Preface

A connected compact complex manifold X is called parallelizable if its (holomorphic) tangent bundle is holomorphically trivial. By a theorem of Wang this is equivalent to the assumption that X is biholomophic to a quotient manifold G/Γ where G is a connected complex Lie group and Γ a discrete, cocompact subgroup.

In the last decades an impressive amount of work has been done on discrete subgroups of Lie groups. However, the emphasis was often on discrete subgroups of real Lie groups. In this paper we are specifically interested in quotients of complex Lie groups by discrete subgroups and in particular in the complexanalytic properties of the (complex) quotient manifold. For instance, we are interested in closed analytic subsets, holomorphic vector bundles, holomorphic mappings from ℂ to X and the structure of the Albanese Alb(X).

Compact complex parallelizable manifolds other than tori are not Kähler. In particular they are not projective. Thus the usual methods and results on Kähler manifolds do not apply to parallelizable manifolds. Nevertheless it turns out that it is possible to deduce a number of results on the complex geometry of a parallelizable manifold by combining techniques and results from several different areas, including complex analysis, algebraic geometry, theory of Lie groups and algebraic groups and number theory.

I wish to express my gratitude and indebtedness to Alan T. Huckleberry for his consistent support throughout my mathematical development.

I also would like to thank Dmitri Ahiezer, Daniel Barlet, Armand Borel, Frederic Campana, Jean-Pierre Demailly, Peter Heinzner, Friedrich Knop, Siegmund Kosarew, Hanspeter Kraft, Francois Lescure, Domingo Luna, Grigori Margulis, Luca Migliorini, Eberhard Oeljeklaus, Karl Oeljeklaus and Georg Schumacher for mathematical cooperation both concerning this paper and other mathematical topics.

I would also like to use the opportunity to thank the Studienstiftung des Deutschen Volkes and the Deutsche Forschungsgemeinschaft for past support.

Compact complex parallelizable manifolds

Summary of results

In the first chapter we recall some basic facts and notation. Furthermore we list a number of examples of compact complex parallelizable manifolds and prove that (up to conjugacy) there exist only countably many non-isomorphic compact complex-parallelizable manifolds with semisimple automorphism group.

We then proceed towards the study of subvarieties, i.e. the closed complex subspaces. Of course this involves those of subvarieties of X which arise as closed orbits of certain canonical Lie subgroups of G, where X is a quotient of a complex Lie group by a discrete cocompact subgroup.

We study the pluricanonical fibration of subvarieties and demonstrate that it takes an especially nice form, analogous to that for subvarieties of abelian varieties. For a closed complex subspace Z in a compact complex parallelizable manifold G/Γ with G semisimple, we show that the tangent space T_{z}Z at every nonsingular point z ∈ Z corresponds to a Lie subalgebra of the Lie algebra of G. This is used to obtain bounds on the possible codimensions of subvarieties. For instance, if Z ⊂ X is a Kähler manifold, then dim(Z) ≤ \frac{1}{3} dim(X). For arbitrary compact complex subspaces Z in X = G/Γ with G simple we obtain codim(Z) ≥ \sqrt{dim(X)}.

Analyzing the cycle space (in the sense of Barlet), we discover that, for X = G/Γ with G non-nilpotent, it always contains non-compact irreducible components.

For certain nilmanifolds every irreducible closed complex subspace must be an orbit. In particular this holds, if Alb(X) is isogenous to a product of simple tori of algebraic dimension zero.

For compact complex parallelizable manifolds a generalization Bloch’s conjecture is true. Namely, given a holomorphic map from ℂ to a compact complex parallelizable manifold X, the smallest closed analytic subset Z ⊂ X containing the image f(ℂ) is again parallelizable. This implies that a subvariety of a compact complex parallelizable manifold is hyperbolic (in the sense of Kobayashi) if and only if it does not contain a compact complex torus.

Chapter B is largely a preparation for later investigations on holomorphic vector bundles. In particular, we investigate the Albanese and the Hodge-Frölicher spectral sequence in the context of compact complex parallelizable manifolds and prove a vanishing result: H^{1}(X, O) = 0 for X = SL_{2}(ℂ)/Γ with Γ/Γ′ finite.

Generalities on parallelizable manifolds

Chapter C deals with holomorphic vector bundles on compact complex parallelizable manifolds. We show that every homogeneous line bundle over a compact complex parallelizable manifold X = G/Γ is flat. For homogeneous vector bundles of rank two the same conclusion holds, provided the radical R_{G} of G is nilpotent. If X = G/Γ with G semisimple without SL_{2}(ℂ)-factors, then a complete classification of homogeneous vector bundles is possible.

Using flat vector bundles over compact complex parallelizable manifolds, we prove that every positive-dimensional compact complex manifold admits a non-trivial holomorphic vector bundle.

For homogeneous vector bundles over arbitrary compact complex parallelizable manifold we prove that the Chern classes are zero.

For X = G/Γ with G semisimple we can show that for every vector bundle E (whether homogeneous or not) c_{1}(E) = 0. Finally we use the Serreconstruction to produce non-homogeneous vector bundles over X = SL_{2}(ℂ)/Γ with Γ/Γ′ finite.

The next Chapter deals with the structure of compact complex nilmanifolds. Any such manifold is a torus principal bundle tower over its Albanese torus. We prove that every simple factor (a torus is simple if it contains no subtorus) in this tower is a simple factor for the Albanese as well. Moreover, if the nilmanifold is non-degenerate in a certain sense (e.g. not a direct product with a torus), then all these simples factors must be tori with complex multiplication. Conversely every torus with complex multiplication occurs in this way. For instance, for every simple torus T with complex multiplication there exists a nilmanifold X = G/Γ with Alb(X) ≃ T × T and Z = G′ where Z denotes the center of G.

Here, a torus T is said to have complex multiplication if the ℚ-endo-morphism algebra End_{ℚ}(T) contains a number field K such that [K : ℚ] = 2 dim_{ℂ}(T). Such a torus is not necessarily an abelian variety. In fact it is an abelian variety if and only if K is a CM -field. We use this to provide an example of a nilmanifold (other than a torus) of algebraic dimension zero.

We also obtain a characterization of nilmanifolds among solvmanifolds: If X = G/Γ is a solvmanifold such that Alb(X) is isogenous to a direct product of tori with complex multiplication, then G is nilpotent. A key point in the proof is the following auxiliary result. Let Λ ∈ ℂ^{g} be a lattice such that Λ ⊂ \overline{ℚ}^{g} (where \overline{ℚ} denotes the algebraic closure of ℚ in ℂ). Then every homomorphism of complex Lie groups ϕ : ℂ^{g} → (ℂ^{∗})^{k} with ϕ(Λ) ⊂ (\overline{ℚ}^{∗})^{k} is constant.

Chapter E is concerned with holomorphic functions on non-compact complex parallelizable manifolds. For a certain class of complex linear-algebraic groups we prove that O(G)^{Γ} = ℂ for every Zariski-dense subgroup.

For a more detailed account of the results we refer to the introduction of the respective chapters.

Generalities

1. Introduction

In this chapter we discuss some basic notions, provide examples of compact complex parallelizable manifolds and present some auxiliary results. These auxiliary results are mainly concerned with the following questions:

(1) Given a complex Lie group G, does there exists a linear-algebraic group isomorphic to G as complex Lie groups?

(2) Given to complex linear-algebraic groups G, H and a homomorphism of complex Lie groups f : G → H, is f necessarily a morphism of algebraic groups?

(3) Given a complex Lie group G, a discrete cocompact subgroup Γ and a homomorphism f : G → GL_{n}(ℂ), does the Zariski-closure of f(G) coincide with the Zariski-closure of f(Γ)? The answer to (3) is always positive, while (2) and (1) have positive

answers if G is simply-connected and G = G′. Under certain additional assumptions such results can be extended to the real category. Of course most of these results are well-known.

2. Basics

We will now introduce the basic notions.

Definition. A connected complex manifold X is called parallelizable if the (holomorphic) tangent bundle is holomorphically trivial.

It is called group-theoretically parallelizable if X is biholomorphic to a quotient G/Γ where G denotes a connected complex Lie group and Γ a discrete subgroup.

Using fundamental vector fields induced by the G-left action on G/Γ, it is evident that a group-theoretically parallelizable complex manifold is parallelizable. Conversely, a result of Wang [20] states that a compact complex manifold being parallelizable implies that it is group-theoretically parallelizable.

If X is a compact complex manifold on which a solvable (resp. nilpotent) complex Lie group acts transitively, then X is called a (compact) complex solvmanifold (resp. nilmanifold). Every compact complex solvmanifold is parallelizable [1].

Let G be a real or complex Lie group (or more generally a locally compact topological group), Γ a (discrete) subgroup. Then Γ is called cocompact or uniform if G/Γ is compact and of finite covolume if there is a finite G-invariant Borel measure on G/Γ. Discrete subgroups of finite covolume are called lattices. In general we will consider only discrete cocompact subgroups, thus avoiding non-cocompact lattices, because the assumption of finite covolume seems at this stage to be of very limited use in complex geometry.

3. Examples of parallelizable manifolds

The simplest examples are naturally compact complex tori, i.e. quotients ℂ^{N}/Π of complex vector spaces by lattices. These are the only Kähler compact complex parallelizable manifolds ([20]).

Another well-known example is the Iwasawa-manifold, which is obtained as X = G_{ℂ}/G_{ℤ⊕iℤ} with

G_{A} = \{( \begin{matrix}1 & x & z \\ & 1 & y \\ & & 1 \end{matrix} ) : x, y, z ∈ A\}

Note that ℤ ⊕ iℤ is just the ring of algebraic integers in the number field k = ℚ[i]. This suggests the following generalization:

Let k be a imaginary quadratic extension field of ℚ, i.e. k = ℚ[\sqrt{q}] for some negative rational number q ∈ ℚ^{−}. Fix an embedding σ : k ↪ ℂ and let A = σ(O_{k}), where O_{k} denotes the ring of algebraic integers in k. Then G_{ℂ}/G_{A} is a compact complex parallelizable manifold.

These methods are generalized in Chapter D. For instance, we show that there exist compact complex nilmanifolds (other than tori) of algebraic dimension zero.

For nilpotent Lie groups there exists a precise criterion which determines whether there exist discrete cocompact subgroups.

Theorem (see [18],[12]). Let G be a simply-connected complex nilpotent Lie group with Lie algebra g.

Then there exists a discrete cocompact subgroup Γ ⊂ G if and only if there exists a ℚ-Lie algebra h such that g ⊕ \overline{g} = h ⊗_{ℚ} ℂ. In other words: if and only if there exists an ℝ-Lie algebra isomorphism between g and h ⊗_{ℚ} ℝ for some ℚ-Lie algebra h.

In a certain sense a ”generic” nilpotent Lie group does not admit a discrete cocompact subgroup. There are only countably many non-isomorphic ℚ-Lie algebras, but there exist uncountably many non-isomorphic real nilpotent Lie algebras admitting a complex structure.

To verify this, let V , W be complex vector spaces of dimension n, m respectively. Let Hom^{∗}(∧^{2}V, W) denote the set of all alternating bilinear maps ϕ : V × V → W for which ϕ(V × V) generates W as a vector space. Then the double coset space

S_{n,m} = GL_{ℝ}(V)\ Hom_{ℂ}^{∗}(∧^{2}V, W)/GL_{ℝ}(W)

enumerates all equivalence classes of real nilpotent Lie algebras n which admit a complex structure and fulfill dim_{ℝ}(n/n′) = 2n, dim_{ℝ}(n′) = 2m and [n, n′] = {0}. Now dim_{ℝ} Hom_{ℂ}^{∗}(∧^{2}V, W) = mn(n−1) while dim_{ℝ} GL_{ℝ}(V)×GL_{ℝ}(W) = 4n^{2}+4m^{2} . Hence for mn(n−1) > 4n^{2}+4m^{2} the set S_{n,m} can not be countable.

For non-nilpotent Lie groups there is no such easy criterion. However, at least one has the following necessary criterion (see e.g. [18]).

Proposition. Let G be a connected (real or complex) Lie group for which there exists a discrete cocompact subgroup.

Then G is unimodular, i.e. Ad(G) ⊂ SL(g), where g denotes the Lie algebra of G.^{1}

Note that every connected nilpotent Lie group is unimodular. Hence it is clear that unimodularity is only necessary but not sufficient for the existence of a discrete cocompact subgroup.

We now present some examples of solvmanifolds, i.e. quotients of solvable complex Lie groups by discrete cocompact subgroups.

Otte-Potters [17]. Let V be a complex vector space, T a maximal torus in SL(V), G = T × V a semi-direct product given by the natural action of T on V . Then G admits a discrete cocompact subgroup Γ. This subgroup is constructed using a totally real number field k with [k : ℚ] = dim_{ℂ}(V). Now V is identified with k ⊗_{ℚ} ℂ and Γ is constructed as a subgroup of finite index of O_{k}^{∗} × (O_{k} ⊕ iO_{k}) embedded into SL(V)× V .

1 In general a locally compact topological group G is called unimodular if there is a left-invariant Haar-measure which is also right-invariant. For a (not necessarily connected) real or complex Lie group G this is equivalent to the assumption that | det Ad(g)| = 1 for all g ∈ G, where Ad : G → GL(Lie(G)) denotes the adjoint representation.

Nakamura [16]. Let A ∈ SL_{2}(ℤ) with T race(A) ∉ {−2, 2}. Then A has a unique eigenvalue α with |α| > 1. Now let E be an elliptic curve (i.e. a onedimensional torus), β ∈ ℂ with e^{β} = α and X the quotient of X_{1} = ℂ^{∗}×E×E by the ℤ-action on X_{1} generated by

(x, s, t) ↦ (βx, A · \binom{s}{t}).

Then X is a compact complex parallelizable manifold.

Semisimple complex Lie groups always admit discrete cocompact subgroups.

Theorem (Borel). Let G be a semisimple complex Lie group. Then there exists a discrete cocompact subgroup.

The construction uses arithmetic subgroups. It yields lattices, and for lattices there is a precise criterion for cocompactness. Here is a precise statement:

Theorem (see e.g. [18]). Let G be a semisimple algebraic group defined over ℚ and f : G → GL_{n} a faithful representation defined over ℚ. Let Γ = f^{−1}(GL_{n}(ℤ)). Then Γ is a discrete subgroup such that G(ℝ)/Γ has finite volume with respect to any G(ℝ)-invariant Borel measure on G(ℝ)/Γ.

The quotient G/Γ is compact if and only if Γ does not contain unipotent elements.

For our purposes, i.e. in the discussing of complex geometry on G/Γ, the finite volume condition is not very useful. In contrast, compactness of G/Γ has strong complex-analytic implications. Hence we usually discuss discrete cocompact subgroups and will not investigate discrete subgroups of finite covolume.

Borel’s proof (as given in [3]) for the existence of discrete cocompact subgroups in every semisimple (real) Lie group may be sketched as follows. Given a real simple Lie group G one can show that there exists a compact simple real Lie group G_{u} and an algebraic group H defined over ℚ such that H is simple over ℚ but isomorphic to G × G_{u} over ℝ. A faithful representation of H over ℚ yields a discrete arithmetic subgroup Γ ⊂ H(ℚ) ⊂ G × G_{u}. ℚ-simplicity of H implies that Γ∩G is finite, hence π_{2} : Γ → G_{u} is almost injective. Therefore Γ has no unipotent elements. Hence (G × G_{u})/Γ is compact. Now the projection map π_{1} : G × G_{u} → G is proper. Thus π_{1}(Γ) is a discrete cocompact subgroup in G. (For details see [3],[18]).

SL_{2}(ℂ) is the only simply-connected simple complex Lie group which admits non-arithmetic discrete cocompact subgroups.

Certain discrete cocompact subgroups in SL_{2}(ℂ) arise from differential geometry. There is a well-known link between real hyperbolic manifolds (hyperbolic in the sense of Riemannian geometry, i.e. constant negative sectional curvature) and complex-parallelizable manifolds of the form X ≃ SL_{2}(ℂ)/Γ.

Theorem. Let M be an oriented connected compact Riemannian manifold, dim_{ℝ}(M) = 3, with constant negative sectional curvature.

Then π_{1}(M) may be embedded into P SL_{2}(ℂ) = SL_{2}(ℂ)/{+/ − I} as discrete cocompact torsion-free subgroup. The preimage τ^{−1}(Γ) in SL_{2}(ℂ) (where τ : SL_{2}(ℂ) → P SL_{2}(ℂ) denotes the natural projection) is again a discrete cocompact group.

Remark. Evidently the preimage τ^{−1}(Γ) is not torsion-free, because ker τ = {+/ − I}. However, every discrete cocompact subgroup in a linear Lie group contains a torsion-free subgroup of finite index In this way the above procedure yields discrete cocompact torsion-free subgroups in SL_{2}(ℂ).

Proof. The universal covering of M is isometrically diffeomorphic to P SL_{2}(ℂ)/K where K is a maximal compact subgroup of P SL_{2}(ℂ). Moreover P SL_{2}(ℂ) is the full group of orientation preserving isometries of M̃. Thus the action of π_{1}(M) on M̃ by deck transformations induces an embedding π_{1}(M) ↪ P SL_{2}(ℂ). The image of this embedding is obviously discrete. The cocompactness follows form the compactness of M and K, because π_{1}(M)\P SL_{2}(ℂ)/K ≃ M. It is torsion-free, because no finite group can act freely on M̃ ≃ ℝ^{3} .

Such considerations produce the following result of Jørgensen, which is a key element in constructing a number of examples involving SL_{2}(ℂ). Similar examples were also constructed by Millson [15].

Proposition (see [10]). There exists a discrete cocompact subgroup Γ ⊂ SL_{2}(ℂ) with a surjective group homomorphism ρ : Γ → ℤ. ^{∗}

Next we discuss Lie groups which are neither solvable nor semisimple. In addition to unimodularity here there exists another necessary condition due to Otte and Potters.

Proposition (Otte-Potters [17]). Let G be a simply-connected complex Lie group, R its radical, and S = G/R. Assume that G is not isomorphic to a direct product S ×R. Furthermore assume that S is simple and not isomorphic to some SL_{n}(ℂ). ^{∗}

∗ Jørgensen has proved more, namely that such a ρ can be choosen in such a way that ker ρ is finitely presentable.

Then G does not admit a discrete cocompact subgroup.

Special arithmetic subgroups in semisimple Lie groups may be used to construct discrete cocompact subgroups in certain ”mixed” Lie groups in the following way.

Proposition 1. Assume that S is a simple complex Lie group, k/ℚ an imaginary quadratic field extension, and s_{k} a k -Lie algebra such that s_{k} ⊗_{k} ℂ is isomorphic to the Lie algebra s of S. Assume in addition that there exists no non-zero element v ∈ s_{k} such that ad(v) is nilpotent. Let V denote the underlying additive group of the Lie algebra s and G = S ×_{ρ} V with ρ induced by the adjoint action on s. Let Λ denote a lattice in V contained in V (k) ≃ s_{k}. Finally define Γ as semidirect product Γ_{0} × Λ with

Γ_{0} = {s ∈ S : Ad(s)(Λ) = Λ}.

Then Γ is a discrete cocompact subgroup in G.

Proof. The absence of nilpotent endomorphisms in ad(s_{k}) implies that S(k) contains no unipotent elements. (This follows from the fact that a unipotent element would be of infinite order, hence the Zariski-closure of the subgroup generated by this element would be a positive-dimensional unipotent k -subgroup.) Observe that k ⊗_{ℚ} ℝ ≃ ℂ. Hence R_{k/ℚ}S is ℝ-isomorphic to R_{ℂ/ℝ}S where R_{K/k} denotes the ”restriction of scalars functor” for a finite separable field extension K/k (See [6]). Thus S(ℂ) ≃ H(ℝ) and S(k) ≃ H(ℚ) for H = R_{k/ℚ}S. Hence we obtain an arithmetic lattice Γ_{0} in H(ℝ) ≃ S(ℂ) which is cocompact, because S(k) contains no unipotent elements. It follows that Γ is likewise cocompact in G.

Naturally this raises the question: Given a simple complex Lie algebra s, does there exists an imaginary quadratic field extension k/ℚ and a k -Lie algebra s_{k} such that s ≃ s_{k} ⊗_{k} ℂ?

The answer is Yes if and only if s is of type A_{n}, i.e. the Lie algebra associated to one of the Lie groups SL_{n}(ℂ).

We will now describe an elementary way to confirm this for SL_{2}(ℂ) (∼ A_{1} ), thereby proving that G = SL_{2}(ℂ) ×_{ρ} ℂ^{3} admits a discrete cocompact subgroup for ρ irreducible.

∗ Otte and Potters assumed in addition that S is not isomorphic to the exceptional simple group of type E_{8} . A later result of Cernousov ([7]) allows to drop this additional assumption.

For a three-dimensional simple Lie algebra over a field k of characteristic zero there is an easy criterion to determine whether there exists an non-zero element v with ad(v) nilpotent: ad(v) is nilpotent if and only if Kill(v, v) = 0.

For every α, β ∈ k^{∗} a simple three-dimensional k -Lie algebra with vector space basis A, B, C is given by

[A, B] = C, [B, C] = αA, [C, A] = βB.

The Killing form is given by

Kill(xA + yB + zC, xA + yB + zC) = −2(βx^{2} + αy^{2} + αβz^{2}).

Hence what we need is an imaginary quadratic field extension k/ℚ and numbers α, β ∈ k such that

αx^{2} + βy^{2} + αβz^{2} = 0  (∗)

has no non-trivial solution.

For instance, let k = ℚ[i], α = i, β = 1 + 2i. If (∗) has a non-trivial solution, then it has a non-trivial solution in O_{k} = ℤ[i] with x, y, z coprime. But calculations modulo 4 show that (∗) implies that all x^{2}, y^{2}, z^{2} equal 2i or 0 modulo 4. This implies that x, y, z are all divisible by 1 + i, contrary to x, y, z coprime. Hence (∗) has no non-trivial solution.

Sometimes it is useful to have examples with torsion-free fundamental group. In this regard a theorem of Selberg [19] implies the following:

Let X be a compact complex parallelizable manifold. Then there exists a finite (unramified) covering τ : X_{1} → X such that π_{1}(X_{1}) contains no element of finite order.

4. On the number of parallelizable manifolds

As mentioned above, for every semisimple complex Lie group there exists a discrete cocompact subgroup. Here we want to show that up to conjugacy there are only countably many such subgroups. This is equivalent to the assertion below.

Proposition 2. Up to biholomorphic equivalence there exist only countably many compact complex parallelisable manifolds X = G/Γ with G semisimple.

Proof. We have to consider compact quotients of semisimple Lie groups S by discrete subgroups Γ. As usual, let us call S/Γ irreducible, if there does not exist any almost direct product decomposition S = S_{1} · S_{2} with S_{1} ∩ S_{2} and Γ/(S_{1} ∩ Γ) · (S_{2} ∩ Γ) finite. Clearly it suffices to consider irreducible S/Γ. Recall that by standard theory of linear algebraic groups there exists only countably many non-isomorphic complex semisimple Lie groups. Thus it is sufficient to consider a fixed semisimple complex Lie group S which we may take to be simply-connected. As a fundamental group of a compact manifold Γ ≃ π_{1}(S/Γ) is finitely presentable [18,Th.6.16]. Up to isomorphism as abstract groups there exist only countably many finitely presentable groups. Thus we may restrict our attention to irreducible lattices in a fixed simplyconnected complex semisimple Lie group S which are isomorphic (as abstract groups) to a fixed finitely presentable group Γ_{0} . Any such lattice may be regarded as image of an injective group homomorphism ρ : Γ_{0} → S. Thus the set of all irreducible lattices corresponds to a subset R of Hom(Γ_{0}, S). As outlined in [21], the set Hom(Γ_{0}, S) may be endowed with the structure of an affine variety. There is a natural S -action by conjugation on Hom(Γ_{0}, S). Now local rigidity results ([18,Th. 7.66 & 6.7][21]) imply that R is a union of open S -orbits in the variety Hom(Γ_{0}, S). Therefore R is just a union of finitely many S -orbits. However conjugate homomorphisms Γ_{0} → S define isomorphic quotients. Thus the proof of countability is completed.

5. Algebraicity results

We will need an algebraicity result for Lie groups.

Proposition 3. Let G be a simply-connected complex Lie group with G = G′. Then G carries the structure of a linear-algebraic group. Moreover, for every complex linear-algebraic group H every homomorphism of complex Lie groups f : G → H is already a morphism of algebraic groups.

Proof. Since G is simply-connected, it is linear and algebraicity follows from G = G′. Let f be a homomorphism of complex Lie groups to some complex linear-algebraic group H. Let I = f(G). Then I = I′ and consequently I is algebraic and moreover the nilradical of I is unipotent. Now every homomorphism of complex Lie groups between unipotent groups is algebraic. Furthermore every homomorphism of complex Lie groups between semisimple groups is algebraic. Using Levi-decomposition this implies that f is algebraic.

Recall that every continuous group homomorphism between Lie groups is real-analytic. Using this fact we can improve the algebraicity result via restriction of scalars.

Proposition 4. Let G, H be complex linear-algebraic groups and assume G = G′.

12 I.5

Then every continuous group homomorphism ρ : G → H is already realalgebraic.

6. Density results

One of the main facts used in the study of compact homogeneous spaces is the Universality of the Tits Fibration.

For our purposes here we state this as follows:

Let X = G/Γ be a connected compact parallelizable complex manifold. Then there is no non-trivial holomorphic G-equivariant map, φ : X → ℙ_{N}, into a projective space. (see [T], [HO])

One of its consequences is the following density result for holomorphic representations.

Proposition (Iwamoto [9]). Let G be a connected complex Lie group and Γ a discrete cocompact subgroup. Let ρ : G → GL_{n}(ℂ) be a homomorphism of complex Lie groups.

Then ρ(G) is contained in the Zariski-closure of ρ(Γ) in GL_{n}(ℂ).

Proof. Let \overline{ρ(G)}, \overline{ρ(Γ)} denote the respective closures of ρ(G) and ρ(Γ) in the linear-algebraic group GL_{n}(ℂ) with respect to the (algebraic) Zariski-topology. Then \overline{ρ(G)}/\overline{ρ(Γ)} is a quotient of linear algebraic groups and therefore equivariantly embeddable in a projective space \overline{ρ(G)}/\overline{ρ(Γ)} ↪^{τ} ℙ_{N} . The induced G-equivariant map G/Γ → \overline{ρ(G)}/\overline{ρ(Γ)} →^{τ} ℙ_{N} has to be constant. It follows that ρ(G) ⊂ \overline{ρ(Γ)}.

This density result is particularly useful, if we can assume that the Lie group G is linear. However, this is not a serious restriction. In fact: Given an action of a complex Lie group G on a complex space X, this action can always be lifted to an action of the universal covering group G̃. The universal covering group is simply-connected and every simply-connected complex Lie group admits an injective homomorphism of complex Lie groups into a general linear group GL_{m}(ℂ).

We would like to remark that in fact there exists a stronger density result due to Borel which in our situation implies the following.

Theorem (Borel [2]). Let G be a simply-connected semisimple complex Lie group, Γ a lattice, ρ : G → GL_{n}(ℂ) a continuous (not necessarily holomorphic) group homomorphism. Then ρ(Γ) and ρ(G) have the same Zariski-closure in GL_{n}(ℂ).

This can be generalized to the following result.

Proposition 5. Let G be a simply-connected complex Lie group with G = G′, Γ a lattice, ρ : G → GL_{n}(ℂ) a continuous (not necessarily holomorphic) group homomorphism.

Then ρ(Γ) and ρ(G) have the same Zariski-closure in GL_{n}(ℂ).

Remark. For G′ ≠ G there often exists a surjective continuous group homomorphism ρ : G → S^{1} = {z ∈ ℂ^{∗} : |z| = 1} with Γ ⊂ ker ρ. Thus the assumption G = G′ is crucial.

Proof. Let H denote the Zariski-closure of ρ(G) in GL_{n}(ℂ). Then H = H′. It follows that the radical R_{H} is unipotent. Now Γ∩R_{G} is cocompact in R_{G} (see A.§3), R_{G} is nilpotent (because G = G′) and ρ|_{R_{G}} is a (real) unipotent representation. By a result of Malcev ([12]) it follows that the Zariski-closure of ρ(Γ∩R_{G}) coincides with the Zariski-closure of ρ(R_{G}). Consider the induced group homomorphism G/R_{G} → H/\overline{ρ(R_{G})} = I. Now I is a linear-algebraic group and the image of G/R is Zariskidense in I. Thereby semisimplicity of G/R implies semisimplicity of I. This is equivalent to the property that ρ(R_{G}) is Zariski-dense in R_{H}. By this argumentation the problem is reduced to the semisimple case and we can invoke the preceding theorem in order to conclude that ρ(G) and ρ(Γ) have the same Zariski-closure.

We present two applications of the density result.

Corollary. Let G be a connected complex Lie group, Γ a discrete subgroup, and H a connected complex subgroup with H/H ∩ Γ compact. Then C_{G}(H) = C_{G}(H ∩ Γ).

Proof. We may assume that G ⊂ GL_{m}(ℂ). Since conjugation is an algebraic operation on \overline{G}, it follows that C_{G}(A) = C_{G}(\overline{A}) for every subgroup A ⊂ G, where \overline{A} denotes the closure in GL_{m}(ℂ) with respect to the Zariski-topology. Hence C_{G}(H) = C_{G}(\overline{H}) = C_{G}(\overline{H ∩ Γ}) = C_{G}(Γ).

Corollary. Let G be a simply-connected complex Lie group, Γ a discrete cocompact subgroup.

Then G is abelian, nilpotent or solvable if and only if Γ is abelian, nilpotent resp. solvable.

Proof. This follows from the density result, because these properties may be defined through the vanishing of certain relations involving commutators.

14 I.6 By passing to the universal covering one can deduce: Let G be a connected complex Lie group and Γ a discrete cocompact

subgroup. Then G is solvable or nilpotent if and only if Γ is solvable resp. nilpotent.

This follows, because the classes of solvable resp. nilpotent groups are stable under central extensions. (The universal covering of a connected Lie group is a central extension.)

However, commutativity is not preserved under central extensions.

Example. Let G̃ the three-dimensional Heisenberg group, i.e. G̃ ≃ ℂ^{3} as manifold with group multiplication given by (x, y, z) · (x′, y′, z′) = (x + x′, y + y′, z + z′ + xy′). Let C = {(0, 0, z) : z ∈ ℤ + iℤ} and Γ̃ = {(x, y, z) : x, y, z ∈ ℤ + iℤ}. Then Γ = Γ̃/C is a discrete cocompact subgroup of G = G̃/C, but Γ is commutative while G is not.

7. Superrigidity

We will need the following special case of Margulis’ superrigidity theorem [13].

Theorem (Superrigidity). Let S be a simply-connected semisimple complex Lie group and Γ a discrete cocompact subgroup. Assume that no factor of S is isomorphic to SL_{2}(ℂ).

Then there exists a simply-connected compact real semisimple Lie group K, a subgroup Γ_{0} ⊂ Γ of finite index and a group homomorphism ξ : Γ_{0} → K such that the following condition is fulfilled:

Given a real linear-algebraic group H and a group homomorphism ρ : Γ → H there exists a subgroup Γ_{ρ} ⊂ Γ_{0} of finite index and a real-algebraic group homomorphism ρ̃: S×K → H such that ρ̃|_{Γ_{0}} coincides with ρ∘(i, ξ) : Γ_{0} → H.

We would like to emphasize that SL_{n}(ℂ) contains cocompact arithmetic subgroups for which K can be choosen to be trivial.

Theorem . Let k/ℚ be an imaginary quadratic field extension, D a central simple k -algebra of dimension n. Then SL_{1}(D) is an anisotropic k -group which is isomorphic to SL_{n} over ℂ. It yields a discrete cocompact subgroup Γ ⊂ SL_{n}(ℂ) such that for every real linear-algebraic group H and every group homomorphism ρ : Γ → H there exists a subgroup Γ_{ρ} ⊂ Γ_{0} of finite index and a real-algebraic group homomorphism ρ̃: SL_{n}(ℂ) → H such that ρ|_{Γ_{0}} coincides with ρ̃: Γ_{0} .

Such a discrete cocompact subgroup of SL_{n}(ℂ) will be called a complexarithmetic subgroup.

We will also make use of the following related result.

Theorem [13]. Let G be a simply-connected complex Lie group without SL_{2}(ℂ)- factors, Γ a discrete cocompact subgroup, Λ a normal subgroup of Γ.

Then either Λ or Γ/Λ is finite.

Another related result is the complete reducibility of finite-dimensional representations of lattices.

Proposition 6. Let S be a simply-connected complex Lie group and Γ a discrete cocompact subgroup (or more generally a lattice). Assume that no factor of S is isomorphic to SL_{2}(ℂ).

Then every finite-dimensional representation of Γ (over ℂ, ℝ or any field of characteristic zero) is completely reducible.

Proof. Let

0 → V_{1} → V_{2} → V_{3} → 0  (∗)

be a short exact sequence of Γ-modules (over a field of characteristic zero). We have to show that it splits. Taking the tensor product with V_{3}^{∗} yields another short exact sequence of G-modules:

0 → V_{3}^{∗} ⊗ V_{1} → V_{3}^{∗} ⊗ V_{2} → V_{3}^{∗} ⊗ V_{3} → 0.

By Margulis [13] H^{1}(Γ, ρ) = 0 for every finite-dimensional representation ρ. Thus the associated sequence of invariant submodules is exact. In particular (V_{3}^{∗} ⊗ V_{2})^{Γ} → (V_{3}^{∗} ⊗ V_{3})^{Γ} is surjective. Hence the identity homomorphism id_{V_{3}} ∈ Hom(V_{3}, V_{3}) lifts to an element in Hom(V_{3}, V_{2})^{G}, i.e. (∗) splits as a sequence of G-modules.

8. Group homomorphisms coinciding on subgroups of finite index

We need some results concerning group homomorphisms which coincide on subgroups of finite index. Just for convenience we state and prove them in an elementary way although they are essentially contained in [13].

Lemma 1. Let Γ, H be groups, ρ, ρ̃ : Γ → H group homomorphisms and assume that ρ and ρ̃ coincide on a subgroup of finite index Γ_{1} in Γ.

Then there exists a normal subgroup Γ_{0} ⊂ Γ of finite index and a map ζ_{0} : Γ/Γ_{0} → H such that

ρ(γ) = ζ(γ) · ρ̃(γ) ∀γ ∈ Γ,

where ζ : Γ → H is the natural lift of ζ_{0} .

Furthermore ζ(γ) and ρ(λ) commute for every γ ∈ Γ, λ ∈ Γ_{0} .

Proof. Consider the Γ-action on the finite set Γ/Γ_{1} . Let Γ_{0} denote the ineffectivity, i.e. the kernel of the natural group homomorphism Γ → P erm(Γ/Γ_{1}), where for a given set S the group of all permutations of S is denoted by P erm(S). Clearly Γ_{0} is a normal subgroup of finite index in Γ.

Define ζ(γ) := ρ(γ)ρ̃(γ)^{−1} . Now ρ(λ) = ρ̃(λ) for λ ∈ Γ_{0} implies ζ(γλ) = ζ(γ) for all γ ∈ Γ, λ ∈ Γ_{0} . Since Γ_{0} is normal in Γ, we obtain furthermore ζ(λγ) = ζ(γ) for all γ ∈ Γ, λ ∈ Γ_{0} .

Now observe that

ρ(λγ) = ζ(λγ)ρ̃(λγ) = ζ(γ)ρ̃(λ)ρ̃(γ)

and

ρ(λγ) = ρ(λ)ρ(γ) = ζ(λ)ρ̃(λ)ζ(γ)ρ̃(γ) = ρ̃(λ)ζ(γ)ρ̃(γ)

for all γ ∈ Γ, λ ∈ Γ_{0} . It follows that ζ(γ) and ρ̃(λ) = ρ(λ) commute for every γ ∈ Γ, λ ∈ Γ_{0} .

Corollary. Let G be a connected complex Lie group with G = G′, Γ a discrete cocompact subgroup, Γ_{0} ⊂ Γ a subgroup of finite index, H a complex linearalgebraic group and ρ : Γ → H a group homomorphism.

Assume that ρ|_{Γ_{0}} extends to a continuous group homomorphism ρ_{0} : G → H (not necessarily holomorphic).

Then there exists a group homomorphism ζ : Γ → H with finite image such that ρ(γ) = ζ(γ)ρ_{0}(γ) for all γ ∈ Γ. Furthermore ρ_{0}(g) and ζ(γ) commute for every γ ∈ Γ and g ∈ G.

Proof. It is clear that there exists a map ζ : Γ → H such that ρ(γ) = ζ(γ)ρ_{0}(γ) for all γ ∈ Γ. Let F = ζ(Γ). Then F is centralized by ρ_{0}(Γ_{0}). Now the centralizer of a finite set in an algebraic group is a closed algebraic group. Therefore the usual density results imply that F is centralized by ρ_{0}(G). Now observe that

ζ(gh)ρ_{0}(g)ρ_{0}(h) = ρ(gh) = ρ(g)ρ(h) = ζ(g)ρ_{0}(g)ζ(h)ρ_{0}(h)

for all g, h ∈ Γ. Since F is centralized by ρ_{0}(G), it follows that ζ(gh) = ζ(g)ζ(h) for g, h ∈ Γ, i.e. ζ is a group homomorphism.

Similarily one obtains the following consequence for the holomorphic case.

Corollary. Let G be a connected complex Lie group, Γ a discrete cocompact subgroup, Γ_{0} ⊂ Γ a subgroup of finite index, H a complex linear-algebraic group and ρ : Γ → H a group homomorphism.

17 I.8

Assume that ρ|_{Γ_{0}} extends to a holomorphic group homomorphism ρ_{0} : G → H.

Then there exists a group homomorphism ζ : Γ → H with finite image such that ρ(γ) = ζ(γ)ρ_{0}(γ) for all γ ∈ Γ. Furthermore ρ_{0}(g) and ζ(γ) commute for every γ ∈ Γ and g ∈ G.

9. Linearity of simply-connected complex Lie groups At many points in our work the following fact is crucial. It is a consequence

of Th. XVIII.4.7 in [8].

Proposition. Let G be a simply-connected complex Lie group. Then there exists an injective homomorphism of complex Lie groups i : G → GL_{n}(ℂ) for some n ∈ ℕ.

For instance, we have the following implications.

Corollary. Let X be a compact complex parallelizable manifold and X̃ its universal covering.

Then X̃ is a Stein manifold.

Proof. Due to Wang’s theorem X is biholomorphic to a quotient of a simply-connected complex Lie group G by a discrete subgroup Γ. Hence X̃ is isomorphic (as complex manifold) to a simply-connected complex Lie group. Now the claim follows the proposition, because every complex Lie subgroup of GL_{n}(ℂ) is a Stein complex manifold [14].

Corollary. Let X be a compact complex parallelizable manifold.

Then the fundamental group π_{1}(X) admits a faithful linear representation. Moreover it is residually finite, i.e. for every element γ ∈ π_{1}(X) there exists a group homomorphism ρ from π_{1}(X) to a finite group such that ρ(γ) ≠ e.

Proof. By Wang’s theorem there is a simply-connected complex Lie group G and a discrete subgroup Γ such that X ≃ G/Γ. This implies π_{1}(X) ≃ Γ. Hence the existence of a faithful representation follows from the proposition. As a fundamental group of a compact manifold Γ is finitely generated. It is a theorem of Malcev that a finitely generated group with a faithful linear representation is necessarily residually finite [11].

References

1. Barth, W.; Otte, M.: Über fast-uniforme Untergruppen komplexer Liegruppen und auflösbare komplexe Mannigfaltigkeiten. Comm. Math. Helv. 44, 269–281 (1969)

2. Borel, A.: Density properties of certain subgroups of semisimple groups. Ann. of Math. 72, 179–188 (1960)

3. Borel, A.: Compact Clifford Klein forms of symmetric spaces. Topology 2, 111–122 (1963)

4. Borel, A.: Introduction aux groupes arithmetiques. Hermann Paris (1969)

5. Borel, A.: Linear algebraic groups. Second Enlarged Edition. Springer New York Berlin Heidelberg (1991)

6. Borel, A.; Tits, J.: Groupes Reductifs. IHES 27, 55–152 (1965)

7. Cernousov, V.I.: On the Hasse-principle for groups of type E_{8} . Dokl. Akad. Nauk. SSSR 306 , 1059–1063 (1989) Translation: Sov. Math. Dokl. 39, 592–596 (1989)

8. Hochschild, G.: The Structure of Lie groups. Holden Day Inc. San Francisco. (1965)

9. Iwamoto, T.: Density properties of complex Lie groups. Osaka J.Math. 23, 859–865 (1986)

10. Jørgensen, T.: Compact 3–manifolds of constant negative curvature fibering over the circle. Ann. Math. 106, 61–72 (1977)

11. Malcev, A.: On isomorphism matrix representations of infinite groups. Mat. Sb. 8, 405-422 (1940)

12. Malcev, A.: On a class of homogeneous spaces. Izvestiya Akad. Nauk SSSR Ser. Math. 13 (1949)/ AMS Transl. no. 39 (1951)

13. Margulis, G.: Discrete subgroups of semi-simple Lie groups. Erg. Math., Springer-Verlag (1990)

14. Matsushima, Y.; Morimoto, A.: Sur certaines espaces fibrés holomorphes sur une variété de Stein. Bull. Soc. Math. France 88, 137-155 (1960)

15. Millson, J.J.: On the first Betti number of a compact constant negatively curved manifold. Ann. of Math. 104, 235–247 (1976)

16. Nakamura, I.: Complex parallisable manifolds and their small deformations. J. Diff. Geo. 10, 85–112 (1975)

17. Otte, M.; Potters, J.: Beispiele homogener Mannigfaltigkeiten. Manu. math. 10, 117–127 (1973)

18. Raghunathan, M.S.: Discrete subgroups of Lie groups. Erg. Math. Grenzgeb. 68, Springer (1972)

19. Selberg, A.: On discontinuous groups in higher-dimensional symmetric spaces. in Contributions to function theory. Bombay 1960, p. 147–164

20. Wang, H.: Complex parallelisable manifolds. Proc. A.M.S. 5, 771–776 (1954)

21. Weil, A.: Remarks on the cohomology of groups. Ann. of Math. 80, 149– 157 (1964)

22. Zimmer, R.: Ergodic Theory and Semisimple Groups. Birkhäuser Boston Basel Stuttgart (1984)