9 1981 Birkh~iuser Verlag, Basel. Ends of complex homogeneous manifolds having non-constant holomorphic functions. By. BRue~ GILLm~*). 1. Introduction.
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 holomorphy over pn (n > 1) this was proved by Behnke [8]. The general result was observed, as a consequence of Theorem B, by Serre [32]. Using this fact together with the Remmert reduction [30], one easily sees that 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 ~ / ~ m a y 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 homogeneous 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.
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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 nonconstant 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 BarthOtte [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 automorphism 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 that 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 that 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 homeomorphic 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 t h a t if a complex homogeneous space has an equivariant compactification 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 t h a t G/H has two ends precisely if there exists a parabolic subgroup P of G containing H such t h a 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
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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 that 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 semisimple it does! Borel [11] pointed out that for every integer k > 2 there exists a discrete s u b g r o u p / ~ of S L (2, R) such t h a t S L (2, R)//~ has k ends. Subgroups/~k of S L (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 homogeneous 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 fibration. Next one has to control the behavior of the ends under fibrations. This is accomplished, as was done by Ahiezer [2] and again in [19], by a Fibration Lemma (Lemma 2, el. w3), 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 that a holomorphically separable complex solvmanifold 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 that the Main Theorem 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 Institut in Miinster. I t is our pleasant d u t y 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 containing 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 embedding in some pk (e.g. the Pliicker embedding). The orbit G/N m a y 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
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closure would be obtained b y adding lower dimensional orbits. B u t 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 a n y connected simply-connected complex Lie group G has a Levi-Malcev decomposition G = S [:> 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 t h a t no component of F \ K n is relatively compact. An explicit proof t h a t this is possible for Riemann surfaces, using the fact t h a 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 b y open connected subsets such t h a t the bundle 4v --> X --> B is trivial over each W~. By renumbering and deleting if necessary, we m a y assume t h a t for every n ~ ~ there exists k (n) ~ ~ such that
.Bn c W1 U ...
U
W~(n).
We m a y furtheI, assume t h a t for every ]~ > k (n), we have W~ n Bn = O. Otherwise, replace W~ b y 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 t h a t the sequence (Un)n 9 ~, given b y
U~:=Uh[I(W~•
),
for
l~kg/c(n),
]e J,
k,]
defines an end of X. For, it is clear t h a t each Un is an open subset of X having none m p t y compact boundary. Further, b y explicit construction, Un+l c Un and r~ Un = 0. I t remains to verify t h a t every Un is connected. To prove this, it suffices to show t h a t for any x ~ h [ l ( W ~ • V]) and a n y y ~ g - I ( B \ B n ) , there exists a path lying in Un which joins x to y. Clearly there is a p a t h in B joining ~(x) to z(y). I f starting at x we can lift to a p a t h in Un then we are done. Over any connected component of . B \ B n and over a n y Wk, this is trivial. The essential point is whether one can "connect u p " such lfftings inside Un. But this follows from the fact t h a t if W~ n Wk' =4=0 and g~k, is the corresponding coordinate transition function of the bundle, then ~ v e n a n y ] there exists a j' such t h a t g ~ , (V]) n V~ =b O. Further details are left to the reader. ~inally we claim t h a 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, contradicting the assumption t h a t (Vn) n 9 ~ is an ends sequence.
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Thus X has precisely one end.
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[]
R e m a r k . The above proof depends only on the definition. As wellH. 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 connected 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 L e m m a 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. B u t the domain has two ends. This example m a y 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 . I n 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 . F r o m L e m m a 1 it follows t h a 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, w2.3]). 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 t h a n X, e.g. the Moebius band can be fibered b y 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 t h a t X has two ends, e.g. take F :---- C \ ( c u b e roots of unity}, B :~- S 1 and "the transition function" to be a nontrivial rotation 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 equivalence class at x e X is called rankx 0(X) and rank 0(X):---- m a x x e x ( r a n k x (P(X)). We say t h a t X satisfies the maximal rank condition if rank (~ (X) : dim c X.
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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 . B y a theorem of Matsushima [26] the line bundle associated to the homogeneous 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 a n y point of a hypersurfaee {~ -----c}, c > O, is identically zero and thus has (n -- 1) zero eigenvalues. 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 b y 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 m a y rewrite G/H as G/I", where 1" : = H/H o is a discrete subgroup of ~ : = G/H ~ Applying the fibration in [24, L e m m a 7] we have ~ / / ' --> G/J, where we m a y assume J/I" connected. Since 0 (J/P) has maximal r a n k and j 0 acts transitively on J/F, it follows b y induction that J / F is C*. B y the Fibration L e m m a 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 r a n k b y L e m m a 3. Thus there are no tori. If G/N is not a point, it is holomorphically separable b y Lie's Flag Theorem (e. g. see [18]). B y the Fibration L e m m a the connected components of N/H are compact. Since ~)(G/H) has maximal rank, G/H -> G/N is a covering. B y 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 t h a 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 r a n k d~(X) ~ 1. P r o o f . Since no discrete uniform subgroup of a connected semi-simple complex Lie group is contained in a n y proper algebraic subgroup [12], it follows t h a t there are no non-constant meromorphic functions on Y and in particular Y has no non-
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trivial divisors. Otherwise, the algebraic dimension of Y would be positive and b y a theorem of Grauert-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 t h a t the bundle X -~ Y is trivial over each Ui and we m a y 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. B u t 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 t h a t G cannot be semi-simple. For, suppose t h a t it were. Now consider the holomorphie separation fibration G/F--> G/F. Since &(G/F) has maximal r a n k , / ~ is discrete. B u t / ~ is algebraic [7] and hence finite. Thus G/~ and also G/F are Stein. Since G/_Fhas more than one end, it must have dimension one [32], contradicting the fact t h a t F is discrete and the dimension of G is at least three. Since G is not semi-simple, we m a y 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 m a y assume the fiber is connected. Since ~)(J/F) has maximal rank, and j0 acts transitively on J / F the fiber is C* b y induction. Since J contains R, we m u s t 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 L e m m a 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 L e m m a and the maximal rank condition imply G/~ and thus G/N are compact. But then G/N is rational [13] and N is connected. The fiber N/H is biholomorphic to C* b y 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 t h e Corollary to Theorem 1 we have the fibration G/H--> G/G'H, which b y the Fibration L e m m a has compact base (a torus!) and its fiber has more than one end. B y the theorem of Ahiezer [2] the fiber G'/G' n H is a C*-bundle over a homogeneous rational. Now since a homogeneous 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.
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B y the Corollary to L e m m a 3 there is no torus. Again G/H is a C*-bundle over a homogeneous rational. Finally to see t h a t we actually have a cone, we note t h a t a semi-simple part S of G in a n y Levi-Malcev decomposition G ----S 9 acts transitively on G/H (e.g. [24, L e m m a 9]). The result now follows as in [19, Theorem 2] since the isotropy subgroup 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 t h a t source for a complete discussion. We only mention t h a 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 v e r t e x " ! 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 t h a n one end and consider the holomorphic separation fibration G/H --> G/J. Using Stein factorization, if necessary, we m a y assume t h a t J/H is connected and O(G/J) has maximal rank. B y the Fibration L e m m a J/H is compact and G/J has the same number of ends as G/H. The result now follows b y 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 f b e r s of the holomorphie separation mapping of G/H are compact and connected and so we have a holomorphie reduction in the sense of R e m m e r 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 arbitrary G/H with two ends when O(G/H) does not have maximal rank. We give an example. J. Snow has shown t h a t i f / ~ is a discrete subgroup of the two-dimensional connected simplyconnected solvable non-abelian complex Lie group G, i.e. C 2 with the solvable group structure given b y (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, nl, 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 b y G/I'--~ G/G'I" and is an elliptic curve bundle over C*. But G/F is not compactifiable. 6. Concluding remarks. I n this final section we gather together a few loose ends. For H a subgroup of G, per usual 57 : = N a (H~
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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 compact and parallelizable and G/I~ is a homogeneous C*-bundle over a homogeneous rational mani/old. P r o o f . Consider the normalizer fibration G/H --> GIN and suppose G/_W is noncompact. L e t / V denote the set of connected components of N which meet H. Then G/H --->G/I~ has connected parallelizable fiber 1V/H. B y the Fibration L e m m a I~/H is compact and G/~ has the same number of ends as G/H. To complete the proof we have to show t h a t there exists a closed complex subgroup J of G containing/~ such t h a t J/l~ is C* and G/J is rational. The proof parallels 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 p a r 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. I n this case we consider a n y minimal Rinvariant algebraic set in GIN and this gives rise to a non-trivial homogeneous fibration with holomorphically separable fiber. Applying the Fibration L e m m a and Theorem 5 we see t h a 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 ibllowing. 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. I n closing we remark t h a 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-cons t a n t meromorphis functions and k ends. I n 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
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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 meromorphically separable and have more t h a n one end, there is no possibility of a classification at this time. Re|erenees
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Ends of homogeneous manifolds
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[26] Y. M~TSUSHn~, Fibres holomorphes sur un tore complexe. Nagoya Math. J. 14, 1--24 (1959). [27] A. MmA~I, and C. REA, Geometry and function algebra on pseudoflat manifolds, Ann. Sc. Norm. Sup. Pisa (3) 26, (1973), 695--709. [28] A. MOR~OTO, I~on-compact complex Lie groups without non.constant holomorphic functions. Proceedings of the Conference on Complex Analysis, Minneapolis, 256--272, 1964. [29] G. ~OSTOW, Some applications of representative functions to solvmanifolds. Amer. J. Math. 93, 1t--32 (1971). [30] R. R ~ R T , Sur les espaees analytiquement s~parables et holomorphiquement eonvexes. C. R. Acad. Sci. Paris 243, 118--121 (t953). [31] R. R ~ r ~ R T and A. v~'~ DE V~', Zur Funktionentheorie homogener komplexer Mannigfaltigkeiten. Topology 2, 137--157 (1963). [32] J. P. SER~, Quelques probl6mes globaux relatifs aux vari6t&s de Stein. Coll. sur les fonctions de plusieurs variables, Bruxelles, 57--68, 1953. [33] J. S~ow, Complex solv-manifolds of dimension two and three. Univ. of Notre Dame, Ph.D. Thesis, 1979. [34] J. TITs, Espaces homog6nes complexes compacts. Comment. ~ath. Helv. 37, 111--120 (1962). Eingegangen am 20. 10. 1980 Anschrift des Autors: B. Gilligan Dept. of Mathematics University of Regina Regina, Saskatchewan Canada $4S 0A2
z.Z. Mathematisches Institut D-4400 Miinster/Westf. RoxelerstraBe 64 Fed. Rep. of Germany
