FLAT VECTOR BUNDLES OVER PARALLELIZABLE MANIFOLDS

JÖRG WINKELMANN

Abstract. We study flat vector bundles over complex parallelizable manifolds.

1. Introduction

Let G be a complex Lie group and Γ a discrete subgroup which is large in a certain sense, e.g. of finite covolume. We are interested in studying holomorphic vector bundles over the quotient manifold X = G/Γ. In general (i.e. if G is noncommutative) these manifolds are not algebraic and not even Kähler.

Thus in order to understand these manifolds it is necessary to employ methods other than those usually applied in algebraic and Kähler geometry. These methods are in particular group-theoretical methods like methode from Lie theory, representation theory, and the theory of algebraic and arithmetic groups. In order to be able to employ these group-theoretical methods we restrict our attention to holomorphic vector bundles which carry a flat holomorphic connection. Any such bundle is induced by a representation of the fundamental group. Special emphasis is put on what we call essentially antiholomorphic representations. For irreducible lattices Γ in semisimple complex Lie groups G of rank at least two it is an easy consequence of the celebrated arithmeticity results of Margulis et al. [7] that every flat holomorphic vector bundle over G/Γ is isomorphic as a holomorphic vector bundle to a flat bundle induced by an essentially antiholomorphic representation. For lattices in arbitrary complex Lie groups there are always non-trivial essentially antiholomorphic representations. Moreover flat vector bundles induced by antiholomorphic representations arise naturally as higher direct images sheaves of the structure sheaf for fibrations of complex parallelizable manifolds. Therefore these results are useful to calculate Dolbeault cohomology groups via Leray spectral sequences associated to certain fibrations. In a separate paper [14] we will exploit this to deduce results on cohomology groups, deformations and the Albanese variety of complex parallellizable manifolds.

We give a precise criterion determining when a flat vector bundle induced by an essentially antiholomorphic representation is trivial (thm. 6.6.) and moreover classify these bundles completely up to isomorphism as holomorphic vector bundles in the case G = G′ (thm. 6.3).

We then proceed to study sections and subbundles of flat vector bundles. Essentially we prove that in general sections exist only to the extent to which the bundle is trivial and that generically every vector subbundle of a flat vector bundle on a complex parallelizable manifold is again flat.

Part of the work was done during the special year on Several Complex Variables at the MSRI.

Research at MSRI is supported in part by NSF grant DMS-9022140. The author also wishes to thank the Harvard University for their invitation.

As a by-product we can show that every compact complex manifold of dimension n > 0 admits a non-trivial holomorphic vector bundle of rank at most n (improving our result in [11]).

The assumption that Γ is of finite covolume is often used in a rather indirect way. For instance we prove that, given a lattice Γ in a complex Lie group G, the group G is always generated by those connected commutative complex Lie subgroups A for which A/(A∩Γ) is compact (see prop. 4.1). We will also use the fact that G/Γ carries no non-constant plurisubharmonic function.

A part of the results for the special case where G/Γ is compact is already contained in the Habilitationsschrift of the author [12].

2. Generalities

Let G be a connected complex Lie group and Γ a discrete subgroup. Then X = G/Γ is a complex parallelizable manifold, i.e. a complex manifold whose tangent bundle is holomorphically trivial. Conversely every compact complex parallelizable manifold with holomorphiclally trivial tangent bundle can be realized as a quotient of a complex Lie group by a discrete subgroup [10]. If we assume G to be simply-connected (this is often convenient), then Γ is isomorphic to the fundamental group of X. For any representation ρ of Γ in a complex Lie group H we obtain a flat H-prinicpal bundle by E = G × H/Γ with Γ acting on G by deck transformations and on H via ρ. Such a bundle admits a trivialization compatible with the flat connection if and only if the representation is trivial. However, we will be concerned whether there exists any trivialization of E as H-principal bundle (not necessarily compatible with the flat connection). Such a trivialization exists if and only if there is a Γ-equivariant holomorphic map ϕ from G to H (i.e. ϕ(gγ) = ϕ(g) · ρ(γ) for all g ∈ G, γ ∈ Γ). More general, two representations ρ, ρ̃: G → H induce bundles which are isomorphic as holomorphic H-principal bundles if and only if there exists a holomorphic map ϕ : G → H such that ϕ(gγ) = ρ(γ)^{−1}ϕ(g)ρ̃(γ) for all g ∈ G and γ ∈ Γ.

If Γ is a “large” discrete subgroup of G there are few holomorphic mappings from X = G/Γ. Especially, if Γ is a lattice in a complex Lie group G the quotient manifold X = G/Γ has the following properties:

1. (Iwamoto’s version of the Borel density theorem ([4])). Every linear representation ρ : G → GL(n, ℂ) the Zariski-closures of ρ(Γ) and ρ(G) coincide. This implies that every G-equivariant holomorphic map from X to a projective space is constant.

2. Every holomorphic and moreover every plurisubharmonic function on X = G/Γ is constant (see e.g. claim 3.3 below).

3. A triviality criterion

Proposition 1. Let G be a connected complex Lie group, H a complex Lie group and Γ ⊂ G a discrete subgroup. Let ρ : Γ → H be a group homomorphism. Assume that every holomorphic function on G/Γ is constant.

Then the H-principal bundle E → G/Γ induced by ρ is holomorphically trivial if and only if ρ extends to a holomorphic group homomorphism ρ̃: G → H.

Proof. Assume that there is a trivializing map ϕ : G → H, i.e. a holomorphic map ϕ : G → H such that ϕ(gγ) = ϕ(g)ρ(γ) for all g ∈ G and γ ∈ Γ. Upon replacing ϕ by ϕ̃(g) =^{def} ϕ(e)^{−1}ϕ(g), we may assume that ϕ(e) = e. Define α : G × G → H by

α(g_{1}, g_{2}) = ϕ(g_{1})ϕ(g_{2})ϕ(g_{1}g_{2})^{−1}.

Then α(g_{1}, e) = α(e, g_{2}) = e and α(g_{1}, g_{2}γ) = α(g_{1}, g_{2}) for all g_{1}, g_{2} ∈ G and γ ∈ G. We consider now the induced maps α_{g} : G/Γ → H given by α_{g} : xΓ ↦ α(g, x). Since α_{e} ≡ e, it is clear that all the maps α_{g} are homotopic to a constant map. Hence these maps may be lifted to the universal covering H̃ of H, i.e. there exist holomorphic maps α̃_{g} : G/Γ → H̃ such that α_{g} = π ∘α̃_{g} where π : H̃ → H is the universal covering map. But H̃ is a simply-connected complex Lie group and therefore Stein. Hence the maps α̃_{g} are constant. It follows that the maps α_{g} are likewise constant. Thus α(g, x) = α(g, e) = e for all g, x ∈ G.

4. Bounded representations

Here we study bounded representations, i.e. representations with relatively compact image. First we want to mention some ways in which bounded representations arise.

1. If Γ is a lattice in a simply-connected complex Lie group G, then Γ is finitely generated and G is linear. In this case a theorem of Malcev [6] implies that Γ is residually finite , i.e. for every element γ ∈ Γ there exists a finite group F and a group homomorphism ρ : Γ → F such that ρ(γ) ≠ e. Since every finite group embeds in some linear group, this yields many representations of Γ with finite image.

2. If H_{1}(G/Γ) ≃ Γ/Γ′ is non-trivial, Γ admits many group homomorphisms to S^{1} = {z ∈ ℂ^{∗} : |z| = 1}.

3. Let K be a number field, O_{K} its ring of algebraic integers, S a semisimple Kgroup and S resp. T the set of all archimedean valuations v such that G is K_{v}-isotropic resp. K_{v}-anisotropic. Assume that K_{v} ≃ ℂ for all v ∈ S and that T is not empty. Then G = Π_{v∈S}S(K_{v}) is a complex Lie group, U = Π_{v∈T} S(K_{v}) is a compact real Lie group and Γ = S(O_{K}) is a lattice in G which admits an injective group homomorphism to U. Now every representation of U induces a representation of Γ with relatively-compact image.

Theorem 1. Let G be a connected complex Lie group, R its radical, Γ a discrete subgroup of G, H a Stein Lie group and ρ_{i} : Γ → H group homomorphisms with relatively compact image.

Assume that Ad(Γ) and Ad(G) have the same Zariski-closure in GL(Lie G) and that Γ ∩ R is cocompact in R.

Let E_{1} and E_{2} denote the H-principal bundles over G/Γ induced by ρ_{1} resp. ρ_{2}. Then E_{1} ∼ E_{2} if and only if ρ_{1} and ρ_{2} are conjugate by an element h ∈ H.

Remark 1. If Γ is a lattice in a complex Lie group G with radical R, then R/(R ∩ Γ) is compact ([8]) and the Zariski-closures of Ad(Γ) and Ad(G) in GL(Lie G) do coincide ([2]).

Proof. Recall that E_{1} ∼ E_{2} holds iff there exists a holomorphic map ϕ : G → H such that

ϕ(gγ) = ρ_{1}(γ)^{−1}ϕ(g)ρ_{2}(γ) (1)

for all g ∈ G, γ ∈ Γ.

Let K_{i} denote the closure of ρ_{i}(Γ) in H. The sets K_{i} are compact subgroups. Since H is Stein, it admits a strictly plurisubharmonic exhaustion function τ. We may assume that τ is invariant under the K_{1} × K_{2}-action on H given by (k_{1}, k_{2}) : h ↦ k_{1}^{−1}hk_{2} (because we can replace τ by the function obtained by averaging τ over the K_{1} × K_{2}-orbits). Then (1) implies τ(ϕ(gγ)) = τ(ϕ(g)). Thus we obtain a plurisubharmonic function on G/Γ.

Claim 1. Under the assumptions of the theorem every plurisubharmonic function on G/Γ is constant.

Proof. Since R/(R∩Γ) is compact, every plurisubharmonic function on G/Γ is a pullback from (G/R)/(RΓ/Γ). Now G/R is semisimple and Γ/(Γ ∩ R) is Zariski-dense in G/R. Hence the assertion follows from a theorem of Berteloot and Oeljeklaus [1].

Thus g ↦ τ(ϕ(g)) is constant. Since τ is strictly plurisubharmonic, this implies that ϕ is constant.

Corollary 1. Let G, Γ and H be as in the above theorem and let ρ : Γ → H be a group homomorphism with relatively compact image.

Then the induced H-principal bundle over G/Γ is holomorphically trivial if and only if ρ ≡ e.

5. Subgroups with a bounded orbit

Given a Lie group G and a lattice Γ in G we will show that there are many Lie subgroups H of G such that the H-orbit through eΓ is relatively compact in G/Γ.

Proposition 2. Let G be a connected complex Lie group, Γ a lattice and G_{0} the subgroup of G generated by all connected commutative complex Lie subgroups A of G with A/(A ∩ Γ) compact.

Then G_{0} = G.

Proof. If G is commutative, then G/Γ is a topological group with finite volume and therefore compact. In the general case we argue by induction over dim(G). Thus let G, Γ and G_{0} be as above and assume that the proposition is valid for all Lie groups of lower dimension. Let γ ∈ Γ\Z (where Z denotes the center of G) and let C^{0}(γ) denote the connected component of the centralizer of γ in G. Then dim C^{0}(γ) < dim G and C^{0}(γ) ∩ Γ is a lattice in C^{0}(γ) ([9],Lemma 1.14). Hence C^{0}(γ) ⊂ G_{0} by the induction hypothesis. This is equivalent to the assertion that V^{Ad(γ)} ⊂ Lie(G_{0}) where V = Lie G. Let H ⊂ GL(V) denote the Zariski-closure of Ad(G). Then H is also the Zariski-closure of Ad(Γ). Let k denote the generic dimension of V^{h} = {v : hv = v} for h ∈ H and Ω the set of all h ∈ H for which dim V^{h} = k. Then Ω is a Zariski-open subset of H. Now V^{h} ⊂ Lie(G_{0}) for all h ∈ Γ ∩ Ω and therefore for all h ∈ Ω. It follows that v ∈ Lie G_{0} for all v ∈ Ad^{−1}(Ω). Since Ad^{−1}(Ω) is a non-empty open subset of Lie G, this implies that G = G_{0}.

We will now deduce another variant of this theme, this time strengthening the assumption on the subgroups (requiring unipotency) while relaxing the assumption on the orbits (only relatively-compact instead of compact).

Proposition 3. Let G be a simply-connected complex Lie group and Γ a lattice.

Let G_{1} denote the subgroup of G generated by all unipotent subgroups U ⊂ G′ for which there exists a compact subset F ⊂ G such that U ⊂ F Γ′.

Then G_{1} = G′.

Remark 2. For a simply-connected complex Lie group G the commutator group G′ carries a unique structure of a complex linear-algebraic group. Hence it makes sense to speak about unipotent subgroups of G′.

Proof. First we note that we may assume R ∩ G′ = {e}. because R ∩ G′ is a normal unipotent subgroup with compact orbits. This assumption implies that R is central and G = S × R with S semisimple. Let π : G → G/R ≃ S be the natural projection.

Claim 2. Let H be a one-dimensional unipotent subgroup of S such that H ∩π(Γ) ≠ {e}.

Then H ⊂ G_{1}.

Proof. Let u ∈ S, r ∈ R such that ur ∈ Γ and π(u) ∈ H \ {e}. Let C denote the connected component of the centralizer of u in S. Clearly, H is contained in the center of C. Now C · R is the centralizer of ur in G. Hence the CR-orbit through eΓ is closed. Let N denote the nilradical of C. Then NR is the nilradical of CR. It follows that the NR-orbit through eΓ is compact. Now H is central in C, hence H ⊂ N and consequently H ⊂ G_{1}.

Next let V denote the subgroup of G/R generated by all one-dimensional unipotent subgroups having non-trivial intersection with π(Γ). Then V is normalized by π(Γ). The Borel Density theorem thus implies that V is a normal subgroup of S. It follows that S = S_{0} × V for some semisimple complex Lie subgroup S_{0} ⊂ S.

Claim 3. The S_{0}-orbit through eΓ in G/Γ is relatively compact.

Proof. By the criterion of Kazdan-Margulis (see [9], Cor. 11.12) we have to show that given a sequence s_{n} ∈ S_{0} there is no sequence γ_{n} ∈ Γ \ {e} such that lim s_{n}γ_{n}s_{n}^{−1} = e. Assume there are such sequences. Let u_{n} ∈ S and r_{n} ∈ R such that u_{n}r_{n} = γ_{n}. Then lim s_{n}u_{n}s_{n}^{−1} = e (because R is central). Since π(Γ) is a lattice in S, it follows that u_{n} is unipotent for n sufficiently large ([9], Cor. 11.18). But in this case u_{n} ∈ V by construction and this implies that s_{n}u_{n}s_{n}^{−1} = u_{n} for all n. Contradiction!

Finally note that S_{0} is a semisimple complex Lie group and therefore generated by its unipotent subgroups.

6. Antiholomorphic maps and actions of unipotent groups

Here we prove an auxiliary result needed later on.

Proposition 4. Let Z be a quasi-affine variety, H̄ a connected commutative linearalgebraic group (both defined over ℂ) and H a connected Zariski-dense complex Lie subgroup of H̄. Assume that there is a regular action μ : H̄ × Z → Z, a map α : H → Z with relatively compact image and an antiholomorphic map ϕ : H → Z such that

α(h) = μ(h)(ϕ(h)) (2)

for all h ∈ H. Let U denote the unipotent radical of H̄.

Then both α(H) and ϕ(H) are contained in a single μ(H̄)-orbit W, U acts trivially on this orbit and the maps α, ϕ are homomorphisms of real Lie groups from H to W with respect to the natural group structure on W with ϕ(e) as neutral element.

Proof. We start with a discussion of semi-invariant functions on Z.

Definition 1. Let Z be a variety, G an algebraic group acting on Z. A regular function f on Z is called semi-invariant if there exists a character χ of G such that f(gz) = χ(g)f(z) for all z ∈ Z and g ∈ G.

Now let f be a semi-invariant for the H̄-action on Z. Then

f(α(h)) = χ(h)f(ϕ(h)) (3)

for a character χ of H̄ and all h ∈ H. This implies

|f(α(h))|^{2} = |χ(h)f(ϕ(h))|^{2} = |χ(h)\overline{f(ϕ(h))}|^{2}

The left side is bounded while the right side is the absolute value of a holomorphic function. Hence there is a constant c such that |f ∘α| ≡ c and χ·\overline{f ∘ ϕ} ≡ c. It follows that f ∘ α and f ∘ ϕ vanish either everywhere or nowhere.

Lemma 1. Let W be a quasi-affine algebraic variety and G a connected solvable linear-algebraic group acting regularly on W. Assume that for every semi-invariant f the zero-set V (f) is either empty or the whole of W.

Then G acts transitively on W.

Proof. Assume the contrary. Then W must contain a proper invariant algebraic subvariety Y. Now the ideal I_{Y} is a non-trivial invariant subvectorspace of the space of regular functions ℂ[W]. Recall that every f ∈ ℂ[W] is contained in a finite-dimensional subvectorspace. Using this fact the theorem of Lie implies that I_{Y} contains a one-dimensional invariant subvectorspace S. Now any f ∈ S \ {0} is a semi-invariant on W vanishing on Y but not vanishing everywhere. Contradiction!

Applying this lemma to our situation we may conclude that both α(H) and ϕ(H) are contained in a single H̄-orbit W.

Since H̄ is commutative, the homogeneous H̄-space W has a canonical structure as a commutative group which is unique up to the choice of the neutral element.

Claim 4. The U-action on W is trivial.

Proof. If not, there is a non-trivial regular U-equivariant map τ : W → ℂ. Using the assumption that H is Zariski-dense in H̄, we may find a one-parameter subgroup γ(t) in H such that τ(μ(γ(t))(x)) = t + τ(x) for x ∈ W. But then t ↦ τ(α(γ(t))) is a bounded function on ℂ which may be represented as

τ(α(γ(t))) = τ(μ(γ(t))(ϕ(t))) = t + \underbrace{τ(ϕ(γ(t)))}_{antiholo.}. (4)

This is a contradiction, because the left side is bounded while the right side is a nonconstant harmonic function on ℂ. Thus the U-action must have been trivial.

Now W is a homogeneous space of the reductive commutative group H̄/U. By choosing a point in W as neutral element, W inherits a structure as reductive commutative algebraic group. Let us choose ϕ(e) = α(e) as neutral element. Then every character of W is an semi-invariant for the H̄-action and consequently our previous considerations imply that χ ∘ ϕ is a real Lie group homomorphism from H to ℂ^{∗} for every character χ of W. It follows that ϕ (and hence also α) are real Lie group homomorphisms.

7. Essentially antiholomorphic representations

Definition 2. Let G and H be complex Lie groups, and Γ ⊂ G a discrete subgroup. A group homomorphism ρ : Γ → H is called essentially antiholomorphic if there exists an antiholomorphic Lie group homomorphism ζ : G → H and a map ξ : Γ → H with relatively compact image ξ(Γ) ⊂ H such that

ρ(γ) = ζ(γ) · ξ(γ) (5)

for all γ ∈ Γ.

Proposition 5. Let G be a connected complex Lie group, Γ a lattice, H a Stein Lie group and ρ : Γ → H an essentially antiholomorphic representation with maps ζ, ξ given as above.

Then

1. ξ : Γ → H is a group homomorphism.

2. Both ξ and ζ are uniquely determined by ρ.

3. ξ(γ) and ζ(g) commute for all g ∈ G, γ ∈ Γ.

Proof. The equation (5) combined with ρ(γδ) = ρ(γ)ρ(δ) implies

ζ(δ)^{−1}ξ(γ)ζ(δ) = ξ(γδ)ξ(δ)^{−1} (6)

for all γ, δ ∈ Γ.

Let A be a connected complex Lie subgroup of G for which there exists a compact set F ⊂ G such that A ⊂ F Γ. For γ ∈ Γ we define an antiholomorphic map ϕ_{γ} : A → H by

ϕ_{γ}(a) = ζ(a)ξ(γ)ζ(a)^{−1}.

If \overline{ξ(Γ)} = K, then ϕ_{γ}(A) ⊂ ζ(F) · K · K^{−1} · ζ(F). Since the latter set is compact, it follows that ϕ_{γ} is constant. This implies that ζ(A) commutes with ξ(Γ). Since G is generated as a group by such subgroups A (prop. 2), this yields the third assertion. The first assertion is an immediate corollary. To check unicity, let ζ, ζ̃ be antiholomorphic representations and ξ, ξ̃ be bounded maps such that ζξ = ζ̃ξ̃. This implies ζ̃^{−1}ζ = ξ̃ξ^{−1}. This is now a bounded antiholomorphic map, hence constant, and in fact constant with value e. Therefore ζ̃ = ζ and ξ̃ = ξ.

Theorem 2. Let G be a connected complex Lie group, H a linear complex Lie group, Γ a lattice and ρ, ρ̃ : Γ → H essentially antiholomorphic representations. Assume that the induced flat bundles are isomorphic as holomorphic H-bundles.

Then there exists a constant c ∈ H such that ρ|_{Γ∩G′} = c (ρ̃|_{Γ∩G′}) c^{−1}.

Proof. There is no loss in generality in assuming that G is simply-connected. The assumption of the induced flat bundles being holomorphically equivalent is equivalent to the statement that there exists a holomorphic map ϕ : G → H such that

ϕ(gγ) = ξ(γ)^{−1}ζ(γ)^{−1}ϕ(g)ζ̃(γ)ξ̃(γ) (7)

for all g ∈ G and γ ∈ Γ where ρ = ζ·ξ resp. ρ̃ = ζ̃·ξ̃are the respective decompositions as products of an antiholomorphic and a bounded factor. We define a map α : G → H by

α(g) = ζ(g)ϕ(g)ζ̃(g)^{−1} (8)

Then

\begin{matrix}α(gγ) = ζ(g)ζ(γ)ξ(γ)^{−1}ζ(γ)^{−1}ϕ(g)ζ̃(γ)ξ̃(γ)ζ̃(γ)^{−1}ζ̃(g)^{−1} \\ = ξ(γ)^{−1}ζ(g)ϕ(g)ζ̃(g)^{−1}ξ̃(γ) = ξ(γ)^{−1}α(g)ξ̃(γ) \end{matrix} (9)

(Recall that ζ(G) and ξ(Γ) commute).

Claim 5. Let A be a connected commutative complex Lie subgroup of G such that there exists a compact subset F_{A} ⊂ G such that A ⊂ F_{A}Γ. Let c = ϕ(e) = α(e) ∈ H and W = μ(G)(c) where μ is the action given by

μ(g) : x ↦ ζ(g)^{−1}xζ̃(g) (10)

Let η denote the Γ-action on H given by

η(γ) : x ↦ ξ(γ)^{−1}xξ̃(γ) (11)

Then α(A) ⊂ W, ϕ(A) ⊂ W and η(A ∩ Γ) stabilizes W. If in addition A is contained in G′, then μ(A) acts trivially on W.

Proof. Fix an antiholomorphic embedding σ : H ↪ GL(n, ℂ) and let μ̂ denote the G-action on GL(n, ℂ) given by

μ̂(g) : x ↦ σ ∘ ζ(g^{−1}) · x · σ ∘ ζ̃(g).

Thus μ̂ embedds G holomorphically in GL(n, ℂ) × GL(n, ℂ) acting on GL(n, ℂ) by left and right multiplication. Now an application of prop. 4 (with Z = GL(n, ℂ)) and the Zariski-closure of μ̂(H) in GL(n, ℂ) × GL(n, ℂ) as Ā ) yields α(A) ⊂ W and ϕ(A) ⊂ W.

By definition of α the inclusion (A∩Γ) ⊂ W implies that η(γ)(c) ∈ W for γ ∈ A∩Γ. Since η(Γ) commutes with μ(G), it follows that η(A ∩ Γ) stabilizes W. Finally, since G was assumed to be simply-connected, this implies that G′ is unipotent. Hence the last assertion also follows from prop. 4.

Claim 6. The group μ(G′) acts trivially on W and in particular stabilizes the point ϕ(e).

This is immediate, because G′ is generated by unipotent subgroups with a bounded orbit (prop. 3).

Next we discuss ’translates’ of ϕ. For x ∈ G let ϕ_{x} denote the map defined by ϕ_{x}(g) = ϕ(xg). Then ϕ_{x} : G → H is a holomorphic map fulfilling (7). Thus we may apply the above considerations and conclude that μ(G′)(ϕ_{x}(e)) = ϕ_{x}(e). Since ϕ_{x}(e) = ϕ(x), it follows that μ(G′) stabilizes ϕ(G) pointwise. Hence

ϕ(gγ) = η(γ)(ϕ(g)) (12)

for g ∈ G, γ ∈ G′ ∩ Γ. By assumption ξ(Γ) and ξ̃(Γ) are contained in compact subgroups K resp. K̃ of H. Since H is Stein, it admits a strictly plurisubharmonic exhaustion function τ. By averaging with respect to Haar measure we may assume that τ is invariant with respect to the given K × K̃-action on H. Then we obtain

τ(ϕ(gγ)) = τ(ϕ(g)) (13)

for g ∈ G, γ ∈ G′ ∩ Γ. Now every plurisubharmonic function on G′/(G′ ∩ Γ) is constant (see [14]). Hence ϕ is constant along the G′-orbits on G. In particular ϕ|_{G′} is constant and this yields the assertion with c = ϕ(e).

Theorem 3. Let G be a connected complex Lie group, Γ ⊂ G a lattice, H a complex linear-algebraic group and ρ : Γ → H an essentially antiholomorphic group homomorphism.

Then the flat bundle over G/Γ defined by ρ is trivial if and only if the following conditions are fulfilled:

1. The Zariski-closure A of ρ(Γ) in H is isomorphic to (ℂ^{∗})^{g} for some g ≥ 0.

2. Let ρ = ζ · ξ denote the decxomposition of ρ into the antiholomorphic part ζ and the bounded part ξ. Let K denote the maximal compact subgroup of A (which is a real form of A) and let τ denote complex conjugation on A with respect to the real form K (i.e. τ(w_{1}, …, w_{g}) = (w̄_{1}^{−1}, …, w̄_{g}^{−1})).

Then ρ(γ) · τ(ζ(γ)) = e for all γ ∈ Γ.

Proof. By prop. 1 triviality of the induced bundle is equivalent to the existence of a holomorphic Lie group homomorphism ϕ : G → H such that ϕ|_{Γ} = ρ. Hence one direction of the statement is clear: If the criterion is fulfilled, then ϕ : g ↦ τ(ζ(g)^{−1}) is a holomorphic Lie group homomorphism from G to A ⊂ H such that ϕ|_{Γ} = ρ.

Let us assume that the bundle is trivial and that ϕ : G → H is the trivializing holomorphic Lie group homomorphism. Then ϕ|_{G′} is constant by the preceding theorem. This implies that ρ(γ) = e for all γ ∈ G′ ∩ Γ. Hence ρ(Γ) is commutative.

Let A denote the Zariski-closure of ρ(Γ) in H. Then H/A is a quasi-affine variety (because A is commutative). Thus ϕ : G → H induces a holomorphic map from G/Γ to H/A which must be constant. We may assume that ϕ(e) = e. Then ϕ(G) ⊂ A. Hence we may assume that H = A, i.e. we may assume that H is a commutative linear-algebraic group. Thus H ≃ (ℂ)^{d} × (ℂ^{∗})^{g} and we may discuss these factors separately.

Let us first discuss the case H = ℂ. There does not exist any relatively compact subgroup in (ℂ, +). Therefore, if ρ = ζ · ξ is the decomposition of ρ into the antiholomorphic part ζ and the bounded part ξ, then ξ ≡ 0, i.e. ρ = ζ. It follows that ϕ − ρ induces a pluriharmonic function on G/Γ. But every pluriharmonic function on G/Γ is constant. This forces both ϕ and ρ to be constant, i.e. ρ ≡ e.

Thus we have seen that the Zariski-closure A of ρ(Γ) cannot contain a factor isomorphic to ℂ, i.e. A must be isomorphic to some (ℂ^{∗})^{g}. Let K ≃ (S^{1})^{g} denote the maximal compaxt subgroup of A. This is a real form, hence there is a map τ given by complex conjugation on A with respect to K. (In coordinates this is just (w̄_{1}^{−1}, …, w̄_{g}^{−1}).) Consider now the map α : g ↦ ϕ(g)τ(ζ(g)). Note that

α(γ) = \underbrace{ζ(γ)τ(ζ(γ))}_{∈K} \underbrace{ξ(γ)}_{∈K}.

Furthermore, Γ being a lattice in the Lie group G implies that the closure (in the euclidean topology) of G′Γ in G is cocompact. This implies that there exists a compact subset C ⊂ G such that C ·G′ ·Γ is dense in G. By the above considerations α maps this dense subset of G into a compact subset of A, namely α(C)·K. Since α is a group homomorphism, it follows that α(G) ⊂ K. But α is also a holomorphic map. Hence α(G) ⊂ K implies that α is constant, i.e. α ≡ e. Thus α(γ) = ρ(γ)·τ(ζ(γ)) = e for all γ ∈ Γ.

In the special case where ρ is bounded (i.e. ζ ≡ e) this reproves cor.1 of thm. 3.1. In an other special case, ρ = ζ|_{Γ} and ξ ≡ e, we obtain the following criterion.

Corollary 2. Let G be a complex Lie group, Γ ⊂ G a lattice, H a complex linearalgebraic group and ρ : G → H an antiholomorphic Lie group homomorphism.

Then the flat bundle over G/Γ defined by ρ|_{Γ} is trivial if and only if there exists a number n and holomorphic Lie group homomorphisms ρ_{1} : G → (ℂ^{∗})^{n} and ρ_{2} : (ℂ^{∗})^{n} → H such that ρ = ρ_{2}∘ζ∘ρ_{1} where ζ denotes complex conjugation on T = (ℂ^{∗})^{n} and ρ_{1}(Γ) ⊂ (ℝ^{∗})^{n}.

8. Vector Bundles

To transfer our results on principal bundles to vector bundles, we need the following generalized Schur lemma.

Lemma 2. Let Γ be a group, V a vector space and ζ, ξ : Γ → GL(V) group homomorphisms such that ζ(γ), ξ(δ) commute for all γ, δ ∈ Γ. Assume that V is an irreducible Γ-module for ρ = ζ · ξ.

Then there exists vector spaces V_{1}, V_{2} and irreducible representations ζ_{0} : Γ → GL(V_{1}), ξ_{0} : Γ → GL(V_{2}) such that (V, ρ) ≃ (V_{1}, ζ_{0}) ⊗ (V_{2}, ξ_{0}).

Lemma 3. Let A ⊂ GL(n, ℂ) be a connected commutative complex Lie subgroup, Λ ⊂ A a lattice, Q a projective manifold with b_{1}(Q) = 0 on which A acts holomorphically and x ∈ Q. Assume that for every λ ∈ Λ there exists a sequence of natural numbers n_{k} such that lim n_{k} = ∞ and lim(λ^{n_{k}}(x)) = x.

Then x is a fixed point for the A-action on Q.

Proof. The assumptions on Q imply that Aut(Q)^{0} is a linear-algebraic group. Let H denote the Zariski-closure of A in Aut(Q)^{0}. Assume that x is not a fixed point for the A-action. The H-orbit through x is a locally closed subset of Q and isomorphic to the quotient H/I where I = {h ∈ h : h(x) = x}. This quotient H/I is again a linear-algebraic group (because H is commutative). Recall that on a complex Lie group A every bounded holomorphic function is constant. It follows that the image of A in H/I under the natural morphism τ : A → H/I can not be relatively compact. Hence there is an element λ ∈ Λ such that the sequence τ(λ^{n}) in H/I contains no convergent subsequence. Since the H-orbits in Q are locally closed, this implies that no subsequence of λ^{n}(x) can converge to x. This contradicts the assumptions of the lemma. Hence x must be an A-fixed point.

Proposition 6. Let G be a connected complex Lie group, Γ a lattice, V a complex vector space and ρ : Γ → GL(V) be an essentially antiholomorphic representation, with antiholomorphic part ζ and bounded part ξ. Assume that V contains a ρ(Γ)- invariant subvectorspace W.

Then W is already invariant under ζ(G) and ξ(Γ).

Proof. Let k = dim W. We consider the induced actions on the Grassmann manifold Q of k-dimensional subvectorspaces of V. For simplicity, they are also denoted by ρ, ζ and ξ. Let x = [W] ∈ Q. Since ξ(Γ) is bounded, it is clear that for every element γ ∈ Γ there exists a sequence of natural numbers n_{k} such that lim n_{k} = ∞ and lim ξ(γ^{n_{k}}) = e. Then ρ = ζ · ξ implies that lim ζ(γ^{−n_{k}})(x) = x. By lemma 3 it follows that x is a fixed point for every connected commutative complex Lie subgroup A ⊂ G with A/(A ∩ Γ) compact. By prop. 2 this implies that x is a fixed point for ζ(G) (and hence for ξ(Γ), too).

Corollary 3. Let G be a connected complex Lie group, Γ a lattice, V a complex vector space, ρ : Γ → GL(V) an essentially antiholomorphic representation and W a ρ(Γ)-stable subvectorspace of V.

Then the restricted representation ρ′ : Γ → GL(W) is likewise essentially antiholomorphic.

Lemma 4. Let Γ be a group, ρ_{i} : Γ → GL(V_{i}) representations on complex vector spaces for i = 1, 1 and assume that

1. Both V_{i} are irreducible Γ-modules with respect to the representations ρ_{i}.

2. The Zariski-closure H of ρ_{1}(Γ) in GL(V_{1}) is connected.

3. The image ρ_{2}(Γ) is finite. Then V_{1} ⊗ V_{2} is an irreducible Γ-module with respect to ρ_{1} ⊗ ρ_{2}.

Proof. Let Γ_{0} = ker ρ_{2}. Since Γ/Γ_{0} is finite and H is connected, it is clear that ρ_{1}(Γ_{0}) is Zariski-dense in GL(V_{1}). Hence V_{1} is an irreducible Γ_{0}-module while Γ_{0} acts trivially on V_{2}. It follows that every Γ_{0}-invariant subvectorspace of V_{1} ⊗ V_{2} has the form V_{1} ⊗ W for some subvectorspace W ⊂ V_{2}. Clearly, such a V_{1} ⊗ W is Γ-invariant only if W = {0} or W = V_{2}. Thus V_{1} ⊗ V_{2} is an irreducible Γ-module.

Corollary 4. Let G be a connected complex Lie group, Γ ⊂ G a lattice, ρ_{1} : G → GL(V_{1}) an antiholomorphic representation, ρ_{2} : Γ → GL(V_{2}) a representation with relatively compact and Zariski-connected image and ρ_{3} : Γ → GL(V_{3}) be a representation with finite image.

Assume that all the representations ρ_{i} are irreducible.

Then (ρ_{1}|_{Γ}) ⊗ ρ_{2} ⊗ ρ_{3} : Γ → GL(V_{1} ⊗ V_{2} ⊗ V_{3}) is likewise irreducible.

9. The classification

We make use of Margulis’ superrigidity theorem in the following form:

Theorem 4. Let S be a simply-connected semisimple complex Lie group and Γ ⊂ S a lattice. Assume that there does not exist a normal Lie subgroup S_{0} ⊂ S such that S_{0} ≃ SL(2, ℂ) and S_{0}/(S_{0} ∩ Γ) is of finite volume.

Then there exists a compact real semisimple Lie group K and a group homomorphism j : Γ → K such that for every complex-algebraic group H and every group homomorphism α : Γ → H there exist continuous group homomorphisms ζ : S → H, ξ : K → H and ν : Γ → H such that

1. α(γ) = ζ(γ) · ξ(j(γ)) · ν(γ) for all γ ∈ Γ.

2. The image ν(Γ) is finite.

3. ζ(s), ξ(k) and ν(γ) commute for every s ∈ S, k ∈ K and γ ∈ Γ.

Note that for every continuous group homomorphism ζ from a complex semisimple Lie group S to a complex algebraic group H there exists a holomorphic group homomorphism ζ_{0} and an antiholomorphic group homomorphism ζ_{1} such that ζ = ζ_{0} · ζ_{1}.

Using Margulis’ theorem and our previous results we obtain the following classification.

Theorem 5. Let S be a simply-connected semisimple complex Lie group and Γ ⊂ S a lattice. Assume that there does not exist a normal Lie subgroup S_{0} ⊂ S such that S_{0} ≃ SL(2, ℂ) and S_{0}/(S_{0} ∩ Γ) is of finite volume.

Then there exists a compact real semisimple Lie group K and a group homomorphism j : Γ → K such that there is a one-to-one correspondance between irreducible holomorphic vector bundles on S/Γ admitting a flat connection and triples (ρ, ζ, ν) where

1. All ρ, ζ and ν are irreducible representations of Γ.

2. ρ extends to an antiholomorphic representation of G.

3. ζ fibers through a representation of K.

4. ν is a representation with finite image.

10. Sections of Homogeneous Vector Bundles

We prove that a homogeneous vector bundle over a quotient X = G/Γ of a connected complex Lie group G by a lattice Γ admits global sections only inasmuch as it is trivial.

Since we do not want to assume G/Γ to be compact, we first have to show that the space of sections is finite-dimensional.

Proposition 7. Let G be a connected complex Lie group, Γ a lattice and E → X = G/Γ a homogeneous vector bundle.

Then Γ(X, E) is finite-dimensional.

Proof. We first need a result on the structure of G/Γ.

Claim 7. Let G be a connected complex Lie group and Γ a lattice. Then there exists a G-equivariant holomorphic surjective map π from X onto a compact complex parallelizable manifold Y such that the algebraic dimension of the fibers equals zero.

Proof. The algebraic reduction of X maps X onto a compact complex torus T. If the fibers have algebraic dimension larger than zero, we continue by considering the algebraic reduction of the fiber. This yields us a holomorphic surjection of X onto a compact complex parallelizable manifold (which is a torus bundle over a torus). We may continue in this way and for dimension reasons we will arrive at a holomorphic surjective map from X onto a compact complex parallelizable manifold for which the fibers have algebraic dimension zero.

We will now discuss E restricted to a fiber of π.

Claim 8. Let F be a complex manifold of algebraic dimension zero (i.e. every meromorphic function on F is constant) and E → F a vector bundle of rank r. Then dim(Γ(F, E)) ≤ r.

Proof. Assume the contrary. Then there exists a number d with 1 ≤ d ≤ r and sections σ_{0}, …, σ_{d} such that

1. The sections (σ_{i})_{0≤i≤d} are linearly independent as elements in Γ(F, E).

2. For a generic point x ∈ F the subvectorspace of E_{x} spanned by (σ_{i}(x))_{1≤i≤d} has dimension d.

3. For every point x ∈ F the subvectorspace of E_{x} spanned by (σ_{i}(x))_{0≤i≤d} has

dimension at most d. But in this case one of the meromorphic functions

f_{i} = (σ_{0} ∧ σ_{1} ∧ … ∧ σ̂_{i} ∧ … ∧ σ_{d})/(σ_{1} ∧ … ∧ σ_{d})

must be non-constant, contradicting the assumption of F having algebraic dimension zero.

Now let X = G/Γ, E → X and π : X → Y as above. Since dim(F, E) ≤ rank(E) for every fiber F of π, we may conclude that the direct image sheaf π_{∗}E is finitely generated as a O_{Y} -module sheaf. For homogeneity reasons it is locally free and therefore coherent. Since Y is compact, it follows that Γ(Y, π_{∗}E) ≃ Γ(X, E) is finitedimensional.

Now we are in a position to prove the following structure theorem on sections of homogeneous vector bundles.

Proposition 8. Let G be a connected complex Lie group, Γ a lattice and E → X = G/Γ a flat vector bundle.

Then E contains a vector subbundle E_{0} which is parallel with respect to the flat connection and trivial as a holomorphic vector bundle such that Γ(X, E) = Γ(X, E_{0}).

Proof. The sections of E generate a coherent subsheaf E_{0} of E. By homogeneity this subsheaf is locally free. Evidently it is invariant under the G-action, hence parallel with respect to the connection. There is an equivariant map from X to the projective space ℙ(Γ(X, E_{0})^{∗}) (which is finite-dimensional due to the preceding proposition). This map is constant and hence E_{0} is trivial as a holomorphic vector bundle.

Together with prop. 3 this enables us to completely determine the global sections in a flat vector bundle on G/Γ which is induced by an antiholomorphic representation of G. We will apply this in [14] in order to calculate the dimension of Dolbeault cohomology groups by use of Leray spectral sequences.

Corollary 5. Let G be a connected complex Lie group, Γ a lattice and ρ be a holomorphic representation of G on a complex vector space V. Let V_{1} = V^{G′} denote the vector subspace of all v ∈ V which are fixed by ρ(G′). Let Σ denote the subset of all v ∈ V_{1} such that v is an eigenvector with real eigenvalue for every ρ(γ) (γ ∈ Γ) and let V_{0} denote the subvectorspace of V spanned by Σ.

Let E and E_{0} denote the flat vector bundles on X = G/Γ which is induced by the representation ρ̄|_{Γ} on V resp. V_{0}.

Then E_{0} is a holomorphically trivial vector bundle and Γ(X, E_{0}) = Γ(X, E).

For antiholomorphic representations we are now able to give a precise description of the global sections of the associated vector bundle.

Proposition 9. Let G be a connected complex Lie group, Γ a lattice, V a complex vector space and ρ : G → GL(V) be an antiholomorphic representation. Let E denote the flat vector bundle over G/Γ which is induced by ρ|_{Γ}.

Let Σ denote the set of all vectors v ∈ V which are invariant under ρ(G′) and a ρ(γ)-eigenvector with real eigenvalue for every γ ∈ Γ.

Then H^{0}(G/Γ, E) ≃< Σ >_{ℂ} where < Σ >_{ℂ} denotes the complex vector space spanned by Σ.

Proof. By construction < Σ >_{ℂ} is the largest ρ(G)-invariant vector subspace of V inducing a holomorphically trivial vector subbundle of E.

11. Subbundles of flat bundles

Theorem 6. Let G be a connected complex Lie group, Γ a lattice and E → G/Γ be a flat vector bundle given by an essentially antiholomorphic representation ρ : G → GL(n, ℂ).

Assume that every meromorphic function on X = G/Γ is constant.

Let L ⊂ E be a vector subbundle. Then L also admits a flat connection.

If G = G′, then L is parallel with respect to the given flat connection on E.

Proof. Let k = rank(L). By passing to Λ^{k}L ⊂ Λ^{k}E we may assume that L is a line bundle. The flat connection on E yields a canonical way to lift the G-action on X to a G-action on E. The subbundle E_{0} = ⊕_{g∈G}g^{∗}L is a G-invariant subbundle of E and therefore parallel with respect to the flat connection on E. Moreover it is given by an essentially antiholomorphic representation (see cor. 3). We may thus assume E_{0} = E. Let d = rank(E_{0}) and choose g_{0}, …, g_{d} ∈ G such that the line subbundles L_{i} = g_{i}^{∗}L are in general position, i.e. such that for every number m with 0 ≤ m ≤ d and every choice of 0 ≤ i_{1} < … i_{m} ≤ d the subsheaf of E spanned by L_{i_{1}} + … + L_{i_{m}} has rank m.

Claim 9. The line bundles L_{0}, …, l_{d} are in general position at every point and therefore yield a trivialization of ℙ(E_{0}).

Proof. Let U be an open neighbourhood of a point such that all the L_{i} admit nowhere vanishing sections σ_{i}. For every k ∈ {0, …, d} we obtain a section α_{k} in ∧^{d}E_{0} by α_{k} = ∧_{i≠k}σ_{i}. Let α = ⊗_{k}α_{k}. Then α is a section in the line bundle ⊗^{d+1}(∧^{d}E_{0}) and is vanishing exactly where the L_{0}, …, L_{d} fail to be in general position. Thus the set of all points in X where the L_{0}, …, L_{d} are not in general position constitute an analytic hypersurface of X. But the assumptions on X imply that X contains no hypersurfaces. Hence L_{0}, …, L_{d} are in general position everywhere.

Thus ℙ(E_{0}) is a holomorphically trivial ℙ_{d−1}-bundle. Observe that ℙ(E_{0}) is defined by an essentially antiholomorphic group homomorphism ρ̃: Γ → P GL(d, ℂ). Using thm. 2 it follows that ρ̃|_{G′∩Γ} is trivial. Hence ρ̃(Γ) constitutes a commutative subgroup of P GL(d, ℂ). This implies that there is a fixed point for the ρ̃(Γ)-action on ℙ(V). Corresponding to this fixed point there exists a sub-line bundle L′ ⊂ E which is G-invariant and therefore parallel to the given flat connection on E.

Claim 10. The line bundles L′ and L are isomorphic (as holomorphic line bundles).

Proof. Recall that ℙ(E) is holomorphically trivial and that every meromorphic function on X is constant. It follows that there is a unique trivialization of ℙ(E) and that every section of ℙ(E) → X is constant with respect to this trivialization. This implies that any two line subbundles of E are holomorphically isomorphic.

Thus L is isomorphic to a flat line bundle. However, it is parallel with respect to the given flat connection on E if and only if it is G-invariant. This in turn is equivalent to the assertion that L corresponds to a fixed point of the Γ-action on ℙ(V). This action is trivial, if G = G′.

References

[1] Berteloot, F.; Oeljeklaus, K.,Invariant Plurisubharmonic Functions and Hypersurfaces on Semisimple Complex Lie Groups. Math. Ann. 281, 513–530 (1988)

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

[3] Huckleberry, A.T.; Margulis, G.A.,Invariant analytic hypersurfaces. Invent. Math. 71, 235–240 (1983)

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

[5] Kazdan, D.A.; Margulis, G.A.,A proof of Selbergs hypothesis. Math. Sbornik 75 , 162–168 (1968)

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

[7] Margulis, G.A., Discrete Subgroups of Semisimple Lie Groups. Springer Berlin Heidelberg New York 1989

[8] Mostow, G.D.,Arithmetic subgroups of groups with radical. Ann.- Math. 93, 409–438 (1971)

[9] Raghunathan, M.S.,Discrete subgroups of Lie groups. Erg. Math. Grenzgeb. 68 , Springer (1972)

[10] Wang, H., Complex parallisable manifolds. Proc. A.M.S. 5, 771–776 (1954)

[11] Winkelmann, J.,Every compact complex manifold admits a holomorphic vector bundle. Revue Roum. Math. Pures et Appl. 38, 743–744 (1993)

[12] Winkelmann, J.,Complex-Analytic Geometry of Complex Parallelizable Manifolds. Habilitationsschrift.

[13] Winkelmann, J.,Holomorphic self-maps of Parallelizable Manifolds. Transformation Groups (1997)

[14] Winkelmann, J.,The Albanese Variety of a Complex Parallelizable Manifold, MSRI preprint 1997-049 (see http://www.msri.org/MSRI-preprints).

Jörg Winkelmann, Math. Institut NA 4/69, Ruhr-Universität Bochum, 44780 Bochum, Germany

E-mail address: jw@cplx.ruhr-uni-bochum.de