Levi's problem for complex homogeneous manifolds

Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup. Then there exists a closed complex subgroup $J$ of $G$ containing $H$ such that the fibration $\pi:G/H \to G/J$ is the holomorphic reduction of $G/H$, i.e., $G/J$ is holomorphically separable and ${\mathcal O}(G/H) \cong \pi^*{\mathcal O}(G/J)$. In this paper we prove that if $G/H$ is pseudoconvex, i.e., if $G/H$ admits a continuous plurisubharmonic exhaustion function, then $G/J$ is Stein and $J/H$ has no non--constant holomorphic functions.

1 Introduction e original Levi problem dealt with the characterization of domains of holomorphy in C n having smooth boundary in terms of conditions on that boundary. Grauert [ ] (resp. Narasimhan [ ]) showed that a complex manifold (resp. complex space) that admits a smooth (resp. continuous) strictly plurisubharmonic exhaustion function is Stein. However, Grauert also pointed out that there exist pseudoconvex domains in tori all of whose holomorphic functions are constant; see e.g., [ ]. is shows that the Levi problem for pseudoconvex domains that are not strongly pseudoconvex fails, in general.
In this paper we restrict our attention to complex homogeneous manifolds X ∶= G H with G a connected complex Lie group and H a closed complex subgroup, and we seek conditions under which its holomorphic function algebra O(G H) is Stein.
yields a closed complex subgroup of G containing H, and, consequently, one has the holomorphic bration π∶ G H → G J, gH ↦ g J, which we call the holomorphic reduction of G H [ ]. By construction, one has O(G H) ≅ π * O(G J). e question then reduces to considerations of when the holomorphically separable complex homogeneous manifold G J is Stein. Now it is known that holomorphic separability implies Steinness for complex Lie groups [ ], for complex nilmanifolds [ ], and for complex solvmanifolds [ ]. But the situation is much di erent when the group acting is semisimple or reductive. For example, C n ∖ { } is not Stein whenever n > . What is known for G reductive is the following: G H holomorphically separable implies H algebraic [ ] and G H is Stein if and only if H is reductive [ , ]. A reductive complex Lie group is the complexi cation of a totally real maximal compact subgroup and the compactness of this subgroup is playing an essential role. An excellent survey of the function theory on G H for G reductive can be found in [ , Chapter ], and we refer the interested reader to that book and the references listed therein.
is paper analyzes the holomorphic function theory of pseudoconvex complex homogeneous manifolds, where a complex manifold is pseudoconvex if it admits a continuous plurisubharmonic exhaustion function. In [ , Main eorem] we proved that the base of the holomorphic reduction of any pseudoconvex complex homogeneous manifold G H is Stein and its ber has no non-constant holomorphic functions if G is solvable (see also [ , eorem . ]) or if G is reductive (see also [ , eorem . ]).
Here we prove that this result holds more generally for pseudoconvex homogeneous manifolds of mixed groups, where by a mixed group we mean that G is a connected, simply connected, complex Lie group that has a Levi-Malcev decomposition G = S ⋉ R, where R is the radical of G, S is a maximal semisimple subgroup, and both dim R > and dim S > .

eorem .
Let G be a connected complex Lie group and H a closed complex subgroup of G such that G H is pseudoconvex. Suppose G H → G J is the holomorphic reduction of G H. en G J is Stein and O(J H) = C.
is note is organized as follows. Section contains some technical results that are needed for the proof. Section presents the proof of eorem . .

The Hirschowitz Annihilator
Every element ξ ∈ g, the Lie algebra of G, can be thought of as a right invariant vector eld on G and, as such, pushes down to a holomorphic vector eld ξ X on any complex homogeneous manifold X ∶= G H of the group G. An inner integral curve in such a homogeneous space X is a non-constant holomorphic map C → X with relatively compact image in X that is the integral curve of some vector eld ξ X assigned to some ξ ∈ g. Hirschowitz [ ] considered such concepts in the context of in nitesimally homogeneous manifolds, where a manifold is in nitesimally homogeneous if every tangent space is generated by global holomorphic vector elds. A complex manifold that is homogeneous under the action of a Lie group of holomorphic transformations is in nitesimally homogeneous.
Hirschowitz showed that a pseudoconvex, in nitesimally homogeneous X that does not contain any inner integral curve is Stein [ , Proposition . ].
is is the starting point of our investigations of pseudoconvex homogeneous manifolds that are not Stein given in [ ]. By the maximum principle, any plurisubharmonic function on a complex manifold X is constant along every inner integral curve in X. One has to determine the "directions of degeneracy" of plurisubharmonic functions in terms of a certain subset of g whose corresponding holomorphic vector elds "kill them". So in [ ] we de ne the Hirschowitz annihilator A to be the connected Lie subgroup of G whose Lie algebra is given by where P(X) is the space of continuous plurisubharmonic functions on X ∶= G H. For continuous functions, the derivative is understood in the sense of distributions. By de nition a is a complex vector subspace of g, and it is also a Lie subalgebra of g that properly contains h, if X is not Stein [ , Lemma . ]. e corresponding (not necessarily closed) connected complex subgroup A of G contains the identity component H of H and de nes a complex foliation F = {F x } x∈X , the Levi foliation of the manifold X. Every leaf F x of this foliation is a relatively compact immersed complex submanifold of X containing every inner integral curve in X passing through the point x and is an orbit of the group A. In general, complex foliations are rather difcult to understand. But here the foliation arises from a subgroup strongly re ecting the underlying geometry of the homogeneous manifold and is related to the existence of a plurisubharmonic exhaustion on it. is allows a su ciently good understanding in order to analyze the structure from the point of view of its holomorphic function algebra.
If the leaves of the foliation are closed, then they are compact and X is holomorphically convex [ , eorem . ]. Indeed, the holomorphic reduction is then given by the Remmert reduction [ ] and has compact ber and Stein base. e main diculty occurs when this is not the case. e following observation is essential in what follows.

Remark .
A question of Serre [ ] asks whether the total space of every holomorphic bration with Stein ber and Stein base is itself Stein. Counterexamples are known, some of which are even homogeneous, e.g., [ ]. Among other things, Lemma . asserts that Serre's question has an a rmative answer in our setting. [ , Remark . ]) Suppose Y is a pseudoconvex complex homogeneous manifold that admits a holomorphic bration Y F →B with both F and B being holomorphically separable manifolds. en Y is Stein.

Lemma . ([ ] and
Proof Note that every complex homogeneous manifold is in nitesimally homogeneous. As neither F nor B can contain any inner integral curve, the same is true of Y.

Tits' Normalizer Fibration
Suppose H is a k-dimensional closed subgroup of an n-dimensional Lie group G. e Lie algebra h of H is a Lie subalgebra of g and can be considered as a point in the Grassman manifold Gr(k, n) of k-dimensional vector subspaces of g. Since ad(G) ⊂ GL(g), there is a natural action of ad(G) on Gr(k, n), and the ad(G)-orbit of the point h can be identi ed with G N, where N ∶= N G (H ), i.e., the normalizer in G of the identity component H of H. Via the Plücker embedding, Gr(k, n) can be realized as a submanifold of some projective space such that the automorphisms of the Grassman are restrictions of those automorphisms of the projective space that stabilize the embedded Grassman. In this way, we realize G acting linearly on G N via the adjoint representation. Since H ⊂ N G (H ), we have the Tits normalizer bration

Remark .
Let I be a complex Lie subgroup of the complex Lie group G. e normalizer N G (I ), where I denotes the connected component of the identity of I is a closed subgroup of G, since it is the isotropy subgroup of a point in a complex projective space under the adjoint action of the group G.

Closure of Orbits
Let G be a (real) Lie group and H a closed subgroup of G. For I a normal Lie subgroup of G, set F ∶= cl X (I.x ), where x is the base point of X ∶= G H.

Lemma .
Let

Remark .
As a consequence, are the bers of the homogeneous bration G H → G J.

Fibrations Over Projective Orbits
We use the following notation for the derived series of the Lie group G:

Lemma .
Let X ∶= G H be an orbit of a connected complex Lie group G acting holomorphically and e ectively on some projective space. Assume that J is a connected, complex subgroup of G that has positive dimensional orbits in G H that are relatively compact. en J is a subgroup of every subgroup G (m) of the derived series of G.
Proof By a result of Chevalley [ ], the image of the commutator group G ′ in the automorphism group of the ambient projective space is algebraically closed. is means that G ′ is acting as an algebraic group on the projective space, and, in particular, that its orbits in G H are closed. us, one has the commutator bration where the bar denotes the Zariski closure. erefore, J.x is contained in the ber of G H → G H ⋅ G ′ by the maximum principle with J.x still being relatively compact in the closed ber x ) and iterate the argument to see that J is a subgroup of every group in the derived series of G.

Lemma .
Let G be a connected complex Lie group, H a closed complex subgroup of G, and I a closed complex subgroup of G containing H with G I equivariantly embedded in the complex projective space P N . If J is a connected, normal, complex Lie subgroup of G whose orbits in G H are relatively compact and cover its orbits in G I, then the J-orbits in G H are ag manifolds.
Proof By Lemma . , the image of J in the automorphism group of G I lies in the image of every subgroup G (m) of the derived series of G. Since G has nite dimension, one has G (k) = (G (k) ) ′ = G (k+ ) = ⋅ ⋅ ⋅ for some k. As the J-orbits have positive dimension, G (k) is a positive dimensional perfect Lie group that is acting algebraically us, R J is a connected complex subgroup of the unipotent group R G (k) and so is acting algebraically on P N with each of its orbits a closed copy of C q for some q ≥ . Since the J-orbits are relatively compact, we must have q = . is implies that R J acts trivially, and thus J is acting algebraically as a semisimple group on G I. Now A ∶= cl G (J ⋅ H) ⊂ K ∶= cl G (J ⋅ I), since H ⊂ I by assumption. By Remark . and the assumption that the J-orbits are relatively compact, we get the diagram e J-orbit through the base point in G I is contained in K I, and the latter space is compact. us, the closure of this J-orbit lies in K I. Since J is acting algebraically as a semisimple complex group, this orbit is Zariski open in its closure, and its boundary consists of J-orbits of strictly lower dimension. Since J ⊲ G, all orbits have the same dimension, and so the boundary is empty; i.e., the J-orbits are closed and thus compact. Compact orbits of a complex Lie group acting holomorphically on a projective space are ag manifolds. Since ag manifolds are simply connected and the J-orbits in G H cover the J-orbits in G I, it follows that the J-orbits in G H are ag manifolds.

Corollary .
Let G be a connected complex Lie group, H a closed complex subgroup of G, and I a closed complex subgroup of G containing H with G I equivariantly embedded in the complex projective space P N and the bers of the bration G H → G I holomorphically separable. If J is a connected, normal, complex Lie subgroup of G whose orbits in G H are relatively compact, then the J-orbits in G H are ag manifolds.
Proof Since I H is holomorphically separable, the J-orbits intersect the bers of the bration G H → G I transversally and so cover the corresponding J-orbits in G I, which are necessarily relatively compact in G I.
e result now follows from Lemma . .

Normality Under Closure and Complexification
For J a connected Lie subgroup of G the complexi cation J C of J is the connected Lie subgroup of G corresponding to the Lie algebra j + ij, where j denotes the Lie algebra of J. Subsequently, we need to consider what happens to normality under closure and complexi cation, and so the following lemma is important.

Lemma .
Suppose I is a connected normal Lie subgroup of a connected Lie subgroup J of a connected complex Lie group G. en In particular, I C ⊲J, whereJ is the smallest connected closed complex subgroup of G that contains J.
Note thatJ can be formed by alternately taking the complexi cation and closure of J. Applying (i) and (ii), as appropriate, completes the proof.

Proposition .
(Personal communication from K. Oeljeklaus) Let X ∶= G Γ, where Γ is a discrete subgroup of a connected, simply connected, complex Lie group G with Levi decomposition G = S ⋉ R and dim R > . en there is a connected, complex, solvable subgroup H of G normalized by Γ, containing R, with H ⋅ Γ a closed subgroup of G. In particular, one has the proper bration (unless S = {e}) that has the connected complex solvmanifold H H ∩ Γ as its typical ber.
Proof If the R-orbits themselves are closed, set H ∶= R. If not, then the Zassenhaus Lemma [ ] is used in [ , eorem ] to show the existence of a minimal connected complex solvable subgroup H ⊂ G normalized by Γ and containing the identity component of cl G (R ⋅ Γ). If H ⋅ Γ is closed in G, one has the desired result with H ∶= H .
Otherwise, let N ∶= N G (H ). Since Γ normalizes the identity component N of N , it also normalizes its radical R . Now H ⊂ R , because H is solvable and normal in N . Either R ⋅ Γ is closed in G, or the identity component of cl G (R ⋅ Γ) is contained in N . Applying the Zassenhaus Lemma again (in N ), we see that this identity component is solvable and normalized by Γ, since N ⋅ Γ = N is closed in G. Let H be the smallest connected closed complex subgroup of G that contains this identity component. en H is solvable, normalized by Γ, and its dimension is strictly greater than the dimension of H . A nite number of steps yields the desired connected complex solvable group H.

Lemma .
Suppose Γ is a cocompact, discrete subgroup of a (positive dimensional) connected solvable Lie group L such that L C Γ is Stein, where L C is the complexi cation of L. en there exists a bration by the center Z of the nilradical of L C that has (C * ) k as ber with k > .
Proof Let N be the nilradical of L and N the nilradical of L C . en N has closed orbits in L Γ by a theorem of Mostow [ ] or [ ], and thus N has closed orbits in L C Γ. Let Z be the center of N (resp. Z of N ). Incidentally, note that dim Z > [ ].
Since N N ∩ Γ is a closed complex submanifold of the Stein manifold L C Γ, we see that N N ∩ Γ is Stein. Hence the subgroup Z ⋅ Γ is closed by a result of Barth-Otte [ ]; see also [ , eorem ]. erefore, we have the bration It follows that the bers of the bration above are (C * ) k -orbits for some positive integer k; see [ , eorem ].

De nition .
A C * power tower of length one is simply the manifold (C * ) p for some positive integer p. For any integer n > , a C * power tower of length n is a (C * ) k -bundle over a C * power tower of length n − .

Remark .
Repeated application of Lemma . shows that the space L C Γ is a C * power tower of length n for some positive integer n.

Remark .
ere is a technical point that can arise in our construction, in that an intermediary bration whose ber is not connected might be involved.
is is handled by a type of Stein factorization for homogeneous brations. Suppose G is a connected Lie group that contains a closed subgroup I containing a closed subgroup H. LetĨ be those connected components of I that meet H. enĨ is a closed subgroup of G containing H, and the bration G H → G Ĩ has connected berĨ H.

Strategy of the Proof
Assume G H is a pseudoconvex homogeneous manifold that is not Stein. In order to prove the theorem we construct a closed complex subgroup I of G containing H with dim I > dim H and I H connected, possibly with the aid of Remark . , such that en if G H → G J is the holomorphic reduction, I is a subgroup of J because of (a), G I is pseudoconvex because of (b) and is either Stein (then I = J and we are done) or not Stein, and one applies the construction until one does reach the holomorphic reduction; see also the last paragraph of the proof below for more details.

Remark .
Here is a list of some complex homogeneous manifolds that do satisfy (b), whenever they occur as the ber of a homogeneous bration G H → G I. For (ii) and (iii), the basic tool to prove this is Kiselman's Minimum Principle [ , eorem . ]: (i) compact complex homogeneous manifolds, (ii) Cousin groups (see [ , Lemma . ( )]), (iii) the bers of certain C * power towers provided the exhaustion function is constant on the underlying circle power tower; see below.

The Proof Itself
Proof Let G H be a non-Stein pseudoconvex homogeneous manifold. en its Hirschowitz annihilator A satis es dim A > dim H. We de ne subgroups by setting Note that in (a), because the A-orbits are positive dimensional and relatively compact, the bration G H → G N G (A) is not a covering and its ber is not Stein. Also because of the fact that A is normal in G , if the bration G H → G N G (H ) is a covering (resp. has a Stein ber), then the A-orbits in G H are ag manifolds by Lemma . (resp. by Corollary . ). We are done, since setting I ∶= A⋅H yields a bration G H → G I with I H compact and thus satisfying (i) and (ii). us, we need only consider the setting where G H is a non-Stein pseudoconvex homogeneous manifold. If the Hirschowitz annihilator A for G H is not normal in G , we apply (a) again setting G ∶= N G (A ). So G = N G (A ) < N G (H ), the latter because G is a subgroup of G . Now setĜ ∶= G H ,Â ∶= A H and Γ ∶= H H and note thatÂ ⊲Ĝ and has positive dimensional orbits inĜ Γ. Furthermore, we can assume thatĜ is not solvable (resp. semisimple) because of [ , eorem . ] (resp. [ , eorem . ]), since each of these settings directly yields the desired subgroup I satisfying (i) and (ii); in the rst case it arises from a bration by a Cousin group and in the second by a compact complex manifold. Hence, we reduce to the case whereĜ is a mixed group. e rest of the construction below produces a subgroupÎ ofĜ containing Γ withÎ Γ satisfying (i) and (ii). Taking the preimage I ofÎ via the quotient homomorphism G → G H gives us the desired bration G H → G I. For notational convenience we suppress the hats from now on and write G instead ofĜ, etc.
Since A ⊲ G, we can apply Lemma . and Remark . . Set L ∶= cl G (A ⋅ Γ) and letL be the smallest connected, closed, complex subgroup of G that contains L. Since the A-orbits are relatively compact, L Γ is compact. As a consequence, O(L Γ) = C by the maximum and identity principles. We claim that we can further reduce to the setting where the group L has a positive dimensional radical R L , a fact that we will use later. IfL is semisimple, then one has the bration G Γ → G L andL Γ is pseudoconvex and thus holomorphically convex [ , eorem . ]. is implies that L Γ is compact, and we are again done with I ∶=L. In particular, in the rest of the proof we assume that dim R L > .
We need to show thatL Γ satis es (ii) in this setting. By Proposition . there is a brationL Γ →L H ⋅ Γ withH ⋅ Γ closed inL, whereH is a connected, solvable, complex Lie group that contains the radical RL ofL and is normalized by Γ. Now we claim that we can further reduce to the setting where the berH ⋅ Γ Γ of the above bration is Stein. Otherwise, H ⋅ Γ Γ would be a connected, pseudoconvex solvmanifold that is not Stein and there would exist a closed complex subgroup I ofH ⋅ Γ containing Γ with I Γ a positive dimensional Cousin group [ , eorem . ]. Clearly, I Γ satis es conditions (i) and (ii). So from here on we can assume thatH ⋅ Γ Γ is Stein. Now in order to nish the proof thatL Γ satis es (ii), we have to analyze the structure of (at least part of) the intersection of the compact orbit L Γ with the Stein or-bitH H ∩ Γ. Consider the connected, real, solvable group B ∶= (L ∩H) . Since R L ⊂ RL ⊂H by Lemma . and R L ⊂ L, it follows that R L ⊂ B and thus dim B > . Let B C be its complexi cation. We have Γ ⊂ NL(B C ), since Γ ⊂ L and Γ normalizesH by Proposition . , and thus we can consider the brationL Γ →L NL(B C ). Now we can assume that NL(B C ) Γ is not Stein, for, otherwise, A would have compact orbits by Corollary . , a case that could be easily handled, as above. e Hirschowitz annihilator for the space NL(B C ) Γ need not be normal inL. So we begin the proof again with the space NL(B C ) Γ and run at most a nite number of times (because dim G H < ∞) through all of its steps until the only situation demanding further attention occurs when NL(B C ) =L. Hence, we can assume that B C is a connected, normal, solvable subgroup ofL. As a consequence, B C ∩ L is normal in L and this implies that B ⊂ R L . But R L ⊂ B and so B = R L . We claim that the R L -orbits inL Γ are compact. As noted above, R L ⊂ RL by Lemma . . So by the construction ofH, one has where, as usual, the superscript denotes the connected component of the identity. It follows that the R L -orbits are closed, and since the L-orbits are compact, the R L -orbits are thus also compact. Note that if r L ∩ ir L = ( ), then the connected complex Lie group corresponding to the complex ideal r L ∩ ir L has positive dimensional orbits in the compact R L -orbits in theH-orbits. By the maximum principle, this contradicts our reduction to the setting where theH-orbits are Stein. us, one must have r L ∩ ir L = ( ), and the R L -orbits inL Γ are totally real. Now consider the complexi cation R C L of R L that has Lie algebra r C L ∶= r L ⊕ ir L . According to a conjecture of Mostow [ ] that was proved by Auslander-Tolimieri [ ] and Mostow [ ], every solvmanifold has the structure of a (real) vector bundle over a compact solvmanifold. In general, the compact base is homogeneous with respect to a group that is not a subgroup of the original solvable Lie group acting on the manifold, but rather lies in a certain algebraic hull of that group. We claim that our setting is special in that R L ⊂ R C L can be taken

Levi's Problem for Pseudoconvex Homogeneous Manifolds
to be that subgroup. Suppose is the vector bundle given by Mostow's conjecture. Since R C L R C L ∩ Γ is Stein, it follows from Serre's homology condition [ ] that dim C R C L ≥ dim R M ≥ dim R R L . On the other hand, we have dim R R L = dim C R C L , as noted above. As a consequence, the compact base M is di eomorphic to R L R L ∩ Γ and R C L ∩ Γ = R L ∩ Γ. So the R C L -orbits are closed inL Γ. We can now apply Lemma . to the triple (R L ∩ Γ, R L , R C L ). Now we have the brationL Γ →L R C L ⋅ Γ. Any continuous plurisubharmonic exhaustion function φ on G Γ is constant on each of the A-orbits, since these orbits lie in the level sets of the exhaustion function. By continuity, φ is then constant on the orbits of the closure L and thus on the orbits of its radical R L . e (S ) k -orbits that arise in bration given by Lemma . are part of the R L -orbits. us, φ is constant on the (S ) k -orbits.
As in [ , Lemma . ( )] one can apply Kiselman's minimum principle [ , eorem . ] and φ induces a continuous plurisubharmonic function onL Z ⋅ Γ. As a consequence,L Z ⋅ Γ is pseudoconvex. We continue in this fashion until a maximal semisimple group is acting transitively on the resulting quotient space Y. But then Y is pseudoconvex and thus holomorphically convex [ , eorem . ]. Since O(L Γ) = C, it follows that Y is compact and we see thatL Γ satis es (ii). is completes the proof thatL Γ satis es (i) and (ii).
In order to complete the proof of the theorem, we assume that G H is a pseudoconvex homogeneous manifold that is not Stein with O(G H) = C, and we let G H → G J be its holomorphic reduction. Note that J H cannot be Stein, since this would imply that G H itself would be Stein by Lemma . . We now choose the maximal I given by the construction above.
en O(I H) = C. Moreover, G I is pseudoconvex due to the fact that I satis es (ii). If G I were not Stein, then there would exist a subgroup I with dim I > dim I with I I satisfying conditions (i) and (ii). But this would imply that I H also satis es these two conditions, contradicting the maximality of I. Consequently, I = J and shows that the holomorphic reduction has the desired properties.