Arch. Math., Vol. 37, 544--555 ( 1 9 8 1 ) 0003-889X/81/3706-0012 $ 01.50 + 0.20/0 �9 1981 Birkh~iuser Verlag, Basel

Ends of complex homogeneous manifolds having non-constant holomorphic functions

By

BRue~ GILLm~*)

1. Introduction. In sharp contrast with what happens in one variable, a Stein space of dimension greater than one has one end. For "finitely sheeted" domains of holo- morphy over pn (n > 1) this was proved by Behnke [8]. The general result was ob- served, as a consequence of Theorem B, by Serre [32]. Using this fact together with the Remmert reduction [30], one easily sees tha t a holomorphically convex space X also has one end, provided that the rank of its function algebra is at least two. One just takes the inverse image under the holomorphic reduction mapping rt: X --> X/.,, (its fibers are compact and connected !) of any sequence defining the one end of the Stein space X/,,~. Of course, if the rank of 0 (X) is one, then X/,~ may have many ends.

Now on any complex analytic space (X, 0) one may introduce the equivalence relation ~ given by: xz ~ x2 for xl, x2 e X if and only if ](x~) = ](x2) for every ] e 0(X). But ~ / ~ may not admit any natural complex structure (see [21] for an example). However if X is complex homogeneous, where for our present purposes this means that a connected complex Lie group G is acting holomorphically and transitively on X, then the holomorphic separation mapTing ~: X--> X/,,~ can in fact be realized as a homogeneous fibration. For, X is biholomorphie to the complex coset space G/H, where H is a closed complex subgroup of G. And since the fibers of the mapping ~ (the equivalence classes of ~ ) are G-equivariant, it follows from a remark of Remmert-van de Ven [31] that these are precisely the fibers of a homo- geneous fibration G/H --> G/J (for details, see [18]). Any fibration of the form G/H --> G/J, for some closed complex subgroup J of G containing H, is called a homogeneous fibration of the given G/H.

Our goal is to understand the function algebra •(G/H) in terms of the fiber J/H and the base G/J of this fibration. In certain cases this can be achieved. For example, if H is normal in G (!~Iorimoto [28]) or ff G is nilpotent [18], then J/H is connected and has no non-constant holomorphie functions and G/J is Stein. But the fiber need not be compact. In general, things are not this well-behaved. For, J/H may be Stein [7] and a holomorphically separable G/H need not be, e.g. C n \ ( 0 ) for n > 1. As

*) Partially supported by I~SERC Grants A-3494 & T 1365.

Vol. 37, 1981 Ends of homogeneous manifolds 545

well we do not know if every holomorphically separable complex homogeneous manifold has an envelope of holomorphy.

In this note we look at the ends of complex homogeneous manifolds having non- constant holomorphic functions. The main purpose is to prove the following:

Theorem. Suppose G is a connected complex Lie group and H is a closed complex subgroup such that (P (G/H) =~ C. Then G/H has either one end or two. Moreover, G/H has two ends precisely i/there exists a closed complex subgroup J o/ G containing H such that J /H is compact and connected and G/J is an a/line homogeneous cone minus its vertex, i.e. a homogeneous C*-bundle over a homogeneous rational mani/old whose associated line bundle is very ample (see [24]).

Holomorphieally separable manifolds G/H, where G is a reductive group which is linear algebraic over C and H is a closed complex subgroup were considered by Barth- Otte [7]. In this setting they showed that H is algebraic and G/H is quasi-affine. What happens if G is not a reductive algebraic group ? I f G/H has two ends and 0(X) has maximal rank, the above theorem provides an ironic answer. There is always a reductive algebraic group acting transitively! Namely, a semi-simple part S of G in any Levi-Malcev decomposition G = S �9 R does the job, except for the already trivial ease of G/H = C*.

Closely related to complex homogeneous manifolds are complex manifolds X which are homogeneous, i.e. given any two points xl, x2 e X there exists an auto- morphism g e Aut o(X) with g (xl) = x2. For X compact there is no distinction, since Aut~ (X) is itseff a complex Lie group (Bochner-Montgomery [10]). However if X is non-compact, then this distinction is essential. For, W. Kaup [25] has pointed out tha t C2\(Z x (0)), which has a countable number of ends, is homogeneous, but not under the action of any connected Lie group. And for any integer k > 2, the argument of Kaup shows that C2\{(zl, 0) . . . . , (zk-1, 0)}, which has k ends, is also homogeneous. Since these spaces ale simply connected, the theorem of Borel, quoted below, rules out the possibility tha t they are homogeneous under the action of any connected Lie group.

Borel [11] showed that a homogeneous space G/H of a (real) Lie group G, where H is connected, has at most two ends. Further it has two ends precisely ff it is homeo- morphic to K/L • R, where K (resp. L) is a maximal compact subgroup of G (resp. of H, contained in K). I f H has a finite number of connected components, then G/H again has at most two ends. In fact Barth-Oeljeklaus [5] used exactly this idea when they showed tha t if a complex homogeneous space has an equivariant compactifica- tion to a K~hler manifold, then it has at most two ends. For, they proved that it suffices to consider the case where G is a linear algebraic group over C and H is an algebraic subgroup. And in the setting of linear algebraic groups, Ahiezer [2] proved tha t G/H has two ends precisely if there exists a parabolic subgroup P of G con- taining H such tha t the fibration G/H--> G/P realizes G/H as a homogeneous C*- bundle over the homogeneous rational manifold G/P. As well in a joint work with Huckleberry [19] we showed that a complex homogeneous manifold G/H, where H has a finite number of connected components, has two ends precisely if there exist

Archiv der Matheraatik 37 35

546 B. GILLIGAN ARCH. MATH.

closed complex subgroups I and J of G, with J containing I containing H, such that G/J is a homogeneous rational manifold and the homogeneous fibrations G/H-> G/1 --~ G/J realize G/H either as a C*-bundle over a torus bundle over G/J or else as a torus bundle over a C*-bundle over G/J. l~ote tha t in the complex setting one eants to display the ends by finding not only the [~ but also the C* and that analy. tically this need not split off as a product.

Can a complex homogeneous manifold G/H have more than two ends ? For G solvable this cannot happen, since every solv-manifold is a vector bundle over a compact solv-manifold (Auslander-Tolimieri [4] and Mostow [29]). But for G semi- simple it does! Borel [11] pointed out tha t for every integer k > 2 there exists a discrete subg r oup /~ of SL (2, R) such tha t SL (2, R) / /~ has k ends. Subgroups/~k of SL (2, C), yielding complex homogeneous manifolds with k ends, exist as well [19].

We now outline the organization of this note. First one needs non-trivial homo- geneous fibrations of the given G/H. Examples are the holomorphie separation fibration, mentioned above, and the normalizer fibration [34], [13]. In section two we describe two others, one for the base and one for the fiber of the normalizer fibra- tion. Next one has to control the behavior of the ends under fibrations. This is ac- complished, as was done by Ahiezer [2] and again in [19], by a Fibration Lemma (Lemma 2, el. w 3), which asserts that if the total space of a locally trivial fiber bundle with connected fiber has more than one end, then either its fiber is compact and its base has more than one end or else its base is compact and its fiber has more than one end. In section four we note tha t a holomorphically separable complex solv- manifold with two ends is biholomorphic to C*. Section five is devoted to proving the Main Theorem (Theorem 6). In the last section we point out tha t the Main Theo- rem permits a classification of G/H with more than one end whenever the base GIN of the normalizer fibration is non-compact.

This note was prepared during a sabbatical stay at the Mathematisches Insti tut in Miinster. I t is our pleasant du ty to thank R. Remmert for his interest and support and A. Huckleberry and E. Oeljeklaus for helpful conversations.

2. Some homogeneous fibrations. Given any complex homogeneous manifold G/H, the normalizer N: --:- Nc (H ~ in G of H ~ is a closed complex subgroup of G con- taining H. The resulting fibration G/H --> GIN is called the normalizer fibration and has proved invaluable in analyzing compact [34], [13] as well as non-compact G/H (e.g. [18], [24]). In this section we describe two fibrations, one for the base G/N and one for the fiber N/H o5 the normalizer.

The base G/_N lies in a Grassmann manifold which admits an equlvariant embed- ding in some pk (e.g. the Pliicker embedding). The orbit G/N may not be closed but G acts linearly on it and this leads to the following.

Theorem 1. Su19pose G is a connected complex Lie group which is acting linearly on P~ and X := G/H is an orbit. Then the G'-orbits are closed in X, i.e. G/H -~ G/G' H is a homogeneous/ibration o/ X.

P r o o f . Since the image of G' under the linear representation is algebraic [15], its orbits are Zariski open in their closures [14]. I f some orbit were not closed, then its

Vol. 37, 1981 Ends of homogeneous manifol4s 547

closure would be obtained by adding lower dimensional orbits. But G' is normal in G and so all orbits are biholomorphic. []

This fibration is also useful when H and h r have the same dimension, for then G/H --~ G/N is a covering.

Corollary. Suppose G is a connected complex Lie group and H is a closed complex subgroup such that H and N :----- N~ (H ~ have the same dimension. Then the G'-orbits in G/H are closed.

Now any connected simply-connected complex Lie group G has a Levi-Malcev decomposition G = S [:><R, where R is the radical of G and S is a semi-simple part. As a general philosophy, we would like to "fiber off the R-orbits", but these are not necessarily closed. However, for discrete isotropy we can form a "hull" which con- tains the R-orbits. In particular, since N / H = N/HO/H/H o with H /H o discrete in N / H o, this will allow us to handle the fiber of the normalizer fibration.

Theorem 2. Suppose G is a connected simply-connected complex Lie group which is not semi-simple and I" is a discrete subgroup o/ G. Then either G is solvable or there exists a proper closed complex subgroup J o/G, containing I ~ and R, i.e. there is a non. trivial homogeneous fibration G / F ---> G / J, whose fibers contain the R-orbits.

P r o o f . I f the R-orbits are closed, then J := /~_~ is a closed complex subgroup. I f J is not proper, then G = J = R is solvable.

I f the R-orbits are not closed, let I : = _FR. I t follows from the Zassenhaus Lemma [3, Prop. 2] tha t I 0 is solvable. I f I ---- G, we are done. Assume I ~ G. Now I is not necessarily a complex subgroup of G. But since the closure and the complexification (the connected subgroup corresponding to the complexification/~ ~- i J of any sub- algebra./~) of any connected solvable subgroup of G are both again solvable, start- flag with I 0 and alternately taking the complexification and the closure yields a connected closed complex solvable subgroup H of G containing I 0. Assume H is a minimal proper subgroup with the above properties and let J : = _N~ (H). Clearly J ~= G. Otherwise, H would be the radical of G. But R does not have closed orbits by assumption. Now g1 o g-1 = I o for any g e I and so I 0 c gHg -1 n H. Since H is minimal, gHg -1 ---- H and thus g e J . Hence I c J . []

R e m a r k . I f G is solvable, bu t non-abelian, there exists a proper homogeneous fibration of G/I" for any discrete subgroup _F [24, Lemma 7]. This also holds if G is abelian and G/F is non-compact.

3. The fibration lemma. First the definition of ends [17].

Definition. Let X be a connected, locally connected topological space. Consider the family ,~" of sequences {U,}ne~ such tha t

1. Un is an open, connected subset of X with non-empty compact boundary,

2. Un+l c Un for every n e ~ ,

35*

548 B. GILLIGAN ARCH. MATH.

In 5 z- we introduce the equivalence relation -~ given by: {Un) ,-" (Vm) if and only if for every m e N there exists n e ~ such tha t U~ c Vm ( ~ is indeed symmet- ric, cf. [17, Satz 2]). By definition, the set of equivalence classes ~ / ~ are the ends of X.

Lemma 1. Suppose F -> X 2~ B is a locally trivial fiber bundle where F, X and B are connected non-compact smooth mani/olds having countable topologies. Then X has one end.

P r o o f . Choose an exhaustion (K~}ne~ of F by compact subsets such tha t no component of F \ K n is relatively compact. An explicit proof tha t this is possible for Riemann surfaces, using the fact tha t a Riemann surface has a countable topology, is given in.[16, pp. 166--71 and carries over word for word to the present setting. Let (V~}jej denote the components of F \ K n . Also let (Bn}neN be an exhaustion of B by compact subsets.

Let ( Wk)~ ~ ~ be a covering of B by open connected subsets such tha t the bundle 4v --> X --> B is trivial over each W~. By renumbering and deleting if necessary, we may assume tha t for every n ~ ~ there exists k (n) ~ ~ such that

.Bn c W1 U ... U W~(n).

We may furtheI, assume tha t for every ]~ > k (n), we have W~ n Bn = O. Other- wise, replace W~ by the connected components of W~ n ( B \ B n ) and renumber and delete if necessary. For each k e [~, choose a trivialization of the bundle hk: ~-I(W~) --> W~ x 2'.

Now we claim tha t the sequence (Un)n �9 ~, given by

U ~ : = U h [ I ( W ~ • ), for l ~ k g / c ( n ) , ] e J , k,]

defines an end of X. For, it is clear tha t each Un is an open subset of X having non- empty compact boundary. Further, by explicit construction, Un+l c Un and r~ Un = 0. I t remains to verify tha t every Un is connected. To prove this, it suffices to show tha t for any x ~ h [ l ( W ~ • V]) and any y ~ g - I ( B \ B n ) , there exists a path lying in Un which joins x to y. Clearly there is a pa th in B joining ~(x) to z(y). I f starting at x we can lift to a pa th in Un then we are done. Over any connected com- ponent of .B \Bn and over any Wk, this is trivial. The essential point is whether one can "connect up" such lfftings inside Un. But this follows from the fact tha t if W~ n Wk' =4= 0 and g~k, is the corresponding coordinate transition function of the bundle, then ~ v e n any ] there exists a j ' such tha t g~ , (V]) n V~ =b O. Further details are left to the reader.

~inally we claim tha t X has no ends sequence which is not equivalent t o ( U n } n e , ~ .

For, suppose {Vm}m~N is such a sequence. Then there exists n e [~ such that Un (~ 17n ~ 0 [17, Satz 3]. But then

k (n) r . c U hT~((~ n ~ ) • K~)

]=1

and thus is compact. As a nested sequence of compacts, one has (~ 17n ~ 0, con- tradicting the assumption tha t (Vn) n �9 ~ is an ends sequence.

Vol. 37, 1981 En.ds of homogeneous manifolds 549

Thus X has precisely one end. []

R e m a r k . The above proof depends only on the definition. As wel lH. Abels has pointed out to us the following proof.For the locally trivial fiber bundle F -> X -> B, there is a spectral sequence with E ~'q~ ~ H~(B; ~ ( F ) ) , where ~/~ is a locally constant sheaf having stalk H~(F). This sequence converges to H*c(X). For F con- nected and non-compact, H~ = 0. For B connected and non-compact, Ho(B; ~ (iv)) ~ 0 as there are no non-trivial sections of the locally constant sheaf ~ (F) having compact support. Thus H~ (X) = 0 and X has one end [20].

R e m a r k . This Lemma is false if one no longer has a local product structure. The projection p: C2\(0} --> C onto the first factor is a surjective holomorphic function having connected fibers. But the domain has two ends. This example may also be interpreted as a holomorphic foliation having non-compact leaves.

Lemma 2 (Fibration Lemma). Suppose F --> X -~ B is a locally trivial fiber bundle, where F, X and B are connected smooth manifolds having countable topologies. I f X has more than one end, then one of the/ollowing holds:

1) .F is compact and B has the same number o/ends as X .

2) B is compact and F has at least as many ends as X . In particular, i/17 has at most two ends, then .F and X both have two ends. Moreover, i / B is simply connected, then 2' has the same number of ends as X .

P r o o f . From Lemma 1 it follows tha t either _~ or B is compact. I f F is compact, then since it is connected, B and X have the same number of ends (e.g. [11, p. 448]).

Now suppose B is compact. Let iV denote the compactification of iv obtained by adding its ends [17, Satz 4]. Then the automorphisms of F extend continuously to

(see [1, w 2.3]). Since these extensions are automorphisms on the open dense subset F, they are automorphisms of 1~ which leave ~ \ F invariant. Consider the locally

trivial fiber bundle P -> X -~ B associated to the given bundle. Since the mapping restricted to X \ X exhibits the ends as a covering space of the base, the assertions

of 2) are clear. []

R e m a r k . I t is possible for 2' to have more ends than X, e.g. the Moebius band can be fibered by the open unit interval over the circle. To be sure, X has only one end in this case. But this example is easily modified so tha t X has two ends, e.g. take F :---- C \ ( c u b e roots of unity}, B :~- S 1 and " the transit ion function" to be a non- trivial rotat ion of F.

4. Complex solv-manifolds. I n this section we determine the structure of certain complex solv-manifolds.

Definition. Let X be a complex manifold. Two points xl and x2 in X are defined to be equivalent if ](xl) = f(x2) for every ] e (~(X). The codimension of an equiv- alence class a t x e X is called rankx 0(X) and rank 0(X):---- maxxex( rankx (P(X)). We say tha t X satisfies the maximal rank condition if rank (~ (X) : dim c X.

550 B. GILLIGAN ARCH. MATH.

Lemma 3. Suppose G is a connected complex Lie group and I and J are closed complex subgroups o/G, with J containing I, such that G/J is a torus and J / I is C*, i.e. X :---- G/I -+ G/J is a homogeneous C*-fibration over a torus. Then rank ~(X) ~ 1.

P r o o f . By a theorem of Matsushima [26] the line bundle associated to the homo- geneous C*-bundle G/I -+ G/J is topologically trivial. Let X denote the total space of this associated line bundle and let n :---- dim c X. Since the torus G/J is K~hler, a construction of Grauert [21] yields a proper exhaustion function ~: X--> ~+, whose Levi form L(~) restricted to the complex tangent space at any point of a hypersurfaee {~ ----- c}, c > O, is identically zero and thus has (n - - 1) zero eigen- values. But any such hypersurface {~ = c}, c > 0, lies in X, and is compact. Thus rank (P (X) =< 1 (e. g. see [27] or [23]). []

Corollary. Suppose X as in the Lemma and suppose ~ (X) has maximal rank. Then X is biholomorphic to C*.

Theorem 3. Suppose G is a connected complex solvable Lie group and H is a closed complex subgroup such that G/H has two ends and &(G/H) has maximal rank. Then G/H is biholomorphic to C*.

P r o o f . We proceed by induction on n : = dim G/H. For n ---- 1 the result is clear. Suppose the result is true for all complex solv-manifolds with two ends of dimension /c < n and consider the normalizer fibration G/H -+ G/N. If N ---- G, then H 0 is normal in G and we may rewrite G/H as G/I", where 1" : = H/H o is a discrete sub- group of ~ : = G/H ~ Applying the fibration in [24, Lemma 7] we have ~ / / ' --> G/J, where we may assume J/I" connected. Since 0 (J/P) has maximal rank and j0 acts transitively on J/F, i t follows by induction that J /F is C*. By the Fibration Lemma G/J is compact and thus a torus tower (BaIth-Otte [6]). I f there were non-trivial tori, then the function algebra of the C*-bundle over the top torus could not have maximal rank by Lemma 3. Thus there are no tori.

If G/N is not a point, it is holomorphically separable by Lie's Flag Theorem (e. g. see [18]). By the Fibration Lemma the connected components of N/H are compact. Since ~)(G/H) has maximal rank, G/H -> G/N is a covering. By the Corollary to Theorem 1 we have the fibration G/H -> G/G'H. Arguing as above, but for this fibration, we are done. Note tha t in fact G' is contained in H. []

5. l~'on-constant holomorphic functions. In this section we prove the Main Theorem. But first we have to eliminate certain C*-bundles over compact S/F.

Lemma 4. Suppose S is a connected semi-simple complex Lie group and 1 ~ is a discrete subgroup of S such that Y :---- S/I" is compact. I] X is the total space o/any holomorphic C*-bundle over Y, then rank d~(X) ~ 1.

P r o o f . Since no discrete uniform subgroup of a connected semi-simple complex Lie group is contained in any proper algebraic subgroup [12], it follows tha t there are no non-constant meromorphic functions on Y and in particular Y has no non-

Vol. 37, 1981 En.ds of homogeneous manifolds 551

trivial divisors. Otherwise, the algebraic dimension of Y would be positive and by a theorem of Grauer t -Remmert [22], there would exist a homogeneous fibration S/F ---> S/L where L is a proper algebraic subgroup of S c o n t a i n i n g / l

Now suppose / e 0(X) and {Ul)t~I is an open covering of Y such tha t the bundle X -~ Y is trivial over each Ui and we may write / as a Laurent series over each U~

/ = ~ a~, m (Y) w~ where ws is a fiber coordinate and y e Y. m~TZ

Here each of the collections {a~,m)ieI forms a section of some power of the line bundle associated to the C*-bundle X --> Y. But since Y has no divisors, each of these sections is constant. Thus / depends only on the fiber coordinate and any two holomorphic functions on X are analytically dependent, i.e. rank d~(X) ~ 1. []

Theorem 4. Suppose G is a connected complex Lie group and I ~ is a discrete subgroup such that G/F has more than one end and r has maximal rank. Then G/1 ~ is biholomorlohic to C*.

P r o o f . First we note tha t G cannot be semi-simple. For, suppose tha t it were. Now

consider the holomorphie separation fibration G/F--> G/F. Since &(G/F) has maxi-

mal r ank , /~ is discrete. B u t / ~ is algebraic [7] and hence finite. Thus G/~ and also G/F are Stein. Since G/_F has more than one end, it must have dimension one [32], contradicting the fact tha t F is discrete and the dimension of G is a t least three.

Since G is not semi-simple, we may apply Theorem 2. For G solvable the result follows from Theorem 3. Otherwise, there exists a non-trivial fibration G/I ~ --> G/J, where we may assume the fiber is connected. Since ~)(J/F) has maximal rank, and j0 acts transit ively on J /F the fiber is C* by induction. Since J contains R, we must have J ----/~R, i.e. the radical has closed orbits which are biholomorphic to C*. But then G/J : S/S n J is a point by Lemma 4. []

Theorem 5. Suppose G is a connected complex Lie group and H is a closed complex subgroup such that G/H has more than one end and O(G/H) has maximal rank. Then G/H is an a/fine homogeneous cone minus its vertex.

P r o o f . Consider the normalizer fibration G/H-->G/N and let _~ be the set of connected components of N meeting H. The fibration G/H ---> G/-~ has connected fibers. As long as _~ ~ H , the Fibration Lemma and the maximal rank condition imply G/~ and thus G/N are compact. But then G/N is rational [13] and N is con- nected. The fiber N/H is biholomorphic to C* by Theorem 4. Therefore G/H is a C*- bundle over a homogeneous rational.

I f ~ - ~ H, then G/H --> G/N is a covering. By the Corollary to Theorem 1 we have the fibration G/H--> G/G'H, which by the Fibration Lemma has compact base (a torus!) and its fiber has more than one end. By the theorem of Ahiezer [2] the fiber G'/G' n H is a C*-bundle over a homogeneous rational. Now since a ho- mogeneous rational bundle over a torus is Kt~hler [9], it is trivial [13]. Then the C*- bundle restricted to any torus inherits the maximal rank condition and has two ends.

552 B. GILLIGAN ARCH. MATH.

By the Corollary to Lemma 3 there is no torus. Again G/H is a C*-bundle over a homogeneous rational.

Finally to see tha t we actually have a cone, we note tha t a semi-simple par t S of G in any Levi-Malcev decomposition G ---- S �9 acts transitively on G/H (e.g. [24, Lemma 9]). The result now follows as in [19, Theorem 2] since the isotropy sub- group of the S-action is algebraic. []

R e m a r k . Affine homogeneous cones were studied in [24] as non-compact almost homogeneous spaces having an isolated point in their exceptional set. We refer the reader to tha t source for a complete discussion. We only mention tha t since any affine homogeneous cone minus its vertex admits an equivariant embedding in some CN, it is holomorphically separable and we have an explicit realization of its envelope of holomorphy, i.e. "add the ver tex"!

Theorem 6. Suppose G is a connected complex Lie group and H is a closed complex subgroup such that ~)(G/H) ~= C. Then G/H has at most two ends. Moreover, G/H has two ends precisely i/ there exists a closed complex subgroup J o/ G containing H such that J/H is connected and comTact and G/J is an a]/ine homogeneous cone minus its vertex.

P r o o f. Assume G/H has more than one end and consider the holomorphic separa- tion fibration G/H --> G/J. Using Stein factorization, if necessary, we may assume tha t J/H is connected and O(G/J) has maximal rank. By the Fibration Lemma J/H is compact and G/J has the same number of ends as G/H. The result now follows by applying Theorem 5 to G/J. []

R e m a r k s . 1) Except when G/J has dimension one, G/H is not holomorphically convex. Nonetheless the fbers of the holomorphie separation mapping of G/H are compact and connected and so we have a holomorphie reduction in the sense of Remmer t [30, Def. 2].

2) An affine homogeneous cone always has an equivariant compactification, namely the associated Pl-bundle. The same is not true for an arbi t rary G/H with two ends when O(G/H) does not have maximal rank. We give an example. J . Snow has shown tha t i f /~ is a discrete subgroup of the two-dimensional connected simply- connected solvable non-abelian complex Lie group G, i.e. C 2 with the solvable group structure given by (al, bl) �9 (a2, b2) :-~ (al -~ as, exp(al) b2 ~- bl), then G/I ~ is equivariantly compactifiable if and only if G ' n f - - - - (e ) [33]. Now let /~:---- {(gikm, nlT1 ~- n22:2)lm, n l , n2 ~ ~}, where k is a fixed odd integer and T1 and ~2 are complex numbers linearly independent over R. Then the holomorphic separation fibration is given by G/I'--~ G/G'I" and is an elliptic curve bundle over C*. But G/F is not compactifiable.

6. Concluding remarks. In this final section we gather together a few loose ends. For H a subgroup of G, per usual 57 : = N a (H~

Vol. 37, 1981 Ends of homogeneous manifold,s 553

Theorem 7. Suppose G is a connected complex Lie group and H is a closed complex subgroup o/G such that G/H has more than one end. I/G/pY is non.compact, then there exists a closed complex subgroup ~ o/G containing H such that 2~]H is connected com- pact and parallelizable and G/I~ is a homogeneous C*-bundle over a homogeneous ra- tional mani/old.

P r o o f . Consider the normalizer fibration G/H --> GIN and suppose G/_W is non- compact. Le t /V denote the set of connected components of N which meet H. Then G/H ---> G/I~ has connected parallelizable fiber 1V/H. By the Fibration Lemma I~/H is compact and G/~ has the same number of ends as G/H.

To complete the proof we have to show tha t there exists a closed complex sub- group J of G containing/~ such tha t J/l~ is C* and G/J is rational. The proof paral- lels the proof of Theorem 4 in [19]. Namely, there are three cases to consider. First, if G/pV is holomorphieally separable, then the result follows from Theorem 5. Next, if a semi-simple par t S of G acts transitively on G/I~, then one can apply the theorem of Ahiezer [2]. Finally, if we are in neither of these settings, then the radical /~ of G acts non-trivially on G/2~ and thus on GIN. In this case we consider any minimal R- invariant algebraic set in GIN and this gives rise to a non-trivial homogeneous fibra- tion with holomorphically separable fiber. Applying the Fibration Lemma and Theorem 5 we see tha t this fiber is an affme homogeneous cone minus its vertex. And, as in [19], the base is a rational. Since a rational bundle over a rational is rational, the result follows. []

The case when GIN is compact is more complicated. However, we note the ibl- lowing.

Theorem 8. Suppose G is a connected complex Lie group and H is a closed complex subgroup such that G/H has more than one end. If GIN is compact, then G/H and N/H have the same number o/ends.

P r o o f . I f G/N is compact, then it is rational [13] and thus simply connected and the result follows from the Fibration Lemma. []

I n certain cases one can say more. For, if H has a finite number of connected components, then, as noted in [19], N/H is a C*-bundle over a torus. Or if N is solvable, then N/H has two ends. Finally, if N/H o is not semi-simple, one can apply Theorem 2, thus obtaining a non-trivial fibration. However, we cannot exclude the possibility of an S/I'~ with k ends, e . g . / ' ~ in SL(2, C) (cf. [19]), from turning up. Thus no complete classification is possible.

In closing we remark tha t one can consider the ends of G/H having non-constant meromorphic functions. However X ~ Y • SL (2, C)/F~, where Y is a compact homogeneous complex manifold with ~ (Y) ~= C a n d / ' ~ is as above, has non-con- s tan t meromorphis functions and k ends. In this case the ends "come from SL(2,C)/I'k", while the functions "live on Y". The right question seems to be whether one can say anything about meromorphically separable G/H with more than one end. As above the interesting case occurs when H is discrete. Recently D. N. Ahiezer

554 B. GILLIGAN ARCH, MATH.

has shown t h a t ~ ' (S/I) = C f o r / ~ a Zariski dense subgroup which is defined over of a semisimple complex Lie group S. As we are ignorant whether S/I' can be

meromorphical ly separable and have more than one end, there is no possibility of a classification a t this t ime.

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Anschrift des Autors:

B. Gilligan Dept. of Mathematics University of Regina Regina, Saskatchewan Canada $4S 0A2

Eingegangen am 20. 10. 1980

z.Z. Mathematisches Institut D-4400 Miinster/Westf. RoxelerstraBe 64 Fed. Rep. of Germany