Flat Vector Bundles over Parallelizable Manifolds
a r
X i
v
:m
a
t h /
9
7
5
2
1
6
v
1
[
m
a
t h
.
A
G ]
5
M
a y
1
9
9
7
FLAT VECTOR BUNDLES OVER PARALLELIZABLE MANIFOLDS J ¨ORG WINKELMANN Abstract.We study ?at vector bundles over complex parallelizable manifolds.1.Introduction Let G be a complex Lie group and Γa discrete subgroup which is large in a certain sense,e.g.of ?nite covolume.We are interested in studying holomorphic vector bundles over the quotient manifold X =G/Γ.In general (i.e.if G is non-commutative)these manifolds are not algebraic and not even K¨a hler.Thus in order to understand these manifolds it is necessary to employ methods other than those usually applied in algebraic and K¨a hler geometry.These methods are in particular group-theoretical methods like methode from Lie theory,represen-tation 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 holomor-phic vector bundles which carry a ?at 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 Γin 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 ?at holomorphic vector bundle over G/Γis isomorphic as a holomorphic vector bundle to a ?at bundle induced by an essentially antiholomorphic representation.For lattices in arbitrary complex Lie groups there are always non-trivial essentially antiholomorphic repre-sentations.Moreover ?at vector bundles induced by antiholomorphic representations arise naturally as higher direct images sheaves of the structure sheaf for ?brations of complex parallelizable manifolds.Therefore these results are useful to calculate Dolbeault cohomology groups via Leray spectral sequences associated to certain ?-
brations.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 ?at 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 ?at vector bundles.Essen-tially we prove that in general sections exist only to the extent to which the bundle
2J¨ORG WINKELMANN
is trivial and that generically every vector subbundle of a?at vector bundle on a complex parallelizable manifold is again?at.
As a by-product we can show that every compact complex manifold of dimension n>0admits a non-trivial holomorphic vector bundle of rank at most n(improving our result in[11]).
The assumption thatΓis of?nite covolume is often used in a rather indirect way. For instance we prove that,given a latticeΓin 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/Γcarries no non-constant plurisubharmonic function.
A part of the results for the special case where G/Γis compact is already contained in the Habilitationsschrift of the author[12].
2.Generalities
Let G be a connected complex Lie group andΓa discrete subgroup.Then X=G/Γis 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),thenΓis isomorphic to the fundamental group of X.For any representationρofΓin a complex Lie group H we obtain a?at H-prinicpal bundle by E=G×H/ΓwithΓacting on G by deck transformations and on H viaρ. Such a bundle admits a trivialization compatible with the?at 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 ?at connection).Such a trivialization exists if and only if there is aΓ-equivariant holomorphic mapφfrom G to H(i.e.φ(gγ)=φ(g)·ρ(γ)for all g∈G,γ∈Γ). More general,two representationsρ,?ρ: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 thatφ(gγ)=ρ(γ)?1φ(g)?ρ(γ)for all g∈G andγ∈Γ.
IfΓis a“large”discrete subgroup of G there are few holomorphic mappings from X=G/Γ.Especially,ifΓis a lattice in a complex Lie group G the quotient manifold X=G/Γhas the following properties:
1.(Iwamoto’s version of the Borel density theorem([4])).Every linear represen-
tationρ:G→GL(n,C)the Zariski-closures ofρ(Γ)andρ(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.claim3.3below).
3.A triviality criterion
Proposition1.Let G be a connected complex Lie group,H a complex Lie group andΓ?G a discrete subgroup.Letρ:Γ→H be a group homomorphism.Assume that every holomorphic function on G/Γis constant.
FLAT VECTOR BUNDLES OVER PARALLELIZABLE MANIFOLDS3 Then the H-principal bundle E→G/Γinduced byρis holomorphically trivial if and only ifρextends to a holomorphic group homomorphism?ρ:G→H.
Proof.Assume that there is a trivializing mapφ:G→H,i.e.a holomorphic map φ:G→H such thatφ(gγ)=φ(g)ρ(γ)for all g∈G andγ∈Γ.Upon replacingφby?φ(g)def=φ(e)?1φ(g),we may assume thatφ(e)=e.De?neα:G×G→H by
α(g1,g2)=φ(g1)φ(g2)φ(g1g2)?1.
Thenα(g1,e)=α(e,g2)=e andα(g1,g2γ)=α(g1,g2)for all g1,g2∈G andγ∈G. We consider now the induced mapsαg:G/Γ→H given byαg:xΓ→α(g,x).Since αe≡e,it is clear that all the mapsαg are homotopic to a constant map.Hence these maps may be lifted to the universal covering?H of H,i.e.there exist holomorphic maps?αg:G/Γ→?H such thatαg=π??αg whereπ:?H→H is the universal covering map.But?H is a simply-connected complex Lie group and therefore Stein.Hence the maps?αg are constant.It follows that the mapsαg 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.IfΓis a lattice in a simply-connected complex Lie group G,thenΓis?nitely
generated and G is linear.In this case a theorem of Malcev[6]implies thatΓis residually?nite,i.e.for every elementγ∈Γthere exists a?nite group F and a group homomorphismρ:Γ→F such thatρ(γ)=e.Since every?nite group embeds in some linear group,this yields many representations ofΓwith ?nite image.
2.If H1(G/Γ)?Γ/Γ′is non-trivial,Γadmits many group homomorphisms to
S1={z∈C?:|z|=1}.
3.Let K be a number?eld,O K its ring of algebraic integers,S a semisimple K-
group 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=Πv∈S S(K v)is a complex Lie group,U=Πv∈T S(K v) is a compact real Lie group andΓ=S(O K)is a lattice in G which admits an injective group homomorphism to U.Now every representation of U induces a representation ofΓwith relatively-compact image.
Theorem1.Let G be a connected complex Lie group,R its radical,Γa discrete subgroup of G,H a Stein Lie group andρi:Γ→H group homomorphisms with relatively compact image.
Assume that Ad(Γ)and Ad(G)have the same Zariski-closure in GL(L ie G)and thatΓ∩R is cocompact in R.
Let E1and E2denote the H-principal bundles over G/Γinduced byρ1resp.ρ2. Then E1～E2if and only ifρ1andρ2are conjugate by an element h∈H.
4J¨ORG WINKELMANN
Remark1.IfΓis 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(L ie G)do coincide([2]).
Proof.Recall that E1～E2holds i?there exists a holomorphic mapφ:G→H such that
(1)
φ(gγ)=ρ1(γ)?1φ(g)ρ2(γ)
for all g∈G,γ∈Γ.
Let K i denote the closure ofρi(Γ)in H.The sets K i are compact subgroups. Since H is Stein,it admits a strictly plurisubharmonic exhaustion functionτ.We may assume thatτis invariant under the K1×K2-action on H given by(k1,k2): h→k?11hk2(because we can replaceτby the function obtained by averagingτover the K1×K2-orbits).Then(1)impliesτ(φ(gγ))=τ(φ(g)).Thus we obtain a plurisubharmonic function on G/Γ.
Claim1.Under the assumptions of the theorem every plurisubharmonic function on G/Γis constant.
Proof.Since R/(R∩Γ)is compact,every plurisubharmonic function on G/Γis a pull-back from(G/R)/(RΓ/Γ).Now G/R is semisimple andΓ/(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.Sinceτis strictly plurisubharmonic,this implies that φis constant.
Corollary1.Let G,Γand H be as in the above theorem and letρ:Γ→H be a group homomorphism with relatively compact image.
Then the induced H-principal bundle over G/Γis holomorphically trivial if and only ifρ≡e.
5.Subgroups with a bounded orbit
Given a Lie group G and a latticeΓin G we will show that there are many Lie subgroups H of G such that the H-orbit through eΓis relatively compact in G/Γ. Proposition2.Let G be a connected complex Lie group,Γa lattice and G0the subgroup of G generated by all connected commutative complex Lie subgroups A of G with A/(A∩Γ)compact.
Then G0=G.
Proof.If G is commutative,then G/Γis a topological group with?nite volume and therefore compact.In the general case we argue by induction over dim(G).Thus let G,Γand G0be as above and assume that the proposition is valid for all Lie groups of lower dimension.Letγ∈Γ\Z(where Z denotes the center of G)and let C0(γ)denote the connected component of the centralizer ofγin G.Then dim C0(γ) FLAT VECTOR BUNDLES OVER PARALLELIZABLE MANIFOLDS5 Zariski-closure of Ad(Γ).Let k denote the generic dimension of V h={v:hv=v} for h∈H and?the set of all h∈H for which dim V h=k.Then?is a Zariski-open subset of H.Now V h?L ie(G0)for all h∈Γ∩?and therefore for all h∈?.It follows that v∈L ie G0for all v∈Ad?1(?).Since Ad?1(?)is a non-empty open subset of L ie G,this implies that G=G0. We will now deduce another variant of this theme,this time strengthening the as-sumption on the subgroups(requiring unipotency)while relaxing the assumption on the orbits(only relatively-compact instead of compact). Proposition3.Let G be a simply-connected complex Lie group andΓa lattice. Let G1denote the subgroup of G generated by all unipotent subgroups U?G′for which there exists a compact subset F?G such that U?FΓ′. Then G1=G′. Remark2.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.Letπ:G→G/R?S be the natural projection. Claim2.Let H be a one-dimensional unipotent subgroup of S such that H∩π(Γ)= {e}. Then H?G1. Proof.Let u∈S,r∈R such that ur∈Γandπ(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 eΓis closed.Let N denote the nilradical of C.Then NR is the nilradical of CR. It follows that the NR-orbit through eΓis compact.Now H is central in C,hence H?N and consequently H?G1. Next let V denote the subgroup of G/R generated by all one-dimensional unipotent subgroups having non-trivial intersection withπ(Γ).Then V is normalized byπ(Γ). The Borel Density theorem thus implies that V is a normal subgroup of S.It follows that S=S0×V for some semisimple complex Lie subgroup S0?S. Claim3.The S0-orbit through eΓin G/Γis relatively compact. Proof.By the criterion of Kazdan-Margulis(see[9],Cor.11.12)we have to show that given a sequence s n∈S0there is no sequenceγn∈Γ\{e}such that lim s nγn s?1n=e. Assume there are such sequences.Let u n∈S and r n∈R such that u n r n=γn.Then lim s n u n s?1n=e(because R is central).Sinceπ(Γ)is a lattice in S,it follows that u n is unipotent for n su?ciently large([9],Cor.11.18).But in this case u n∈V by construction and this implies that s n u n s?1n=u n for all n.Contradiction! Finally note that S0is a semisimple complex Lie group and therefore generated by its unipotent subgroups. 6J¨ORG WINKELMANN 6.Antiholomorphic maps and actions of unipotent groups Here we prove an auxiliary result needed later on. Proposition4.Let Z be a quasi-a?ne variety,ˉH a connected commutative linear-algebraic group(both de?ned over C)and H a connected Zariski-dense complex Lie subgroup ofˉH.Assume that there is a regular actionμ:ˉH×Z→Z,a map α:H→Z with relatively compact image and an antiholomorphic mapφ:H→Z such that (2) α(h)=μ(h)(φ(h)) for all h∈H.Let U denote the unipotent radical ofˉH. Then bothα(H)andφ(H)are contained in a singleμ(ˉH)-orbit W,U acts trivially on this orbit and the mapsα,φare homomorphisms of real Lie groups from H to W with respect to the natural group structure on W withφ(e)as neutral element. Proof.We start with a discussion of semi-invariant functions on Z. De?nition1.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 characterχof G such that f(gz)=χ(g)f(z)for all z∈Z and g∈G. Now let f be a semi-invariant for theˉH-action on Z.Then (3) f(α(h))=χ(h)f(φ(h)) for a characterχofˉH and all h∈H.This implies |f(α(h))|2=|χ(h)f(φ(h))|2=|χ(h) f?φ≡c.It follows that f?αand f?φvanish either everywhere or nowhere. Lemma1.Let W be a quasi-a?ne 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 ?nite-dimensional https://www.wendangku.net/doc/fc17419530.html,ing 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.Contradic-tion! Applying this lemma to our situation we may conclude that bothα(H)andφ(H) are contained in a singleˉH-orbit W. SinceˉH is commutative,the homogeneousˉH-space W has a canonical structure as a commutative group which is unique up to the choice of the neutral element. FLAT VECTOR BUNDLES OVER PARALLELIZABLE MANIFOLDS 7 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 ˉH ,we may ?nd 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 +τ(φ(γ(t ))) antiholo. .(4)This is a contradiction,because the left side is bounded while the right side is a non-constant harmonic function on C .Thus the U -action must have been trivial. Now W is a homogeneous space of the reductive commutative group ˉH/U .By choosing a point in W as neutral element,W inherits a structure as reductive com-mutative algebraic group.Let us choose φ(e )=α(e )as neutral element.Then every character of W is an semi-invariant for the ˉH -action and consequently our previous considerations imply that χ?φis a real Lie group homomorphism from H to C ?for every character χof W .It follows that φ(and hence also α)are real Lie group homomorphisms. 7.Essentially antiholomorphic representations De?nition 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,Γa lattice,H a Stein Lie group and ρ:Γ→H an essentially antiholomorphic representation with maps ζ,ξgiven as above. Then 1.ξ:Γ→H is a group homomorphism. 2.Both ξand ζare 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)?1ξ(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∈Γwe de?ne an antiholomorphic map φγ:A →H by φγ(a )=ζ(a )ξ(γ)ζ(a )?1. 8J¨ORG WINKELMANN If FLAT VECTOR BUNDLES OVER PARALLELIZABLE MANIFOLDS9 and the Zariski-closure of?μ(H)in GL(n,C)×GL(n,C)asˉA)yieldsα(A)?W and φ(A)?W. By de?nition ofαthe inclusion(A∩Γ)?W implies thatη(γ)(c)∈W forγ∈A∩Γ. Sinceη(Γ)commutes withμ(G),it follows thatη(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. Claim6.The groupμ(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’ofφ.For x∈G letφx denote the map de?ned by φx(g)=φ(xg).Thenφx:G→H is a holomorphic map ful?lling(7).Thus we may apply the above considerations and conclude thatμ(G′)(φx(e))=φx(e).Since φx(e)=φ(x),it follows thatμ(G′)stabilizesφ(G)pointwise.Hence (12) φ(gγ)=η(γ)(φ(g)) for g∈G,γ∈G′∩Γ.By assumptionξ(Γ)and?ξ(Γ)are contained in compact subgroups K resp.?K of H.Since H is Stein,it admits a strictly plurisubharmonic exhaustion functionτ.By averaging with respect to Haar measure we may assume thatτis invariant with respect to the given K×?K-action on H.Then we obtain (13) τ(φ(gγ))=τ(φ(g)) for g∈G,γ∈G′∩Γ.Now every plurisubharmonic function on G′/(G′∩Γ)is constant(see[14]).Henceφis constant along the G′-orbits on G.In particularφ|G′is constant and this yields the assertion with c=φ(e). Theorem3.Let G be a connected complex Lie group,Γ?G a lattice,H a complex linear-algebraic group andρ:Γ→H an essentially antiholomorphic group homo-morphism. Then the?at bundle over G/Γde?ned byρis trivial if and only if the following conditions are ful?lled: 1.The Zariski-closure A ofρ(Γ)in H is isomorphic to(C?)g for some g≥0. 2.Letρ=ζ·ξdenote the decxomposition ofρinto the antiholomorphic partζand the bounded partξ.Let K denote the maximal compact subgroup of A(which is a real form of A)and letτdenote complex conjugation on A with respect to the real form K(i.e.τ(w1,...,w g)=(ˉw?11,...,ˉw?1g)). Thenρ(γ)·τ(ζ(γ))=e for allγ∈Γ. Proof.By prop.1triviality of the induced bundle is equivalent to the existence of a holomorphic Lie group homomorphismφ:G→H such thatφ|Γ=ρ.Hence one direction of the statement is clear:If the criterion is ful?lled,thenφ:g→τ(ζ(g)?1) is a holomorphic Lie group homomorphism from G to A?H such thatφ|Γ=ρ. Let us assume that the bundle is trivial and thatφ:G→H is the trivializing holomorphic Lie group homomorphism.Thenφ|G′is constant by the preceding the-orem.This implies thatρ(γ)=e for allγ∈G′∩Γ.Henceρ(Γ)is commutative. 10J¨ORG WINKELMANN Let A denote the Zariski-closure ofρ(Γ)in H.Then H/A is a quasi-a?ne variety (because A is commutative).Thusφ:G→H induces a holomorphic map from G/Γto H/A which must be constant.We may assume thatφ(e)=e.Thenφ(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?rst discuss the case H=C.There does not exist any relatively compact subgroup in(C,+).Therefore,ifρ=ζ·ξis the decomposition ofρinto the anti-holomorphic partζand the bounded partξ,thenξ≡0,i.e.ρ=ζ.It follows that φ?ρinduces a pluriharmonic function on G/Γ.But every pluriharmonic function on G/Γis constant.This forces bothφandρto be constant,i.e.ρ≡e. Thus we have seen that the Zariski-closure A ofρ(Γ)cannot contain a factor isomorphic to C,i.e.A must be isomorphic to some(C?)g.Let K?(S1)g denote the maximal compaxt subgroup of A.This is a real form,hence there is a mapτgiven by complex conjugation on A with respect to K.(In coordinates this is just (ˉw?11,...,ˉw?1g).)Consider now the mapα:g→φ(g)τ(ζ(g)).Note that α(γ)=ζ(γ)τ(ζ(γ)) ∈K ξ(γ) ∈K . Furthermore,Γbeing a lattice in the Lie group G implies that the closure(in the euclidean topology)of G′Γin G is cocompact.This implies that there exists a compact subset C?G such that C·G′·Γis dense in G.By the above considerations αmaps this dense subset of G into a compact subset of A,namelyα(C)·K.Sinceαis a group homomorphism,it follows thatα(G)?K.Butαis also a holomorphic map. Henceα(G)?K implies thatαis constant,i.e.α≡e.Thusα(γ)=ρ(γ)·τ(ζ(γ))=e for allγ∈Γ. In the special case whereρis bounded(i.e.ζ≡e)this reproves cor.1of thm.3.1. In an other special case,ρ=ζ|Γandξ≡e,we obtain the following criterion. Corollary2.Let G be a complex Lie group,Γ?G a lattice,H a complex linear-algebraic group andρ:G→H an antiholomorphic Lie group homomorphism. Then the?at bundle over G/Γde?ned byρ|Γis trivial if and only if there exists a number n and holomorphic Lie group homomorphismsρ1:G→(C?)n andρ2: (C?)n→H such thatρ=ρ2?ζ?ρ1whereζdenotes complex conjugation on T=(C?)n andρ1(Γ)?(R?)n. 8.Vector Bundles To transfer our results on principal bundles to vector bundles,we need the following generalized Schur lemma. Lemma2.LetΓbe a group,V a vector space andζ,ξ:Γ→GL(V)group ho-momorphisms such thatζ(γ),ξ(δ)commute for allγ,δ∈Γ.Assume that V is an irreducibleΓ-module forρ=ζ·ξ. Then there exists vector spaces V1,V2and irreducible representationsζ0:Γ→GL(V1),ξ0:Γ→GL(V2)such that(V,ρ)?(V1,ζ0)?(V2,ξ0). FLAT VECTOR BUNDLES OVER PARALLELIZABLE MANIFOLDS11 Lemma3.Let A?GL(n,C)be a connected commutative complex Lie subgroup,Λ?A a lattice,Q a projective manifold with b1(Q)=0on which A acts holomor-phically and x∈Q.Assume that for everyλ∈Λthere exists a sequence of natural numbers n k such that lim n k=∞and lim(λn k(x))=x. Then x is a?xed point for the A-action on Q. Proof.The assumptions on Q imply that Aut(Q)0is a linear-algebraic group.Let H denote the Zariski-closure of A in Aut(Q)0.Assume that x is not a?xed 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 morphismτ:A→H/I can not be relatively compact. Hence there is an elementλ∈Λsuch that the sequenceτ(λn)in H/I contains no convergent subsequence.Since the H-orbits in Q are locally closed,this implies that no subsequence ofλn(x)can converge to x.This contradicts the assumptions of the lemma.Hence x must be an A-?xed point. Proposition6.Let G be a connected complex Lie group,Γa lattice,V a complex vector space andρ:Γ→GL(V)be an essentially antiholomorphic representation, with antiholomorphic partζand bounded partξ.Assume that V contains aρ(Γ)-invariant subvectorspace W. Then W is already invariant underζ(G)andξ(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,ζandξ.Let x=[W]∈Q.Sinceξ(Γ)is bounded,it is clear that for every elementγ∈Γthere exists a sequence of natural numbers n k such that lim n k=∞and limξ(γn k)=e.Thenρ=ζ·ξimplies that limζ(γ?n k)(x)=x.By lemma3it follows that x is a?xed point for every connected commutative complex Lie subgroup A?G with A/(A∩Γ)compact.By prop.2this implies that x is a?xed point for ζ(G)(and hence forξ(Γ),too). Corollary3.Let G be a connected complex Lie group,Γa 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 representationρ′:Γ→GL(W)is likewise essentially antiholo-morphic. Lemma4.LetΓbe a group,ρi:Γ→GL(V i)representations on complex vector spaces for i=1,1and assume that 1.Both V i are irreducibleΓ-modules with respect to the representationsρi. 2.The Zariski-closure H ofρ1(Γ)in GL(V1)is connected. 3.The imageρ2(Γ)is?nite. Then V1?V2is an irreducibleΓ-module with respect toρ1?ρ2. Proof.LetΓ0=kerρ2.SinceΓ/Γ0is?nite and H is connected,it is clear that ρ1(Γ0)is Zariski-dense in GL(V1).Hence V1is an irreducibleΓ0-module whileΓ0acts 12J¨ORG WINKELMANN trivially on V2.It follows that everyΓ0-invariant subvectorspace of V1?V2has the form V1?W for some subvectorspace W?V2.Clearly,such a V1?W isΓ-invariant only if W={0}or W=V2.Thus V1?V2is an irreducibleΓ-module. Corollary4.Let G be a connected complex Lie group,Γ?G a lattice,ρ1:G→GL(V1)an antiholomorphic representation,ρ2:Γ→GL(V2)a representation with relatively compact and Zariski-connected image andρ3:Γ→GL(V3)be a represen-tation with?nite image. Assume that all the representationsρi are irreducible. Then(ρ1|Γ)?ρ2?ρ3:Γ→GL(V1?V2?V3)is likewise irreducible. 9.The classification We make use of Margulis’superrigidity theorem in the following form: Theorem4.Let S be a simply-connected semisimple complex Lie group andΓ?S a lattice.Assume that there does not exist a normal Lie subgroup S0?S such that S0?SL(2,C)and S0/(S0∩Γ)is of?nite volume. Then there exists a compact real semisimple Lie group K and a group homomor-phism j:Γ→K such that for every complex-algebraic group H and every group homomorphismα:Γ→H there exist continuous group homomorphismsζ:S→H,ξ:K→H andν:Γ→H such that 1.α(γ)=ζ(γ)·ξ(j(γ))·ν(γ)for allγ∈Γ. 2.The imageν(Γ)is?nite. 3.ζ(s),ξ(k)andν(γ)commute for every s∈S,k∈K andγ∈Γ. Note that for every continuous group homomorphismζfrom a complex semisimple Lie group S to a complex algebraic group H there exists a holomorphic group homo-morphismζ0and an antiholomorphic group homomorphismζ1such thatζ=ζ0·ζ1. Using Margulis’theorem and our previous results we obtain the following classi?-cation. Theorem5.Let S be a simply-connected semisimple complex Lie group andΓ?S a lattice.Assume that there does not exist a normal Lie subgroup S0?S such that S0?SL(2,C)and S0/(S0∩Γ)is of?nite volume. Then there exists a compact real semisimple Lie group K and a group homomor-phism j:Γ→K such that there is a one-to-one correspondance between irreducible holomorphic vector bundles on S/Γadmitting a?at connection and triples(ρ,ζ,ν) where 1.Allρ,ζandνare irreducible representations ofΓ. 2.ρextends to an antiholomorphic representation of G. 3.ζ?bers through a representation of K. 4.νis a representation with?nite image. FLAT VECTOR BUNDLES OVER PARALLELIZABLE MANIFOLDS13 10.Sections of Homogeneous Vector Bundles We prove that a homogeneous vector bundle over a quotient X=G/Γof a con-nected complex Lie group G by a latticeΓadmits global sections only inasmuch as it is trivial. Since we do not want to assume G/Γto be compact,we?rst have to show that the space of sections is?nite-dimensional. Proposition7.Let G be a connected complex Lie group,Γa lattice and E→X= G/Γa homogeneous vector bundle. ThenΓ(X,E)is?nite-dimensional. Proof.We?rst need a result on the structure of G/Γ. Claim7.Let G be a connected complex Lie group andΓa lattice.Then there ex-ists a G-equivariant holomorphic surjective mapπfrom X onto a compact complex parallelizable manifold Y such that the algebraic dimension of the?bers equals zero. Proof.The algebraic reduction of X maps X onto a compact complex torus T.If the?bers have algebraic dimension larger than zero,we continue by considering the algebraic reduction of the?ber.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 ?bers have algebraic dimension zero. We will now discuss E restricted to a?ber ofπ. Claim8.Let F be a complex manifold of algebraic dimension zero(i.e.every mero-morphic 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 with1≤d≤r and sectionsσ0,...,σd such that 1.The sections(σi)0≤i≤d are linearly independent as elements inΓ(F,E). 2.For a generic point x∈F the subvectorspace of E x spanned by(σi(x))1≤i≤d has dimension d. 3.For every point x∈F the subvectorspace of E x spanned by(σi(x))0≤i≤d has dimension at most d. But in this case one of the meromorphic functions f i=(σ0∧σ1∧...∧ σi∧...∧σd)/(σ1∧...∧σd) must be non-constant,contradicting the assumption of F having algebraic dimension zero. Now let X=G/Γ,E→X andπ:X→Y as above.Since dim(F,E)≤rank(E) for every?ber F ofπ,we may conclude that the direct image sheafπ?E is?nitely generated as a O Y-module sheaf.For homogeneity reasons it is locally free and therefore coherent.Since Y is compact,it follows thatΓ(Y,π?E)?Γ(X,E)is?nite-dimensional. 14J¨ORG WINKELMANN Now we are in a position to prove the following structure theorem on sections of homogeneous vector bundles. Proposition8.Let G be a connected complex Lie group,Γa lattice and E→X= G/Γa?at vector bundle. Then E contains a vector subbundle E0which is parallel with respect to the?at connection and trivial as a holomorphic vector bundle such thatΓ(X,E)=Γ(X,E0). Proof.The sections of E generate a coherent subsheaf E0of 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,E0)?)(which is?nite-dimensional due to the preceding proposition). This map is constant and hence E0is trivial as a holomorphic vector bundle. Together with prop.3this enables us to completely determine the global sections in a?at vector bundle on G/Γwhich 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. Corollary5.Let G be a connected complex Lie group,Γa lattice andρbe a holo-morphic representation of G on a complex vector space V.Let V1=V G′denote the vector subspace of all v∈V which are?xed byρ(G′).LetΣdenote the subset of all v∈V1such that v is an eigenvector with real eigenvalue for everyρ(γ)(γ∈Γ)and let V0denote the subvectorspace of V spanned byΣ. Let E and E0denote the?at vector bundles on X=G/Γwhich is induced by the representationˉρ|Γon V resp.V0. Then E0is a holomorphically trivial vector bundle andΓ(X,E0)=Γ(X,E). For antiholomorphic representations we are now able to give a precise description of the global sections of the associated vector bundle. Proposition9.Let G be a connected complex Lie group,Γa lattice,V a complex vector space andρ:G→GL(V)be an antiholomorphic representation.Let E denote the?at vector bundle over G/Γwhich is induced byρ|Γ. LetΣdenote the set of all vectors v∈V which are invariant underρ(G′)and a ρ(γ)-eigenvector with real eigenvalue for everyγ∈Γ. Then H0(G/Γ,E)?<Σ>C where<Σ>C denotes the complex vector space spanned byΣ. Proof.By construction<Σ>C is the largestρ(G)-invariant vector subspace of V inducing a holomorphically trivial vector subbundle of E. 11.Subbundles of flat bundles Theorem6.Let G be a connected complex Lie group,Γa lattice and E→G/Γbe a?at vector bundle given by an essentially antiholomorphic representationρ:G→GL(n,C). Assume that every meromorphic function on X=G/Γis constant. Let L?E be a vector subbundle.Then L also admits a?at connection. If G=G′,then L is parallel with respect to the given?at connection on E. FLAT VECTOR BUNDLES OVER PARALLELIZABLE MANIFOLDS15 Proof.Let k=rank(L).By passing toΛk L?Λk E we may assume that L is a line bundle.The?at connection on E yields a canonical way to lift the G-action on X to a G-action on E.The subbundle E0=⊕g∈G g?L is a G-invariant subbundle of E and therefore parallel with respect to the?at connection on E.Moreover it is given by an essentially antiholomorphic representation(see cor.3).We may thus assume E0=E.Let d=rank(E0)and choose g0,...,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 with0≤m≤d and every choice of0≤i1<...i m≤d the subsheaf of E spanned by L i 1+...+L i m has rank m. Claim9.The line bundles L0,...,l d are in general position at every point and there-fore yield a trivialization of P(E0). Proof.Let U be an open neighbourhood of a point such that all the L i admit nowhere vanishing sectionsσi.For every k∈{0,...,d}we obtain a sectionαk in∧d E0by αk=∧i=kσi.Letα=?kαk.Thenαis a section in the line bundle?d+1(∧d E0)and is vanishing exactly where the L0,...,L d fail to be in general position.Thus the set of all points in X where the L0,...,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 L0,...,L d are in general position everywhere. Thus P(E0)is a holomorphically trivial P d?1-bundle.Observe that P(E0)is de?ned by an essentially antiholomorphic group homomorphism?ρ:Γ→P GL(d,C).Using thm.2it follows that?ρ|G′∩Γis trivial.Hence?ρ(Γ)constitutes a commutative subgroup of P GL(d,C).This implies that there is a?xed point for the?ρ(Γ)-action on P(V). Corresponding to this?xed point there exists a sub-line bundle L′?E which is G-invariant and therefore parallel to the given?at connection on E. Claim10.The line bundles L′and L are isomorphic(as holomorphic line bundles). Proof.Recall that P(E)is holomorphically trivial and that every meromorphic func-tion 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?at line bundle.However,it is parallel with respect to the given?at connection on E if and only if it is G-invariant.This in turn is equivalent to the assertion that L corresponds to a?xed point of theΓ-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) 16J¨ORG WINKELMANN [5]Kazdan,D.A.;Margulis,G.A.,A proof of Selbergs hypothesis.Math.Sbornik75,162–168 (1968) [6]Malcev,A.,On isomorphism matrix representations of in?nite groups.Mat.Sb.8,405-422 (1940) [7]Margulis,G.A.,Discrete Subgroups of Semisimple Lie Groups.Springer Berlin Heidelberg New York1989 [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.Habilita- tionsschrift. [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 https://www.wendangku.net/doc/fc17419530.html,/MSRI-preprints). J¨o rg Winkelmann,Math.Institut NA4/69,Ruhr-Universit¨a t Bochum,44780Bochum, Germany E-mail address:jw@cplx.ruhr-uni-bochum.de