Math. Nachr. 189 (1998), 145-156

Globalization of Holomorphic Actions on Principal Bundles

By BRUCE GILLIGAN of Regina, and PETER HEINZNER of Bochum

(Received August 10, 1995) (Revised Version October 28, 1995)

Abstract . Suppose G is a Lie group acting as a group of holomorphic automorphisms on a holomorphic principal bundle P --t X . We show that if there is a holomorphic action of the com- plexification GC of G on X , this lifts to a holomorphic action of GC on the bundle P --t X . Two applications are presented. We prove that given any connected homogeneous complex manifold G / H with more than one end, the complexification GC of G acts holomorphically and transitively on G / H . We also show that the ends of a homogeneous complex manifold G / H with more than two ends essentially come from a space of the form S/r, where r is a Zariski dense discrete subgroup of a semisimple complex Lie group S with S and r being explicitly constructed in terms of G and H .

1. Introduction

Every Lie group G has a universal complexification. This consists of a continuous homomorphism 4 : G + G" , where GQ: is a complex Lie group, which has the property that for every continuous homomorphism f of G into a complex Lie group H there exists a unique holomorphic homomorphism F : G" + H such that f = F o 4. For any Lie group G the corresponding complex Lie group G" is always Stein. But the map Cp need not be an injection of G into G" . If G is a compact Lie group, then the map 4 is injective and the group G" is a reductive complex Lie group. And in the important case when G is a Lie subgroup of a complex Lie group, the map 4 is also injective. For the construction and basic properties of complexifications we refer the reader to Chapter XVII.5 in [14] and to section 1 in the paper [13].

Now suppose G is a (real) Lie group acting as a group of holomorphic automorphisms on a complex space X. There are several settings where the complexification G" of G acts holomorphically on some complex space X" into which X can be G- equivariantly embedded as an open subset. One variant of this problem is to try to choose XQ: = X . The automorphism group of any compact complex space is a complex Lie group and

1991 Mathematics Subject Classification. Primary 32M05; Secondary 32M10. Keywords and phrases. Complexifications of Lie groups, principal bundles, ends of homogeneous

complex manifolds.

146 Math. Nachr. 180 (1998)

one can use this fact to solve the above problem in the compact case. However, if X is non-compact, then this problem has no solution, in general. For, it could happen that the automorphism group of X is a positive dimensional Lie group, but no positive dimensional complex Lie group can act effectively on X , e. g., this is the case for certain X which are hyperbolic.

In this paper we consider a Lie group G acting as a group of holomorphic automor- phisms on a holomorphic principal bundle P + X. We prove that if there exists a holomorphic action of the complexification G" of G on X, then this lifts to a holo- morphic action of G" on the bundle P -+ X. The proof of this basically amounts to showing that if the complexification of the vector fields corresponding to the G- action on X are globally integrable, then the complexification of the vector fields corresponding to the G - action on P are also globally integrable.

Two applications of this result are presented here. First we show that if X = G / H is a complex manifold homogeneous under the holomorphic action of a Lie group G and X has more than one end (in the sense of H. FREUDENTHAL [7]), then the complexification G" of G acts (necessarily transitively) on X . (This is also proved under the related assumption dx = 1, see 83.) Then we use this to give a structure result for homogeneous complex manifolds G / H which have more than two ends, by observing that the ends in this setting "essentially come from" a space of the form S/r, where I' is a Zariski dense discrete subgroup of the semisimple complex Lie group S. (There is an explicit construction of S and I', given G and H.) Other applications appear elsewhere, e. g., [ll].

2. Vector fields on principal bundles

Let X be a complex manifold and suppose 7r : P + X is a holomorphic principal bundle, where S denotes the (complex) Lie group which is the structure group of the bundle. We assume that S acts from the right on P and for g E S we use the same symbol rg to denote the maps P -+ P , p + p . g and S + S, x + x . g . Let V be an S-invariant holomorphic vector field on P. For every p E P set w(7r(p)) := .,V(p). Then w is a vector field on X , because w(.(p. 9)) = 7r*V(p. g) = 7r,V(p) = v ( ~ ( p ) ) . Let 5 denote the Lie algebra of S.

Lemma 2.1. Let U be an open subset of X and let Q : U x S + PIU be a tm'vialization of PIU. Then for every S -invariant holornorphic vector field V on P, there exists a unique holomorphic map ( : U + s such that

V(P) = Q*(v(.UL (rz)*C(U)>

for all p = Q(u,x) E PIU.

Proof. For each (u,x) E U x S we define the map w : U x S + TS by

QEIIV(Q(U, X I ) = (V(U), 4% x)) * Since V is S-invariant, the function 5 is given by ((u) := (r,-~),w(u,z). 17

Let I be a compact interval in R and W a vector field on a manifold X. A map y

Gilligan/Heinzner, Globalization of Holomorphic Actions 147

which is defined on an open neighborhood of I is said to be an integral curve of W on I if

i. ( t ) = W ( y ( t ) ) for all t E I.

Proposition 2.2. Let y be an integral curwe of w which i s defined on IR. Then for every to E IR and every p E a - l ( r ( t o ) ) there exists a unique lifting rp : IR + P of y with r p ( t 0 ) = p such that rP is an integral curve of V .

Proof. Since locally the lifting over X is unique, it suffices to prove this in the following case. Let U be an open neighborhood in X such that PIU is trivializable. Suppose y is an integral curve of w on a compact interval I c IR such that y ( I ) c U. We claim that for every to E I and p E a - l ( r ( t 0 ) ) there exists an integral curve rp of V on I with r(t0) = p . In order to see this note that the map rplI is necessarily a lifting of yll. We may assume that P = U x S and consequently the vector field V is of the form V ( v , x ) = (w(u), ( T ~ ) * ( ( u ) ) , where c : U -+ s is a holomorphic map, see Lemma 2.1. There exists a differentiable map A : R + s such that A = ( o y on some neighborhood of I. Now let $ be the unique differentiable map from some open neighborhood of I into S which is the solution of the equation

( Q ( t ) - l ) * $ ( t ) = A ( t )

satisfying $ ( t o ) = e; for the proof of the existence and uniqueness of such a map, see p. 69 in [l?]. Given a point p = (y(to),z) , where 2 E S, we define rp by

r P ( 4 := ( r ( t ) , $ ( t ) x ) *

It follows that rp(to) = (r(to),s) and

fiom the definition of the vector field v we have y = ?r o rp. 0

By an action of a group G on a holomorphic principal bundle P we mean a group homomorphism p from G into the group of S - equivariant biholomorphic automor- phisms of the manifold P. If G is a Lie group, then the action given by the map p is said to be differentiable if the map G x P -+ P, ( g , w) + p(g) (v) is differentiable. If, in addition, the group G is a complex Lie group and the map G x P + P, (9 , w) + p(g) (v) is holomorphic, then the action of G on P is said to be holomorphic.

By an infinitesimal action of a Lie group G on a holomorphic principal bundle P over X with structure group S we mean a Lie algebra homomorphism from the Lie

148 Math. Nachr. 189 (1998)

algebra g of right -invariant vector fields on G into the Lie algebra of S-invariant holomorphic vector fields on P . If G is a complex Lie group, then an infinitesimal holomorphic action means a Q: -linear representation of g into the Lie algebra of S - invariant holomorphic vector fields on P . We say that a G - action on P is a lifting of a given G-action on X if the diagram

G x P -+ P -4 -4

G x X -+ X

is commutative, where the horizontal arrows denote the maps given by the G- actions. We will also use similar terminology for infinitesimal G - actions.

Note that an action on P determines a unique action on X. As a consequence of Proposition 2.2 we have the following partial converse to this.

Corollary 2.3. Every R -action on X which lifts as an infinitesimal action to the principal bundle P lifts to a n R -action on P .

This can be extended to a similar statement about general Lie group actions.

Corollary 2.4. Every infinitesimal action of a simply connected Lie group G on P such that the corresponding infinitesimal G -action on X is globally integrable, i. e., is given b y an action G x X + X , is globally integrable.

Proof . This is a consequence of the following fact. Suppose g is a finite dimensional Lie subalgebra of the algebra of vector fields on the manifold P such that every vector field which belongs to g is complete. Then g determines a unique action on P of the

0 simply connected group G which corresponds to g, see p. 95 in [20].

Remark 2.5. Note that the analogous statement also holds for differentiable ac- tions.

Let X be a complex manifold endowed with an action of a Lie group G. Let GQ: be the universal complexification of G, see [14], and assume that the G-action on X extends to a holomorphic G" -action on X . Let ?r : P + X be a holomorphic principal bundle with structure group S.

Proposition 2.6. I f the G -action on X lifts to a G -action on the bundle P , then the holornorphic G" --action on X lifts to a holomorphic action on the bundle P.

Proof . Without loss of generality we may assume that G is connected. Let be the universal covering of G. The Lie algebra representation of g into the complex Lie algebra of holomorphic S - invariant vector fields on P extends to a C -linear repre- sentation of the complexified Lie algebra 8". By Corollary 2.4 the complexification 6" of 6 acts on P. To see that_the @-action induces a G"-action on P , let I? denote the kernel of the covering G + G and denote by N the smallest closed complex normal subgroup of @ which contains the image of r in G" . Since I' acts trivially

Gilligan/Heinzner, Globalization of Holomorphic Actions 149

on P, it follows that N also acts trivially on P. And by the construction of G" we have G" = GC IN. Therefore, the a" - action induces a holomorphic G" -action on

0 P which is a lifting of the given G" - action on X.

3. Homogeneous spaces with more than one end

In this section we consider a complex manifold X = G / H which is homogeneous under the action of a Lie group G and which satisfies dx = 1, where for a connected manifold X with dimR X = n we define

dx := min { r I H,-,(X, Z,) # 0).

We show that there is a complex Lie group which is acting holomorphically and tran- sitively on X.

Theorem 3.1. Suppose G is a connected Lie group acting transitively and almost effectively as a group of holomorphic transformations on a complex manifoold X = G / H with dx = 1. Then the complexification G" of G acts holomorphically and transitively on x.

Corollary 3.2. Suppose G as a connected Lie group acting ~ ~ a ~ i t ~ v e l y and almost effectively as a group of holomorphic transfornations o n a complex manifold X = G / H . Assume that X has more than one end. Then the complexification G" of G acts holomorphically and transitively on X .

Proof. By means of Poincar6 duality one can easily see that a result proved in [l] (see 2.8) implies that any connected manifold X with more than one end satisfies

0 dx = 1. The Corollary now follows from the Theorem.

Remark 3.3. Let G/H + G / J be the g-anticanonical fibration of GIH. Set J := H . J". Note that up to a covering the g -anticanonical fibration is a holomorphic principal bundle. A. T. HUCKLEBERRY and E. OELJEKLAUS proved in Proposition 8 in [16] that if X = G/H is a homogeneous space with more than one end and the fiber J / H is compact, then the complexification of G acts holomorphically and transitively on G I J . This is a special case of the above. As well, it was noted in [lo] that in some instances a homogeneous complex manifold X = G / H has more than one end exactly when dx = 1.

P roof of Theorem 3.1. Let G / H + G / J be the g-anticanonical fibration of GIH. If J = G, then G/H" is a complex Lie group, see [16], which is acting transitively on G / H and H" is normal in G. Since we assume that G is acting almost effectively on GIH, it follow that Ha = {e} and G itself is a complex Lie group. So we may assume J is a proper subgroup of G. By construction there is a holomorphic representation of G" into Aut(pN) such that G/ J is an open orbit in the orbit of the complexification G" in PN, i.e., GIJ C ) G"/J", see [16]. If we show that G" acts transitively on G I J , then the result will follow from Proposition 2.6. If G / J is compact, then

150 Math. Nachr. 180 (1998)

G I J = G" / J" and it is clear that the complexification GQ: of G acts transitively on G/ J. So we assume G/ J is not compact.

We begin with the assumption that the radical of G acts ineffectively on G / J , i.e., we assume that the transitive action of G on G f J is given by a semisimple subgroup of G. In this case the action of G" on G" /Jc is also given by a semisimple complex subgroup of G" . By abuse of language in these next three paragraphs we proceed as if G and G" themselves were semisimple. Let M be the complex normal subgroup in G" corresponding to the ideal g n ig in the Lie algebra g" of G". Note that M c G and M is a complex Lie group. The group M has closed orbits in GQ: /Jc, because it is a product of some of the simple factors of G". As well, the fibers of the fibrations of G / J and G" /Jc induced by the M -orbits are equal. If we show G/M . J = G" / M . J" =: Y, then it is clear that G / J = G" / J c and thus G" acts on G/ J . If GIM J is compact, then the above equality is clear. So assume G/M - J is not compact. Note that we are now in the setting where the real semisimple Lie group which is acting effectively on G/M . J is a real form of the complex semisimple Lie group acting effectively on GQ: / M . J". Let J consist of those components of J which meet H , i,e., set 3 := J" H. It follows from Lemma 2 in [3], applied to the fibration GIH + G/M . j, that dGIM.j = 1. The isotropy subgroup M . J has a finite number of connected components, since M - j is a (real) algebraic subgroup of G. Thus MOSTOW'S theorem [la] says that G/M * j is a homogeneous real vector bundle over a K - orbit in G/M . J , where K is a maximal compact subgroup of G. Now, because dG,M,j = 1, this bundle is a real line bundle, i. e., the K -orbits in G/M. 3 have (real) codimension cne. Let k be a maximal compact subgroup of G" containing K . Then the minimal K - orbit in Y has (real) codimension one or zero.

We first rule out the latter. If Y were compact, then Y would be a homogeneous projective rational manifold and one could apply results of J . WOLF [22] concerning flag domains in order to get a contradiction. Since we are in a situation where a real form acts, by Theorem 5.4 in [22] the space G / M J would be simply connected. Thus M . J would be connected. By MOSTOW'S theorem [18] again it would follow that G/M . 5 has two ends. But this would contradict the fact that the minimal G-orbit in Y lies in the closure of all other G-orbits in Y , see Corollary 3.4 in [22].

-orbits have (real) codimension one. Since going to a finite covering does not change the dimension of these orbits, without loss of gen- erality we may assume that M e J", the isotropy of the G" -action on Y, is connected. Applying MOSTOW'S theorem again, we see that_Y has two ends. Then by a Theorem of AKHIEZER [2] there is a parabolic subgroup P of G" containing M . J" such that

Hence Y is not compact and all

realizes Y as a-, homogeneous C! * -bundle over the homogeneous projective rational manifold G" /P. (This also follows from the fact that G" is a connected linear group. Then there is a parabolic subgroup F of G" containing M . Jc such that F / M . J" is affine. Since G"/P is simply connected, F / M . J" has two ends. But a Stein homogeneous space with two ends is biholomorphic to C *.) Consider the induced fibration G/M . J + G/P, where P := G n ?. Clearly the fiber P /M J cannot be compact. Let P := Po . J . Since dG,M.j = 1, it follows from the inequality

I -

Gilligan/Heinzner, Globalization of Holomorphic Actions 151 -

d G / M . j 2 d F I M , j + d G I F (see Lemma 2 in [3]) that G/P is compact. Hence G / P is

compact and so G / P = G" /$. Because P/M . J is homogeneous and 1 -dimensional, it follows that P/M * J = C * = p / M J". This implies G/M . J = G" / M . J". Since the M -orbits in G/ J and G" / J c are equal, G" acts transitively on G/ J , finishing the proof in this case.

We now show how to handle the situation when G is not semisimple. By CHEVAL- LEY'S theorem [S] the group (G") acts as an algebraic group on projective space. Since (Ge )' is a normal subgroup of G" , its orbits in G" / J c are closed and one has the fibration G" / J c + G" / J " - (G" )', where G" / J c - (G")' is a complex Lie subgroup of a linear algebraic abelian group (see [15]) and so is Stein. Now there is an induced fibration GIJ -+ G/U, where U := G n J c (G")'. Note that G/U is an open subgroup of the abelian complex Lie group G" / J c . (%") I and thus these two groups are equal, i.e., G" acts transitively on GIU. Set U := U" - H . Then 6 / H is connected. Now G I 6 is a connected abelian Lie group and thus is homotopy equivalent to its maximal compact subgroup. By Lemma 2 in [3] it follows that

Since GI6 covers a Stein Lie group, GI6 is not compact, if its dimension is positive. Thus c / H is compact. It follows that f i / j is compact and thus U / J = J". (Ge) ' /JC, the fiber of the fibration of the complex orbit. 'Since G" acts on the base, Ge also acts on G/J . This finishes the case when (Ge) is not transitive on G" / J c .

Now we assume that U = G, i.e., that (G")' is transitive on G"/J". We claim that R(Gc If acts ineffectively on G" /J" and thus we can reduce to the case where semisimple subgroups of G and G" act, as described above. Since (G")' acts alge- braically on G@/J" , the orbits of its radical R(Gc)' are closed m (G")'/(G")'fl Jc . Suppose that R ( G c ) ~ has positive dimensional orbits. Because R(Gc)' is acting as a unipotent group, these orbits are biholomorphic to C k and thus are holomorphically separable. There is an induced fibration of G / J which we denote by G / J -+ GIL, where L := G n JCR(Gc ) I . Let i := Lo . H and note that J = J" - H C i. Consider the induced fibration G/H + G / i . Because i/j is holomorphically separable and so cannot be compact, one sees from Lemma 2 in [3] that G / i must be compact and dilH- =- 1. We claim that dim i/j > 0 is impossible. In order to see this suppose dim L/J > 0. Due to the fact that there exists an R ( G ~ ) l -equivariant fibration of R(Gc l t /R(Gc 1 1 n J c with positive dimensional fiber and base, it is easy to see (cf. the proof of Lemma 5 in [ll]) that there is a fibration L / j -+ i / V , where dime i / V = 1. Clearly L/V can be retracted onto a compact submanifold. Since we can choose V so that V / H is connected, by Lemma 2 in [3], applied to the fibration i / H + i / V , we get

1 = ~ E / H 2 dV/H -k d E / v - By construction, L/V is an open orbit in Q: and so cannot be compact. The above inequality implies dElv = 1 and thus i / V is biholomorphic to C*. But, as is noted

152 Math. Nachr. 189 (1998)

in the proof of Lemma 5 in [ll], this is not possible. Thus R ( G c ) ~ c J" and we may assume that there are semisimple subgroups of G and G" which are acting transitively on G/ J and G" / Jc, respectively. The proof can now be completed by using the observations made above. 0

Remark 3.4. Note that in the above proof when dimGI6 > 0 it follows from equality (3.1) that dG,c = 1 and hence G I 6 = C*. So by Lemma 6 on p. 75 in [16] the radical RGC of G" is one-dimensional and central and the group J c RGC is parabolic in GQ: . Hence the fibration

G" /J" + GQ: /Jc * RGC

is a C * -bundle over a homogeneous projective rational manifold. If this bundle is not trivial, then a maximal semisimple subgroup of GC acts transitively on the total space G" / Jc of the bundle, contradicting the assumption that the commutator subgroup (G')' does not act transitively on G"/J". Thus, in fact, this bundle is GQ: -equivariantly trivial and G / j is a product of C * with a homogeneous projective rational manifold.

Remark 3.5. Note that the theorem implies that, without loss of generality, we may assume that G is embedded in its complexification G" .

4. Homogeneous spaces with more than two ends

In the following e ( X ) denotes the number of ends of the (connected) topological space X . From Corollary 3 we know that if X is a complex manifold which is ho- mogeneous under the holomorphic action of a Lie group and satisfies e ( X ) > 1, then X = G/H, where G is a complex Lie group and H is a closed complex subgroup of G. We now consider the structure of homogeneous complex manifolds X with e ( X ) > 2 and fix a Klein form G/H for such a space. The following gives an explicit description of such spaces and thus provides an improvement on a result in [9]. For a connected complex Lie group L we denote by RL the radical of L and let ?TL : L + L / R L be the natural map.

Theorem 4.1. Assume X = G / H is a connected homogeneous complex manifold, where G is a connected complex Lie group and H is a closed complex subgroup, and assume e ( X ) > 2. Then

a) H" is normal in P , where P is the smallest parabolic subgroup of G containing H. Let L denote the connected complex subgroup P I H " .

b) The RL -orbits in P I H are compact and c) I? := x L ( H / H o ) is a Zariski dense discrete subgroup of SL := LIRL. Moreover,

e(G/H) = e ( S t / r ) . Thus there is a closed complex subgroup I of P containing H (the stabilizer in P of the typical RL -orbit) such that I/H = RL/RL n ( H I H " ) and P/I = SL/r and hence

Gilligan/Heinzner , Globalization of Holomorphic Actions 153

one has the fibrutions Sr, /r GIH 9 GII + G / P ,

where GIP is a homogeneous projective mtional manifold and I IH as a compact com- plex solvmanifold. Furthermore, any homogeneous complex manifold GIH which fibers in this way satisfies e(G/H) = e(SL/I').

In the proof we will make repeated use of the fact that certain types of homogeneous spaces cannot have more than two ends. For our purposes we note the following:

1) Suppose X = G / H , where G is a connected Lie group and H is a closed subgroup having a finite number of connected components. Then by the Theorem of MOSTOW [18] X can be realized as a real vector bundle over a compact submanifold. Clearly X has at most two ends. We apply this in the case X itself is a connected Lie group or else G is an algebraic group and H is an algebraic subgroup of G.

2) Suppose X = G I N , where G is a connected solvable Lie group. Then GIH is a vector bundle over a compact manifold, see [4] or [19]. In particular, X has at most two ends.

The following is a direct consequence of the proof of Lemma 2 in [8].

F Lemma 4.2. Suppose X + B is a fiber bundle, where X , F and B are connected

a) If F is a space with at most two ends, then F as compact and e (B) = e ( X ) . b) If B is a space with at most two ends, then B is compact and e (F) 5 e ( X ) . If,

smooth manifolds. Assume e ( X ) > 2.

an addition, B is simply connected, then e (F) = e ( X ) .

Proof. Let GIH + GIN be the normalizer fibration of G I H , i.e., N := NG(H") is the normalizer in G of go. One sees that N is parabolic in G in the following way. Let fi := No H . Then NIH is connected. Now the commutator subgroup G' of G has closed orbits in GIN and thus also in its covering space G / f i , yielding a fibration G / E + G / f i . G' with connected fibers biholomorphic to G'IG' n N. Since G / f i . G' is a connected Lie group, it follows that e ( G / f i . G') 5 2. Thus Lemma 4.2, applied to the fibration GIH + G / f i . GI, implies G / f i G' is compact. Since G / f i . G' is a Stein abelian Lie group, see [15], fi G' = G and thus G' is transitive on GIN. But then G / f i can be written as a quotient of algebraic groups and it follows from Lemma 4.2, applied to the fibration G / H + G/@, that G / E and thus also GIN is compact, i.e., N is parabolic in G. Now let P be the minimal parabolic subgroup of G which contains H. Since P c N, it follows that H" is normal in P . Hence PIH = ( P / H " ) / ( H / H " ) , where HI := HIH" is a discrete subgroup of the connected complex Lie group L := PIH", proving a). Without loss of generality we may assume L is simply connected.

First we claim that the RL -orbits are closed in LIHI. Suppose not. Then . r r~ (Hl ) is not closed in SL := LIRL. Let U := x ~ ( H 1 ) be its closure in the (euclidean) topology of SL. Since Hi' is solvable, U" is also solvable, e. g., see Theorem A in the Appendix in [21]. (In the present setting I f 1 is, in fact, discrete and one could make do with simpler versions of the AUSLANDER- ZASSENHAUS Lemma,) In general, U is a real Lie group and so we consider (Uo)c . Since the adjoint representation of SL

154 Math. Nachr. 189 (1998)

is algebraic, the normalizer j? := NsL ( (V")") of (U")" is an algebraic subgroup of SL. By assumption U is positive dimensional and hence so is ( U o ) e . Thus j? # SL, since the semisimple group SL cannot contain the positive dimengonal connected complex solvable subgroup (V")" as a normal subgroup. Since S L / N is the quotient of algebraic groups, by Lemma 4.2 it follows that S~/fi must be compact. But then p-1 0 iTL -'(j?) is a proper parabolic subgroup of P , and thus also of G , containing H , where p : P + L = P / H " is the natural map. This contradicts the minimality of P. Thus the R~-orbi ts are closed in P / H , i.e., ( H I H " ) . RL is closed in L. Set I := P - ~ ( ( H / H " ) . RL). Since L = P / H " 3 ( H / H " ) RL 3 HI = H/H", it is clear that P 3 I 3 H . Consider the fibration G / H + G / I . Now

I / H = p ( I ) / p ( H ) = (HIH") * R L / ( H / H " ) = RL/RL n ( H / f i " )

is a connected solvmanifold. Because e ( G / H ) > 2 , it follows that I / H is compact by Lemma 4.2, i. e., the orbits of the radical RL of L = P / H " in P / H are compact. This proves b).

As a consequence of the fact that the RL -orbits are closed in L/H1, it follows that I' := T L ( H / H " ) is a discrete subgroup of SL. We now check that SL /I? has the same number of ends as G / H . Because G / P is a homogeneous projective rational manifold, and thus is simply connected, it follows from Lemma 4.2 that e ( G / H ) = e ( P / H ) . Now I / H , as an orbit of the connected solvable group RL, is a connected solvmanifold and thus by applying Lemma 4.2 again, we see that e ( P / H ) = e ( P / I ) . Also note that P / I = ( P / H " ) / ( H / H " ) . RL = S L / ? T L ( H / H " ) = SL/I'. Hence

e ( G / H ) = e ( P / H ) = e ( P / I ) = e(SL/I')

Finally, one can easily see that I' is Zariski dense in SL, by again using the minimality of P. For, suppose M := rz , the Zariski closure of in SL, were a proper subgroup of SL. Since SLIM is the quotient of two algebraic group, Lemma 4.2, applied to the fibration S L / r + S L / M , implies SLIM must be compact. But then the inverse image of M in P would be a parabolic subgroup of G containing H and properly contained in P, contrary to the minimality of P . Therefore, I' is Zariski dense in SL. This proves c)

Remark 4.3. A compact complex solvmanifold is a torus tower, see [5] .

Remark 4.4. For every integer k > 2 there exists a Zariski dense discrete subgroup r k of SL(2 , C) with e (SL(2 , C) / rh ) = k , see [12].

Acknowledgements

We thank A . T. HUCKLEBERRY and E. OELJEKLAUS for helpful discussions concerning this paper. Also the first author was partially supported by NSERC and SFB 237 of the Deutsche Forschungsgemeanschaft and is grateful for having received this support.

Gilligan/Heinzner, Globalization of Holomorphic Actions

References

155

ABELS, H.: Some Topological Aspects of Proper Group Actions; Non-Compact Dimension of Groups, J. London Math. SOC. (2) 26 (1982), 525-538

AKHIEZER, D. N.: Dense Orbits with Two Ends, Izv. Acad. Nauk SSSR Ser. Mat. 41 (1977),

AKHIEZER, D. N., and GILLIGAN, B.: On Complex Homogeneous Spaces with Top Homology in Codimension Two, Canad. J. Math. 46 (1994), 897-919 AUSLANDER, L., and TOLIMIERI, R.: Splitting Teorems and the Sructure of Solvmanifolds, Ann. of Math. (2) 92 (1970), 164-173 BARTH, W., and OTTE, M.: Uber fast -miforme Untergruppen komplexer Liegruppen und auflosbare komplexe Mannigfaltigkeiten, Comment. Math. Helv. 44 (1969), 269 - 281 CHEVALLEY, C.: Thborie des Groupes de Lie 11: Groupes Algbbriques. Hermann, Paris, 1951 FREUDENTHAL, H.: Uber die Enden topologischer Raume und Gruppen, Math. 2. 33 (1931),

GILLIGAN, B .: Ends of Complex Homogeneous Manifolds Having Nonconstant Holomorphic Fhctions, Arch. Math. 37 (1981), 544-555 GILLIGAN, B.: On Homogeneous Complex Manifolds Having more than Two Ends, C.R. Math. Rep. Acad. Sci. Canada 16 (1993), 29-34

308 - 324.

692 - 713

[lo] GILLIGAN, B.: Comparing Two Topological Invariants of Homogeneous Complex Manifolds, C.R. Math. Rep. Acad. Sci. Canada 16 (1994), 155-160

(111 GILLIGAN, B.: Complex Homogeneous Spaces of Real Groups with Top Homology in Codimen- sion Two, Ann. Global Anal. Geom. 13 (1995), 303-314

[12] GILLIGAN, B., and HUCKLEBERRY, A. T.: Complex Homogeneous Manifolds with Two Ends, Michigan Math. J. 28 (1981), 183 - 198

[13] HEINZNER, P.: Equivariant Holomorphic Extensions of Real Analytic Manifolds, Bull. SOC. Math. France 121 (1993), 445-463

[14] HOCHSCHILD, G.: The Structure of Lie Groups, Holden-Day Inc., San Francisco-London- Amsterdam, 1965

[15] HUCKLEBERRY, A. T., and OELJEKLAUS, E.: Homogeneous Spaces from a Complex Analytic Viewpoint, Manifolds and Lie groups. (Papers in honor of Y. MATSUSHIMA), pp. 159-186, Progress in Math., 14, Birkhauser, Boston (1981)

[16] HUCKLEBERRY, A. T., and OELJEKLAUS, E.: Classification Theorems for Almost Homogeneous Spaces, Publ. Inst. E. Cartan 9, Nancy, 1984

[17] KOBAYASHI, S., and NOMIZU, K.: Foundations of Differential Geometry. I. Interscience Publish- ers, New York London, 1963

1181 MOSTOW, G.D.: On Covariant Fiberings of Klein Spaces, I, 11, Amer. J. Math. 77 (19551,

[19] MOSTOW, G. D.: Some Applications of Representative Functions to Solv-Manifolds, Amer. J.

[20] PALAIS, R.: A Global Formulation of the Lie Theory of Transformation Groups. Mem. of the

[21] WANG, H-C.: On the Deformation of Lattices in a Lie Group, Amer. J. Math. 86 (1963),

[22] WOLF, J. A,: The Action of a Real Semisimple Group on a Complex Flag Manifold. I: Orbit

247 - 278; 84 (1962), 466 - 474

Math. 93 (1971), 11-32

A.M. S., 1955

189-212

Structure and Holomorphic Arc Components, Bull. Amer. Math. SOC. 75 (1969), 1121 - 1237

156

Dept. of Math. and Stats. University of Regina Regina, Canada S4S OA2 e -mail: gilliganOmath.uregina. ca e -mail:

Fakultat f i r Mathematik Ruhr - Wniversitat, Bochum D -44780 Bochum Federal Republic of Germany

heinznerOcplz.ruhr - uni - bochum.de

Math. Nachr. 180 (1998)