following fact. Letπ_P :P −→M be a K-principle fibre bundle and letQ=P ×_χV be an associated vector bundle for a representationχ:K −→Gl(V). Then the space of sections fromM intoQ is isomorphic to the K-χ-equivariant C^∞-maps fromP to V (cf. [5]).

This isomorphism is explicitly given by:

C^∞(P, V)^χ −→ Γ(M,Q)

f 7−→ n

x7→[px, f(px)]_χ

o (26)

wherep_x lies in the fibre overx∈M, i.e. π_P(p_x) =x. Proof of Proposition 3.14.

If we assume the existence of the first isomorphism of Proposition 3.14, i.e.

ψ: Γ(M,W^(λ;∗)) −→˜ Eig_λ ^[0,∗]_V

⊂A^(0,∗)_V (E,L),

it can be extended to the spaceA^(0,s)(M,W^(λ;t))∼=A^(0,s)(M)⊗Γ(M,W^(λ;t))as follows.

Takeα∈A^(0,s)(M)and s∈Γ(M,W^(λ;t)) now the ψ-morphism extends via

ψ(α⊗s) := (π^∗_Eα)∧ψ(s). (27)

This extension respects the Eigenspace structure because for anyα∈A^(0,s)(M) and any β∈A^(0,t_H ¹⁾(E)∧A^(0,t_V ²⁾(E,L), we obtain:

^[s+t_V ^1,t2]

(π^∗_Eα)∧β

= (π_E^∗α)∧

^[t_V^1,t2]β which has been shown in Corollary 2.28.

Thus, we have to construct the first isomorphism only, i.e.

ψ: Γ(M,W^(λ;∗)) −→ Eig_λ(^[0,∗]_V )⊂A^(0,∗)_V (E,L) Letα be inA^(0,∗)_V (E,L).

We apply the Corollaries 3.17 and 3.21 and obtain a unique map ˆ˜

α∈C^∞

P,A^(0,t)(F,L)˜ ρˇ

corresponding to α.

Lemma 3.22 now implies thatα is a λ-Eigenform ofV if and only ifα(p)ˆ˜ is a λ-Eigenform ofL˜ for every p∈P.

Consequently, the map b

e restricts to

be : Eig_λ(^[0,∗]_V ) −→ C^∞(P,Eig_λ(L˜))^ρˇ.

However, the bundle ^(λ;∗) is associated to , i.e. ^(λ;∗) (compare

Im Dokument Equivariant holomorphic torsion for a fibre bundle (Seite 62-73)