Classification of Kaehler homogeneous manifolds of non-compact dimension two

0.25 MB·English

by  Seyed Ruhallah Ahmadi

CLASSIFICATION OF KA¨HLER HOMOGENEOUS
MANIFOLDS OF NON–COMPACT DIMENSION TWO
5
1
SEYEDRUHALLAHAHMADIANDBRUCEGILLIGAN
0
2
n Abstract. Suppose G is a connected complex Lie group and H is a closed
a complexsubgroupsuchthatX :=G/H isK¨ahlerandthecodimensionofthe
J top non–vanishing homology group of X with coefficients in Z2 is less than
or equal to two. We show that X is biholomorphic to a complex homoge-
9
neous manifoldconstructed usingwell–known“basicbuildingblocks”,i.e.,C,
2
C∗, Cousin groups, and flag manifolds. For H discrete the classification was

presented inthefirstauthor’sdissertation[4].
]
V
C
.
h 1. Introduction
t
a
In this paper we consider complex homogeneous manifolds of the form G/H,
m
where G is a connected complex Lie group and H is a closed complex subgroup of
[
G. The existence of complex analytic objects on such a G/H, like non–constant

1 holomorphic functions, plurisubharmonic functions and analytic hypersurfaces, is
v related to when G/H could be K¨ahler. So the first question one might consider
8
concernstheexistenceofK¨ahlerstructuresandwerestrictourselvestothatquestion
5
here. ThestructureofcompactK¨ahlerhomogeneousmanifoldsisnowclassical[28]
5
7 and[13]andthe structureinthe caseofG–invariantmetrics isalsoknown[15]. As
0 aconsequence,ourinvestigationshereconcernnon–compactcomplexhomogeneous
. manifolds having a K¨ahler metric that is not necessarily G–invariant.
1
0
Some resultsareknownunder restrictionsonthe type ofgroupGthatis acting.
5
ThebaseoftheholomorphicreductionofanycomplexsolvmanifoldisalwaysStein
1
: [24],wheretheproofusessomefundamentalideasin[27]. ForGasolvablecomplex
v
Lie group and G/H K¨ahler the fiber of the holomorphic reduction of G/H is a
i
X Cousin group, see [32] and the holomorphic reduction of a finite covering of G/H
r is a principal Cousin group bundle, see [19]. If G is semisimple, then G/H admits
a
a K¨ahler structure if and only if H is algebraic [11]. For G reductive there is the
characterizationthatG/H is K¨ahlerifandonlyif S·H is closedinGandS∩H is
analgebraicsubgroupofS,amaximalsemisimplesubgroupofG,see[20,Theorem
5.1]. There is also a result if G is the direct product of its radical and a maximal
semisimple subgroup under some additional assumptions on the isotropy subgroup
and so on the structure of X [31].
One way to proceed is to impose some topological restraints on X. In [17] we
classifiedK¨ahlerhomogeneousmanifoldsX havingmorethanoneendbyshowingX
iseitheraproductofaCousingroupandaflagmanifoldorX admitsahomogeneous
Date:January30,2015.
1991 Mathematics Subject Classification. Primary32M10;Secondary32Q15.
Key words and phrases. homogeneous K¨ahlermanifold,non-compact dimensiontwo.
1
2 SEYEDRUHALLAHAHMADIANDBRUCEGILLIGAN
fibration as a C∗–bundle over the product of a compact complex torus and a flag
manifold. Now in the setting of proper actions of Lie groups Abels introduced the
notion of non–compact dimension, see [2] and [3, §2]. We do not wish to assume
that our Lie groupactions are necessarily proper ones, so we take a dual approach
and define the non–compact dimension d of a connected smooth manifold to be
X
the codimension of its top non–vanishing homology group with coefficients in Z ,
2
see §2. Our goal in this paper is to classify K¨ahler homogeneous manifolds G/H
withd =2. Allsuchspacesareholomorphicfiberbundles wherethe fibers and
G/H
the bases of the bundles involved consist of Cousin groups, flag manifolds, C, and
C∗. We now present the statement of our main result, where T denotes a compact
complex torus, C a Cousin group, and Q a flag manifold.
Theorem 1.1 (Main Theorem). Suppose X := G/H is K¨ahler, where G is a
connected complex Lie group and H is a closed complex subgroup of G. Then
d =2 if and only if X is one of the following:
X
Case I: H discrete: A finite covering of X is biholomorphic to a product C×A,
with C a Cousin group, A a Stein Abelian Lie group and d +d =2.
C A
Case II: H is not discrete:
(1) Suppose O(X)=C and let G/H →G/N be its normalizer fibration.
(a) X is a (C∗)k–bundle over C×Q with d +k =d =2.
C X
(b) X is T ×G/N with O(G/N) = C and G/N fibers as a C–bundle
over a flag manifold; there are two subcases depending on whether S acts
transitively on G/N or not.
(2) Suppose O(X)6=C and let G/H →G/J be its holomorphic reduction.
(a) G/J is Stein with d =2 and X =T ×Q×G/J, where
G/J
(i) G/J =C.
(ii) G/J is the 2–dimensional affine quadric
(iii) G/J is the complement of a quadric curve in P
2
(iv) G/J =(C∗)2
(b) G/J is not Stein with d =2; then G/J is a C∗–bundle over an
G/J
affine cone minus its vertex and X =T ×G/J.
(c) d =1 with G/J an affine cone minus its vertex and d =1
G/J J/H
(i) O(J/H) =C: X is a C∗–bundle over T ×Y, where Y is a flag
manifold bundle over the affine cone minus its vertex.
(ii) O(J/H)6=C: this case does not occur.
Thepaperisorganizedasfollows. Insectiontwowegatheranumberoftechnical
tools. Inparticular,wenote thatProposition2.10deals withthe setting wherethe
fiber of the normalizer fibration is a Cousin group and its base is flag manifold. It
isessentialforCaseII (1)(a)inthe MainTheoremandcanbe usedtosimplifythe
proofwhend =1,seeRemark2.11. Sectionthreeisdevotedtothecasewhenthe
X
isotropysubgroupis discrete. Sections four andfive dealwith generalisotropyand
contain the proof of the main result when there are no non–constant holomorphic
functions andwhen there arenon–constantholomorphicfunctions, respectively. In
the last section we present some examples.
This work was partially supported by an NSERC Discovery Grant. We thank
Prof. A. T. Huckleberry for his comments that led to improvements in the work.
WealsothanktheArbeitsgruppeTransformationsgruppenattheRuhr–Universit¨at
CLASSIFICATION OF KA¨HLER G/H WITH dG/H=2 3
Bochumfortheirkindhospitalityduringasojournatwhichtimepartofthispaper
was written.
2. Technical Tools
The purpose of this section is to gather together a number of definitions and
basic tools that are subsequently needed.
2.1. Basic Notions.
Definition 2.1. A Cousin group is a complex Lie group G with O(G) = C. The
terminology toroidal group is also found in the literature. Every Cousin group is
Abelian and is the quotient of Cn by a discrete subgroup having rank n+k for
some k with 1≤k ≤n. For details, we refer the reader to [1].
Definition 2.2. A flag manifold (the terminology homogeneous rational manifold
isalsoincommonusage)isa homogeneousmanifoldofthe formS/P,where S is a
connected semisimple complex Lie group and P is a parabolic subgroup of S. One
source concerning the structure of flag manifolds is [8, §3.1].
Definition 2.3. For X a connected (real) smooth manifold we define
dX := dimRX − min{ r | Hk(X,Z2) = 0 ∀ k > r },
i.e., d is the codimension of the top non–vanishing homology group of X with
X
coefficients in Z . We call d the non–compact dimension of X.
2 X
Proposition 2.4. Suppose X is a connected Stein manifold. Then dimCX ≤dX.
Proof. For X Stein one has Hk(X)=0 for all k >dimCX by [34] . (cid:3)
2.2. Fibration Methods. Throughoutwemakeuseofanumberoffibrationsthat
are now classical.
(1) NormalizerFibration: GivenG/H letN =N (H0)bethenormalizerin
G
GoftheconnectedcomponentoftheidentityH0 ofH. SinceH normalizes
H0, we have H ⊂N and the normalizer fibration G/H →G/N.
(2) Holomorphic Reduction: Given G/H we set J := { g ∈ G | f(gH) =
f(eH) for all f ∈ O(G/H) }. Then J is a closed complex subgroup of G
containing H and we call the fibration p : G/H → G/J the holomorphic
reduction of G/H. By construction G/J is holomorphically separable and
O(G/H)∼=p∗(O(G/J)).
F
Suppose a manifold X admits a locally trivial fiber bundle X →B with F and
B connected smooth manifolds. One would then like to know how d and d are
F B
relatedto d whenever possible. The following result was provedin §2 in [7] using
X
spectral sequences.
Lemma 2.5 (The Fibration Lemma). Suppose X →F B is a locally trivial fiber
bundle with X,F,B smooth manifolds. Then
(1) if the bundle is orientable (e.g., if π (B)=0), then d =d +d .
1 X F B
(2) if B has the homotopy type of a q–dimensional CW complex, then d ≥
X
d +(dimB−q).
F
(3) if B is homotopy equivalent to a compact manifold, then d ≥d +d .
X F B
4 SEYEDRUHALLAHAHMADIANDBRUCEGILLIGAN
Remark 2.6. IfB ishomogeneous,thenoneknowsthatB ishomotopyequivalent
to a compact manifold if:
(1) the isotropy subgroup of B has finitely many connected components [30];
e.g., in an algebraic setting,
(2) if B is a solvmanifold [29]; indeed, every solvmanifold is a vector bundle
over a compact solvmanifold [9].
2.3. Special case of a question of Akhiezer. Later we will need a result that
is based on [7, Lemma 8]. Since that Lemma was stated in a way suitable for its
particular application in [7], we reformulate it in a form suitable for the present
context.
Lemma 2.7 (Lemma 8 [7]). Let G be a connected, simply connected complex Lie
group with Levi-Malcev decomposition G=S⋉R with dimCR=2 and Γ a discrete
subgroup of G such that X =G/Γ is K¨ahler. Then Γ is contained in a subgroup of
G of the form A⋉R, where A is a proper algebraic subgroup of S.
This has the following consequence which we use later.
Theorem 2.8. Suppose G is a connected, simply connected, complex Lie group
with Levi-Malcev decomposition G = S ⋉R with dimCR = 2. Let Γ be a discrete
subgroupofGsuchthatX =G/ΓisK¨ahler, Γisnotcontainedinaproperparabolic
subgroup of G and O(G/Γ)≃C. Then S ={e}, i.e., G is solvable.
Proof. The radical orbits are closed, e.g., see [18]. By lemma 2.7 the subgroup Γ
is contained in a proper subgroup of G of the form A⋉R, where A is an algebraic
subgroup of S. Thus there are fibrations
G/Γ −→ G/R·Γ −→ S/A,
where the base G/R·Γ = S/Λ with Λ := S ∩R.Γ. If A is reductive, then S/A is
Stein and we get non-constant holomorphic functions on X as pullbacks using the
above fibrations. But this contradicts the assumption that O(X)≃ C. If A is not
reductivethen[25,Theorem,§30.1]appliesandAiscontainedinaproperparabolic
subgroup of S. But this implies Γ is too, thus contradicting the assumption that
this is not the case. (cid:3)
2.4. The algebraicsettingrevisited. Throughoutwerepeatedlyusetworesults
of Akhiezer concerning the invariantd in the setting where X =G/H and G is a
X
connected linear algebraic group over C and H is an algebraic subgroup of G. For
the convenience of the reader we now state these here.
Theorem 2.9 (d = 1 [5]; d = 2 [6]). Suppose G is a connected linear algebraic
group over C and H is an algebraic subgroup of G and X :=G/H.
(1) d =1=⇒H is contained in a parabolic subgroup P of Gwith P/H =C∗.
X
(2) d =2=⇒H is contained in a parabolic subgroupP of G with P/H being:
X
(a) C
(b) the affine quadric Q
2
(c) the complement of a quadric curve in P
2
(c) (C∗)2
CLASSIFICATION OF KA¨HLER G/H WITH dG/H=2 5
2.5. Cousin group bundles over flag manifolds. In this section we prove a
resultconcerningthestructureofK¨ahlerhomogeneousmanifoldswhosenormalizer
fibrations are Cousingroupbundles overflag manifolds, where there is no assump-
tion about the invariantd. In order to do this we will show that one can reduce to
the case where a complex reductive group is acting transitively and employ some
nowclassicaldetailsaboutthestructureofparabolicsubgroups,e.g.,see[8]or[16].
A crucialpoint in the setting ofinterest is the fact that all S–orbitsare closed and
have algebraic isotropy [20, Theorem 5.1].
Proposition 2.10. Suppose G/H is a K¨ahler homogeneous manifold whose nor-
malizer fibration G/H → G/N has fiber C := N/H a Cousin group and base
Q := G/N a flag manifold. Then there exists a closed complex subgroup J of N
containing H such that the fibration G/H → G/J realizes X as a (C∗)k–bundle
over a product G/J =Q×Y, where Y is a Cousin group with d =d −k.
Y X
Proof. Our first task is to show that there is a reductive complex Lie group acting
holomorphicallyandtransitivelyonX. Write C =Ck/Γandnotethatthereexists
a subgroup Γ < Γ such that C := Cq/Γ is isomorphic to (C∗)q and is a covering
group of C,be.g., see [1, §1.1b]. In parbticular, the reductive complex Lie group
G:=S×C acts transitively on X. We dropthe hats from now on and assume, by
gboing to abfinite covering, if necessary, that G= S×Z is a reductive complex Lie
group,whereZ ∼=(C∗)q isthecenterofGandS isamaximalsemisimplesubgroup.
Let x ∈ X be the neutral point, z ∈ G/N be its projection in the base, and
0 0
F be the fiber over the point z . Now the S–orbit S·H/H =S/S∩H is closed
z0 0
in X and S ∩H is an algebraic subgroup of S [20, Theorem 5.1]. So there is an
induced fibration
S/S∩H ֒→ G/H
(2.1) A↓ ↓F ∼=C
z0
S/P = G/N
Since the center normalizes any subgroup, we have N =P ×Z with P a parabolic
subgroupof S. Now any parabolicgroupP canbe written as a semidirect product
of its unipotent radicalU anda subgroupL . Further, L is the centralizerin S
P P P
of a subgroup C of L . In particular, it follows that C is the center of L , e.g.,
P P P P
see[8]. Inpassing,wealsonotethatthe commutatorsubgroupofL issemisimple
P
and thus L is a complex reductive Lie group.
P
The bundle in (2.1) is defined by a representation ρ : N −→ Aut0(C) ∼= C and
thegroupA=ρ(P)liesintheconnectedcomponentoftheidentityoftheautomor-
phism groupof C because P is connected. Since C is Abelian, ρ| factors through
P
the canonical projection from P to P/P′. Because every parabolic subgroup con-
tains a maximal torus, every root space in the parabolic is in its commutator
subgroup. AlsoC isnotinthe commutatorsubgroup,whilethe commutatorsub-
P
groupofL is. Asaconsequence,P/P′ ∼=(C∗)p,withp=dimC ,e.g.,seealso[8,
P P
Proposition8, §3.1]. Now the S–orbits in G/H are closed and the C –orbit in the
P
typical fiber N/H through the neutral point is the intersection of N/H with the
correspondingS–orbit. So this C –orbit is closed and complex and is an algebraic
P
6 SEYEDRUHALLAHAHMADIANDBRUCEGILLIGAN
quotientof the groupC biholomorphic to (C∗)k for some k ≤p. In this reductive
P
settingS isanormalsubgroupofGandsoallS–orbitsarebiholomorphicandfiber
overG/N with fiber biholomorphic to (C∗)k. If k=0, then X =C×G/N, by the
argument given in the last paragraphof the proof.
Assume k > 0. Let N −π→1 N/H0 −π→2 N/H ∼= C, where π is the canonical
1
homomorphism with H0 normal in N and π is a covering homomorphism, since
2
H/H0 is a (normal) discrete subgroup of the Abelian group N/H0. Set
J := π−1◦π−1(ρ(C )).
1 2 P
It follows that J is a closed complex Lie subgroupof G contained in N, since both
π and π are holomorphic Lie group homomorphisms, that J contains H by its
1 2
definition, and that the J–orbit through the neutral point is the C –orbit. This
P
yields an intermediate fibration
(C∗)k
G/H −→ G/J −→ G/N
with the first fiber (C∗)k ∼=A.
WeclaimthatthebundleG/J →G/N isholomorphicallytrivial. Inordertosee
thisnotethattheP–actionontheneutralfiberofthebundleG/J →G/N istrivial.
Otherwise,thedimensionoftheP–orbitinF wouldbebiggerthank,asassumed
z0
above,andthiswouldbeacontradiction. Asaconsequence,allS–orbitsinG/J are
sections of the bundle G/J → G/N. Since G/N is simply connected, this bundle
is holomorphicallytrivial, i.e., G/J =(C/A)×S/P. Because C is a Cousin group,
Y := C/A is also a Cousin group. The statement about the topological invariant
follows because d =d , since S/P is compact, and d =d −k. (cid:3)
C/A G/J G/J X
Remark 2.11. The case d = 1 is treated in [17, Proposition 5], where it is
X
assumedthatX hasmorethanoneend. ForX K¨ahlerthisisequivalenttod =1.
X
3. The Discrete Case
ThroughoutthissectionweassumethatX =G/ΓisK¨ahlerwithd =2,where
X
Gisaconnected,simplyconnected,complexLiegroupandΓisadiscretesubgroup
ofG. WefirstshowthatGissolvable. Thenweprovethatafinite coveringofsuch
an X is biholomorphic to a product C×A, where C is a Cousin group and A is a
holomorphically separable complex Abelian Lie group.
3.1. The reduction to solvable groups. We first handle the case when the
K¨ahler homogeneous manifold has no non–constant holomorphic functions.
Lemma 3.1. Assume Γ is a discrete subgroup of a connected simply connected
complex Lie group G that is not contained in a proper parabolic subgroupof G, with
X :=G/Γ K¨ahler, O(X)=C, and d ≤2. Then G is solvable.
X
Proof. Assume G=S⋉R is a Levi decomposition. Since the R–orbits are closed,
we have a fibration
G/Γ −→ G/R·Γ = S/Λ,
where Λ := S ∩ R · Γ is Zariski dense and discrete in S, e.g., see [18]. Now if
O(R·Γ/Γ)=C, then the result was proved in [18]. Otherwise, let
R·Γ/Γ −→R·Γ/J =: Y
CLASSIFICATION OF KA¨HLER G/H WITH dG/H=2 7
be the holomorphic reduction. Then Y is holomorphically separable and since R
acts transitively on Y, it follows that Y is Stein [24]. But 2=dX ≥dY ≥dimCY.
Further J0 is a normal subgroup of G. As a consequence, the quotient group
R:=R/J0 has complex dimension one or two. If dimR=1, then G:=S×R is a
pbroductandthis implies S ={e} by [31]. If dimR=2,bthen Gis eibther a probduct,
see [31] again, or is a non–trivial semidirect prodbuct. In theblatter case the result
now follows by Theorem 2.8. (cid:3)
InthenextPropositionwereducetothesettingwheretheLevifactorisSL(2,C).
We first prove a technical Lemma in that setting.
Lemma 3.2. Suppose G/Γ is K¨ahler and d ≤2, where Γ is a discrete subgroup
G/Γ
of a connected, complex Lie group of the form G=SL(2,C)⋉R with R the radical
of G. Then Γ is not contained in a proper parabolic subgroup of G.
Proof. Assumethecontrary,i.e.,thatΓiscontainedinaproperparabolicsubgroup
and let P be a maximal such subgroup of G. Note that P is isomorphic to B⋉R,
where B is a Borel subgroup of SL(2,C). Let P/Γ → P/J be the holomorphic
reduction. ThenP/ΓisaCousingroup[32]andP/J isStein[24]. NotethatJ 6=P,
since otherwise P would be Abelian, a contradiction. The Fibration Lemma and
Proposition2.4 imply dimCP/J =1 or 2. So P/J is biholomorphic to C,C∗,C∗×
C∗, or the complex Klein bottle.
InthefirsttwocasesP/J isequivariantlyembeddableinP andby[26]itfollows
1
thatG/J isK¨ahler. InthelattertwocasesthefiberJ/ΓiscompactbytheFibration
Lemma and we can push down the K¨ahler metric on X to obtain a K¨ahler metric
on G/J, e.g., see [12]. In particular, the S–orbit S/(S ∩J) in G/J is K¨ahler and
so its isotropy S∩J is algebraic [11]. Now consider the diagram
F Y
G/Γ −→ G/J −→ G/P
||
∪ ∪ P
1
||
S/S∩Γ −F→S S/S∩J −Z→ S/B
Note that because Y := P/J is noncompact and d = 2, it follows from the
G/Γ
Fibration Lemma that either d = 1 or F is compact. Since F is an Abelian Lie
F
group, it is clear that d ≤d .
FS F
The following enumerate, up to isomorphism, the algebraic subgroups of B and
in each case we derive a contradiction.
(1) dimCS∩J =2. Then S∩J =B. This yields the contradiction dS/S∩Γ ≤
dFS +dS/B =1+0=1<3=dS/S∩Γ, since S∩Γ is finite.
(2) dimCS∩J =1.
(a)IfS∩J =C∗,thenS/S∩J isanaffinequadricorthe complement
of a quadric curve in P and thus Z = C. So P/J 6= C∗ and is either C
2
or (C∗)2, i.e., d = 2. Then the Fibration Lemma implies F is compact
P/J
and, sinceF is closedinF, itmustalsobe compact. Butthis forcesS∩Γ
S
to be an infinite subgroup of S∩J which is a contradiction.
(b) If S∩J =C, then S/S∩J is a finite quotient of C2\{(0,0)} and
so Z =C∗. Now P/J =C,(C∗)2 or C∗. In the first two instances F would
8 SEYEDRUHALLAHAHMADIANDBRUCEGILLIGAN
be compact and we get the same contradiction as in (a). In the last case
d = 1 by the Fibration Lemma and F is either compact or C∗. Again
F S
S∩Γ is infinite with the same contradiction as in (a).
(3) dimCS ∩J = 0. Here S ∩J is finite, since it is an algebraic subgroup
of B. Then dimS/S ∩ J = 3 and we see that dimG/J = 3, since we
know dimG/P = 1 and dimP/J ≤ 2. Then P/J = (C∗)2 and, since
S/S ∩J is both open and closed in G/J, it follows that S/S ∩J = G/J
and dS/S∩J = 2. But F is compact and thus so is FS and we get the
contradiction that dS/S∩Γ =2<3=dS/S∩Γ.
As a consequence, Γ is not contained in a proper parabolic subgroup of G. (cid:3)
Proposition 3.3. Suppose G/Γ is K¨ahler with d ≤2. Then G is solvable.
G/Γ
Proof. First note that G cannot be semisimple. If that were so, then Γ would
be algebraic, hence finite and thus G/Γ would be Stein. But then 2 = d ≥
G/Γ
dimCG/Γ=dimCG which is a contradiction, since necessarily dimCG≥3 for any
complex semisimple Lie group G.
SoassumeG=S⋉Rismixed. TheproofisbyinductiononthedimensionofG.
Now if a proper parabolic subgroupof G contains Γ, then a maximalone does too,
is solvable by induction and so has the special form B⋉R, where B is isomorphic
to a Borel subgroup of S =SL(2,C). But this is impossible by Lemma 3.2.
Lemma 3.1 handles the case O(G/Γ) = C. So we assume O(G/Γ) 6= C with
holomorphicreductionG/Γ→G/J. The MainResultin[7]givesthe followinglist
of the possibilities for the base G/J:
(1) C
(2) affine quadric Q
2
(3) P \Q, where Q is quadric curve
2
(4) an affine cone minus its vertex
(5) C∗–bundle over an affine cone minus its vertex
In case (1) the bundle is holomorphically trivial, with compact fiber a torus
and the group that is acting effectively is solvable. In cases (2) and (3) we have
fibrations G/Γ → G/J → G/P = P and so Γ is contained in a proper parabolic
1
subgroup of G, contradicting what was shown in the previous paragraph.
Inordertohandlecases(4)and(5)werecallthatanaffineconeminusitsvertex
fibers equivariantly as a C∗–bundle over a flag manifold. Thus we get fibrations
G/Γ −→ G/J −→ G/P.
Note that it cannot be the case that G6=P, since then Γ would be contained in a
proper parabolic subgroup, a possibility that has been ruled out. So G = P and
G/J (or a 2–1 covering) is biholomorphic to C∗ or (C∗)2. In the second case the
fiber J/Γ is compact and thus a torus and thus G is solvable. If G/J = C∗, then
J/Γ is K¨ahler with dimJ < dimG and d = 1 by the Fibration Lemma. By
J/Γ
induction J is solvable and so G is solvable too, because G/J =C∗. (cid:3)
CLASSIFICATION OF KA¨HLER G/H WITH dG/H=2 9
3.2. A product decomposition. Inordertoprovethe classificationwe willhave
need of the next splitting result.
Proposition 3.4. Suppose G is a connected, simply connected solvable complex
Lie group that contains a discrete subgroup Γ such that G/Γ is K¨ahler and has
holomorphic reduction G/Γ → G/J with base (C∗)2 and fiber a torus. Then a
finite covering of G/Γ is biholomorphic to a product.
Proof. First assume that J0 is normal in G and let α:G→G/J0 be the quotient
homomorphism with differential dα : g → g/j. Then G/J0 is a two dimensional
complex Lie group that is either Abelian or solvable. In the Abelian case G :=
0
α−1(S1×S1) is a subgroup of G that has compact orbits in X, since these orbits
fiberastorusbundlesoverS1×S1inthebase. Theresultnowfor(Γ,G ,G)follows
0
from [?, Theorem 6.14], since this triple is a CRS.
Next assume that G/J0 is isomorphic to the two dimensional Borel group B
with Lie algebra b. Then n := dα−1(nb) is the nilradical of G. Let N denote the
corresponding connected Lie subgroup of G. Now choose γ ∈ Γ := N ∩Γ such
N N
thatα(γ )6=e. Thereexistsx∈nsuchthatexp(x)=γ . LetU betheconnected
N N
Lie group corresponding to hγNiC. Since Γ centralizes J0 (see [19, Theorem 1]), it
followsthatn=u⊕jandN =U×J0 isAbelian. SetΓ :=Γ∩U andΓ :=Γ∩J0.
U J
Then N/Γ =U/Γ ×J0/Γ .
N U J
Since Γ/Γ = Z, we may choose γ ∈ Γ such that γ projects to a generator of
N
Γ/ΓN. AlsosetA:=exp(hwiC)forfixedw ∈g\n. SinceAiscomplementarytoN,
we have G= A⋉N. Now there exist γ ∈A and γ ∈N such that γ =γ ·γ .
A N A N
Both γ and γ centralize J0 and thus γ does too. Also γ = exp(h) for some
N A A
h=sw with s∈C. Therefore,
(3.1) [h,j] = 0.
Since a+u is isomorphic to b = g/j as a vector space, there exists e ∈ u such
that
[dα(h),dα(e)] = 2dα(e).
Let {e1,...,en−2} be a basis for j. There exist structure constants ai so that
n−2
[h,e] = 2e + a e
X i i
i=1
andthe remainingstructureconstantsareall0by(3.1). Notethat, conversely,any
choice of the structure constants a determines a solvable Lie algebra g and the
i
corresponding connected simply–connected complex Lie group G=A⋉N.
We now compute the action ofγ ∈A onN by conjugation. In orderto do this
A
note that the adjoing representation restricted to n, i.e., the map ad : n → n, is
h
expressed by the matrix
2 0 ... 0
 
a 0 ... 0
1
M := [adh] =  ... ... ... ... 
 
 an−2 0 ... 0 
10 SEYEDRUHALLAHAHMADIANDBRUCEGILLIGAN
So the action of A on N is through the one parameter group of linear transforma-
tions t7→etM for t∈C. For k ≥1
1
(tM)k = (2t)kM,
2
and it follows that
1
etM = (e2t−1)M + Id.
2
Nowtheprojectionoftheelementγ actstriviallyonthebaseY =G/J,sot=πik
A
where k ∈ Z. Hence γ acts trivially on U. Also γ acts trivially on N, since N
A N
is Abelian. Thus γ acts trivially on N and as a consequence, although G is a
non-Abelian solvable group the manifold X =G/Γ is just the quotient of Cn by a
discreteadditivesubgroupofrank2n−2. Its holomorphicreductionis the original
torus bundle which, since we are now dealing with the Abelian case, is trivial.
Now assume J0 is not normal in G, set N :=N (J0), and let G/J −N−−/→J G/N
G
be the normalizer fibration. Since the base G/N of the normalizer fibration is an
orbit in some projective space, G/N is holomorphically separable and thus Stein
[24]. Since we also have 2 ≥ dG/N ≥ dimCG/N we see that G/N ∼= C, C∗ or
C∗ ×C∗. We claim that we must have G/N = C∗, i.e., we have to eliminate the
other two possiblities. Assume G/N ∼= C. Since d ≤ 2 the Fibration Lemma
X
implies d = 0, i.e., N/J is biholomorphic to a torus T. Thus G/J = T ×C.
N/J
However, we assume that G/J = C∗ ×C∗ and this gives us a contradiction. Now
assume G/N ∼= C∗ ×C∗. By Chevalley’s theorem [14] the commutator group G′
acts algebraically. Hence the G′–orbits are closed and one gets the commutator
C ′ ′
fibration G/N −→ G/G ·N. Since G is solvable, it follows that G is unipotent
and the G′–orbits are cells, i.e., G′·N/N ∼= C. By the Fibration Lemma the base
of the commutator fibration is a torus. But it is proved in [23] that the base of
a commutator fibration is always Stein which is a contradiction. This proves the
claimthatG/N ∼=C∗ andbythe FibrationLemmad =1andhenceN/J =C∗
N/J
Since G is simply connected, G admits a Hochschild-Mostow hull ([22]), i.e.,
there exists a solvable linear algebraic group
G = (C∗)k⋉G
a
thatcontainsG as aZariskidense, topologicallyclosed, normalcomplex subgroup.
By passing to a subgroup of finite index we may, without loss of generality, as-
sume the Zariski closure G (Γ) of Γ in G is connected. Then G (Γ) ⊇ J0 and
a a a
G (Γ) is nilpotent [19]. Let π : G (Γ) → G (Γ) be the universal covering and set
a a a
Γ:=π−1(Γ). Since G (Γ) is asimcply connected, nilpotent, complex Lie group,the
a
ebxponentialmapfromctheLiealgebrag (Γ)toG (Γ)isbijective. Foranysubsetof
a a
G (Γ) and, in particular for the subgroup Γ, thce smallest closed, connected, com-
a
pclex (resp. real)subgroupG (resp. G ) ofbG (Γ) containing Γ is well-defined. By
1 0 a
construction G /Γ is compcact, e.g., scee [33,cTheorem 2.1]. Sbet G := π(G ) and
0 1 1
G := π(G )candbconsider the CRS manifold (G ,G ,Γ). Note that the hcomoge-
0 0 1 0
neousCRc–manifoldG0/ΓfibersasaT–bundleoverS1×S1. Inordertounderstand