Homogeneous holomorphic hermitian principal bundles over hermitian symmetric spaces

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by  Indranil Biswas

HOMOGENEOUS HOLOMORPHIC HERMITIAN
PRINCIPAL BUNDLES OVER HERMITIAN
6 SYMMETRIC SPACES
1
0
2 INDRANIL BISWAS AND HARALD UPMEIER

n
a Abstract. We give a complete characterization of invariant in-
J
tegrable complex structures on principalbundles defined overher-
2
mitian symmetric spaces, using the Jordan algebraic approach for
1
the curvature computations. In view of possible generalizations,

] the general setup of invariant holomorphic principal fibre bundles
G
is described in a systematic way.
D
.
h
t
a
m Contents
[
1. Introduction 1

1
2. Homogeneous H-bundles 2
v
9 3. Connexions and complexions 6
5 4. Homogeneous connexions and complexions 13
8
5. Curvature and Integrability 17
2
0 6. The symmetric case 20
.
1 7. Concluding remarks 26
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References 27
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:
v
i
X 1. Introduction
r
a The classification of hermitian holomorphic vector bundles, or more
general holomorphic principal fibre bundles, over a complex manifold
M is a central problem in algebraic geometry and quantization the-
ory, e.g. for a compact Riemann surface M. In geometric quantization,
where M = G/K is a co-adjoint orbit, G-invariant principal fibre bun-
dles have been investigated from various points of view [Bo, Ra, OR].
In case M is a hermitian symmetric space of non-compact type, a
complete characterization of invariant integrable complex structures
2010 Mathematics Subject Classification. 32M10, 14M17, 32L05.
Key words and phrases. Irreducible hermitian symmetric space; principal bun-
dle; homogeneous complex structure; hermitian structure.
1
2 INDRANILBISWAS ANDHARALDUPMEIER
on principal bundles over M was obtained in [BM] (for the unit disk)
and [Bi] (for all bounded symmetric domains).
The main objective of this paper is to treat the dual case of com-
pact hermitian symmetric spaces, and to show that the compact case
(as well as the flat case) leads to exactly the same characterization,
resulting in an explicit duality correspondence for invariant integrable
complex principal bundles (with hermitian structure) between the non-
compact type, the compact type and the flat type as well. The proof
is carried out using the Jordan theoretic approach towards hermitian
symmetric spaces [Lo, FK]. Of course, the traditional Lie triple system
approach could be used instead, but Jordan triple systems (essentially
the hermitian polarization of the underlying Lie triple system) make
things more transparent and somewhat more elementary.
More importantly, the Jordan triple approach leads in a natural way
tomoregeneralcomplexhomogeneous(non-symmetric)manifoldsG/C
where C is a proper subgroup of K. These manifolds are fibre bundles
over G/K withcompact fibresgivenbyJordantheoreticflagmanifolds.
Again, there exists a duality between such spaces of compact, non-
compact and flat type, and in a subsequent paper [BU] the duality
correspondenceforinvariantfibrebundles, provedhereinthehermitian
symmetric case C = K, will be studied in the more general setting. In
view of these more general situations, the current paper describes the
general setup for homogeneous holomorphic principal fibre bundles in a
careful way, specializing to the symmetric case only in the last section.
As a next step beyond the classification, its dependence on the un-
derlying complex structure on M is of fundamental importance. While
the hermitian symmetric case G/K has a unique G-invariant complex
structure, the more general flag manifold bundles G/C have an inter-
esting moduli space of invariant complex structures. It is a challenging
problem whether this moduli space carries a projectively flat connex-
ion (with values in the vector bundle of holomorphic sections) similar
to the case of abelian varieties, [Mu], [We], or Chern-Simons theory
[ADW], [Wi].
Fromthegeometricquantizationpoint ofview, itisalsoofinterest to
describethespacesofholomorphicsections, givenbysuitableDolbeault
operators, in an explicit way.
2. Homogeneous H-bundles
Let M be a manifold and G a connected real Lie group, with Lie
algebra g, acting smoothly on M. Denoting the action G M M by
× →
HOMOGENEOUS BUNDLES ON HERMITIAN SYMMETRIC SPACES 3
(a,x) a(x), we define RM : G M, x M, by
7→ x → ∈
RM(a) = a(x) (a,x) G M.
x ∀ ∈ ×
LetH beaconnectedcomplexLiegroup; itsLiealgebrawillbedenoted
by h. Fix a maximal compact subgroup L H; all such subgroups are
⊂
conjugate in H. The Lie algebra of L is denoted by l. Let Q be a C
∞
principal H-bundle over M, with projection π : Q M = Q/H. The
→
free action Q H Q is written as (q,b) qb. Define RQ : Q Q
× → 7→ b →
and LQ : H Q by
q →
RQq = LQb = qb (q,b) Q H.
b q ∀ ∈ ×
Then π RQ = π and ker(dπ) T Q is the “vertical” subspace of the
◦ b q ⊂ q
tangent space T Q at q Q. We call Q an equivariant H-bundle if
q
∈
there is a C action G Q Q, denoted by (a,q) aq, such that
∞
× → 7→
π(aq) = aπ(q) and
a(qb) = (aq)b.
Define LQ : Q Q and RQ : G Q by
a → q →
LQq = RQa = aq (a,q) G Q.
a q ∀ ∈ ×
Then LQ RQ = RQ LQ and π LQ = LM π for all a G, b H.
a ◦ b b ◦ a ◦ a a ◦ ∈ ∈
A hermitian structure on a principal H-bundle Q is a principal
subbundle P Q with structure group L, i.e., we have
⊂
RQ : P P b L
b → ∀ ∈
and the action of L on the fibre P is transitive for all x M. We also
x
∈
say that (Q,P) is a hermitian H-bundle. An equivariant hermitian
H-bundle is an equivariant H-bundle Q endowed with a hermitian
structure P Q such that G P = P, i.e., we have
⊂ ·
LQP = P a G.
a ∀ ∈
In this case P becomes an equivariant L-bundle. When G acts transi-
tivelyonM, wecallQahomogeneousH-bundle. Inthehomogeneous
case, fix a base point o M and put
∈
K = k G k(o) = o .
{ ∈ | }
Then M = G/K. The Lie algebra of K will be denoted by k.
The following proposition is straight-forward to prove.
Proposition 2.1. Let f : K H be a (real-analytic) homomorphism.
→
Consider the quotient manifold
Q := G H
K,f
×
4 INDRANILBISWAS ANDHARALDUPMEIER
consisting of all equivalence classes
g h = gk 1 f(k)h , (2.1)
−
h | i h | i
with g G, h H and k K. (This bracket notation is used to avoid
∈ ∈ ∈
confusion with the commutator bracket.) Then Q becomes a homoge-
neous H-bundle, with projection π(g h) = g(o) = RM(g). The action
| o
of (a,b) G H is given by
∈ ×
a g h b = ag hb .
h | i h | i
If in addition, f(K) L, we obtain a hermitian homogeneous principal
⊂
H-bundle
P := G L Q := G H.
K,f K,f
× ⊂ ×
In the set-up of Proposition 2.1, the maps LQ : Q Q and RQ :
a → gh
G Q have the form |
→
LQ g h = RQ (a) = ag h .
ah | i gh h | i
|
For any ℓ H, let
∈
IHh = ℓhℓ 1
ℓ −
be the inner automorphism induced by ℓ. Then d IH = Adh. The con-
e ℓ ℓ
jugate homomorphism
IH f : K H
ℓ ◦ →
induces the H-bundle isomorphism
G H G H
×K,f → ×K,IℓH◦f
mapping the equivalence class g h to g IHh .
h | if h | ℓ iIℓH◦f
Theorem 2.2. Every homogeneous principal H-bundle Q on M =
G/K is isomorphic to G H for a homomorphism f : K H,
K,f
× →
which is unique up to conjugation by elements ℓ H. Similarly, every
∈
hermitian homogeneous H-bundle (Q,P) is isomorphic to the pair
G L G H
K,f K,f
× ⊂ ×
for a homomorphism f : K L H, which is unique up to conju-
gation by an element in L. M→ore p⊂recisely, for any base point o Q
o
(respectively, o Po) there exists a unique homomorphism fo : K ∈ H
∈ →
(respectively, fo : K L) such that
→
ko = ofo(k) k K, (2.2)
∀ ∈
and
g h goh (2.3)
h | i 7→
defines an isomorphism G H Q of equivariant H-bundles (re-
×K,fo →
spectively, an isomorphism G L P of hermitian equivariant
×K,fo →
HOMOGENEOUS BUNDLES ON HERMITIAN SYMMETRIC SPACES 5
H-bundles). Another base point o = oℓ 1, with ℓ H (respectively,
′ −
∈
ℓ L), corresponds to the homomorphism fo′ = Iℓ fo.
∈ ◦
Proof. Let Q be a homogeneous H-bundle. Choose o Q with π(o) =
∈
o. Since the fibre Q is preserved by K, and H acts freely on Q , there
o o
exists a unique map fo : K H such that (2.2) holds for all k K.
→ ∈
Then
ofo(k1k2) = (k1k2)o = k1(k2o) = k1(ofo(k2)) = (k1o)fo(k2)
= (ofo(k1))fo(k2) = o(fo(k1)fo(k2))
for all k ,k K. Since H operates freely on Q, this implies that
1 2
∈
fo(k1k2) = fo(k1)fo(k2), and hence fo : K H is a (real-analytic)
→
homomorphism. For all b H we have
∈
kob = (ofo(k))b = o(fo(k)b).
The resulting identity
(ak)ob = a(kob) = a(o(fo(k)b)) = ao(fo(k)b)
shows that (2.3) defines an isomorphism G H Q of equivariant
H-bundles, mapping the “base point” e e×tKo,foo. If→o Q is another
′ o
base point, there exists a unique ℓ Hhs|ucih that o =∈oℓ 1. It follows
′ −
∈
that
ko′ = koℓ−1 = ofo(k)ℓ−1 = o′ℓfo(k)ℓ−1.
Thus the new base point o corresponds to the conjugate homomor-
′
phism
foℓ−1(k) = ℓfo(k)ℓ−1 = IℓHfo(k).
In the hermitian case, for any base point o P Q the defining
o o
identity (2.2) implies fo(k) L for all k K. A∈noth⊂er base point o′ =
oℓ 1 P differs by a uniqu∈e element ℓ ∈L. In both cases, since G acts
− o
∈ ∈
transitively on M, the G-invariance condition implies that the entire
construction is independent of the choice of base point o M. (cid:3)
∈
Inviewof (2.2), thehomomorphismfo couldbedenotedbyfo = Io−1,
i.e., ko = oI 1(k). In this notation, the identity I 1 = I I 1 is ob-
o− o−ℓ−1 ℓ ◦ o−
vious. It will be convenient to express the tangent spaces in an explicit
manner using equivalence classes. Notethatthedifferential d f : k h
e
→
of f at the unit element e K is a Lie algebra homomorphism.
∈
Lemma 2.3. For a given class g h Q := G H the tangent
K,f
h | i ∈ ×
space T Q consists of all equivalence classes
gh
|
g˙ h˙ = (d RG )g˙ (d LG )κ (d LH )h˙ +(d RH )(d f)κ ,
h | i h g k−1 − e gk−1 | h f(k) e f(k)h e i
where g˙ T G, h˙ T H, k K and κ k.
g h
∈ ∈ ∈ ∈
6 INDRANILBISWAS ANDHARALDUPMEIER
Here we regard TG and TH as the disjoint union of the respec-
tive tangent spaces, so that the first expression is evaluated at g h
h | i
whereas the second expression is evaluated at the same class written
as gk 1 f(k)h . The identity follows from differentiating the relation
−
h | i
g h = g k 1 f(k )h at t = 0, where k K satisfies k = k and
h t| ti h t t− | t ti t ∈ 0
∂0k = κ. As special cases we obtain
t t
g˙ h˙ = (d RG )g˙ (d LH )h˙ = g˙ (d LG)κ h˙ +(d RH)(d f)κ
h | i h g k−1 | h f(k) i h − e g | e h e i
(2.4)
putting κ = 0 or k = e, respectively. The projection π has the differ-
ential
d
(d π) g˙ h˙ = ( (g )(0))(o) = (d RM)g˙.
g|h h | i dt t g o
Thus the vertical subspace is given by
ker(d π) = g˙ h˙ g˙ ker(d RM) T G, h˙ T H .
g|h {h | i | ∈ g o ⊂ g ∈ h }
For β h, the fundamental vector field ρQ has the form
∈ β
(ρQ) = ∂0 g hb = 0 (d LH)β = 0 (d RH)AdH β .
β g|h t h | ti h g| e h i h g| e h h i
3. Connexions and complexions
A connexion on a principal H-bundle Q is a smooth distribution
q TΘQ of “horizontal” subspaces of T Q such that T Q = TΘQ
7→ q q q q ⊕
ker(d π) and
q
TΘQ = (d RQ)(TΘQ) (q,b) Q H.
qb q b q ∀ ∈ ×
We use the same symbol for the associated connexion 1-form Θ :
q
T Q h on Q, uniquely determined by the condition that X T Q
q q
→ ∈
has the horizontal projection
XΘ = X (d LQ)(Θ (X)).
− e q q
A connexion Θ on an equivariant H-bundle Q is called invariant if
(d LQ)(TΘQ) = TΘQ a G.
q a q aq ∀ ∈
In this case the associated connexion 1-form satisfies
Θ = Θ (d LQ).
q aq q a
Let Q h denote the associated bundle of type Ad , with fibres
H H
×
(Q h) = [q : β] = [qh : Ad 1β] q Q ,β h
×H x { −h | ∈ x ∈ }
for x M, with h H being arbitrary. By [KN, Section II.5] every
∈ ∈
tensorial i-form on Q is given by
(CQ d π)(X1, ,Xi) := CQ((d π)X1, ,(d π)Xi)
q ◦ q q ··· q q q q ··· q q
HOMOGENEOUS BUNDLES ON HERMITIAN SYMMETRIC SPACES 7
for X1, ,Xi T Q, where
q ··· q ∈ q
C (v v ) = [q : CQ(v v )] q Q
x 1 ∧···∧ i q 1 ∧···∧ i ∀ ∈ x
i
isani-formoftypeAd (onM), withhomogeneousliftCQ : T M
H q V x →
h having the right invariance property
CQ = AdH CQ b H.
qb b−1 q ∀ ∈
An i-form C of type Ad is called invariant if
H
CQ(d LM) = CQ (a,q) G Q. (3.1)
aq x a q ∀ ∈ ×
Proposition 3.1.
(i) Let Θ0 be a connexion on Q. If C is a 1-form of type Ad , then
H
Θ = Θ0 +CQ d π q Q (3.2)
q q q ◦ q ∀ ∈
is a connexion 1-form on Q. Every connexion 1-form Θ on Q
arises this way.
(ii) In the equivariant case, let Θ0 be an invariant connexion on Q.
Then Θ is invariant if and only if C is invariant, i.e.,
CQ(d LM) = CQ (a,q) G Q. (3.3)
aq x a q ∀ ∈ ×
Proof. Part (i) is well-known. For part (ii) let the connexion Θ be
invariant. The condition π LQ = LM π implies that (d π)(d LQ) =
◦ a a ◦ aq q a
(d LM)(d π), and hence we have
x a q
CQ(d π) = Θ Θ0 = (Θ Θ0 )(d LQ)
q q q − q aq − aq q a
= CQ(d π)(d LQ) = CQ(d LM)(d π).
aq aq q a aq x a q
Since d π issurjective, (3.3)follows. The converse is proved ina similar
q
(cid:3)
way.
If (3.2) holds, we say that Θ is related to Θ0 via C. For a hermitian
H-bundle (Q,P) a connexion Ξ on P is called invariant if
(d LP)TΞP = TΞP (a,p) G P.
p a p ap ∀ ∈ ×
In this case the associated connexion 1-form satisfies
Ξ = Ξ (d LP).
p ap p a
By [KN, Proposition II.6.2] every connexion Ξ on P has a unique ex-
tension to a connexion ιΞ on Q such that
TΞP = TιΞQ p P Q.
p p ∀ ∈ ⊂
Equivalently, the connexion forms satisfy
(ιΞ) = Ξ p P.
p|TpP p ∀ ∈
8 INDRANILBISWAS ANDHARALDUPMEIER
A connexion Θ on Q is called hermitian if Θ = ιΞ for a (unique)
connexion Ξ on P. Thus hermitian connexions on Q are in 1-1 corre-
spondence with connexions on P. A connexion Ξ on P is invariant if
and only if its extension ιΞ is invariant.
ForhermitianH-bundles(P,Q),wehave i-formsoftypeAd written
L
as
A (v v ) = [p : AP(v v )] p P ,
x 1 ∧···∧ i p 1 ∧···∧ i ∀ ∈ x
i
with homogeneous lift AP : T M l having the right invariance
p V x →
property
AP = AdL AP ℓ L. (3.4)
pℓ ℓ−1 p ∀ ∈
Let l be the Lie algebra of L. For each x M, there is a linear injection
∈
of fibres
ι : (P l) (Q h) , [p,β] [p,ιβ],
x L x H x
× → × 7→
where p P ,β l and ι : l h is the inclusion map. The map ι is
x x
∈ ∈ →
well-defined because
[p ,β ] = [p ,β ] (P l)
1 1 2 2 L x
∈ ×
implying that p = p ℓ and β = AdL β for some ℓ L. Therefore,
2 1 2 ℓ−1 1 ∈
we also have ιβ = AdH ιβ . To show that ι is injective, suppose
2 ℓ−1 1 x
that [p ,ιβ ] = [p ,ιβ ] (Q h) . Then we have p = p h and
1 1 2 2 H x 2 1
∈ ×
ιβ = AdH ιβ for some h H. Since p ,p P it follows that h L
2 h−1 1 ∈ 1 2 ∈ x ∈
and hence β = AdL β .
2 h−1 1
As a consequence, an i-form A of type Ad induces an i-form ιA of
L
type Ad , with homogeneous lift
H
i
(ιA)Q := AdH AP : T M h (p,h) P H.
ph h−1 p ^ o → ∀ ∈ ×
Then A is invariant in the sense that
AP (d LM) = AP (a,p) G P
ap x a p ∀ ∈ ×
if and only if ιA is invariant as in (3.1).
Proposition 3.2.
(i) For a hermitian bundle (P,Q), let Ξ0 be a connexion on P. If
A is a 1-form of type Ad , then
L
Ξ = Ξ0 +AP d π p P (3.5)
p p p ◦ p ∀ ∈
is a connexion 1-form on P. Every connexion 1-form Ξ on P
arises this way. We also have
(ιΞ) = (ιΞ)0 +(ιA)Q d π q Q.
q q q ◦ q ∀ ∈
HOMOGENEOUS BUNDLES ON HERMITIAN SYMMETRIC SPACES 9
(ii) In the equivariant case, let Ξ0 be an invariant connexion on P.
Then Ξ is invariant if and only if A is invariant, i.e.,
AP (d LM) = AP (a,p) G P. (3.6)
ap x a p ∀ ∈ ×
If (3.5) holds, we say that Ξ is related to Ξ0 via A. In this case, ιΞ
is related to ιΞ0 via ιA. Now suppose that (M,j) is an almost complex
manifold, with almost complex structure j End(T M), x M,
x x
∈ ∈
having the left invariance property
(d LM)j = j (d LM) (g,x) G M.
x g x gx x g ∀ ∈ ×
Let H be a complex Lie group. Consider the bi-invariant complex
structure i End(T H) on H such that i End(h) is multiplication
h h e
∈ ∈
by √ 1. See [At], [Kos] for complex structures on principal bundles.
−
Definition 3.3. An almost complex structure J End(T Q) on an
q q
∈
H-bundle Q is called a complexion if
(d π)J = j (d π) (3.7)
q q x q
and the map Q H Q is almost-holomorphic. Writing qb = LQ(b) =
× → q
RQ(q) for q Q, b H, this means that
b ∈ ∈
(d RQ)J = J (d RQ) (3.8)
q b q qb q b
J (d LQ) = (d LQ)i . (3.9)
qb b q b q b
In the equivariant case, a complexion J is called invariant if in addi-
tion
(d LQ)J = J (d LQ) a G.
q a q aq q a ∀ ∈
By right invariance the condition (3.9) is equivalent to
J (d LQ)β = (d LQ)(√ 1β) β h.
q e q e q − ∀ ∈
Since the fibre (Q h) is a complex vector space, the notion of
H x
×
(p,q)-forms of type Ad makes sense.
H
Proposition 3.4.
(i) Let J0 be a complexion on Q. If B is a (0,1)-form of type Ad ,
H
then
J = J0 +(d LQ)(BQ d π) (3.10)
q q e q q ◦ q
defines a complexion J on Q. Every complexion J on Q arises
this way.
(ii) Let J0 be an invariant complexion on an equivariant bundle Q.
Then J is invariant if and only if B is invariant, i.e.,
BQ(d LM) = BQ (a,q) G Q. (3.11)
aq x a q ∀ ∈ ×
10 INDRANILBISWAS ANDHARALDUPMEIER
Proof. By (3.7) we have
(d π)(J J0) = (j j )(d π) = 0.
q q − q x − x q
Thus Image(J J0) Ker(d π). Since d LQ : h Ker(d π) is an
q − q ⊂ q e q → q
isomorphism, there exists a 1-form Ψ : T Q h such that
q q
→
J J0 = (d LQ)Ψ .
q − q e q q
For b H we have RQ LQ = LQ IH , and hence (d RQ)(d LQ) =
∈ b ◦ q qb ◦ b−1 q b e q
(d LQ)AdH . It follows that
e qb b−1
(d LQ)Ψ (d RQ) = (J J0)(d RQ) = (d RQ)(J J0)
e qb qb q b qb − qb q b q b q − q
= (d RQ)(d LQ)Ψ = (d LQ)AdH Ψ .
q b e q q e qb b−1 q
Therefore, we have Ψ (d RQ) = AdH Ψ , so Ψ is pseudo-tensorial.
qb q b b−1 q
For β h we have
∈
J (d LQ)β = (d LQ)(√ 1β) = J0(d LQ)β.
q e q e q − q e q
Therefore, we have (J J0) = 0, which implies that Ψ is
q − q |Ker(dqπ)
tensorial. Hence there exists a unique 1-form B of type Ad such that
H
Ψ = BQ(d π). We also have (J J0)2 = 0 and hence
q q q q − q
(d LQ)(BQj +√ 1BQ)(d π) = (d LQ)BQj (d π)+(d LQ)√ 1BQ(d π)
e q q x − q q e q q x q e q − q q
= (d LQ)BQ(d π)J0 +J0(d LQ)BQ(d π) = (J J0)J0 +J0(J J0)
e q q q q q e q q q q − q q q q − q
= J2 (J J0)2 (J0)2 = J2 (J0)2 = id+id = 0.
q − q − q − q q − q −
Since d LQ is invertible, we obtain
e q
BQj +√ 1BQ = 0.
q x − q
Thus B is of type (0,1).
For part (ii), let J be invariant. Using LQ LQ = LQ and LM π =
a ◦ q aq a ◦
π LQ we obtain
◦ a
(d LQ)BQ(d LM)(d π) = (d LQ)BQ(d π)(d LQ) = (J J0 )(d LQ)
e aq aq x a q e aq aq aq q a aq− aq q a
= (d LQ)(J J0) = (d LQ)(d LQ)BQ(d π) = (d LQ)BQ(d π).
q a q − q q a e q q q e aq q q
Since d LQ is invertible, (3.11) follows. The converse is proved in a
e aq
(cid:3)
similar way.
If (3.10) holds, we say that J is related to J0 via B. There is a close
relationship between (invariant) connexions and complexions: Every
connexion Θ on Q induces a unique complexion JΘ which is “horizon-
tal” in the sense that
JΘ : TΘQ TΘQ q Q.
q q → q ∀ ∈