τW(0;2t)(ˇγW(0;2t))−τW(0;2t+1)(ˇγW(0;2t+1)) .
4.3 Special case: isolated non-degenerated fixed points
For a special case of a legitimate action~γ, there is a further simplification of the expression of theζ-function given in Theorem 4.1.
In order to state it, we have to recall the definition of an isolated and non-degenerated fixed point of an isometry.
4.3 Special case: isolated non-degenerated fixed points
Definition 4.7:
LetM be a Riemannian manifold and letγM be an isometry of M.
• A fixed pointx∈M ofγM is calledisolated if there is anε >0 such that there is no other fixed point of γM in anε-neighbourhood ofx.
• A fixed point x∈M of γM is callednon-degenerated if there is no vectorX in TxM such that TxγM
(X) =X.
In particular, every non-degenerated fixed point is isolated.
Now, in the setting from Theorem 4.1, we look at the special case where the~γ-action on the lowest level, i.e. γM :M →M, has only non-degenerated fixed points.
Here, we can use the Atiyah-Bott fixed point formula (compare [6, Ch. 6, Thm 6.6]) to calculate the termind(γW(λ;t),W(λ;t)).
In order to make it easier for the reader follow, we will not recite the whole theorem but a corollary (compare [6, Ch. 6, Cor 6.8]) tailored to the situation at hand.
We explain the notations used in this corollary in the subsequent remark.
Theorem (Corollary following from Atiyah-Bott):
"IfM is a compact complex manifold with holomorphic vector bundle W →M, andγ is a holomorphic transformation of W →M, then γ acts on the∂¯-cohomology spaces H0,i(M,W). If the action of γ on M has only isolated non-degenerate fixed points, then
X
i
(−1)iTr(γ, H0,i(M,W)) = X
x0∈Mγ
Tr(γxW0) detT1,0
x0 M(1−γx−10 )."
Remark 4.8:
We now clarify the notations above.
• The cohomology H0,i(M,W) is defined to be the kernel of(0,i)W and the action of γ on H0,i(M,W) is the actionγˇL (compare Definition 2.40) onA(0,∗)(M,W) restricted to the kernel of (0,i)W .
• The symbolMγ denotes the fixed point set of γM.
• For any fixed pointx0 ∈Mγ the map γx0 denotes the restriction ofTx0γM to the space TxC0M. Similarly,γxW
0 is the restriction of γW to the fibreWx0.
• In the expression above, the determinant of1−γ−1x0 is taken on the restriction of 1−γx−10 to the invariant subspaceTx(1,0)0 M.
These identifications in mind, we observe that the left hand side equalsind(γW,W) (compare Definition 2.42).
We apply the Atiyah-Bott theorem and obtain the following corollary.
Corollary 4.9:
In the setting from Theorem 4.1, suppose that γM :M →M is a biholomorphic isometry which has only isolated, non-degenerated fixed points.
Then the formula for the equivariant holomorphic Zeta function can be simplified to:
ZˇγLL(z) = X
x0∈Mγ
h detT1,0
x0 M(1−γx−10 )i−1
·ZLx0
ˇ
γLx0(z) +X
t
(−1)tZγW(0;t)(z) (45)
where we use the notations from above.
In particular, the equivariant torsion is now given by:
τL(ˇγL) = X
x0∈Mγ
h detT1,0
x0M(1−γx−10)i−1
·τLx0(ˇγLx0)
+X
t
τW(0;2t)(ˇγW(0;2t))−τW(0;2t+1)(ˇγW(0;2t+1)) .
This finishes the first part of this thesis.
The last section is dedicated to apply the theory, we evolved so far, to a specific example.
We study the equivariant holomorphic torsion of a flat line bundle over a compact Lie group.
5 Equivariant torsion for Lie groups
In this section, we apply Theorem 4.1 to an explicit example.
Every example has to fulfil a lot of prerequisites. There has to be a holomorphic fibre bundle in the sense of Definition 2.13, a compatible holomorphic, Hermitian line bundle (compare Definition 3.1) and a legitimate group action, described in Definition 3.26.
Therefore, it requires a lot of preparing to apply Theorem 4.1 to an example.
However, there is one class of examples that seems to be the most manageable for this kind of investigation, the case where the total space of the holomorphic fibre bundle is given by a compact, even-dimensional Lie group. We chose the holomorphic fibre bundle to be the Lie group over the homogeneous space which is given by factorising a maximal torus out of the Lie group.
This section is dedicated to analyse this example.
At first, in Section 5.1, we apply known results for compact Lie groups to show that every compact, even-dimensional Lie group induces a natural holomorphic fibre bundle structure in the sense of Definition 2.13.
In Section 5.2, we recall classical results about the set of flatable smooth complex line bundles over an even-dimensional, compact Lie groupG. In particular, we recall that every even-dimensional, compact semi-simple Lie group admits only flatable complex line bundles.
Afterwards, in Subsection 5.3, we recall a commonly known result about holomorphic structures on a complex line bundle which is associated to a principle fibre bundle with discrete fibre.
Later on, in Section 5.4, we examine which holomorphic and Hermitian structures we can endow on our line bundles in order to make them compatible with the holomorphic fibre bundle structure ofG→G/T.
In Sub-Subsection 5.4.1, we show thatG˜×χC becomes a smooth vector bundle over the fibre bundleG→G/T, i.e. L= ˜G×ρ˜L˜ →G/T.
Further on, in Sub-Subsections 5.4.2 as well as 5.4.3, we show that the natural holomorphic structure and Hermitian metric onL˜ are respected by the actionρ.˜
Additionally, we derive the implications for the possible holomorphic structures onL in Sub-Subsection 5.4.4.
At last, in Sub-Subsection 5.4.5, we generalise the result for the Laplace splitting property, shown by Stanton, to the more general holomorphic structures onL.
In Section 5.5, we restate Theorem 4.1 tailored to the situation at hand, i.e. for the set of holomorphic line bundles overGwith flatable, smooth complex line bundle structure.
Here, we apply known facts about the kernel of the LaplacianL˜ for flat line bundles over the torus.
Finally, in Section 5.6, we look at an easy example for a legitimate action. We takeγL to beL˜g0, i.e. the left transition with an elementg˜0 ∈G˜.
Here, we apply classical facts about the left action ofGon G/T to obtain a very simple
result for the equivariantζ-function for this special case of the legitimate action.
Its result for general˜g0∈G˜ is summarised in Theorem 5.2.
At last, for the special case whereπ1,G˜(˜g0)∈G generates a maximal torus, we deduce a very convenient result for the equivariant holomorphic torsion.
It is stated in Corollary 5.49 as well as in Corollary 5.50.