Equivariant holomorphic torsion - An example of legitimate group actions

5.6 An example of legitimate group actions

5.6.2 Equivariant holomorphic torsion

In this sub-subsection, we present our results for the equivariant holomorphic torsion for the legitimate action described above, i.e. ~γ = γ^G/T, γ^G, γ^L

withγ^L=L˜g0 for an arbitrary but fixed elementg˜₀∈G˜.

We summarise the result for an unconditioned choice of˜g0 inG˜ in the following theorem.

5.6 An example of legitimate group actions

Theorem 5.2:

Let G be a compact, connected, real even-dimensional Lie group and let T ⊂ G be a maximal torus ofG. And equipGwith a holomorphic fibre bundle structure, as described in Subsection 5.1.

Let furthermore χ :π₁(G) → U(1) be a character of π₁(G) and let L = ˜G×_χC be the Hermitian line bundle overG associated via this character.

Furthermore, equip L with a holomorphic structure ∂¯_L, given by ∂¯_L = ¯∂_L_χ +ε(π^∗_G(ω)), where ∂¯_L_χ denotes the natural holomorphic structure on L_χ = ˜G×_χC, we described in Lemma5.10 and whereω is a∂-closed form in¯ A^(0,1)(G/T).

At last, let g˜0 be an element of G˜ such that for g0 = π_1,G˜(˜g0) the (0,1)-form ω is left invariant under the pullback withL^G/Tg0 .

Let~γ = (γ^G/T, γ^G, γ^L) = (L^G/Tg0 , L_g₀, L_˜_g₀) denote the induced legitimate action of ˜g₀ on L, given by

γ^L :=Lg˜0 : L −→ L [˜g, z]_χ 7−→ [˜g₀g, z]˜ _χ.

Then we obtain the following expression for the equivariant holomorphic ζ-function for sufficiently large Re(z):

Z_ˇ_γ^LL(z) =−X

λ6=0

λ^−zX

t(−1)^tind(γ^W^(λ;t),W^(λ;t)).

Proof.

We start using Theorem 5.1 to get the following expression for the equivariantζ-function.

Z_ˇ_γ^LL(z) =−X

λ6=0

λ^−zX

t(−1)^tind(γ^W^(λ;t),W^(λ;t))

χγ(T)Z_γ^W^(0;0)(z) if χ≡1

0 if χ6≡1

The second term is0because χ_γ(T) is0 for γ^L=L_˜_g₀ which we show now.

Recall thatH^0,q(T, T ×C) = Λ^q t^(0,1)∗

and that the~γ action on the cohomology of T×C is given byγ =L^∗_g₀ : Λ^q t^(0,1)∗

→Λ^q t^(0,1)∗

But since the Lie algebra consists of left invariant vector fields the~γ action on H^0,q(T, T×C) is trivial.

We conclude:

χγ(T) =χ(T) =

k=0

(−1)^k n

= (1−1)ⁿ= 0.

Naturally, we get a corollary out of Theorem 5.2 that holds for isolated, non-degenerated fixed points ofγ^G/T.

Corollary 5.48:

In the setting from Theorem 5.2.

If g₀=π_1,_G_˜(g₀) generates a maximal torus inG.

Then the expression for the equivariantζ-function simplifies to:

Z_ˇ_γ^LL(z) =

det_T1,0

[x0]G/T(1−γ_[x⁻¹

0]) −1

· X

[n]∈W(T)

Z^˜^L

ˇ γ_[x^˜^Lχ

0·n]

(z).

Here, [x0]∈(G/T)^γ denotes an arbitrary chosen fixed point of L^G/Tg0 that always exists (compare Lemma 5.40). Further on, W(T) denotes the Weyl group of T in G.

The equivariant ζ-function on the right hand side is that of the holomorphic line bundle π˜Lχ: L˜_χ:= ˆT×_χC −→ T

with a group action γ_[x^˜^L

0·n]:=L^L^˜_ˆ

Ω([x0·n]): L˜_χ −→ L˜_χ

t, zˆ

χ 7−→ h

Ω([xˆ 0·n])·ˆt, z i

where Ωˆ is constructed as above (compare Corollary 5.47).

Proof.

Starting at the formula for the equivariantζ-function of Theorem 5.2 and compare this with the formula from Corollary 5.36, we obtain:

Z_ˇ_γ^LL(z) = X

[x]∈(G/T)^γ

det_T1,0

[x]G/T

1−γ_[x]⁻¹−1

·Z^L^[x]

ˇ γ^L^[x](z)

Recall that every fixed point[x]differs from [x0] by an element of W(T), i.e. there is an [n]∈W(T) such that

[x] =δ(nT,[x0]) = [x0]·[n]

(compare Equation (60)).

In the proof of Lemma 5.43 we saw, thatT_[x](G/T)is isomorphic to mas complex vector space and furthermore, that the mapγ_[x]⁻¹ corresponds to the map Ad(Ω([x])) via that isomorphism.

5.6 An example of legitimate group actions

It follows, using theΩ-δ-equivariance, i.e. Equation (61), that we obtain:

det_T1,0

Thus, it does not depend on the choice of the fixed point[x0].

At last, we simplify the expressionZ

L|π−1

Now, look at the map

[˜x0·n]˜_ρ_˜: L˜ −→ L|_π⁻¹

G ([x0·n])

[ˆs, z]_χ 7−→ [˜x0·n˜·s, z]ˆ _χ. (62) With the notations for the local trivialisations from Remark 5.23, take ani∈I such that [x₀·n]∈U_i.

In particular, Equation (62) defines a biholomorphic isometry becauseρ(ˆ˜t) is one and so is(φi,x)^∗◦Φi,x (compare Lemma 5.24).

The actionγ^L restricted to L|

In the situation of Corollary 5.48, ifG has rank (which is defined as the real dimension of the maximal torusT) greater than2 than the equivariant holomorphic ζ-function simplifies to

Z_ˇ_γ^LL(z) =

χ_γ(T)Z_γ^W^(0;0)(z) if χ≡1

0 if χ6≡1

because here the equivariant holomorphic torsion of a holomorphic line bundle vanishes.

In particular, the equivariant holomorphic torsion vanishes, i.e.

τ^L(ˇγ^L) = 0.

Starting at the definition of the equivariantζ-function (compare Def. 2.43), we obtain for largeRe(z):

Note that the Eigenspace decomposes as follows Eig_λ(^(0,q)_L_x

and note furthermore thatˇγ_[x^L^˜^χ_·n] covers a left transition onT that acts trivially on the

5.6 An example of legitimate group actions

The determinant can be expressed using weights of the adjoint representation.

Corollary 5.50:

In the situation of Corollary 5.48 where the rank ofG equals 2, we compute the equivariant torsion via:

where the product goes over all the positive roots of the Lie groupG and wheree^−2πiα denotes the global root corresponding to −α∈R⁻:

e^−2πiα : T −→ U(1)

t= exp(X) 7−→ e^−2πiα(X).

Proof. Corollary 5.48 directly implies the following formula for the equivariant holomorphic torsion ofL:

τ^L(ˇγ^L) = det_T^1,0

Now, observe thatγ_[x

0]=T_[x₀_]L^G/Tg0 and furthermore, that

Usingg₀·x₀ =x₀·Ω([x₀]), we now compute:

T_[x₀_]L^G/T_g₀ ◦(Tx0πG)◦(TeLx0) (X) =

T_x₀·Ω([x₀])πG

◦Tx0Lg0 ◦(TeLx0) (X)

T_x₀_·Ω([x₀_])π_G

◦T_x₀L_g₀ ◦(T_eL_x₀) (X)

= (Tx0π_G)◦(TeLx0)◦Ad (Ω([x0])) (X) Consequently, we obtain:

det_T^1,0

[x0]G/T

1−γ_[x⁻¹

= det^L

α∈R+g_α

1−Ad (Ω([x0]))⁻¹ We apply the root space composition and finally obtain:

det_T^1,0

[x0]G/T

1−γ_[x⁻¹

= Y

α∈R⁺

1−e^2πiα(Ω([x₀])⁻¹) which finishes the proof.

This finally reduces the computation of the equivariant holomorphic torsion for the holomorphic, Hermitian line bundleL= ˜G×_χC→Gwith natural Hermitian metric and a holomorphic structure∂¯_L= ¯∂_L_χ+ε(π_G^∗(ω))for the action Lg˜0 :L→L to a

computation of the equivariant torsion of the Hermitian, holomorphic line bundle L˜_χ →T with holomorphic structure ∂¯L˜χ : ˜L→L˜ for different actionsγ_[x]^˜^L .

In particular, the result does not depend on the(0,1)-form ω.

A Appendix: Linear algebra

In this part of the appendix, we introduce the linear algebra setting that we need.

In particular, give a definition of an almost complex vector space and recall some useful definitions and properties of Hermitian vector spaces and operators thereon.

Furthermore, we state two identities for the Hodge-Star-operator that we need throughout this thesis.

This part of the appendix is more or less common knowledge. Nonetheless, some conventions have to be made. Furthermore, in order to simplify the process of

understanding for the reader, we choose to repeat some of the necessary definitions and results.

Throughout this thesis we will constantly use the fact, that the fibre of the (co-)tangent space over a point in a complex manifold is a complex vector space and that some operators like the Hodge-Star operator can be described by linear algebra purely.

Therefore, it sometimes suffices to understand these objects on such a low level.

The first object of interest will be an almost complex vector space.

Definition A.1:

• The tuple(V, JV) whereV is a real even dimensional vector space andJV is an automorphism ofV such thatJ_V² =−id_V is calledalmost complex vector space.

The map JV is called almost complex structure onV.

• The triple(V, JV, g_V) is calledHermitian, almost complex vector space if (V, J_V) is an almost complex vector space andg_V is a Euclidean metric which is compatible with the almost complex structureJ_V, i.e. J_V is an isometry of (V, g_V).

Remark A.2:

• Every almost complex vector space (V, JV)has a natural orientation given by a basee1, J_V(e1), . . . , en, J_V(en).

• We may define an almost complex structureJ_V^[ on the dual space V^∗, via

J_V^[(α)

(v) :=α(J_V(v)) for any α∈V^∗ and v∈V.

• In particular, for a Hermitian, almost complex vector space, ife₁, . . . , e_n is

orthonormal with respect tog_V and if furthermoreJ_V(e_k)does not lie in the linear

hull of e₁, . . . , e_nfor any k= 1. . . n, thene₁, J_V(e₁). . . , e_n, J_V(e_n)forms an orthonormal base of V.

In particular, this base defines an orientation onV that does not depend on the choice of the e₁, . . . , e_n.

Further on, its dual base given bye¹, J_V^[(e¹), . . . , eⁿ, J_V^[(en) is orthonormal as well.

• For a Hermitian, almost complex vector space, we obtain thecanonical volume form:

dvolV :=e¹∧JV(e¹)∧. . .∧eⁿ∧JV(eⁿ)∈Λ²ⁿV^∗.

Let from now on for the time being(V, J_V, g_V)be a Hermitian, almost complex vector space.

Remark A.3:

• The complexified vector space V_C:=V ⊗_RC splits into itsholomorphic part, V^(1,0) := Ker(J_V −i),

and its antiholomorphicpart, V^(0,1) := Ker(JV +i), i.e. we obtain:

V_C=V^(1,0)⊕V^(0,1). Likewise, its dual spaceV^∗

C :=V^∗⊗_RCsplits, too:

V_C^∗ =V^∗(1,0)⊕V^∗(0,1).

This splitting may be extended to the entire exterior algebra ofV_C^∗: Λ^qV_C^∗= M

r+s=q

Λ^rV^∗(1,0)∧Λ^sV^∗(1,0) =: M

r+s=q

Λ^(r,s)V^∗

• OnV_C, we have a Hermitian product h_V induced by g_V given through h_V(v⊗_Rz,vˆ⊗_Ry) :=g_V(v,ˆv)·zy.¯

We denote its analog onV^∗

C by h^[_V and in abuse of notation,h^[_V will also be used for the extended Hermitian form on the exterior algebra Λ^·V^∗

The operator we define right now helps us to compute the dual of a certain operator, namely the∂¯-operator, on complexified differential forms with coefficients in a Hermitian vector bundle.

Definition A.4:

Let(W, hW) be a finite dimensional Hermitian vector space.

There is a natural operator

¯∗_V_⊗W : Λ^(p,q)V^∗

⊗W −→ Λ^{(n−p,n−q)}V^∗

⊗W^∗, theHodge-Star operator.

It is given implicitly through:

h^[_V ⊗hW

(α⊗w, β⊗w⁰)dvol^g_V^V =α∧ ιw¯∗_V_⊗W(β⊗w⁰) extended linearly onto the whole tensor product.

For w∈W, the operator ιw denotes the mapping W^∗

C to C putting winto the first component, i.e. (ι_wκ)(w₁, . . . , wq−1) :=κ(w, w₁, . . . , wq−1).

Remark A.5:

• Recall that

h^[_V ⊗hW

(α⊗w, β⊗w⁰) =h^[_V(α, β)·hW(w, w⁰) and therefore

¯∗_V_⊗W(β⊗w⁰) = (¯∗_Vβ)⊗h_W(·, w⁰) (63) for an operator

∗_V : Λ^(p,q)V^∗ →Λ^{(n−p,n−q)}V^∗.

• ¯∗_V_⊗W and ¯∗_V areC-antilinear maps.

Now, that we have defined, what the Hodge-Star operator is, we will proof a small Lemma, that we use for the splitting of the∂¯^∗ operator into a horizontal and a vertical part. Actually it will be used directly, when we introduce the induced holomorphic structure of a line bundle, if we restrict it to a complex submanifold, in our case this submanifold is the fibre of a holomorphic fibre bundle.

Lemma A.6:

Let (V, JV, gV), (U, JU, gU) be two Hermitian, almost complex vector spaces and let ϕ:U ,→V be an isometric embedding that is compatible with the almost complex structures, i.e. ϕ(J_U(u)) =J_V(ϕ(u)).

Then for any Hermitian vector space (W, hW) the following two statements hold.

∗_U⊗W ◦ϕ^∗ = ϕ^∗◦¯∗_V_⊗W ◦ε(dvol_ϕ(U)^⊥)

∗_V_⊗W |_Λ^·_ϕ(U)^∗ = ε

dvol_ϕ(U)^⊥

◦¯∗_V_⊗W ◦ε

dvol_ϕ(U)^⊥

|_Λ^·_ϕ(U)^∗

Here, ε(α) denotes the wedge-product with α from the left hand side and ϕ(U)^⊥ is the orthogonal complement of ϕ(U) in V regarding g_V.

Proof.

Because of Equation (63), the first statement is true if and only if it is true for¯∗_U and¯∗_V without theW part.

Observe now thatV =ϕ(U)⊕^⊥ϕ(U)^⊥ which leads to a splitting Λ^·V^∗ = Λ^·(ϕ(U)^∗)∧Λ^·

ϕ(U)^⊥∗ ,

using the musical isomorphismv7→g_V(·, v) fromV to V^∗. In particular, we obtaindvolV = dvol_ϕ(U)∧dvol_ϕ(U)^⊥. Now letω be inΛ^p(ϕ(U)^∗⊗_RC)∧Λ^q

ϕ(U)^⊥∗

⊗_RC

with non-zeroq.

We observe that, on the left hand side, the pullbackϕ^∗ω= 0 and, on the right hand side, ε(dvol_ϕ(U)^⊥)ω= 0.

Therefore, the statement holds for these forms.

So let from now onq equal zero, i.e. ω∈Λ^p(ϕ(U)^∗⊗_RC).

Now, look at the defining equation for the Hodge-Star-operator¯∗_V on V. The expression:

h^[_V(β, ω∧dvol_ϕ(U)^⊥)dvol_V =β∧¯∗_V

ω∧dvol_ϕ(U)^⊥

is zero ifβ is not of the typeβ =β⁰∧dvol_ϕ(U)⊥ for a β⁰ inΛ^·(ϕ(U)^∗⊗_RC). On the other hand, we get forβ =β⁰∧dvol_ϕ(U)^⊥:

h^[_V(β, ω∧dvol_ϕ(U)⊥)dvol_V =β⁰∧¯∗_V

ω∧dvol_ϕ(U)⊥

∧dvol_ϕ(U)⊥. (64) Further on, computing the left hand side, we obtain:

h^[_V(β, ω∧dvol_ϕ(U)^⊥)dvol_V =h^[_V(β⁰, ω)dvol_V

=h^[_U(ϕ^∗(β⁰), ϕ^∗(ω))dvol_ϕ(U)∧dvol_ϕ(U)^⊥. (65)

Now, we compare Equation (64) and Equation (65) and derive:

ϕ^∗

β⁰∧¯∗_V

ω∧dvol_ϕ(U)^⊥

=h^[_U(ϕ^∗(β⁰), ϕ^∗(ω))ϕ^∗dvol_ϕ(U)

=ϕ^∗(β⁰))∧¯∗_U(ϕ^∗ω)

which, holding for anyβ⁰, completes the proof of the first statement.

The second statement can be easily seen, using an orthonormal frame.

B Fréchet space of sections in a vector bundle

The content of this section is to define and understand the Fréchet structure of the vector space of smooth sections from a manifold into a vector bundle.

First of all, we define what a Fréchet space is (compare [30]).

Definition B.1:

A Fréchet space is a topological vector spaceF, with the following properties:

• F is metrizable.

• F is complete.

• F is locally convex, i.e. there is a basis{B_k}_k∈K of the topology ofF, such that each base setBk is a convex subset of F.

For maps into a Fréchet spaceF, there is a concept of differentiability.

Definition B.2:

LetF be a Fréchet space,Ω⊂Rⁿ be an open subset andh: Ω−→F be a map.

• h ispartially differentiable in x∈Ωif for eachl= 1, . . . , nthere is an element

∂h

∂x_l(x)∈Fsuch that

06=t→0lim t⁻¹(h(x+t·e_l)−h(x)) = ∂h

∂x_l(x).

• h isdifferentiable in x∈Ωif it is partially differentiable in x and if the following equation holds:

y→xlim h(y)−h(x)−

l=1

(y_l−x_l)· ∂h

∂x_l(x)

= 0.

• h is(partially) differentiable onΩif it is (partially) differentiable in x for every x inΩ.

LetM be a compact Riemannian manifold andQ −→M be a smooth vector bundle with Euclidean metric and a metric connection∇.

Lemma B.3:

Let V ⊂M be an open subset.

Then the space of smooth sections from V into Q denoted byΓ(V,Q |_V) becomes a Fréchet space.

In particular, the space Γ(M,Q) is a Fréchet space.

Proof. First, we have to define a topology on Γ(V,Q |_V).

Take therefore the following set of semi-norms onΓ(V,Q |_V). For each compact subset K⊂V and eachl∈N, define for s∈Γ(V,Q |_V):

ksk_l,K := sup

t≤l

sup

x∈K

k∇^tsk_x

, where∇^t denotes the map

∇^t: Γ(V,Q |_V) −→ Γ V, Nt

i=1T^∗M

⊗ Q |_V ,

induced be the metric connection∇as well as by the Levi-Civita connection on T M. The norm onΓ(V,Nt

i=1T^∗M⊗ Q |_V)|_x is given by the metric on M as well as the Euclidean metric onQ.

ThatΓ(V,Q |_V) is a real topological vector space is obvious.

The local convexity can be seen by takingεballs inΓ(V,Q |_V).

ThatΓ(V,Q |_V) is metrizable follows from the fact that every open set U ⊂V can be approximated by a sequence of compact sets{K_i}_i∈_N such thatKi ⊂int(Ki+1)⊂U for everyi∈Nand the fact that

ksk_l,K_i ≤ ksk_k,K_i+1

for everyi∈N(compare [30, Ch. 10.3] for the analogous result for C^∞(V,R^m)).

The completeness follows now from the theorem of uniform convergence (compare again [30, Ch. 10.3] for theC^∞(V,R^m) case).

Now,Q →M is a vector bundle. Hence, we may chose a finite covering{V_j}_j∈J of M such thatQ −→M becomes locally trivial, i.e. such that there are smooth maps

ψ_j : Q |_V

j −→ V_j ×R^m. Lemma B.4:

Let now Ω⊂Rⁿ be an open subset.

A map

ϕ: Ω −→ Γ(M,Q)

is differentiable if and only if the induced maps ϕ|_V

j: Ω −→ Γ(V_j,Q |_V

j) are differentiable for all j∈J.

Proof. "⇒": Letϕ: Ω−→Γ(M,Q)be differentiable.

Consequently, there are _∂x^∂ϕ_k(x)∈Γ(M,Q) for every x∈Ω such that

Almost the same observation for the continuity of ^∂ϕ|_∂x^Vj_k (x).

y→xlim

We now use the sheave property of the vector space of sections, i.e. we use that if two sectionss_i ∈Γ(V_i,Q |_V

Observe that we have not made use of the fact thatM is compact in the "⇒"-direction of this proof.

We conclude that by "⇒" the following equation holds.

∂ϕ|_V

Now, we show the two limit formulae for the differentiability ofϕ:

This property extends to higher differentiabilities by induction.

Corollary B.5:

Let Ω⊂Rⁿ be an open subset.

A map

ϕ: Ω −→ Γ(M,Q)

isC^∞-smooth if and only if the induced maps ϕ|_V

Without restrictions to generalityVj is a chart, i.e. there is a diffeomorphism φ_j :V_j −→ U_j

withU_i⊂R^p open.

We identify nowVj withUj to make it less difficult.

The space ^∞ ^m becomes a Fréchet space if we apply the standard topology, i.e.

LetK⊂V_j be a compact subset andl∈N, define the semi-norm k · k_l,K for f ∈C^∞(V_j,R^m) to be:

kfk_l,K := sup

|~t|≤l

sup

x∈K

∂^~^tf

∂x(x)

! ,

where for a tuple~t= (t0, . . . , tp)⊂Nⁿ the degree of~tis given by |~t|:=t1+. . .+tn and the operator _∂x^∂^~^t is given by

∂^~^tf

∂x :=

i=1

∂^tⁱ

∂x^t_iⁱ

! f.

Now, take as∈Γ(Vj,Q |_V

j) and take aK ⊂Vj, then there is are positive constants C1, C2 such that:

C1ksk_l,K ≤ kproj₂◦ψj(s)k_l,K ≤C2ksk_l,K.

This is due to the fact that the Euclidean metrics onN

T^∗M⊗V_j differ from the standard metric onN

(R^p)^∗⊗R^m by smooth functions. These functions have onK a maximum as well as a minimum.

Corollary B.6:

Let Ω⊂Rⁿ be an open subset.

A map

ϕ: Ω −→ Γ(M,Q)

isC^∞-smooth if and only if the induced maps proj₂◦ψj◦ϕ|_V

j: Ω −→ C^∞(Vj,R^m) areC^∞-smooth for all j∈J.

REFERENCES

References

[1] M. F. Atiyah. K-theory. Lecture notes by D. W. Anderson. W. A. Benjamin, Inc., New York-Amsterdam, 1967.

[2] M. F. Atiyah, N. J. Hitchin, and I. M. Singer. Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. London Ser. A, 362(1711):425–461, 1978.

[3] M. F. Atiyah and G. B. Segal. The index of elliptic operators. II. Ann. of Math. (2), 87:531–545, 1968.

[4] M. F. Atiyah and I. M. Singer. The index of elliptic operators. IV. Ann. of Math.

(2), 93:119–138, 1971.

[5] H. Baum. Eichfeldtheorie. Springer London, Limited, 2010.

[6] N. Berline, E. Getzler, and M. Vergne. Heat kernels and Dirac operators, volume 298 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1992.

[7] C. Birkenhake and H. Lange. Complex Abelian Varieties, volume 302 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, second edition, 2004.

[8] J.-M. Bismut. The Atiyah-Singer index theorem for families of Dirac Operators:

Two heat equation proofs. Invent. math, 83:91–151, 1986.

[9] A. Borel. Topology of Lie groups and characteristic classes. Bull. Amer. Math. Soc., 61:397–432, 1955.

[10] T. Bröcker and T. tom Dieck. Representations of compact Lie groups, volume 98 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995. Translated from the German manuscript, Corrected reprint of the 1985 translation.

[11] J. Cheeger and D. G. Ebin. Comparison theorems in Riemannian geometry.

North-Holland Publishing Co., Amsterdam, 1975. North-Holland Mathematical Library, Vol. 9.

[12] W. Franz. Über die Torsion einer Überdeckung. J. Reine Angew. Math. 173, 1935.

[13] W. Fulton and J. Harris. Representation theory, volume 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. A first course, Readings in Mathematics.

[14] P. Griffiths and J. Harris. Principles of algebraic geometry. Wiley Classics Library.

John Wiley & Sons Inc., New York, 1994. Reprint of the 1978 original.

[15] A. Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002.

[16] M. W. Hirsch. Differential topology. Springer-Verlag, New York, 1976. Graduate Texts in Mathematics, No. 33.

[17] D. Huybrechts. Complex geometry. Universitext. Springer-Verlag, Berlin, 2005. An introduction.

[18] A. W. Knapp. Lie groups beyond an introduction, volume 140 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1996.

[19] S. Kobayashi. Differential geometry of complex vector bundles, volume 15 of Publications of the Mathematical Society of Japan. Princeton University Press, Princeton, NJ, 1987. Kanô Memorial Lectures, 5.

[20] S. Kobayashi. Transformation groups in differential geometry. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the 1972 edition.

[21] S. Kobayashi and K. Nomizu. Foundations of differential geometry. Vol I.

Interscience Publishers, a division of John Wiley & Sons, New York-Lond on, 1963.

[22] K. Köhler and D. Roessler. A fixed point formula of Lefschetz type in Arakelov geometry. I. Statement and proof. Invent. Math., 145(2):333–396, 2001.

[23] D. B. Ray and I. M. Singer. R-torsion and the Laplacian on Riemannian manifolds.

Advances in Math., 7:145–210, 1971.

[24] D. B. Ray and I. M. Singer. Analytic torsion for complex manifolds. Ann. of Math.

(2), 98:154–177, 1973.

[25] K. Reidemeister. Homotopieringe und Linsenräume. Abh. Math. Sem. Univ.

Hamburg, 11(1):102–109, 1935.

[26] H. Samelson. Topology of Lie groups. Bull. Amer. Math. Soc., 58:2–37, 1952.

[27] H. Samelson. A class of complex-analytic manifolds. Portugaliae Math., 12:129–132, 1953.

[28] C. Soulé. Lectures on Arakelov geometry, volume 33 ofCambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1992. With the collaboration of D. Abramovich, J.-F. Burnol and J. Kramer.

[29] N. K. Stanton. HolomorphicR-torsion for fiber bundles. Amer. J. Math., 100(3):547–578, 1978.

[30] F. Trèves. Topological vector spaces, distributions and kernels. Dover Publications Inc., Mineola, NY, 2006. Unabridged republication of the 1967 original.

[31] J. H. Whitehead. Convex regions in the geometry of paths. Oxford Journals, os-3,

Erklärung

Die hier vorgelegte Dissertation habe ich eigenständig und ohne unerlaubte Hilfe angefertigt.

Die Dissertation wurde in der vorgelegten oder in ähnlicher Form noch bei keiner anderen Institution eingereicht.

Ich habe bisher keine erfolglosen Promotionsversuche unternommen.

Thomas Ueckerdt Düsseldorf, 21. Januar 2014

Im Dokument Equivariant holomorphic torsion for a ﬁbre bundle (Seite 150-171)