Proper actions of high-dimensional groups on complex manifolds

We explicitly classify all pairs $(M,G)$, where $M$ is a connected complex manifold of dimension $n\ge 2$ and $G$ is a connected Lie group acting properly and effectively on $M$ by holomorphic transformations and having dimension $d_G$ satisfying $n^2+2\le d_G<n^2+2n$. These results extend -- in the complex case -- the classical description of manifolds admitting proper actions of groups of sufficiently high dimensions. They also generalize some of the author's earlier work on Kobayashi-hyperbolic manifolds with high-dimensional holomorphic automorphism group.


Introduction
Let M be a connected C ∞ -smooth manifold and Diff(M) the group of C ∞smooth diffeomorphisms of M endowed with the compact-open topology. A topological group G is said to act continuously on M by diffeomorphisms, if a continuous homomorphism Φ : G → Diff(M) is specified. The continuity of Φ is equivalent to the continuity of the action map Φ : G × M → M, (g, p) → Φ(g)(p) =: gp.
We only consider effective actions, that is, assume that the kernel of Φ is trivial.
The action of G on M is called proper, if the map is proper, i.e. for every compact subset C ⊂ M × M its inverse image Ψ −1 (C) ⊂ G × M is compact as well. For example, the action is proper if G is compact. The properness of the action implies that: (i) G is locally compact, hence by [BM1], [BM2] (see also [MZ]) it carries the structure of a Lie group and the action mapΦ is smooth; (ii) Φ is a topological group isomorphism between G and Φ(G); (iii) Φ(G) is a closed subgroup of Diff(M) (see [Bi] for a brief survey on proper actions). Thus, one can assume that G is a Lie group acting smoothly and properly on the manifold M, and that it is realized as a closed subgroup of Diff(M). Suppose now that M is equipped with a Riemannian metric G, and let Isom(M, G) be the group of all isometries of M with respect to G. It was shown in [MS] that Isom(M, G) acts properly on M (and so does its every closed subgroup). Conversely, by [Pal] (see also [Al]), for any Lie group acting properly on M there exists a C ∞ -smooth G-invariant metric G on M. It then follows that Lie groups acting properly and effectively on the manifold M by diffeomorphisms are precisely closed subgroups of Isom(M, G) for all possible smooth Riemannian metrics G on M.
If G acts properly on M, then for every p ∈ M its isotropy subgroup G p := {g ∈ G : gp = p} is compact in G. Then by [Bo] the isotropy representation is continuous and faithful, where T p (M) denotes the tangent space to M at p and dg(p) is the differential of g at p. In particular, the linear isotropy subgroup LG p := α p (G p ) is a compact subgroup of GL(R, T p (M)) isomorphic to G p . In some coordinates in T p (M) the group LG p becomes a subgroup of the orthogonal group O m (R), where m := dim M. Hence dim G p ≤ dim O m (R) = m(m − 1)/2. Furthermore, for every p ∈ M its orbit Gp := {gp : g ∈ G} is a closed submanifold of M, and dim Gp ≤ m. Thus, setting d G := dim G, we obtain d G = dim G p + dim Gp ≤ m(m + 1)/2.
It is a classical result (see [F], [C], [Ei]) that if G acts properly on a smooth manifold M of dimension m ≥ 2 and d G = m(m + 1)/2, then M is isometric (with respect to some G-invariant metric) either to one of the standard complete simply-connected spaces of constant sectional curvature R m , S m , H m (where H m is the hyperbolic space), or to RP m . Next, it was shown in [Wa] (see also [Eg], [Y1]) that a group G with m(m−1)/2+1 < d G < m(m+1)/2 cannot act properly on a smooth manifold M of dimension m = 4. The exceptional 4-dimensional case was considered in [Ish]; it turned out that a group of dimension 9 cannot act properly on a 4-dimensional manifold, and that if a 4-dimensional manifold admits a proper action of an 8-dimensional group G, then it has a G-invariant complex structure. Invariant complex structures will be discussed below in detail.
We study proper group actions in the complex setting with the general aim to build a theory for group dimensions lower than (m − 1)(m − 2)/2 + 2, thus extending -in this setting -the classical results mentioned above. In our setting real Lie groups act by holomorphic transformations on complex manifolds. Thus, from now on, M will denote a complex manifold of complex dimension n (hence m = 2n) and G will be a subgroup of Aut(M), the group of all holomorphic automorphisms of M. We will be classifying pairs (M, G), but we will not be concerned with determining G-invariant Riemannian metrics on M.
Proper actions by holomorphic transformations are found in abundance. A fundamental result due to Kaup (see [Ka]) states that every closed subgroup of Aut(M) that preserves a continuous distance on M acts properly on M. Thus, Lie groups acting properly and effectively on M by holomorphic transformations are precisely those closed subgroups of Aut(M) that preserve continuous distances on M. In particular, if M is a Kobayashi-hyperbolic manifold, then Aut(M) is a Lie group acting properly on M (see also [Ko1]).
In the complex setting, in some coordinates in T p (M) the group LG p becomes a subgroup of the unitary group U n . Hence dim G p ≤ dim U n = n 2 , and therefore d G ≤ n 2 + 2n.
We note that n 2 + 2n < (m − 1)(m − 2)/2 + 2 for m = 2n and n ≥ 5. Thus, the group dimension range that arises in the complex case, for n ≥ 5 lies strictly below the dimension range considered in the classical real case and therefore is not covered by the existing results. Furthermore, overlaps with these results for n = 3, 4 and n = 2, d G = 6 occur only in relatively easy situations and do not lead to any significant simplifications in the complex case. The only interesting overlap with the real case occurs for n = 2, d G = 5 (see [Pat]), but we do not discuss it in this paper. Note that in the situations when overlaps do occur, the existing classifications in the real case do not necessarily immediately lead to classifications in the complex case, since the determination of all G-invariant complex structures on the corresponding real manifolds may be a non-trivial task. It was shown by Kaup in [Ka] that if d G = n 2 +2n, then M is holomorphically equivalent (in fact, holomorphically isometric with respect to some Ginvariant metric) to one of B n := {z ∈ C n : |z| < 1}, C n , CP n , and an equiva- , respectively, one of the groups Aut(B n ), G(C n ), G(CP n ). Here Aut(B n ) ≃ P SU n,1 := SU n,1 /(center) is the group of all transformations where U ∈ U n , a ∈ C n (we usually write G (C) instead of G (C 1 )); and G (CP n ) ≃ P SU n+1 := SU n+1 /(center) is the group of all holomorphic automorphisms of CP n of the form where ζ is a point in CP n written in homogeneous coordinates, and U ∈ SU n+1 (this group is a maximal compact subgroup of the complex Lie group Aut(CP n ) ≃ P SL n+1 (C) := SL n+1 (C)/(center)). In the above situation we say for brevity that F transforms G into one of Aut(B n ), G(C n ), G(CP n ), respectively, and, in general, if F : M 1 → M 2 is a biholomorphic map, G j ⊂ Aut(M j ), j = 1, 2, are subgroups and We remark that the groups Aut(B n ), G(C n ), G(CP n ) are the full groups of holomorphic isometries of the Bergman metric on B n , the flat metric on C n , and the Fubini-Study metric on CP n , respectively, and that the above result due to Kaup can be obtained directly from the classification of Hermitian symmetric spaces (cf. [Ak], pp. 49-50).
We are interested in characterizing pairs (M, G) for d G < n 2 + 2n, where G ⊂ Aut(M) acts on M properly. In [IKra], [I1], [I2], [I3] we considered the special case where M is a Kobayashi-hyperbolic manifold and G = Aut(M), and explicitly determined all manifolds with n 2 − 1 ≤ d Aut(M) < n 2 + 2n, n ≥ 2 (see [I4] for a comprehensive exposition of these results). The case d Aut(M) = n 2 −2 represents the first obstruction to the existence of an explicit classification, namely, there is no good description of hyperbolic manifolds with n = 2, d Aut(M) = 2 (see [I3], [I4]); it is possible, however, that a reasonable classification exists in this case for n ≥ 3. Our immediate goal is to generalize these results to arbitrary proper actions on not necessarily Kobayashi-hyperbolic manifolds by classifying all pairs (M, G) with n 2 − 1 ≤ d G < n 2 + 2n, n ≥ 2, where G is assumed to be connected. This classification problem splits into two cases: that of G-homogeneous manifolds and that of non-G-homogeneous ones (note that due to [Ka] Ghomogeneity always takes place for d G > n 2 ). While the techniques that we developed for non-homogeneous Kobayashi-hyperbolic manifolds seem to work well for general non-transitive proper actions, there is a substantial difference in the homogeneous case. Indeed, due to [N] every homogeneous Kobayashi-hyperbolic manifold is holomorphically equivalent to a Siegel domain of the second kind, and therefore such manifolds can be studied by using techniques available for Siegel domains (see e.g. [S]). This is how homogeneous Kobayashi-hyperbolic manifolds with n 2 −1 ≤ d Aut(M) < n 2 +2n, n ≥ 2, were determined in [IKra], [I1], [I3], [I4]. This approach cannot be applied to general transitive proper actions, and one motivation for the present work is to re-obtain the classification of homogeneous Kobayashi-hyperbolic manifolds without using the non-trivial result of [N].
For general G-homogeneous manifolds we have Hence for n 2 − 1 ≤ d G < n 2 + 2n we have n 2 − 2n − 1 ≤ dim G p < n 2 . The starting point of our method of studying G-homogeneous manifolds with compact isotropy subgroups within the above dimension range is describing connected subgroups of the unitary group U n of respective dimensions, thus determining the connected identity components of all possible linear isotropy subgroups. In the present paper we deal with manifolds equipped with proper actions for which n 2 + 2 ≤ d G < n 2 + 2n. Due to [Ka], all such manifolds are G-homogeneous, and our proofs use the description of connected closed subgroups H ⊂ U n with n 2 − 2n + 2 ≤ dim H < n 2 obtained in [IKra] (see also [I4]). The first step towards a general classification for proper actions with d G < n 2 +2n was in fact made in [IKra] where we observed that if d G ≥ n 2 +3, then, as in the case d G = n 2 + 2n, the manifold must be holomorphically equivalent to one of B n , C n , CP n . However, in [IKra] we did not investigate the question what groups (if any) are possible for each of these three manifolds within the dimension range n 2 + 3 ≤ d G < n 2 + 2n. We resolve this question in Theorem 1.1 (see Section 1). Furthermore, in Theorem 2.1 we give a complete classification of all pairs (M, G) with d G = n 2 + 2 (see Section 2).
In the proofs of Theorems 1.1 and 2.1 we do not use the existing structure theory for actions of Lie groups on complex manifolds (see e.g. [HO], [Wo]). Neither do we use the classification of Hermitian symmetric spaces due to E. Cartan (see [H]), a reference to which can significantly simplify the proof of Part (ii) of Theorem 1.1 and that of Theorem 2.1 (see Remark 2.2). We deliberately do not refer to these general facts and give proofs based on elementary calculations involving holomorphic fundamental vector fields of the G-action.
Working with lower values of d G requires, in particular, further analysis of subgroups of U n . For example, for the case d G = n 2 +1 one needs a description of closed connected (n − 1) 2 -dimensional subgroups. A description of such subgroups was given in Lemma 2.1 of [IKru], and we will attempt to deal with the case d G = n 2 + 1 in our future work. There are a large number of examples of actions with d G = n 2 +1, and at this stage it is not clear whether all such actions can be classified in a reasonable way.
1 The case n 2 + 3 ≤ d G < n 2 + 2n In this section we prove the following theorem.
THEOREM 1.1 Let M be a connected complex manifold of dimension n ≥ 2 and G ⊂ Aut(M) a connected Lie group that acts properly on M and has dimension d G satisfying n 2 + 3 ≤ d G < n 2 + 2n. Then one of the following holds: (i) M is holomorphically equivalent to C n by means of a map that transforms G into the group G 1 (C n ) which consists of all maps of the form (0.2) with U ∈ SU n (here d G = n 2 + 2n − 1); (ii) n = 4 and M is holomorphically equivalent to C 4 by means of a map that transforms G into the group G 2 (C 4 ) which consists of all maps of the Choose coordinates in T p (M) so that LG p ⊂ U n . Then Lemma 2.1 in [IKra] (see also Lemma 1.4 in [I4]) implies that the connected identity component LG 0 p of LG p either is SU n or, for n = 4, is conjugate in U 4 to e iR Sp 2 . In both cases, it follows that LG p acts transitively on directions in T p (M), that is, for any two non-zero vectors v 1 , v 2 ∈ T p (M) there exists h ∈ LG p such that hv 1 = λv 2 for some λ ∈ R * (observe that the standard action of Sp 2 on C 4 is transitive on the sphere S 7 = ∂B 4 ). Now the result of [GK] gives that if M is non-compact, it is holomorphically equivalent to one of B n , C n , and an equivalence map can be chosen so that it maps p into the origin and transforms G p into a subgroup of U n ⊂ G(C n ); it then follows that one can find an equivalence map that transforms G 0 p either into SU n or, for n = 4, into e iR Sp 2 . Furthermore, the result of [BDK] gives that if M is compact, it is holomorphically equivalent to CP n .
Suppose first that LG 0 p = SU n . In this case d G = n 2 + 2n − 1. If M is holomorphically equivalent to B n , then the equivalence map transforms G into a closed subgroup of codimension 1 in Aut(B n ). However, the Lie algebra of Aut(B n ) is isomorphic to su n,1 , and it was shown in [EaI] that for n ≥ 2 this algebra does not have codimension 1 subalgebras. Thus, M cannot be equivalent to B n . Next, if M is equivalent to CP n , the group G is compact. Therefore, the equivalence map transforms G into a closed codimension 1 subgroup of a maximal compact subgroup in Aut(CP n ). It then follows that one can find an equivalence map that transforms G into a closed codimension 1 subgroup of G(CP n ). Since G(CP n ) is isomorphic to P SU n+1 , we obtain that SU n+1 has a closed codimension 1 subgroup, which contradicts Lemma 2.1 in [IKra] (see also Lemma 1.4 in [I4]). Thus, M cannot be equivalent to CP n either.
Assume finally that M is equivalent to C n and let F be an equivalence map that transforms G 0 p into SU n ⊂ G(C n ). We will show that F transforms G into G 1 (C n ). We only give a proof for n = 2 (hence d G = 7); the general case follows by considering copies of SU 2 lying in SU n , and we omit details.
Denote by (z, w) coordinates in C 2 and let g be the Lie algebra of holomorphic vector fields on C 2 that are fundamental vector fields of the action of G F := F • G • F −1 , that is, g consists of all vector fields X on C 2 for which there exists an element a of the Lie algebra of G F such that for all (z, w) ∈ C 2 we have Since G F acts on C 2 transitively, the algebra g is generated by su 2 (realized as the algebra of fundamental vector fields of the standard action of SU 2 on C 2 ), and some vector fields Here the functions f j , g j , j = 1, 2, 3, 4, are holomorphic on C 2 and satisfy the conditions To prove that G F = G 1 (C n ), it is sufficient to show that Y j can be chosen as follows: We fix the following generators in su 2 :

It then follows that
This identity yields Representing the functions f 1 and g 1 as power series around the origin, plugging these representations into (1.3) and collecting terms of fixed orders, we obtain Adding to Y 1 an element of su 2 if necessary, we can assume that Y 1 has no linear terms, that is and the application of an analogous argument to Y 2 yields that Y 2 can be chosen to have the form We also observe that considering [Y 3 , X 1 ] and [Y 4 , X 1 ] yields that Y 3 , Y 4 can be chosen to have the forms Further, computing [Y j , X 2 ] for j = 1, 2, 3, 4 and collecting terms of orders 2 and greater, we obtain Y 4 ] and see that α 1 = δ 1 = 0. Thus, Y j chosen as above (that is, not having linear terms) are in fact given by (1.1), and we have obtained (i) of the theorem. Suppose next that n = 4 and LG 0 p is conjugate in U 4 to e iR Sp 2 . In this case d G = n 2 + 3 = 19. If M is equivalent to CP 4 , the group G is compact. Therefore, one can find an equivalence map that transforms G into a closed 19-dimensional subgroup of G(CP 4 ). Since G(CP 4 ) is isomorphic to P SU 5 , we obtain that SU 5 has a closed 19-dimensional subgroup, which contradicts Lemma 2.1 in [IKra] (see also Lemma 1.4 in [I4]). Thus, M cannot be equivalent to CP 4 .
It then follows from (1.15) that ( 1.17) hence the vector fields in the right-hand side of formulas (1.17) lie in g. Since have no linear terms and vanish at the origin, they vanish identically and we obtain (1.18) Thus, (1.8) holds, and we have obtained (ii) of the theorem. The proof is complete.
2 The case d G = n 2 + 2 In this section we obtain the following result.  −1 ), and G ′′ is one of Aut(B 1 ), G(C), G(CP 1 ), respectively; (ii) n = 4 and M is holomorphically equivalent to C 4 by means of a map that transforms G into the group G 3 (C 4 ) which consists of all maps of the form (0.2) for n = 4 with U ∈ Sp 2 .
Proof: Fix p ∈ M. It follows from (0.3) that dim LG p = n 2 −2n+2. Choose coordinates in T p (M) so that LG p ⊂ U n . Then Lemma 2.1 in [IKra] implies that the connected identity component LG 0 p of LG p either is conjugate in U n to U n−1 × U 1 or, for n = 4, is conjugate in U 4 to Sp 2 .
Suppose first that LG 0 p is conjugate to U n−1 × U 1 . By Bochner's linearization theorem (see [Bo]) there exist a G p -invariant neighborhood V of p in M, an LG p -invariant neighborhood U of the origin in T p (M) and a biholomorphic map F : V → U, with F (p) = 0, such that for every g ∈ G p the following holds in V: where α p is the isotropy representation at p (see (0.1)). Let g M be the Lie algebra of fundamental vector fields of the action of G on M, and g V the Lie algebra of the restrictions of the elements of g M to V. Denote by g the Lie algebra of vector fields on U obtained by pushing forward the elements of g V by means of F . Observe that g M , g V , g are naturally isomorphic, and we denote by ϕ : g M → g the isomorphism induced by F . Choose coordinates (z 1 , . . . , z n ) in T p (M) so that in these coordinates LG 0 p is the group of matrices of the form LG 0 p | U and since G acts transitively on M, the algebra g is generated by u n−1 ⊕ u 1 and some vector fields for j = 1, . . . , n, where f k j , g k j are holomorphic functions on U such that f k j (0) = δ k j , g k j (0) = iδ k j . Here u n−1 ⊕ u 1 is realized as the algebra of vector fields on U of the form n−1 j=1 (a j 1 z 1 + . . . + a j n−1 z n−1 ) ∂/∂z j + iaz n ∂/∂z n , and a ∈ R.
Observe that g contains the vector field Therefore, due to Hilfssatz 4.8 of [Ka], every vector field in g is polynomial and has degree at most 2. Next, considering necessary, we can assume that V j , W j , j = 1, . . . , n, have no linear terms. Furthermore, g contains the vector fields Z k := iz k ∂/∂z k , k = 1, . . . , n.
Since for each j = 1, . . . , n the commutators [V j , Z k ] and [W j , Z k ], with k = j, vanish at the origin and do not contain linear terms, they vanish identically, which gives for some α k j , β k j ∈ C (cf. the proof of Proposition 4.9 of [Ka]). Next, [V j , Z j ] has no linear terms and [V j , Z j ](0) = (0, . . . , 0, i, 0, . . . , 0), where i occurs in the jth position. Hence [V j , Z j ] = W j which implies for all j, k. Now, for j = 1, . . . , n − 1 consider the commutator [V j , V n ]. Clearly, the linear part L j of this commutator must be an element of u n−1 ⊕u 1 . It is straightforward to see that L j = α j n z n ∂/∂z j − α n j z j ∂/∂z n , which can lie in u n−1 ⊕ u 1 only if α n j = α j n = 0 (see (2.2)). Thus, we have shown that for j = 1, . . . , n − 1 the vector fields V j , W j do not depend on z n and the vector fields V n , W n do not depend on z j . Accordingly, we have g = g 1 ⊕ g 2 , where g 1 is the ideal generated by u n−1 and V j , W j , for j = 1, . . . , n − 1, and g 2 is the ideal generated by u 1 and V n , W n .
Let G j be the connected normal (possibly non-closed) subgroup of G with Lie algebrag j := ϕ −1 (g j ) ⊂ g M for j = 1, 2. Clearly, for each j the subgroup G j contains α −1 p (L j p ) ⊂ G 0 p , where L 1 p ≃ U n−1 and L 2 p ≃ U 1 are the subgroups of LG 0 p given by α = 0 and A = id in formula (2.1), respectively. Consider the orbit G j p, j = 1, 2. Clearly, for each j there exists a neighborhood W j of the identity in G j such that for some neighborhood U ′ ⊂ U of the origin in T p (M). Thus, each G j p is a complex (possibly non-closed) submanifold of M, and the idealg j consists exactly of those vector fields from g M that are tangent to G j p at some point (and hence at all points).
Furthermore, for the isotropy subgroup G j p of the point p with respect to the G j -action we have G 0 j p = α −1 p (L j p ), j = 1, 2. Since L j p acts transitively on real directions in T p (G j p) for j = 1, 2, by [GK], [BDK] we obtain that G 1 p is holomorphically equivalent to one of B n−1 , C n−1 , CP n−1 and G 2 p is holomorphically equivalent to one of B 1 , C, CP 1 .
We will now show that each G j is closed in G. We assume that j = 1; for j = 2 the proof is identical. Let U be a neighborhood of 0 in g M where the exponential map into G is a diffeomorphism, and let V := exp(U). To prove that G 1 is closed in G it is sufficient to show that for some neighborhood W of e ∈ G, W ⊂ V, we have G 1 ∩ W = exp(g 1 ∩ U) ∩ W. Assuming the opposite we obtain a sequence {g j } of elements of G 1 converging to e in G such that for every j we have g j = exp(a j ) with a j ∈ U \g 1 . Observe now that there exists a neighborhood V ′ of p in M foliated by complex submanifolds holomorphically equivalent to B n−1 in such a way that the leaf passing through p lies in G 1 p. Specifically, we take V ′ := F −1 (U ′ ) for a suitable neighborhood U ′ ⊂ U of the origin in T p (M), and the leaves of the foliation are then given as F −1 (U ′ ∩ {z n = const}). For every s ∈ V ′ we denote by N s the leaf of the foliation passing through s. Observe that for every s ∈ V ′ vector fields fromg 1 are tangent to N s at every point. Let p j := g j p. If j is sufficiently large, we have p j ∈ V ′ . We will now show that . We also assume that V ′ is chosen so that N p ⊂ exp(g 1 ∩U ′′ )p. Suppose that p j ∈ N p . Then p j = sp for some s ∈ exp(g 1 ∩U ′′ ) and hence t := g −1 j s is an element of G 1 p . For large j we have g −1 j ∈ exp(U ′′ ). Condition (a) now implies that t ∈ exp(U ′ ) and hence by (c), (d) we have t −1 ∈ exp(g 1 ∩ U ′ ). Therefore, by (b) we obtain g j ∈ exp(g 1 ∩ U) which contradicts our choice of g j . Thus, for large j the leaves N p j are distinct from N p . Furthermore, they accumulate to N p ⊂ G 1 p. At the same time, since vector fields fromg 1 are tangent to every N p j , we have N p j ⊂ G 1 p for all j, and thus the orbit G 1 p accumulates to itself. Below we will show that this is in fact impossible thus obtaining a contradiction. Clearly, we only need to consider the case when G 1 p is non-compact, that is, equivalent to one of B n−1 , C n−1 .
Since G 0 1 p acts on G 1 p effectively, by the result of [GK], the orbit G 1 p is holomorphically equivalent to one of B n−1 , C n−1 by means of a map that takes p into the origin and transforms G 0 1 p into U n−1 ⊂ G(C n−1 ). Consider the set S := G 1 p ∩ G 2 p. The orbit G 1 p accumulates to itself, and therefore S contains a point other than p. Note that S does not contain any curve. Since G 0 1 p preserves each of G 1 p, G 2 p, it preserves S. However, the G 0 1 p -orbit of every point in G 1 p other than p is a hypersurface in G 1 p diffeomorphic to the sphere S 2n−3 . This contradiction shows that in fact S consists of p alone, and hence G 1 is closed in G.
Thus, G j is closed in G for j = 1, 2. Hence G j acts on M properly and G j p is a closed submanifold of M for each j. Recall that G 1 p is equivalent to one of B n−1 , C n−1 , CP n−1 and G 2 p is equivalent to one of B 1 , C, CP 1 , and denote by F 1 , F 2 the respective equivalence maps. Let K j ⊂ G j be the ineffectivity kernel of the G j -action on G j p for j = 1, 2. Clearly, K j ⊂ G j p and, since G 0 j p acts on G j p effectively, K j is a discrete normal subgroup of G j for each j (in particular, K j lies in the center of G j for j = 1, 2). Since d G 1 = n 2 − 1 = (n − 1) 2 + 2(n − 1) and d G 2 = 3, the results of [Ka] yield that F 1 can be chosen to transform G 1 /K 1 into one of Aut(B n−1 ), G(C n−1 ), G(CP n−1 ), respectively, and F 2 can be chosen to transform G 2 /K 2 into one of Aut(B 1 ), G(C), G(CP 1 ), respectively, where G j /K j is viewed as a subgroup of Aut(G j p) for each j.
We will now show that the subgroup K j is in fact trivial for each j = 1, 2. We only consider the case j = 1 since for j = 2 the proof is identical. Clearly, K 1 \ {e} ⊂ G 1 p \ G 0 1 p , and if K 1 is non-trivial, the compact group G 1 p is disconnected. Observe that any maximal compact subgroup of each of Aut(B n−1 ) ≃ P SU n−1,1 and G(C n−1 ) ≃ U n−1 ⋉ C n−1 is isomorphic to U n−1 and therefore, if G 1 /K 1 is isomorphic to either of these two groups, it follows that G 1 p is a maximal compact subgroup of G 1 . Since G 1 is connected, so is G 1 p , and therefore K 1 is trivial in either of these two cases. Suppose now that G 1 /K 1 is isomorphic to G(CP n−1 ) ≃ P SU n . Then the universal cover of G 1 is the group SU n , and let Π : SU n → G 1 be a covering homomorphism. Then Π −1 (G 0 1 p ) 0 is a closed (n − 1) 2 -dimensional connected subgroup of SU n . It follows from Lemma 2.1 of [IKru] that Π −1 (G 0 1 p ) 0 is conjugate in SU n to the subgroup of matrices of the form where B ∈ U n−1 . This yields that Π −1 (G 0 1 p ) 0 contains the center of SU n , hence G 0 1 p contains the center of G 1 . In particular, K 1 ⊂ G 0 1 p which implies that K 1 is trivial in this case as well. Thus, G 1 is isomorphic to one of Aut(B n−1 ), G(C n−1 ), G(CP n−1 ) and G 2 is isomorphic to one of Aut(B 1 ), G(C), G(CP 1 ).
We remark here that since M is G-homogeneous and G j is normal in G, the discussion above remains valid for any point q ∈ M in place of p; in particular, all G j -orbits are pairwise holomorphically equivalent, for j = 1, 2.
Next, since g = g 1 ⊕ g 2 , the group G is a locally direct product of G 1 and G 2 . We claim that H := G 1 ∩ G 2 is trivial. Indeed, H is a discrete normal subgroup of each of G 1 , G 2 . However, every discrete normal subgroup of each of Aut(B k ), G(C k ), G(CP k ) for k ∈ N is trivial, since the center of each of these groups is trivial. Hence H is trivial and therefore G = G 1 × G 2 .
We will now show that for every q 1 , q 2 ∈ M the orbits G 1 q 1 and G 2 q 2 intersect at exactly one point. Let g ∈ G be an element such that gq 2 = q 1 . It can be uniquely represented in the form g = g 1 g 2 with g j ∈ G j for j = 1, 2, and therefore we have g 2 q 2 = g −1 1 q 1 . Hence the intersection G 1 q 1 ∩ G 2 q 2 is non-empty. We will now prove that G 1 q ∩ G 2 q = {q} for every q ∈ M. Suppose that for some q ∈ M the intersection G 1 q ∩ G 2 q contain a point q ′ = q. Let g 1 ∈ G 1 be an element such that g 1 q = q ′ . Clearly, g 1 preserves G 2 q. Since g 1 ∈ G 1 and G = G 1 × G 2 , the element g 1 commutes with every element of G 2 . Consider the restriction g ′ 1 := g 1 | G 2 q . LetF be a biholomorphic map from G 2 q onto one of B 1 , C, CP 1 that transforms G 2 into one of Aut(B 1 ), G(C), G(CP 1 ), respectively. ThenF transforms g ′ 1 into a holomorphic automorphism of one of B 1 , C, CP 1 that lies in the centralizer of the corresponding group. In each of the three cases we immediately see that g ′ 1 is the identity, which is a contradiction. Thus, the intersection G 1 q ∩ G 2 q consists of q alone for every q ∈ M.
Let, as before, F 1 be a biholomorphic map from G 1 p onto M ′ , where M ′ is one of B n−1 , C n−1 , CP n−1 , that transforms G 1 into G ′ , where G ′ is one of Aut(B n−1 ), G(C n−1 ), G(CP n−1 ), respectively, and let F 2 be a biholomorphic map from G 2 p onto M ′′ , where M ′′ is one of B 1 , C, CP 1 , that transforms G 2 into G ′′ , where G ′′ is one of Aut(B 1 ), G(C), G(CP 1 ), respectively. We will now construct a biholomorphic map F from M onto M ′ ×M ′′ . For q ∈ M consider G 2 q and let r be the unique point of intersection of G 1 p and G 2 q. Let g ∈ G 1 be an element such that r = gp. Then we set F (q) := (F 1 (r), F 2 (g −1 q)). Clearly, F is a well-defined diffeomorphism from M onto M ′ × M ′′ . Since the foliation of M by G j -orbits is holomorphic for each j, the map F is in fact holomorphic. By construction, F transforms G into G ′ × G ′′ . Thus, we have obtained (i) of the theorem.
Suppose now that n = 4 and LG 0 p is conjugate in U 4 to Sp 2 . In this case LG p acts transitively on directions in T p (M). Now the result of [GK] gives, as before, that if M is non-compact, it is holomorphically equivalent to one of B 4 , C 4 , and an equivalence map can be chosen so that it maps p into the origin, transforms G p into a subgroup of U 4 ⊂ G(C 4 ), and transforms G 0 p into Sp 2 . Furthermore, the result of [BDK] gives, as before, that if M is compact, it is holomorphically equivalent to CP 4 .
If M is equivalent to CP 4 , arguing as in the proof of Theorem 1.1, we obtain that SU 5 has a closed 18-dimensional subgroup. This contradicts Lemma 2.1 in [IKra] (see also Lemma 1.4 in [I4]), and therefore M cannot be equivalent to CP 4 .
Assume now that n = 4, the manifold M is equivalent to one of B 4 , C 4 and let F be an equivalence map that transforms G 0 p into Sp 2 ⊂ G(C 4 ). Let g be the Lie algebra of fundamental vector fields of the action of G F := F •G•F −1 on one of B 4 , C 4 , respectively. Since G F acts transitively on one of B 4 , C 4 , the algebra g is generated by sp 2 (where, as before, sp 2 denotes the Lie algebra of Sp 2 realized as the algebra of fundamental vector fields of the standard action of Sp 2 on C 4 ), and some vector fields (1.6), where f k j , g k j , j, k = 1, 2, 3, 4 are functions holomorphic on one of B 4 , C 4 , respectively, and satisfying (1.7). We will show that F maps M onto C 4 and transforms G into G 3 (C 4 ). To obtain this, it is sufficient to prove that one can choose for some a ′ , b ′ , c ′ , α ′ , β ′ , ε ′ ∈ C (cf. (1.11)). Plugging (2.10), (2.12) into either of identities (2.5) we obtain (2.13) Then [V 1 , W 1 ] = −2i 2αz 1 ∂/∂z 1 + βz 2 + (aβ − bα)z 1 z 2 z 3 ∂/∂z 2 + αz 3 ∂/∂z 3 + εz 4 + (aε − cα)z 1 z 3 z 4 ∂/∂z 4 .
Hence W 2 , V 3 , W 3 , V 4 , W 4 can be chosen as in formula (1.18), and we have obtained (ii) of the theorem. The proof is now complete.
Remark 2.2 In the situations arising in Part (ii) of Theorem 1.1 and in both parts of Theorem 2.1 the group LG 0 q contains the map −id for every q ∈ M. Therefore, M equipped with a G-invariant Hermitian metric becomes a Hermitian symmetric space. Then, with some extra work, Part (ii) of Theorem 1.1 as well as all of Theorem 2.1 follow from E. Cartan's classification of Hermitian symmetric spaces (see [H]). The same applies to Part (i) of Theorem 1.1 if n is even. We also remark that Part (i) of Theorem 1.1 for all n follows from the results of [Wo] (see Theorem 13.1 therein). Our proofs of Theorems 1.1 and 2.1 given above are elementary and do not refer to this general theory.