The aim of this subsection is to specify the holomorphic fibre bundle of our example.
Therefore, we apply common knowledge about compact Lie groups in order to obtain the necessary structures.
LetE:=Gbe a compact, real even-dimensional Lie group with bi-invariant metricgG. Such a metric always exists sinceGis compact. In particular, we still have a degree of freedom left because we still may chose(gG)h at one point of h∈G.
Furthermore, letF :=T ⊂Gbe a maximal torus. We denote the Lie algebra ofGwithg and the Lie algebra ofT witht.
We obtain a smooth principle fibre bundleT ,→G→G/T over the homogeneous space G/T.
At first, we have to show that this fibre bundle naturally induces a holomorphic fibre bundle structure in the sense of Definition 2.13.
Remark 5.1:
By prerequisite, Gis compact. Therefore,G/T becomes a reductive homogeneous space, i.e. there is an Ad|T invariant complementmoft, s.t. g=t⊕m. This can be seen by the subsequent argument.
Chosemto be the orthogonal complement of t ing for the bi-invariant metricgG. Now, gG is bi-invariant and thereforemis Ad|T-invariant.
As a direct consequence, we obtain the following facts.
• For a reductive homogeneous space, the theory of principle fibre bundles and associated bundles (cf. [5]) now gives us the tangent space of the base space, i.e.
T(G/T), as an associated vector bundle to the principle fibre bundleG→G/T, T(G/T)∼=G×Ad,T m.
In particular, every Ad|T-invariant structure onmdirectly induces a corresponding structure on T(G/T).
• For example, we obtain a metricgG/T induced by the bi-invariant metricgG.
• Furthermore, we obtain a smooth horizontal distribution THG⊂T Gby left
5.1 General setting
Denote, forg∈G, the left transition withg with Lg: G −→ G
h 7−→ g·h.
We define the distribution TgHG:=TeLg(m)for every g inG.
Observe now that Lg :G−→Gcovers a diffeomorphism LG/Tg , i.e. the following diagram commutes.
G
πG
Lg //G
πG
G/T
LG/Tg
//G/T
Furthermore, note thatTeπG:m−→T[e]G/T is an isomorphism.
Consequently,THGbecomes a horizontal distribution.
So far, we have a fibre bundleT ,→G→G/T, a connection THGof the fibre bundle G→G/T and a Riemannian, bi-invariant metricgG onG. The latter one is fixed up to a choice of a scalar product ong.
Now, we construct a complex structure for the manifoldG.
A known fact for a compact Lie groupGis, that its Lie algebrag is a product of an Abelian Lie algebrahand a semi-simple Lie algebra gs (cf. [18, Ch. 4 Cor. 4.25.]).
Furthermore, the the semi-simple partgs possesses a maximal Abelian sub-Lie algebrahs such thatt=h⊕hs.
Using representation theory (cf. [13]) we obtain the following splitting of the complexified semi-simple Lie algebra.
gs⊗RC= (hs⊗RC)⊕ M
α∈R+
(gs,α⊕gs,−α).
Here,R+ denotes an arbitrary, but fixed, set of positive roots,gα denotes the root space for the rootα and hs⊗RC denotes the complexified Cartan algebra.
Using this decomposition, we define the almost complex structureJg on the Lie algebra g via:
(g)1,0 = (h⊕hs)1,0⊕ L
α∈R+
gs,α (g)0,1 = (h⊕hs)0,1⊕ L
α∈R−
gs,α (46)
where we choose an arbitrary almost complex structure onh⊕hs=t.
LetJG be the left translated almost complex structure onGinduced by Equation (46), i.e. for anyX˜ ∈TgGgiven byX˜ =TeLg(X)we define
JG( ˜X) :=TeLg(Jg(X)).
Samelson shows in [27] thatJG is integrable, i.e. G becomes a complex manifold.
Furthermore, he proves thatJG naturally induces a complex structureJG/T on M =G/T.
To be more explicit, Samelson showed thatG,T and G/T are complex manifolds with complex structuresJG,JT :=JG |t and JG/T induced byJG|m.
Note thatJG/T is a well defined map due to the fact that the root spaces gs,α are invariant underAd|T. It follows that the projection πG is a holomorphic map.
The metricgG as well as the complex structureJGmay be chosen compatible ong⊂TeG (in the sense of Definition A.1, i.e. JG is an isometry of(g, gG)).
Now, because they are left invariant, they stay compatible on all ofG.
This compatibility extends to(T, JT, gT) wheregT is given bygG through restriction.
Furthermore,JG/T and gG/T are compatible becausegG/T as well asJG/T are given by restriction ofgG or JG to the subspace m⊂TeG.
We now check that(G, πG,(G/T, gG/T),(T, gT), THG) fulfils the prerequisites of a holomorphic fibre bundle (Definition 2.13).
• The map πG :G−→G/T defines a smooth (principle) fibre bundle whose fibretype is the maximal torus T. X
• Furthermore,(T, gT) and (G/T, gG/T) are complex manifolds with compatible Riemannian metrics. X
• The setG has a complex manifold structure andπG is a holomorphic map whose differential has constant maximal rank.
Now, the implicit function theorem for holomorphic functions implies that there are local holomorphic sections
qk:Uk−→G∩π−1G (Uk).
Consequently, the local trivialisations φ−1k : Uk×T −→ π−1G (Uk) (x, t) 7−→ qk(x)·t are holomorphic maps.
Therefore, so are the transition functions
φk◦φ−1l : (Uk∩Ul)×T −→(Uk∩Ul)×T.
Now, we apply Lemma 2.16 and obtain that the maps
5.1 General setting
as well as
φ\i◦φ−1j : Ui∩Uj → Hol(F) are holomorphic. X
• Additionally, we note that φi,x◦φ−1k,x is an isometry for eachiand eachk.
• We have a direct sum decompositionT G=TVG⊕THG, i.e. we have a connection m=THGon G→G/T. X
• The connectionTHGisJG invariant because misJG|T
eG invariant by construction and because the JG is given by left transition of JG|T
eG. X
• The connectionmis of type (1,1)because
[gs,α,gs,β]⊂gs,α+β (47)
and because α+β ∈R− ifα and β are negative roots (cf. [13]).
Finally, our chosen metricgG fulfils the properties of Remark 2.17 which is shown in the subsequent consideration.
• By construction, the horizontal space and the vertical space are orthogonal with respect to gG. X
• Furthermore,gG is left invariant. Hence, every inclusion of the fibre t7→g·t is an isometric immersion. X
• The projectionπG is a Riemannian submersion which follows directly from the definition of gG/T. X
We summarise the information we collected so far in the following corollary.
Corollary 5.2:
The tuple G, πG, G/T, gG/T
,(T, gT), THG
, with notations from above, is a holomorphic fibre bundle in the sense of Definition 2.13.
From now on, throughout this section G, πG, G/T, gG/T
,(T, gT), THG
denotes the holomorphic fibre bundle defined in the preceding subsection.
We have shown thatG→G/T becomes a holomorphic fibre bundle in a natural way.
In the next subsection, we recall some well known facts about the set of smooth complex line bundles over Lie groups.
5.2 Line bundles over even-dimensional Lie groups