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ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
BRUCE GILLIGAN, KARL OELJEKLAUS
Two remarks on Kähler homogeneous manifolds
Tome XVII, no 1 (2008), p. 73-80.
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Annales de la Faculté des Sciences de Toulouse Vol. XVII, n◦ 1, 2008
pp. 73–80
Two remarks on Kähler homogeneous manifolds(∗)
Bruce Gilligan(1), Karl Oeljeklaus(2)
ABSTRACT. — We prove that every Kähler solvmanifold has a finite cov-
ering whose holomorphic reduction is a principal bundle. An example is
given that illustrates the necessity, in general, of passing to a proper cover-
ing. We also answer a stronger version of a question posed by Akhiezer for
homogeneous spaces of nonsolvable algebraic groups in the case where the
isotropy has the property that its intersection with the radical is Zariski
dense in the radical.
RÉSUMÉ. — Nous montrons que chaque variété homogène résoluble ad-
met une revêtement finie dont la réduction holomorphe est un fibré prin-
cipal. On donne un exemple qui montre qu’il est nécessaire en général de
passer à un revêtement fini. Nous donnons de plus une réponse à une ver-
sion plus forte d’une question d’Akhiezer pour un espace homogène d’un
groupe algébrique non-résoluble dans le cas où l’isotropie est telle que
l’intersection avec le radical est dense au sens de Zariski dans le radical.
1. Introduction
Let G be a complex Lie group, H ⊂ G a closed complex subgroup and
X := G/H the corresponding complex homogeneous manifold. One main
question in the theory of complex homogeneous manifolds is to specify con-
ditions on G and H under which the manifold X admits a (not necessarily
G-invariant) Kähler metric. For X compact the theorem of Borel-Remmert
(cf. [4]) gives a complete answer to the problem. In the non-compact case
answers are known under certain assumptions on G, e.g. G semisimple [3]
or solvable [11]. In this paper we make two further contributions.
(∗) Reçu le 21 octobre 2005, accepté le 14 mars 2007
(1) Dept. of Mathematics and Statistics, University of Regina, Regina, Canada S4S
0A2.
gilligan@math.uregina.ca
(2) Centre de Mathématiques et d’Informatique, CNRS-UMR 6632 (LATP), 39, rue
Joliot-Curie, Université de Provence, 13453 Marseille Cedex 13 France.
Karl.Oeljeklaus@cmi.univ-mrs.fr
– 73 –
Bruce Gilligan, Karl Oeljeklaus
We first consider X = G/H → G/I = Y the holomorphic reduction
of X, i.e., the unique G-equivariant fibration such that all holomorphic
functions on X come from the holomorphically separable base Y , see [5].
It has the universality property that every holomorphic map of X into a
holomorphically separable complex manifold factors through this fibration.
Using known results, see [11], we prove that if G is solvable and X is Kähler
then the holomorphic reduction of X is, up to a finite covering, a principal
bundle with a Cousin group as fiber. A Cousin group is a complex abelian
Lie group admitting no non constant holomorphic functions.
Finally, we consider the following stronger version of a question posed
by Akhiezer [1]: Suppose G is a connected complex Lie group, H a closed
complex subgroup of G and X = G/H with O(X) ∼= C and there is no G-
equivariant fibration of X over a positive dimensional homogeneous rational
manifold. Is it true that X Kähler implies that X is a Cousin group?
We prove that if G is non-solvable algebraic and if the intersection of the
radical R ⊂ G with H is Zariski dense in R then X is not Kähler.
2. Kähler solvmanifolds
Suppose X = G/H is a Kähler solvmanifold, i.e., G is a complex solvable
Lie group. We consider its holomorphic reduction
G/H −→ G/I
and prove that it is always a principal bundle up to a finite covering. It is
already known that its base G/I is Stein [7] and that its fiber C := I/H
is connected and is, as a complex manifold, biholomorphic to the Cousin
group I◦/H ∩ I◦ [11].
Theorem 2.1. — Suppose G is a connected, solvable, complex Lie group
and H is a closed complex subgroup of G with X := G/H a Kähler manifold.
Let
G/H −→ G/I
be the holomorphic reduction. Then there is a subgroup of finite index Î ⊂ I
such that the bundle
X̂ := G/Î ∩ H −→ G/Î
is a holomorphic I◦/H∩I◦-principal bundle and represents the holomorphic
reduction of X̂.
Proof. — Without loss of generality we may assume that G is simply
connected. Then G admits a Hochschild-Mostow hull, see [6], i.e., there
– 74 –
Two remarks on Kähler homogeneous manifolds
exists a solvable linear algebraic group
Ga = (C∗)k  G
and Ga contains G as a Zariski dense, topologically closed, normal complex
subgroup.
Part I: We first assume that the isotropy is discrete. In order to underline
this fact, we will write the isotropy subgroup as Γ in this part of the proof.
Let
X = G/Γ −→ G/J · Γ
be the holomorphic reduction of X with J connected. Because X is Kähler,
the fiber J ·Γ/Γ of its holomorphic reduction is (as a complex manifold) iso-
morphic to a Cousin group [11]. We remark now that Ga/Γ is biholomorphic
to (C∗)k × (G/Γ). As a consequence, Ga/Γ is Kähler and its holomorphic
reduction is given by
Ga/Γ −→ Ga/J · Γ ∼= (C∗)k × (G/J · Γ),
because the base of this fibration is holomorphically separable and its fiber
is J · Γ/Γ and is, as a complex manifold, biholomorphic to a Cousin group.
Let Ga(Γ) denote the algebraic closure of Γ in Ga. The manifold Ga/Ga(Γ)
is Stein, as a quotient of algebraic solvable Lie groups. In view of the uni-
versality property of the holomorphic reduction one has the following com-
mutative diagram
Ga/Γ −→ Ga/J · Γ
↘ ↓
Ga/Ga(Γ)
In particular, J · Γ ⊂ Ga(Γ).
Now consider the fibration
Ga(Γ)/Γ −→ Ga(Γ)/J · Γ.
This fibration is the holomorphic reduction of Ga(Γ)/Γ, because its fiber is,
as a complex manifold a Cousin group, and its base, as an orbit in Ga/J ·Γ,
is holomorphically separable and hence Stein.
Let Ga(Γ)◦ be the connected component of the identity of Ga(Γ) and define
Γ̂ := Γ ∩ Ga(Γ)◦. Then Γ̂ is a Zariski dense subgroup of Ga(Γ)◦. Because
the quotient Ga(Γ)◦/Γ̂ is Kähler and Γ̂ is Zariski dense in Ga(Γ)◦, it follows
that Ga(Γ)◦ is a unipotent, see Lemma 3, p. 412 in [11].
Therefore
Ga(Γ)◦/Γ̂ −→ Ga(Γ)◦/J · Γ̂ (†)
– 75 –
Bruce Gilligan, Karl Oeljeklaus
is the holomorphic reduction of the homogeneous Kähler nilmanifold Ga(Γ)◦/Γ̂
the fiber of which is still biholomorphic to J/Γ ∩ J .
Let us introduce the following notation: let ga denote the Lie algebra
of the connected unipotent group Ga(Γ)◦, and Z denote its center, with
its Lie algebra being z. Since the exponential map exp : ga → Ga(Γ)◦ is
bijective we may consider the real Lie algebra gR spanned by exp−1(Γ̂) and
its maximal complex ideal m := gR ∩ igR.
The construction of the holomorphic reduction of a complex nilmanifold
given in [5] shows that the group J is the smallest connected, closed, complex
subgroup of Ga(Γ)◦ that contains the connected group corresponding to the
ideal m and such that J ·Γ̂ is closed in Ga(Γ)◦. We note that Z is a group such
that Z · Γ̂ is closed, as follows from a result of Barth-Otte; see Theorem 4 in
[5]. By using the Kähler assumption and detailed Lie algebra calculations
it is shown in the proof of Theorem 2’, p. 409 in [11] that
m ⊂ z.
As a consequence, J ⊂ Z. In particular, J centralizes Γ̂ in the nilpotent
group Ga(Γ)◦. Hence Γ̂  J · Γ̂ and thus J · Γ̂/Γ̂ is a group. We note also
that Î := J · Γ̂ is a subgroup of finite index of J · Γ and Î ∩ Γ = Γ̂. Thus
the holomorphic reduction (†) is a J · Γ̂/Γ̂ = Î/Γ̂-principal bundle and the
theorem is proved in this case.
Part II: We now assume that the isotropy subgroup H is not discrete in
G. Let
G/H −→ G/I
be the holomorphic reduction of G/H. We would like to show that H is a
normal subgroup of I. Here again Ga/H is biholomorphic to (C∗)k × G/H
and is therefore Kähler. Also Ga/H → Ga/I is clearly the holomorphic re-
duction of Ga/H. Set N := (NGa(H
◦))H , i.e., those connected components
of the normalizer of H◦ in Ga that meet H. Note that N/H is connected
and N is an algebraic subgroup of Ga. Therefore, N◦ has finite index in N .
We remark that Ga/N is holomorphically separable by Lie’s Flag Theorem.
Again by the universality property of the holomorphic reduction one has
the following diagram
Ga/H −→ Ga/I
↘ ↓
Ga/N
In particular, I ⊂ N . Since N/I is an orbit in Ga/I and the latter space
is Stein, it follows that N/I is holomorphically separable. But we are in
– 76 –
Two remarks on Kähler homogeneous manifolds
a Kähler setting and thus the fiber I/H of the holomorphic reduction of
G/H is biholomorphic to a (connected) Cousin group. As a consequence,
N/H → N/I is the holomorphic reduction of N/H. Now define Î := I∩N◦.
Then Î ∩H = N◦∩H. Moreover, Î ⊂ G and Î ∩H ⊂ G. With this notation
we get as in the first case the fibration
N/H  N◦/(N◦ ∩ H) → N◦/Î  N/I
which is the holomorphic reduction of N◦/(N◦ ∩H) with fiber Î/(N◦ ∩H).
Now we have the following
(N◦/H◦)/(N◦ ∩ H/H◦)  N◦/N◦ ∩ H −→ N◦/Î  (N◦/H◦)/(Î/H◦)
and Part I above applies. It follows that N◦ ∩ H/H◦ is a normal subgroup
of (N◦∩H/H◦) · (I◦/H◦) and that N◦∩H is a normal subgroup of Î which
in turn is a subgroup of finite index of I. Thus the holomorphic reduction
of the Kähler solvmanifold G/H is, up to a finite covering, a Cousin group
principal bundle. 
The following examples show that, in general, passing to a subgroup of
finite index is unavoidable in order to get a principal bundle and that the
Kähler condition is necessary.
Example 1. — Let
G :=
{(
a b
0 1
)
| a ∈ C∗, b ∈ C
}
and
Γ :=
{( √
−1n k +
√
−1l
0 1
)
| n, k, l ∈ Z
}
.
Then X := G/Γ is Kähler, Γ is not nilpotent, and the holomorphic reduction
is not a principal bundle. But of course all this is true, if one replaces Γ by
the obvious subgroup of index 4.
Example 2. — We also note that the Kähler condition is necessary. We
give the following example due to J.J. Loeb [8]. Let K = Z, C and GK =
K  K2 be defined by:
(z, b) ◦ (z′, b′) := (z + z′, eAz
′
b + b′),
where z, z′ ∈ K, b, b′ ∈ K2 and A is the real logarithm of the matrix
( 2 11 1 ). The complex solvmanifold X = GC/GZ is not Kähler and has as
holomorphic reduction
GC/GZ
C
∗×C∗−→ GC/G′C · GZ = C
∗
– 77 –
Bruce Gilligan, Karl Oeljeklaus
No finite covering of X = GC/GZ has a holomorphic reduction that is a
principal bundle since in this case X would be Stein and thus Kähler.
3. A non-Kähler criterion
For a complex Lie group G we let G = S · R be a Levi-Malcev decom-
position, where S is semisimple and R is the radical of G.
Theorem 3.1. — Suppose G = S · R is a Levi-Malcev decomposition of
an algebraic group with both factors having positive dimension. Let H be a
closed complex subgroup of G and X := G/H such that O(X) ∼= C. Assume
H is not contained in any proper parabolic subgroup of G, i.e. there is no G-
equivariant fibration of X over a positive dimensional homogeneous rational
manifold. Suppose R ∩ H is Zariski dense in R. Then G/H is not Kähler.
Proof. — Without loss of generality we may assume that the G-action
on X is almost effective. Let N := NG(H◦) be the normalizer of the identity
component of H in G. Then G/N is an orbit in some projective space and
our assumptions force N = G, i.e. H is discrete. We shall use Γ instead of
H for this reason. We assume that X is Kähler and derive a contradiction.
It is known that the orbits of R are closed in our situation, see [12], and so
we may consider the fibration
G/Γ −→ G/R · Γ = S/(S ∩ R · Γ).
Note that Λ := S ∩ R · Γ is a discrete subgroup of S. It follows from the
assumptions on X that Λ is a Zariski dense subgroup of S, since every
algebraic subgroup of a semisimple complex Lie group is contained in either
a maximal reductive or a maximal parabolic subgroup of G. Recall that a
homogeneous manifold of a semisimple Lie group is Stein if and only if the
isotropy group is reductive [9], [10].
As a consequence, we can find a semisimple element λ ∈ Λ of infinite order.
The Zariski closure of the infinite cyclic subgroup generated by λ is an
algebraic torus A := 〈λ〉Z ∼= (C∗)l for some l > 0. Without loss of generality,
we may assume A is connected, since one can replace λ by a finite power,
if necessary. Note that A/〈λ〉 is a Cousin group and therefore A/A ∩ Λ is
also Cousin. Now let S = S1 · · ·Sn be a decomposition of S into a product
of simple factors and let pj : S → Sj denote the projection onto the j-th
factor Sj . It was noted in the Lemma on p. 328 in [2] that one may choose
λ so that
pj(A) = {e} for all j = 1, . . . , n. (∗)
– 78 –
Two remarks on Kähler homogeneous manifolds
Set G∗ := A  R which is an algebraic solvable subgroup of G and
Γ∗ := G∗ ∩ Γ. Let N be the Zariski closure of Γ∗ in G∗. Then N ⊂ G∗ and
N◦ is nilpotent by Lemma 4.3, p. 412, in [11].
We claim N = G∗. Consider the fibration π : G∗/Γ∗ → A/A ∩ Λ and
its restriction to the N◦-orbit N◦/Γ∗ ∩ N◦. The fiber of the restriction
π|N◦/Γ∗∩N◦ is equal to R/(R ∩ Γ) by our assumption that Γ ∩ R is Zariski
dense in R. This implies R ⊂ N◦. Now the projection p : G → S ∼= G/R is
an algebraic homomorphism and thus the image of an algebraic subgroup of
G containing R is an algebraic subgroup of S. Hence p(N◦) is an algebraic
subgroup of S that contains 〈λ〉Z. As a consequence, p(N◦) = A. These two
observations imply N◦ = G∗. Thus G∗ is nilpotent.
Let φ denote the representation of A on R in the semidirect product
G∗ = A φ R. Since A ∼= (C∗)l, it follows that φ(A) is reductive. But φ(A)
is also nilpotent and so must be trivial. Thus A acts trivially on the radical
R. Therefore G∗ = A×R is a direct product. But this implies that the group
G is a group theoretic direct product S ×R by the choice of A, see (∗), and
the fact, just proved, that the subgroup A is contained in the kernel of the
representation that builds the semidirect product S  R. This yields the
desired contradiction, since S cannot have positive dimension in the case of
a Kähler quotient of a direct product S × R, see [12]. 
Acknowledgements. — The first author gratefully acknowledges that
he was partially supported by an NSERC Discovery Grant and was an invité
du LATP à l’Université de Provence.
Bibliography
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groups, Ann. Global Anal. Geom., 2, p. 129-140 (1984).
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[3] Berteloot (F.). — Existence d’une structure kählérienne sur les variétés ho-
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Bruce Gilligan, Karl Oeljeklaus
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complex solvmanifolds, Ann. Inst. Fourier (Grenoble) 36, p. 57-65 (1986).
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– 80 –


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