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       Some remarks on automorphism groups oftoroidal/(H,C) groups
Dedicated to Professor Akihiko Morimoto on his 80 th birthday,by
Shigeru Takeuchi
Department of Mathematics, Faculty of Education, Gifu University
Yanagido 1-1, Gifu, Japanemail: takeuchi@gifu-u.ac.jp(received January 7th and revised February 8th 2008)
Abstract
The purpose of the present note is to discuss the automorphisms of an arbitrary (H,C) space. For the case of group manifold, (H,C) space is toroidal if
we follow the deﬁnition/naming of toroidal group by Vogt [15] (see also AbeKopfermann [1] and Kazama-Umeno [3]); if a complex Lie group G has no nonconstant (global) holomorphic functions, it is said to be a toroidal group, which
is ﬁrst deﬁned and named (H,C) group by Morimoto [10] (also see [11] by thepresent author) in the more general setting. As a generalization of compact
complex torus, toroidal group G is generically isomorphic with holomorphic automorphism group Aut(G) up to connectivity. We consider the problem related
to this result. 1

1

Introduction

First we give a short summary of fundamentals of complex Lie groups for the readers’convenience. A group G is a complex Lie group if it is a complex manifold whosegroup operation μ : G × G → G, deﬁned by μ(g, h) := gh−1 , is holomorphic. For thepresent moment we assume G to be connected unless otherwise stated. As was statedin [11](see also [7, 8, 10]) we have the two fundamental structure theorems of complex
Lie group G as follows:
Theorem 1.1 Let K be a maximal compact subgroup of G. Then the complexiﬁcation K c of K (see [7] for details) is a closed reductive complex subgroup G and thefactor space G/K c is biholomorphic to a complex aﬃne space Cα . Hence G is biholomorphic to the product complex manifold K c × Cα (see [11] for the precise deﬁnitionof the complex characteristic index α = α(G) of G). If G itself is a reductive complex
Lie group then G contains a unique maximal Stein subgroup S, such that S is normal
and the factor group G/S is a complex torus. Thus G has the structure of holomorphic
ﬁber bundle over complex torus of dimension β = β(G) with Stein ﬁber S × Cα .
Theorem 1.2 There is a maximal toroidal subgroup T of G, contained in the center
Z = Z(G) of G, such that the factor group G/T is a Stein group. Then T is equal
to the Zariski closure (in G) of the maximal complex subgroup Ko of K. The complex
1

Key Words and Phrases: no non-constant holomorphic functions, complex Lie group, automorphism group, (H,C) group, toroidal group

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analytic invariant index β = β(G) is deﬁned to be the complex dimension δ(Ko ) of Ko
and is equal to the completeness of G, denoted com(G) which is independent of the
choice of K [11].
The readers are referred to [10] for a detailed proof of the second theorem and we
instead give some basic characteristics of S, Ko and T , since it would be useful forthem to understand our main result. A complex Lie group G is a Stein group if it
is a Stein manifold as a group manifold. A subgroup S of an arbitrary complex Liegroup G is a maximal Stein subgroup if it is a Stein group, and there are no larger
Stein subgroups containing S. As for the second theorem we should mention that T
is deﬁned as follows. Let X be a complex space, and let O(X) be the set of global
holomorphic functions on X and we set Xx := {x ∈ X; f (z) = f (x), f ∈ O(X)}. Then
it is observed that Xx is a complex analytic subvariety which is invariant under the
biholomorphic automorphism group Aut(X) of X [11]. Morimoto [?] proved T = Ge
is a central complex subgroup of G and the quotient complex Lie group G/T is a Steingroup, such that O(G) = O(G/T ). Thus as a corollary to the above theorem we ﬁndO(G) is a Stein algebra for every complex Lie group G.

2

The automorphism group of a complex space

We can formulate some problems in the holomorphic symmetry of a complex space
X. First we consider the set of all biholomorphic self-mappings of X, which forms agroup, called the automorphism group of X, denoted Aut(X), with the multiplication
deﬁned by the composition of two automorphisms. Then it would be natural to ask
the following
Question What is Aut(X) ? Is it a Lie transformation group (with the compact
open topology) which acts holomorphically on X ? If so, determine (the dimension of)Aut(X), e.g.,, its group structure.
In some well known cases this task has been done by several authors (see [4] for classicalresults as a bibliography). There are answers in the most general setting of complex
space X. The one is for the bonded domain case, which says Aut(X) is a totally real
CR Lie group. Another one is for the compact case, which says Aut(X) is a totallycomplex CR Lie group with the invariant complex structure J, in the sense that the
canonical action Aut(X) × X → X is holomorphic with respect to J. In these twocases some concrete results are obtained for speciﬁc objects X. As far as the present
author knows there are few results beside these cases even in the CR category [12, 13].
For compact cases there is a general result that compact complex surface X of general
type doesn’t admit any non-trivial automorphisms. We may roughly say that X admits
abundant automorphisms if X is akin to homogeneous. Precisely saying we should givethe following
Deﬁnition 2.1 A complex space X is (holomorphically) heterogeneous if X admits nonon-trivial holomorphic automorphisms.
The opposite extreme is the well known2     
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Deﬁnition 2.2 A complex space X is homogeneous if Aut(X) acts transitively on X.
There are no deﬁnitions for the intermediate cases. Before giving the intermediatedeﬁnitions we prepare some preliminary notions. Take a point x ∈ X. We consider theorbit Ox of x by Aut(X); Ox = {gx|g ∈ |Aut(X)}. In general it is not known whether
Ox is a complex analytic subvariety or not, except the special cases deﬁned by some
additional assumptions. Among others, we should mention the case where Aut(X) is
the Lie transformation group G acting smoothly on X. Then we can smoothly identify
the subspace Ox with the coset space G/Gx equipped with the quotient topology,
where Gx is the isotropy subgroup of G at x. In general G is a CR Lie transformationgroup with the invariant CR structure and we can consider every orbit Ox to be a
CR submanifold with the induced CR structure [12, 13, 14]. In this direction we haveno results on the CR property(e.g.,, CR codimension etc.,) of the orbits for speciﬁc
examples. To treat more general cases we consider the Zariski closure Ox of each orbit
Ox in X. For simplicity we denote it by Zx , which is a complex analytic subvariety by
deﬁnition.
We deﬁne the (local complex) homogeneity hx at a point x ∈ X by the complex dimension of Zx .
hx := dimC Zx
Then we deﬁne the (global complex) homogeneity h(X) of X by the maximum of hx .
h(X) := max{hx |x ∈ X}
Now we should give the following
Deﬁnition 2.3 A complex space X is of homogeneity h or X is homogeneous of degree
h iﬀ h(X) is equal to h.
Thus in this terminology if X is homogeneous then obviously δ(X) = h(X), whereδ(X) denotes the complex dimension of X. But the converse is not true in general.
Some people say X is almost homogeneous if there exists an open orbit Ox which is(topologically) dense in X. In this article we adopt the terminology that X is quasihomogeneous (with respect to Aut(X)), if X is homogeneous of degree δ = δ(X);
h(X) = δ(X). This is equivalent to the existence of a Zariski dense orbit Ox , which issometimes expressed a generic orbit.
Remark 2.1 It should be mentioned even if a generic orbit exists, it is not unique ingeneral as will be shown in the next section.
The category of complex spaces is deﬁned naturally; a morphism of two objects X → Y
is a holomorphic mapping. Then taking the holomorphic function algebra O(X) we
obtain the contravariant functor F to the category of complex commutative algebras
with the usual morphisms. For those spaces of high homogeneity we could get lots of
information from their function algebras. For an automorphism f : X → Y we havethe induced automorphism f ∗ : O(Y ) → O(X); f ∗ = F(f ), and O(X) = F(X),etc.,.
Especially for Stein object X, this functor is contravariantly isomorphic and we cantreat the homogeneity equivalently in both categories. But in the (H,C) cases this
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functor cannot play any role, since it is a trivial functor. But there are inﬁnitely many
(H,C) spaces which are homogeneous. So for this class we should directly attack thespaces themselves and their automorphisms. We will try this task in the next section.

3(H,C) space

In the previous section we gave an idea to classify complex spaces by the homogeneity.
Here in the present section we would like to classify complex spaces by the abundanceof its holomorphic functions O(X). The most abundant one is the class of Stein spaces,
and the opposite extreme is the class of (H,C) spaces due to Morimoto [?]. The compactness of X is a suﬃcient condition to belong to (H,C) class. By Hartogs phenomena
it is not a necessary condition, since a Zariski open subset Y of a compact complex
space X belongs to (H,C) class, if the complementary subset X − Y is an analytic
subvariety of codimension greater than one. In order to simplify the expression fromnow on we use the term (H,C) not only as an attributive but also a predicate adjective;
instead of saying X belongs to (H,C) class we say X is (H,C). So we are mainly interested in holomorphically non-compact cases; if Y has a compactiﬁcation X such thatcomplex codimension of (X −Y ) is less than 2, Y is said to be holomorphically compact.
In order to systematically study (H,C) spaces it would be convenient to introduce thefollowingDeﬁnition 3.1 An (H,C) space X is reducible if there exist positive dimensional closed
subspaces X1 , X2 such that X is biholomorphically equivalent to the product space
X1 × X2 . If it is not reducible then it is said to be irreducible. If X is expressed as a
product space Πki=1 Xi of irreducible subspaces {Xi |i = 1.2, . . . k} we say each Xi is anirreducible component of X.
Remark 3.1 Each irreducible component is obviously (H,C) and hence may be called
irreducible (H,C) component or factor.
Remark 3.2 At the present moment we have no general results on the uniqueness ofthese components up to order.
Remark 3.3 Abe [1] proved a quasi-abelian variety is a quasi-projective variety, whichsuggests the similar philosophy as ours in the (H,C) case, since both object have aprojective algebraic compactiﬁcation.
As for an (H,C) space X in general we will develop systematic approach not here butin the forthcoming paper [14]. Instead we treat an (H,C) complex Lie group G in
the next section. Owing to the pioneer work of Morimoto [?] it is an abelian groupwhose universal covering group G̃ is a complex vector spaceCn . We can take R linearly
independent vectors {e1 , e2 , . . . , en } so that they are C linearly independent, hence forma base of complex vector space G̃ = Cn . Let π be the canonical projection G̃ → G.
Then its kernel Γ is a discrete subgroup of the complex vector space Cn , that is referredto the (associated) lattice of G. We can take a ﬁnite set of R-linearly independentvectors {γi |i = 1, 2, . . . r} together with {e1 , e2 , . . . , en } form a base of G̃ over R. Thus
1
n−r
. So
as a real Lie group G is isomorphic to the direct product group Πn+r
i=1 S × R
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if r = n, G is the ordinary (compact) complex torus of complex dimension n. If
0 < r < n, then G is a non-compact (H,C) group or toroidal group. If r = 0, then G isholomorphically equivalent to the algebraic torus (C∗ )n of complex dimension n , hence
not (H,C). By Kazama Umeno [3] or Vogt [15] we know as cohomological conditions foran (H,C) group G, given in terms of the linear independence of the lattice generators
over the rational number ﬁelds Q.

4

The automorphism group of a toroidal group

Now we follow the notation in the preceding section. Let G be a an (H,C) group ofcomplex dimension n with the lattice Γ of rank m = n + r. Here we investigate thestructure of Aut(G), where we consider G to be a complex space forgetting it is acomplex Lie group. Thus an element f ∈ Aut(G) has nothing to do with the group
operation of G by deﬁnition. But an element g ∈ G is naturally considered an elementof Aut(G) by translation. Thus, before stating a theorem, we prepare the following
Deﬁnition 4.1 (translation) Translation t of G is a holomorphic transformation of
G that is given by the following mapping:t(x) = x + afor some element a ∈ G. To emphasize the dependence of t on an element a wedenote t by ta . The group of translations forms a subgroup of Aut(G), called thetranslation group of G; t : G → Aut(G) deﬁned as above is obviously an injective
group homomorphism.
The natural question is whether it is surgective. Now we state
Theorem 4.1 (Abe [?] Above homomorphism t is generically an isomorphism, or
equivalently every holomorphic automorphism of G is given by some translation ta for
an element a ∈ G up to ﬁnite symmetries.
Proof.The readers are kindly referred to the forthcoming paper for the a generalization([14]). The proof given here is essentially the same with the compact case, but weinclude it here for completeness. Anyway let us take an arbitrary automorphism f of
G. Then taking the universal covering G̃, we get the lift f˜ of f . By deﬁnition f˜ isa holomorphic self-mapping of G̃. Since f is an automorphism f has a holomorphic
inverse f −1 . Thus f −1 has a lift f˜−1 , which is a holomorphic self mapping of tdG. From
the uniqueness of the covering mapping we have the identity (f˜)(f˜−1 ) = idG̃ , whichshows the existence of the canonical homomorphism π∗ : Aut(G) → Aut(G̃); π ∗ (f ) =f˜. Since f˜ : G̃ → G̃ is invariant under the action of the lattice Γ, it satisﬁes thefollowing equations:f˜(z + γ) = f˜(z) + γz ,z ∈ G̃, γ, γz ∈ Γ,

where γz is the additive automorphic factor with respect to the lattice Γ, which is notco-compact since we have only to consider a non-compact (H,C) group G. We can
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express the mapping f˜ : Cn → Cn in terms of the coordinate functions z1 , z2 , . . . , zn
as follows:f˜(z) = (f˜1 (z), f˜2 (z), . . . , f˜n (z)),

z = (z1 , z2 , . . . , zn ) ∈ Cn .

We have the following commutative diagram:f˜G̃ = Cn→ G̃ = Cn↓π↓πfn
G = C /Γ → G = Cn /Γwhose rows are isomorphism. The commutativity is shown above. Then each entire holomorphic function f˜j on Cn is not pushed down to the quotient holomorphic
functionfj on the lower row so that the following diagram be commutative:f˜jG̃ = Cn↓π→ C
fj

G = Cn /Γ 
But passing through to the respective partial diﬀerential of the function which can bewell deﬁned by using the additive automorphic factor which is reduced to the respective
constant due to the discreteness of Γ. Then all of them are constant since the basespace is (H,C). Thus we deduce f is a linear mapping, from which there follows f is atranslation up to the signature of the respective partial derivative of each coordinate
representation fj . Q.E.D..

5

Final remarks on automorphisms of holomorphic principal
bundlesMorimoto [9] obtained a result on automorphisms of holomorphic principal bundles:
Theorem 5.1 (Morimoto [9]) Let P := P (M, G) be a holomorphic principal bundle
over a compact complex manifold M as a base space, with the ﬁber complex Lie group
G. Then the automorphism group AutF (P ) is a complex Lie transformation groupacting holomorphically on P . Furthermore we have the canonically induced complex
Lie group homomorphism π∗ : AutF (P ) → Aut(M ).
He also proved the surjectivity of the above homomorphism under some additional
conditions ;the one is the existence of a holomorphic connection of P (M, G) and theother is the vanishing of the ﬁrst Betti number of M .
Remark 5.1 In our setting the (H,C) group G is a holomorphically nontrivial holomorphic principal bundle P (S, T ) as was stated in the beginning theorem, where S is
the maximal Stein subgroup which is normal in G since G itself is abelian and T is
a compact complex torus obtained as a quotient group G/S. Then we have an exact
sequence of complex Lie group as follows:
0→S→G→T →0

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. Then we have the associated sequence
0 → ker π → AutF (G) → Aut(T ) → 0.
Obviously the second condition above is not satisﬁed but the ﬁrst one is not examinedyet. Anyway above theorem is a partial example of Morimoto’s theorem and propositionin [9], except the connectivity of Aut(T ), since the second exact sequence is genericallythe same with the ﬁrst one where ker(π) is equal to the typical ﬁber S, π denoting thenatural surjection G → T = G/S.

References[1] Abe,Y., Kopfermann, ”Toroidal groups”, Lecture Notes in Mathematics 1729, Springer,
2001.
[2] Kazama,H., On pseudoconvexity of complex abelian Lie groups, J. Math. Soc. Japan,
25 (1973),pp.329-333.
[3] Kazama,H.,and Umeno, Complex abelian groups with ﬁnite dimensional cohomology
groups, J. Math. Soc. Japan, 36,(1984),pp. 91-106.
[4] Kobayashi,S., and Nomizu, K., ” Foundations of Diﬀerential Geometry I,II”, Interscience
Tracts in Pure and Applied Mathematics 15, John Wiley and Sons, .
[5] Kodama,A., and Shimizu, S., A characterization of C k × (C ∗ ) from the viewpoint of
biholomorphic automorphism groups J. Korean Math. Soc. 40 (2003), 563-575.
[6] — Standardization of certain compact group actions and the automorphism group of
the complex Euclidean space, to appear in Complex Variables and Elliptic Equations[7] Matsushima,Y., Sur la Structure du groupe d’homeomorphismes analytique d’une certaine variété kaehlerienne, Nagoya Math.J.11(1957),145-150.
[8] Matsushima,Y., and Morimoto,A., Sur certains espaces ﬁbrés holomorphes sur une
variété de Stein, Bull. Soc. Math. France 88 (1960),pp.137-155.
[9] Morimoto,A., Sur le group d’automorphismes d’un espace ﬁbré principal analytiquecomplexe, Nagoya Math. Jour. 13(1958),pp.157-188.
[10] Morimoto,A.,Non-compact complex Lie groups without non-constant holomorphic functions, Proc.Conf. Complex Analysis at Univ. Minn., Springer Verlag, Berlin,1965,pp.257272.
[11] Takeuchi,S., On completeness of holomorphic principal bundles, Nagoya Mathematical
Journal vol.56, pp.121-138,(1974).
[12] Takeuchi,S., On the universal reciprocity law of the evaluation functor, Proc. 12th ICFIDCA, Kyushu Univ. Press,pp.367-371,(2005).
[13] Takeuchi,S., On the commutative unitary Fréchet algebra and Stein algebra, in preparation.
[14] Takeuchi,S., in preparation.
[15] Vogt,C., Line bundles on toroidal groups, Journal fdie reine und angewandte Mathematik band 335(1982),pp.197-225.

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