Vector bundles

C. Vector Bundles over Parallelizable Manifolds

1. Survey

We study vector bundles over complex parallelizable manifolds. Naturally the most easily accessible vector bundles are those which are homogeneous, i.e. vector bundles E → X = G/Γ such that g^{∗}E ≃ E for all g ∈ G = Aut_{O}(X)^{0} . This is equivalent to the assumption that there exists some extension group of G whose action on X can be lifted to the action of E. Furthermore, over a compact complex parallelizable manifold, a vector bundle is homogeneous if and only if it admits a holomorphic connection.

A vector bundle is flat if there exists a holomorphic connection with vanishing curvature. This is equivalent to the assumption that it can be defined by a representation of the fundamental group. For a compact complex parallelizable manifold X it is furthermore equivalent to the assumption that the action of the universal covering group of Aut_{O}(X)^{0} on X lifts to an action on the vector bundle.

Certain homogeneous vector bundles are automatically flat.

Theorem 1. Let X be the quotient manifold of a simply-connected complex Lie group G and a discrete cocompact subgroup Γ. Let E be a homogeneous holomorphic vector bundle over X

Then E is flat if one of the following conditions are fulfilled:

(i) G is semisimple; (ii) E is a vector bundle of rank two and the radical R_{G} of G is nilpotent; (iii) E is a line bundle.

For the special case where X is a torus this is a classical result of Matsushima [22].

For X = G/Γ with G semisimple without SL_{2}(ℂ)-factors we have a classification of homogeneous vector bundles.

Theorem 2. Let G be a simply-connected semisimple complex Lie group, Γ a discrete cocompact subgroup, X = G/Γ. Assume that no direct factor of G is isomorphic to SL_{2}(ℂ).

Then there is a one-to-one correspondence between indecomposable homogeneous vector bundles over X and pairs (ϕ, ζ) where ϕ is an irreducible holomorphic representation of G and ζ : Γ → GL(V) is an irreducible representation of Γ such that ζ(Γ) is a relatively compact subgroup of GL(V).

Compact complex parallelizable manifolds

The correspondence is given in the following way: The vector bundle associated to (ϕ, ζ) is the flat bundle given by the representation ρ = (ϕ̄|_{Γ}) ⊗ ζ of the fundamental group π_{1}(X) ≃ Γ. (ϕ̄ is the antiholomorphic representation obtained from ϕ by complex conjugation.)

An arbitrary homogeneous vector bundle E is in a a unique way a direct sum E = ⊕_{i}E_{i} of indecomposable homogeneous vector bundles E_{i}.

Of course, the set of irreducible representations of Γ with relatively compact image depends on Γ. It includes in particular the irreducible representations with finite images. There are always plenty such representations, because Γ is residually finite as a finitely generated linear group. If Γ is complexarithmetic as defined in I.7, then every representation of Γ with relatively compact image has in fact finite image. In general, superrigidity results ([21]) allow a good description of representations with relatively compact image.

One can prove that for fixed Γ and fixed n ∈ ℕ there exist only finitely many non-conjugate representations ζ : Γ → GL_{n}(ℂ) with relatively compact image. It follows that for fixed X = G/Γ and fixed n ∈ ℕ there exists only finitely many homogeneous vector bundles of rank ≤ n over X up to isomorphism.

For arbitrary G we have a criterion to determine which representations of the fundamental group yield holomorphically non-trivial bundle.

Here some word of caution is in order. Given a compact manifold, any nontrivial representation of the fundamental group yields a bundle with flat connection. There exists a trivialization preserving the given flat connection if and only if the representation is trivial. Nevertheless for non-trivial representations it is possible that there exists a trivialization of the induced bundle as holomorphic bundle. For example, consider line bundles over non-compact Riemann surfaces X. For every non-compact Riemann surface P ic(X) = H^{1}(X, O^{∗}) = 0, although Hom(π_{1}(X), ℂ^{∗}) ≠ 1 for every Riemann surface which is not simplyconnected.

For parallelizable manifolds there is an exact criterion determining whether a flat bundle is holomorphically trivial. Let E be a vector bundle E over a compact complex parallelizable manifold X = G/Γ given by a representation of the fundamental group ρ : π_{1}(X) ≃ Γ → GL(V). Then E is trivial as holomorphic vector bundle if and only if ρ can be extended to a holomorphic representation of G.

This implies the expected statement for line bundles. Let X = G/Γ be a compact complex parallelizable manifold. Assume that no direct factor of G/R_{G} is isomorphic to SL_{2}(ℂ). Then there is a one-to-one correspondence between homogeneous line bundles and Hom(Γ, S^{1}) (with S^{1} = {z ∈ ℂ^{∗} : |z| = 1).

For general compact complex manifolds, vector bundles arising from representations of the fundamental group having bounded image are of a particularly simple nature. Let X be a compact complex manifold, ρ_{i} : π_{1}(X) → GL(V) (i = 1, 2) representations such that the images ρ_{i}(π_{1}(X)) are relatively compact subsets in GL(V). Then the induced vector bundles are isomorphic as holomorphic vector bundles if and only if the ρ_{i} are conjugate (by an element in GL(V)).

In particular it follows that, if ρ : π_{1}(X) → GL(V) a representation of the fundamental group with image ρ(π_{1}(X)) ≠ {e} finite, then the induced vector bundle is not holomorphically trivial.

It is natural to ask whether finite image representations always exists. For arbitrary compact complex manifolds X there need not exist a non-trivial representation of π_{1}(X) with finite image. However for compact complex parallelizable manifolds such representations always exist. Let X be a compact complex parallelizable manifold. Then π_{1}(X) is infinite and for every element γ ∈ π_{1}(X) there exists a representation ρ : π_{1}(X) → GL(V) with ρ(π_{1}(X)) finite such that ρ(γ) ≠ 1. As a corollary we obtain the following positive answer to a question posed by Bănică. For every positive-dimensional compact complex manifold X there exists a non-trivial holomorphic vector bundle.

Flat vector bundles are close to being topologically trivial. The presence of a flat connection for a vector bundle E implies that all the Chern classes c_{k}(E) ∈ H^{2k}(X, ℝ) vanish. Furthermore, for every flat line bundle L over a compact complex manifold X, there exists a finite unramified covering π : X_{1} → X such that π^{∗}L is topologically trivial. We prove similar results for flat vector bundles over certain parallelizable manifolds. These results even extend to homogeneous bundles, because (as mentioned above) in many cases homogeneous bundles over parallelizable manifolds are automatically flat.

Theorem 3. Let E be a homogeneous vector bundle over a compact parallelizable manifold X = G/Γ. Then the Chern classes c_{k}(E) ∈ H^{2k}(X, ℂ) are zero.

Next we study vector bundles which are not necessarily homogeneous.

Theorem 4. Let X = G/Γ be a compact complex parallelizable manifold, where G is semisimple.

Then every line bundle is topologically equivalent to a homogeneous line bundle. Moreover NS(X) = P ic(X)/P ic^{0}(X) is finite and therefore the first Chern class c_{1}(E) vanishes for every holomorphic vector bundle E → X.

If G has no SL_{2}(ℂ)-factors, then P ic(X) itself is finite.

For certain quotients X = SL_{2}(ℂ)/Γ we can prove the existence of nonhomogeneous vector bundles.

Theorem 5. Let Γ be a torsion-free discrete cocompact subgroup of SL_{2}(ℂ), X = SL_{2}(ℂ)/Γ, C an elliptic curve in X. Assume that Γ/Γ′ is finite.

Then there exists a holomorphic vector bundle E of rank two with a section σ ∈ H^{0}(X, E) such that C is the zero-set of σ.

Moreover, this vector bundle E is not homogeneous.

Recall that such a quotient manifold SL_{2}(ℂ)/Γ always contains elliptic curves by the results of Chapter A.

We also prove that elliptic curves are the only smooth curves in SL_{2}(ℂ)/Γ which can arise as zero-sets of sections in holomorphic rank two vector bundles.

Instead of vector bundles we usually will consider the associated GL_{n}(ℂ)- principal bundle, since usually the relevant properties hold for a vector bundle if and only if they hold for the associated GL_{n}(ℂ)-principal bundle.

Many of the above results generalize to principal bundles with a structure group other than GL_{n}(ℂ).

2. Generalities on connections and homogeneous vector bundles

Here we collect some well-known basic facts on connections.

A connection D on a vector bundle is a rule for differentiating sections. A precise definition may be given in the following way:

Definition. Let E be a holomorphic vector bundle, E the sheaf of holomorphic sections in E and Ω^{1} the sheaf of holomorphic one-forms. A holomorphic connection D on a holomorphic vector bundle E → X is a morphism of sheaves D : E → Ω^{1} ⊗ E such that D(fσ) = (df) ⊗ σ + f(Dσ) for every open subset U and f ∈ O(U), σ ∈ E(U).

For every holomorphic connection D on the trivial bundle there exists a r × r-matrix of one-forms A_{i,j} such that for a section s = (s_{1}, …, s_{r}), D(s) = ds + A(s) (with A(s) = (∑_{j} A_{1j} ⊗ s_{j}, … ∑_{j} A_{rj} ⊗ s_{j})). Given a holomorphic vector bundle with local trivialization on an open covering U_{i} this implies that there is a correspondence between holomorphic connections and a collection of r ×r-matrices of one-forms A^{i} on U_{i} fulfilling certain transition relations. This yields that there is an obstruction to the existence of a holomorphic connection given by an element in the cohomology group H^{1}(X, Ω^{1} ⊗ Hom(E, E)). For a line bundle L given by a cocycle ϕ_{ij} ∈ O^{∗}(U_{ij}) this obstruction is given by the element in H^{1}(X, Ω^{1} ⊗ Hom(L, L)) ≃ H^{1}(X, Ω^{1}) given by (1/ϕ_{ij})dϕ_{ij} = d log ϕ_{ij}.

Connections and Group Actions

A connection D on a vector bundle E over a manifold X induces a direct sum decomposition of the tangent bundle T E = V E ⊕HE, where V E consists of the vertical tangent vectors (those tangent to the fibers of E → X) and HE consists of the horizontal tangent vectors. Here a tangent vector v ∈ T_{p}E is called horizontal if there exists an open neighbourhood U of x = π(p) in X and a section σ ∈ E(U) such that D_{π_{∗}v}σ = (Dσ)(π_{∗}v) = 0 and v is tangent to {σ(x) : x ∈ U}.

This decomposition yields a natural way of lifting vector fields from the base manifold X to horizontal vector fields on E. Moreover the construction implies that horizontal vector fields induce local 1-parameter groups of vector bundle automorphisms of E.

Next we note

Lemma 1. Let M, N be manifolds, X, Y vector fields on M resp. N. Let f be a function on M. Assume that X and Y are globally integrable.

Then the vector field X + fY on M × N is globally integrable.

(For notational simplicity the function f and the vector fields X and Y are identified with there respective lifts to M × N.)

Proof. Let μ : I × M → M, ν : I × N → N (with I = ℝ or I = ℂ) be the induced one-parameter groups, i.e. μ_{∗}\frac{∂}{∂t} = X and ν_{∗}\frac{∂}{∂t} = Y . Then η : I × M × N → M × N defined by η = (μ(t, m), ν(f(n)t, n)) yields the desired one-parameter group.

Now consider a vector field X lifted to a horizontal vector field X̃ on a vector bundle E with respect to a connection. Locally the vector bundle is trivial and with respect to any trivialization X̃ = X + fY where f is a function on the base manifold and Y a fundamental vector field for the action of GL_{n}(ℂ) on the fiber ℂ^{n} of the vector bundle. Thus using the preceding lemma one can deduce Proposition 1. Let M be a manifold, E → M a vector bundle, D a connection and X̃ the lift of a vector field X on M with respect to D.

Then X̃ is globally integrable on E if and only if X is globally integrable on M. Furthermore if X̃ is globally integrable, then the induced one-parameter subgroup acts on E by vector bundle automorphisms.

We will frequently need the fact that, for a principal bundle over compact complex manifold, the group of bundle automorphisms is a Lie group.

Proposition (Morimoto, [25]). Let X be a compact complex manifold, H a complex Lie group, E → X a (holomorphic) H -principal bundle. Let G denote the group of bundle automorphism, i.e. the group of all biholomorphic self-maps of E which commute with the H -principal right action.

Then G is a complex Lie group with respect to compact-open topology.

Definition. Let E → X be a vector bundle, G a group acting on X. The bundle E is called homogeneous with respect to the G-action, if for every g ∈ G the induced automorphism λ_{g} : X → X can be lifted to a bundle automorphism g̃ : E → E.

We do not require any uniqueness of g̃ and in fact the lack thereof may imply that the G-action can not be lifted to E.

Using the above proposition, the ”lifting” property can be formulated in various ways.

Lemma 2. Let X be a complex manifold, E → X a holomorphic vector bundle and G ⊂ Aut_{O}(X).

Then the following conditions are equivalent

(i) E is homogeneous (with respect to the G-action); (ii) For every g ∈ G the bundle L_{g}^{∗}E → M obtained by pull-back via the

translation L_{g} : X → X is isomorphic to E; (iii) Let G̃ denote the group of all vector bundle automorphisms of E which

induce on X an automorphism contained in G. Then G̃ → G is surjective.

Let L denote the group of all vector bundle automorphisms inducing the identity map on X. Then condition (iii) states that we have a short exact sequence of groups

1 → L → G̃ →^{α} G → 1

In order to lift the G-action on X to an action on E we need a splitting of this sequence. In general this sequence need not be split. For instance, let E = O(1) be the ample generator of the Picard group on ℙ_{1} . With G = Aut_{O}(ℙ_{1}) = P SL_{2}(ℂ) we obtain L ≃ ℤ_{2} × ℂ^{2} and G̃ ≃ SL_{2}(ℂ)× ℂ^{2} and the sequence 1 → L → G̃ → G → 1 is not split.

The following two propositions are well-known. Nevertheless we sketch the proofs for completeness.

Proposition 2. Let X be a complex manifold on which a connected complex Lie group G acts holomorphically. Let E be a holomorphic vector bundle which admits a holomorphic connection.

Then E is homogeneous.

Proof. Using the connection one can lift vector fields and thereby one-parameter groups of automorphisms. Since G is connected, it is generated by its one-parameter subgroups.

Proposition 3. Let X be a compact parallelizable manifold, G = Aut_{O}(X)^{0} , E a homogeneous vector bundle. Then E admits a holomorphic connection.

Proof. Let F denote the connected component of the automorphism group of E. By definition of homogeneity the natural group homomorphism F → G is surjective. Let H ⊂ Lie(F) denote a subvectorspace of Lie(F) such that the induced projection map H → Lie(G) is a vector space isomorphism. Then the fundamental vector fields on E associated to elements X ∈ H yield a horizontal distribution HE ⊂ T E which determines a connection.

3. Factors of automorphy

Factors of automorphy are used to study fiber bundles E → X which become trivial after pulling back to the universal covering. For our purposes this theory is particularly useful, because many holomorphic bundles over compact complex parallelizable manifolds become trivial after pulling back to the universal covering which is a Stein Lie group.

Proposition 4. Let G be a simply-connected complex Lie group. Let E be a holomorphic vector bundle over G. Assume that at least one of the following conditions is fulfilled.

(1) E is topologically trivial;

(2) G is solvable;

(3) G/R ≃ SL_{2}(ℂ), where R denotes the radical of G;

(4) E is a line bundle.

(5) E admits a holomorphic connection;

(6) E is homogeneous.

Then E is holomorphically trivial.

Proof. By a result of Matsushima and Morimoto [23] a simply-connected complex Lie group is Stein as a complex manifold. Hence Grauert’s Oka-principle implies that every topologically trivial vector bundle is already holomorphically trivial.

The topologically classification of vector bundles is given by homotopy classes of continuous mappings to classifying spaces. Since every simply-connected solvable Lie group is contractible as a topological space, this implies (2).

In case (3), G is homotopy-equivalent to the 3-sphere S^{3} on which every vector bundle is topologically trivial.

A topological classification of complex line bundles over a manifold X is given by H^{2}(X, ℤ). Now for a simply-connected Lie group G we have π_{1}(G) = π_{2}(G) = {e} [9]. Thus H_{1}(G, ℤ) = {0} = H_{2}(G, ℤ) by the Hurewicz isomorphism. In particular Ext(H_{1}(G, ℤ), ℤ) = {0} = Hom(H_{2}(G, ℤ), ℤ). Therefore H^{2}(G, ℤ) = {0} by a universal coefficient theorem.

Finally for the case (5) ⟺ (6) note that in this case the G-action on G may be lifted to an action of an extension Lie group of G on E. Let S be a maximal connected semisimple Lie subgroup of G. Simply-connectedness of G forces S to be simply-connected. It follows that every Lie group extension of S is split. Therefore the S -action on G lifts to an action of S on E and similarily to an action on the associated GL_{r}(ℂ)-principal bundle P . Now the S -orbits in P yield sections in P |_{S(x)} . Hence for x ∈ G the restricted principal bundle P |_{S(x)} is trivial. Since S is a deformation-retract of G, this implies that P (hence E) is topologically and therefore holomorphically trivial.

We now summarize basic facts on factors of automorphy. This theory is valid both in the differentiable and the holomorphic category.

Let X be a (differentiable or complex) manifold, π : X̃ → X the universal covering and Γ = π_{1}(X), acting on X̃ from the right on X̃ by deck transformations. Furthermore let H be a Lie group, acting from the left on a manifold F. Let E → X denote a fiber bundle with F as fiber and H as structure group. Assume that π^{∗}E → X̃ is trivial.

Then there exists a map f : Γ × X̃ → G, called factor of automorphy such that E ≃ (X̃ × F)/ ∼ with (x, z) ∼ (x′, z′) if and only if (x′, z′) = (xγ^{−1}, f(γ, x)z) for some γ ∈ Γ. Conversely, given an arbitrary map f : Γ × X̃ → G one can define a bundle in this way, provided ∼ is a equivalence relation. This is equivalent to the condition

f(γ̃γ, x) = f(γ̃, xγ^{−1}) · f(γ, x)  (Z)

82 C.3 for all γ̃, γ ∈ Γ, x ∈ X̃.

Two factors of automorphy f, f′ define isomorphic bundles if and only if there exists a map g : X̃ → G such that

g(xγ^{−1}) · f′(γ, x) = f(γ, x) · g(x)  (B)

for all x ∈ X̃, γ ∈ Γ.

Thus the set of all bundles with fiber F and structure group G over X becoming trivial after pulling back to the universal covering may be identified with equivalence classes of maps f : Γ × X̃ → G fulfilling (Z) with f ∼ f′ if and only if there exists a map g : X̃ → G such that (B) is fulfilled.

Using group cohomology this may be reformulated as follows: There is a one-to-one correspondence of bundles over X becoming trivial after pulling back to the universal covering and the Galois cohomology set H^{1}(Γ, M) where M denotes the Γ-module of mappings from X̃ → G with the Γ-action induced by the usual action by deck transformations on X̃.

Flat bundles

The simplest case of factors of automorphy f : Γ×X̃ → G are those which are constant in the second variable, i.e. f(γ, x) = f_{0}(γ). In this case condition (Z) reduces to f_{0}(γ̃γ) = f_{0}(γ̃) · f_{0}(γ).

Thus these special factors of automorphy simply correspond to group homomorphisms f_{0} ∈ Hom(π_{1}(X), G). Bundles arising in this way are called bundles given by a representation of the fundamental group or flat bundles.

A vector bundle is given by a representation of the fundamental group if and only if it admits a connection with vanishing curvature ([4]).

It is a natural idea to construct non-trivial bundles via representations of the fundamental group. In this case the equivalence for factors of automorphy (B) takes a special form.

Lemma 3. Let ρ, ρ̃: π_{1}(X) → G be a group homomorphisms. Then the induced bundles E, Ẽ are equivalent if and only if there exists a mapping g : X̃ → G with

g(xγ^{−1}) = ρ(γ^{−1}) · g(x) · ρ̃(γ)

for all γ ∈ π_{1}(X), x ∈ X̃.

In other words: Two G-principal bundles defined by group homomorphisms ρ, ρ̃ : π_{1}(X) → G are equivalent as G-principal bundles if and only if there exists an π_{1}(X)-equivariant map from the universal covering to G where the π_{1}(X)-action on G is given by

γ : g ↦ ρ(γ^{−1}) · g · ρ̃(γ).

Warning. Every G-principal bundle defined by a group homomorphism ρ : π_{1}(X) → G admits a flat connection in a canonically way. Here we discussed only equivalence as G-principal bundles. Instead, one can discuss equivalence as flat G-principal bundles, i.e. ask for bundle isomorphisms which are compatible with the flat connection. It turns out that two G-principal bundles defined by group homomorphisms ρ, ρ̃ are isomorphic as flat G-principal bundles if and only if ρ = ρ̃.

4. Sections in homogeneous vector bundles

As an elementary consequence of the universality of Tits-fibration, it follows that homogeneous vector bundles over compact complex parallelizable manifolds have sections only insofar as they are trivial.

Lemma 4. Let X = G/Γ be a compact complex parallelizable manifold, E → X a homogeneous vector bundle.

Then E admits a holomorphic subbundle E_{0} such that

(1) E_{0} is a holomorphically trivial vector bundle;

(2) Γ(X, E) ≃ Γ(X, E_{0}) ≃ ℂ^{r} with r = rank(E_{0}).

Proof. Consider the sheaf E of holomorphic sections in E. Let E_{0} denote the subsheaf spanned by global sections. Homogeneity implies that E_{0} is again locally free, i.e. corresponds to a subbundle E_{0} of E which is spanned by global sections. Now we obtain a holomorphic map from X to a Grassmann manifold by

ϕ : x ↦ V_{x} = {σ ∈ Γ(X, E_{0}) : σ(x) = 0}

This map is evidently equivariant. Hence it must be constant due to universality of Tits fibration. It follows that E_{0} is holomorphically trivial.

5. Group extensions

We have already emphasized that, given a homogeneous bundle E over a homogeneous manifold X = G/H, one can in general not lift the G-action.

Instead of lifting the action of G one has to pass to an extension group G̃. Naturally one would like to keep this extension as small as possible. In particular if G̃ contains a subgroup such that the natural projection to G is still surjective, G̃ should be replaced be this subgroup. We now formalize this.

Definition. Let

1 → L → G̃ → G → 1

be a short exact sequence of connected Lie groups.

Such an extension is called essential if for every Lie subgroup H ⊂ G̃ either H = G̃ or H · L ≠ G.

This implies some restriction on L.

Lemma 5. Let

1 → L → G̃ → G → 1

be an essential extension of connected Lie groups.

If G is semisimple, then L = {e}.

Proof. This is a consequence of Levi-Malcev-decomposition applied to G̃.

Proposition 5. Let

1 → L → G̃ → G → 1

be an essential extension of connected Lie groups.

Then L ⊂ R′ where R′ denotes the commutator group of the radical R of G̃.

In particular L is nilpotent.

Proof. Let S ·R be a Levi-Malcev-decomposition for G̃, i.e. S is a maximal connected semisimple Lie subgroup and R the radical. Let S_{L} denote a maximal connected semisimple Lie subgroup of L. Using conjugation we may assume that S_{L} ⊂ S. Now S_{L} is a normal Lie subgroup of the semisimple Lie group S. Due to semisimplicity there exists a normal Lie subgroup S_{0} ⊂ S such that S_{0} · S_{L} = S and S_{0} ∩ S_{L} is discrete. Now G_{0} = S_{0} · R constitutes a Lie subgroup of G̃ with G_{0} · L = G̃. Hence G_{0} = G̃ by the assumption of essentiality. Therefore S_{L} = {e}, i.e. L is solvable.

We consider the adjoint action on Lie(G̃). As a linear representation of a semisimple group Ad(S) is completely reducible. Clearly Lie(L), Lie(R) and Lie(R′) are all stable under Ad(S). It follows that there is a direct sum decomposition as a vector space Lie(R) = V ⊕ W , where V and W are Ad(S)-stable vectorspaces such that Lie(L) = V ⊕ Lie(R′ ∩ L) and Lie(R′) ⊂ W . Now Lie(R′) ⊂ W implies that

85 C.5

W is actually a Lie subalgebra. Since it is Ad(S)-stable, it follows that Lie(S)⊕W is a Lie subalgebra of Lie(G̃). Now the assumption of (∗) being an essential extension implies that Lie(S)⊕W = Lie(G̃). It follows that V = {0}, hence Lie(L) ⊂ Lie(R′).

Proposition 6. Let

1 → L → G̃ → G → 1

be an essential extension of connected Lie groups.

Assume that the radical R(G) of G is nilpotent. Then the radical R = R(G̃) of G̃ is also nilpotent.

Proof. By arguments similar to those in the preceding proof, there is a direct sum decomposition Lie(R) = Lie(R ∩ G′) ⊕ V of Ad(S)-stable vectorspaces, where S is a maximal connected semisimple Lie subgroup of G̃. In order to show that R is nilpotent, it suffices to prove that ad(v)|_{Lie R} is nilpotent for every v ∈ Lie(R) (Engel’s theorem). Clearly ad(v)|_{Lie R} is a nilpotent for every v ∈ Lie(G′ ∩R). Thus it suffices to prove that ad(v) is nilpotent for v ∈ V . For a given v ∈ V set

Φ_{v} = {w ∈ Lie(G̃) : ∃n : ad(v)^{n}(w) = 0}

Thus Φ_{v} is the largest subvectorspace of Lie(G̃) on which ad(v) is nilpotent. We claim that Φ_{v} is a Lie subalgebra. Indeed, if ad(v)^{n}(x) = 0 and ad(v)^{m}(y) = 0 for some n, m ∈ ℕ, x, y ∈ Lie(G̃), repeated application of the Leibnitz rule yields ad(v)^{n+m−1}[x, y] = 0. Recall that V is Ad(S)-stable and V ∩ Lie(G′) = {0}. Hence [v, s] = 0 for s ∈ Lie(S). It follows that Lie(S) ⊂ Φ_{v}. Furthermore nilpotency of R/L implies that Lie(R) ⊂ Lie(L) + Φ_{v}. Therefore Φ_{v} + Lie(L) = Lie(G̃). Since the extension was assumed to be essential, it follows that Φ_{v} = Lie(G̃), i.e. ad(v) is nilpotent.

These two propositions imply that a number of properties of Lie groups are preserved under essential extensions.

Corollary. The following properties of a connected Lie group are preserved by essential extensions: α) G is solvable resp. nilpotent; β) G is perfect, i.e. G = G′; γ) G/R contains no simple factor isomorphic to SL_{2}(ℂ) or P SL_{2}(ℂ), where

R denotes the radical. δ) The radical of G is nilpotent. ϵ) G is semisimple.

86 C.5

So far we deduced some general results on essential extension of connected Lie groups. We will now provide a more specialized result needed later in order to prove that every homogeneous line bundle is flat.

Proposition 7. Let

1 → L → G̃ → G → 1  (∗)

be an extension of connected complex Lie groups.

Assume that L is a non-compact central Lie subgroup of G̃ and that G̃ admits a discrete subgroup Λ with L ∩ Λ = {e} finite, L · Λ closed and G̃/LΛ compact.

Then R̃′ ∩ L is discrete, where R̃ denotes the radical of G̃. In particular, (∗) is not essential.

Proof. Observe that R̃Λ is closed in G̃, because for a compact complex parallelizable manifold such as G̃/LΛ the radical orbits are closed. Hence without losing generality we may assume that G̃ is solvable.

Now L is a connected commutative Lie group. Hence there exists a discrete subgroup Π ⊂ L with L/Π compact. Let Γ = Π · Λ. This is a group, because Π is central. It is easily verified that Γ is discrete and G̃/Γ compact. Solvability of G̃ implies that G̃′ ∩ Γ is a subgroup of finite index in Γ′. But Γ′ = Λ′ due to centrality of Π. Hence Γ′ ∩ L = {e} implying finiteness of (G̃′ ∩ Γ ∩ L). Observe that the G̃′ -orbits in G̃/Γ are closed and that L/(Γ ∩ L) is compact. Therefore it follows that G̃′ ∩ L is discrete. Using proposition 6 it follows that (∗) is not essential.

6. Flatness theorems

We will now translate the results on essential Lie group extensions to theorems stating that certain homogeneous bundles are automatically flat. First we develop some auxiliary results which establish the relationship between essential Lie group extensions and non-flatness of homogeneous bundles.

Lemma 6. Let G be a simply-connected complex Lie group, Γ a discrete cocompact subgroup, X = G/Γ, S a complex Lie group and E → X a homogeneous S -principal bundle.

Assume that the G-action on X can be lifted to a G-action on E. Then E → X admits a flat holomorphic connection.

Proof. A connection on a principal bundle can be described in terms of a subbundle H ⊂ T E of ”horizontal” vector fields. This subbundle has to satisfy the following conditions:

(1) T E = H ⊕ V where V denotes the subbundle of vertical tangent vectors, i.e. the subbundle of all tangent vectors which are tangent to the fibers of the projection E → X.

(2) H must be invariant under the S -principal right action on E.

If the G-action on X lifts to a G-action on E, then such a subbundle H is given by H_{x} = T_{x}(G · x). Furthermore in this case H is clearly involutive, because it is the tangent bundle of a foliation. This implies that the curvature of the associated connection is zero.

Corollary. Let X = G/Γ be a compact complex parallelizable manifold, S a complex Lie group and E → X a homogeneous S -principal bundle.

Then either E admits a flat holomorphic connection or there exists an essential extension of connected Lie groups

1 → L → G̃ →^{τ} G → 1

such that dim_{ℂ}(L) > 0 and the G̃-action on X induced by τ lifts to an action on E.

Corollary. Let X = G/Γ be a compact complex parallelizable manifold, S a complex Lie group and E → X a homogeneous S -principal bundle Assume that G is semisimple.

Then E admits a flat connection and consequently is given by a representation of the fundamental group ρ : π_{1}(X) → S.

This follows, because a semisimple Lie group does not admit any essential extension.

For a homogeneous bundle E without flat connection we can say more about the action of the extension group G̃ on E.

Lemma 7. Let X = G/Γ be a compact complex parallelizable manifold, H a complex Lie group, E → X a homogeneous H -principal bundle.

Let 1 → L → G̃ → G → 1 be an extension of connected Lie groups such that the G̃-action on X lifts to E. Assume that G̃ acts almost effectively on E.

Then L acts almost freely on E, i.e. all the isotropy groups are discrete.

Proof. For every x ∈ X let V_{x} denote the set of all g ∈ L which act trivially on the fiber E_{x}. Then V_{x} is a complex Lie subgroup of L and we obtain a holomorphic map x ↦ Lie(V_{x}) which is a G-equivariant holomorphic map from X to a Grassmann manifold, hence constant. Thus V_{x}^{0} = V_{y}^{0} for all x, y ∈ X. Since G̃ acts almost effectively on E, it follows that V_{x}^{0} = 1, i.e. L acts almost effectively on every fiber.

Now observe that L consists of bundle automorphisms, i.e. commutes with the principal H -right action on E. For any group H the only self-maps f : H → H commuting with all right translations are the left translations. Hence for every fiber E_{x} there is a group homomorphism ρ : L → H and an isomorphism E_{x} ≃ H such that L acts by left translations via ρ. Since L acts almost effectively on every fiber, it follows that L acts almost freely on every fiber.

Corollary. Under the assumptions of the proposition dim_{ℂ}(G̃) ≤ dim_{ℂ}(G) + dim_{ℂ}(H) = dim_{ℂ}(E).

Theorem 6. Let H = ℂ^{∗} or H = ℂ, X a compact complex parallelizable manifold.

Then every homogeneous H -principal bundle over X is flat.

Proof. Let E be a homogeneous H -principal bundle. Write X as a quotient G/Γ of a connected complex Lie group G by a discrete cocompact subgroup Γ. Due to homogeneity of E there is an extension of connected complex Lie groups

1 → L → G̃ → G → 1

such that the G̃-action on X lifts to an action on E. We may require that G̃ acts effectively on E.^{∗} Now the corollary to lemma 8 implies that dim_{ℂ}(G̃) ≤ dim_{ℂ}(G) + 1. If dim_{ℂ}(G̃) = dim_{ℂ}(G), the bundle is flat. Hence we may assume that dim_{ℂ}(G̃) = dim_{ℂ}(G) + 1. Again due to lemma 8 this implies that G̃ acts transitively on E. We have to study the H -action in detail. Since H is commutative, right and left translation on H coincide. Therefore every element g ∈ L determines a map ϕ_{g} : X → H such that g is acting on the fiber E_{x} as right translation by ϕ_{g}(x). Now X is compact and H Stein, hence ϕ_{g} is constant. This implies in particular that L acts freely on every fiber. Thus we are now able to invoke proposition 8 with an isotropy group of the G̃-action on E as Λ. It follows that the extension 1 → L → G̃ → G → 1 can not be essential. Hence E is flat.

Corollary. Let X = G/Γ be a compact parallelizable manifold, L a homogeneous line bundle.

Then L is flat, i.e. admits a flat holomorphic connection.

∗ To obtain an effective action of G̃ we have to drop our usual assumption that G is simply-connected.

89 C.7 7. Homogeneous vector bundles of rank two

In many cases a homogeneous vector bundle of rank two is automatically flat. The key reason for this is that GL_{2}(ℂ) contains no interesting nilpotent subgroup and essential extensions always involve nilpotent groups.

Theorem 7. Let G be a simply-connected complex Lie group, Γ a discrete cocompact subgroup, E a homogeneous vector bundle of rank two over X = G/Γ. Assume that the radical of G is nilpotent.

Then E is flat.

As preparation for the proof we present several technical results on Lie groups.

Lemma 8. Let N be a nilpotent connected Lie group, H a closed subgroup with N/H compact.

Then N′/H′ is compact, too.

Proof. There is no loss in generality, if we replace N by the universal covering Ñ and H by its pre-image under the projection π : Ñ → N.

By a criterion of Malcev, compactness of N/H is equivalent to the existence of a continuous faithful unipotent representation ρ : N → GL_{n}(ℝ) such that ρ(N) and ρ(H) have the same Zariski-closure in GL_{n}(ℂ) ([20],[32]). Let I denote this Zariskiclosure. Then both ρ(N)′ and ρ(H)′ are Zariski-dense in I′. Thus ρ(N′) and ρ(H′) have the same Zariski-closure, and Malcev’s criterion implies that N′/H′ is compact.

Proposition 8. Let N be a connected nilpotent Lie group and H a closed Lie subgroup such that N/H is compact and H^{0} ⊂ N′.

Then H^{0}/(H^{0} ∩ H′) is compact, too.

Proof. By the preceding lemma, compactness of N/H implies that N′/H′ is compact. Now H/H^{0} is discrete, hence H^{0} ⊂ H′H^{0} ⊂ H implies that H′H^{0} is closed. Thus H′H^{0}/H′ is a closed subset of N′/H′ and therefore compact. Finally note that H′H^{0}/H′ ≃ H^{0}/(H^{0} ∩ H′).

Corollary. Let N be a connected nilpotent complex Lie group and H a closed complex Lie subgroup with N/H compact and H^{0} ⊂ N′. Let ρ : H → GL_{2}(ℂ) be a holomorphic group homomorphism.

Then H^{0} ⊂ ker ρ.

Proof. Let I denote the Zariski-closure of ρ(H) in GL_{2}(ℂ). Then I is a nilpotent algebraic subgroup of GL_{2}(ℂ). For any such I the connected component I^{0} is commutative. Hence H admits a subgroup H_{1} of finite index such that ρ(H_{1}) is commutative. It follows that H′_{1} ⊂ ker ρ. Now H′_{1} is of finite index in H′, hence H^{0}/(H^{0} ∩ H′_{1}) is compact. Thus ρ(H^{0}) is compact.

Since ρ was assumed to be holomorphic, ρ(H^{0}) must be a complex Lie subgroup of GL_{2}(ℂ). Therefore compactness of ρ(H^{0}) implies H^{0} ⊂ ker ρ.

Proof of the theorem. For any such vector bundle E there exists an extension of connected Lie groups

1 → L → G̃ →^{τ} G → 1

and a holomorphic group homomorphism ρ : H = τ^{−1}(Γ) → GL_{2}(ℂ) such that E arises as a fibered product G̃×_{H} ℂ^{2} . Furthermore we may assume that L ⊂ R̃′ where R̃ denotes the radical of G̃. Moreover R̃ is nilpotent. Recall that the R-orbits on G/Γ are closed. Hence R̃/(R̃∩H) is compact and we may apply the preceding lemma to R̃ ∩ H ⊂ R̃ and conclude that H^{0} = L ⊂ ker ρ. This implies that E is given by a representation of the fundamental group

π_{1}(G/Γ) = π_{1}(G̃/H) → H/H^{0} →^{ρ} GL_{2}(ℂ)

91 C.7

Flat bundles

Our goal here is to classify flat vector bundles or more generally flat principal bundles. As we previously pointed out, every flat bundle is induced by a representation of the fundamental group ρ : π_{1}(X) → GL_{n}(ℂ) (resp. ρ : π_{1}(X) → G for flat G-principal bundles.) Two representations ρ_{i} : π_{1}(X) → G yield the same holomorphic G-principal bundle if and only if there exists a holomorphic map f from the universal covering X̃ to the structure group G which is π_{1}(X)-equivariant in the following sense:

f(xγ) = ρ_{1}(γ^{−1}) · f(x) · ρ_{2}(γ).

Naturally one desires to obtain a more accessible criterion. One easily shows that a representation ρ : Γ → H induces a trivial holomorphic H -principal bundle on G/Γ if and only if ρ can be extended to a holomorphic group homomorphism ρ̃ : G → H. It is another matter to obtain a criterion which determines whether or not two given non-trivial flat bundles are isomorphic.

For three special cases we are able to provide such a criterion.

First, a special result covers commutative structure groups. In particular we prove the following: Let X = G/Γ, assume G/R contains no SL_{2}(ℂ)- factor. Let H be a commutative complex Lie group and H_{ℝ} a real form which contains the maximal compact subgroup of H. Then there is a one-to-one correspondence between flat H -principal bundles over X and Hom(Γ, H_{ℝ}). In particular this implies that homogeneous line bundles are parametrized by Hom(Γ, S^{1}).

Second, we consider representations with bounded images. If the images ρ_{i}(π_{1}(X)) are both relatively compact in G, then the induced holomorphic principal bundles are equivalent if and only if ρ_{1} and ρ_{2} are conjugate by an element in G. (This is true for every compact complex manifold X, parallelizable or not.)

Thirdly, we deduce the main result of this section which gives a complete classification of homogeneous principal bundles over X = S/Γ with S semisimple without SL_{2}(ℂ)-factors. Superrigidity applies here. Hence for every complex linear-algebraic group H and every group homomorphism ρ : Γ → H there exists a unique continuous group homomorphism ρ̃: S → H such that ρ̃|_{Γ} and ρ coincide on a subgroup of finite index. We call ρ essentially antiholomorphic if ρ̃ is antiholomorphic. We prove that there is a one-to-one correspondence between homogeneous H -principal bundles and conjugacy classes of essentially antiholomorphic representations of Γ → H.

Subsequently we use these results in order to prove the following statement: Every positive-dimensional compact complex manifold admits a nontrivial holomorphic vector bundle.

We study in some detail the question how many representations with finite image exist and obtain two results in different directions: First, for every compact complex parallelizable manifold X = G/Γ and every element γ ∈ Γ ≃ π_{1}(X) there exists a number n ∈ ℕ and a group homomorphism ρ : Γ → GL_{n}(ℂ) with finite image such that ρ(γ) ≠ I. Second, for G semisimple without SL_{2}(ℂ)-factors there is a result in the converse direction: For fixed X = G/Γ and a fixed connected Lie group H there exist only finitely many group homomorphisms from π_{1}(X) to H with finite image (up to conjugacy).

Finally we study the topological nature of flat bundles. In a certain sense flat bundles are automatically almost topologically trivial. For instance, it is easy to see that for flat vector bundles all the Chern classes must vanish. We prove the following statement: Let X = G/Γ and assume that G/R has no SL_{2}(ℂ)-factors. Then for every homogeneous H -principal bundle E over X there exists a finite covering τ : X_{1} → X such that τ^{∗}E → X_{1} is topologically trivial. (This is stronger than the statement on Chern classes, because τ^{∗} : H^{•}(X, ℂ) → H^{•}(X_{1}, ℂ) is injective for every finite covering τ : X_{1} → X.)

8. A non-triviality criterion

The goal is to prove the following criterion which determines whether a flat bundle is holomorphically trivial as principal bundle.

Proposition 9. Let G be a connected complex Lie group, Γ a discrete cocompact subgroup, H a complex Lie group, E a H -principal bundle over X = G/Γ induced by a group homomorphism ρ : Γ → H.

Then E is holomorphically trivial if and only if ρ can be extended to a holomorphic group homomorphism:ρ̃: G → H.

This is a direct consequence of the following (more technical) result.

Lemma 9. Let G be a connected complex Lie group, Γ a discrete cocompact subgroup, H a complex Lie group, ρ : Γ → H a group homomorphism.

Assume that there exists a holomorphic map F : G → H such that F (e) = e and F (gγ) = F (g)ρ(γ) for all g ∈ G, γ ∈ Γ. Then F is a group homomorphism.

Proof. We define an auxiliary map ζ : G × G → H which measures to which extent F fails to be a group homomorphism. This is done by

ζ(g_{1}, g_{2}) = F (g_{1})^{−1}F (g_{1}g_{2})F (g_{2})^{−1}.

Thus ζ(g_{1}, g_{2}) = e if and only if F (g_{1}g_{2}) = F (g_{1})F (g_{2}). Note that ζ(g_{1}, g_{2}γ) = ζ(g_{1}, g_{2}) for all g_{i} ∈ G, γ ∈ Γ. Thus ζ is induced by a holomorphic map ζ_{0} : G × X → H. Furthermore ζ_{0}(g, eΓ) = e and ζ_{0}(e, gΓ) = e for all g ∈ G. It follows that ζ_{0} maps {e} × X to {e}. Let V be a small open heighbourhood of e in H which is biholomorphic to an open subset in ℂ^{n}. Then there exists an open neighbourhood W of {e} × X in G × X such that ζ_{0}(W) ⊂ V . Since X is compact and connected, every holomorphic function on X is constant. With V ↪ ℂ^{n} it follows that ζ_{0}({g} × X) = ζ_{0}(g, e) = {e} for every g ∈ G with {g} × X ⊂ W . Since X is compact, there exists an open neighbourhood U of e in G such that U ×X ⊂ W . Hence ζ_{0} is constant on an open subset of g × X. By the identity principle it follows that ζ_{0} is constant on the whole manifold G × X.

Remark. For the special case where G is a compact complex torus and Γ = {e}, this yields the well-known fact that holomorphic maps between compact complex tori are affine-linear.

9. Abelian representations of the fundamental group

In this section, we classify flat principal bundles with commutative structure group over compact parallelizable manifolds. The key fact is that for a commutative group H (and any group G) the set of group homomorphisms Hom(G, H) is again a (commutative) group in a natural way, viz. by (ρ_{1} · ρ_{2})(g) = ρ_{1}(g) · ρ_{2}(g). Let X be a complex manifold, H a complex Lie group and ρ_{i} : π_{1}(X) → H (i = 1, 2) group homomorphism defining H -principal bundles E_{i} → X. Then E_{1} ≃ E_{2} if and only if the H -principal bundle defined by ρ_{1} ·ρ_{2}^{−1} is trivial. Therefore the above triviality criterion yields the following classification of flat H -principal bundles for commutative structure group H.

Proposition 10. Let G be a simply-connected complex Lie group, Γ a discrete cocompact subgroup, H a connected commutative complex Lie group, Hom(G, H) (resp. Hom(Γ, H)) the group of all homomorphisms of complex Lie groups. Let r : Hom(G, H) → Hom(Γ, H) denote the natural restriction homomorphism.

Then there is a one-to-one correspondence between flat H -principal bundles over X = G/Γ and the quotient group Hom(Γ, H)/r Hom(G, H).

(Murakami [27] proved this for the case where G/Γ is a torus.)

If H has a real form which contains its maximal compact subgroup and SL_{2}(ℂ)-factors are not present, we may improve this result.

Proposition 11. Let G be a simply-connected complex Lie group, Γ a discrete cocompact subgroup, H a connected commutative complex Lie group. Assume that H_{ℝ} is a real form of H such that H_{ℝ} contains the maximal compact real Lie subgroup K of H. Assume moreover that no factor of G/R_{G} is isomorphic to SL_{2}(ℂ), where R_{G} denotes the radical of G.

Then there is a one-to-one correspondence between H -principal bundles on X = G/Γ admitting a flat connection and Hom(Γ, H_{ℝ}).

Proof. The assumptions imply in particular that Γ/(G′ ∩ Γ) is discrete in G/G′ and that (Γ ∩ G′)/Γ′ is finite (theorem A.1).

Note that Γ/Γ′ ≃ H_{1}(X, ℤ) is a finitely generated abelian group, hence isomorphic to A × ℤ^{2n} for some finite abelian group A. Now Hom(Γ, H) = Hom(A, H) ⊕ Hom(ℤ^{2n}, H). Note that Hom(A, H) = Hom(A, H_{ℝ}), because H_{ℝ} is required to contain the maximal compact subgroup of H. Moreover there is a one-to-one correspondence between Hom(ℤ^{2n}, H) and Hom_{ℝ}(G/G′, H). Now Hom_{ℂ}(G/G′, H) ∩ Hom_{ℝ}(G/G′, H_{ℝ}) = {0}, because a holomorphic mapping with totally real image is constant. Using this, dimension reasons imply

Hom_{ℝ}(G/G′, H) = Hom_{ℂ}(G/G′, H) ⊕ Hom_{ℝ}(G/G′, H_{ℝ}).

It follows that Hom(Γ/Γ′) ≃ Hom(Γ/Γ′, H_{ℝ}) ⊕ Hom_{ℂ}(G/G′, H).

Corollary. Let G be a simply-connected complex Lie group and Γ a discrete cocompact subgroup. Assume that no factor of G/R_{G} is isomorphic to SL_{2}(ℂ), where R_{G} denotes the radical of G.

Then there is a one-to-one correspondence between homogeneous complex line bundles over G/Γ and Hom(Γ, S^{1}).

Proof. Every homogeneous line bundle over G/Γ is flat and for H = ℂ^{∗} the subgroup S^{1} = {z ∈ ℂ^{∗} : |z| = 1} is evidently a real form H_{ℝ} containing the maximal compact subgroup.

For G ≃ SL_{2}(ℂ) it is possible that rank_{ℤ}(Γ/Γ′) > 0 for a discrete cocompact subgroup Γ. In this case Hom(Γ, S^{1}) ≠ Hom(Γ, ℂ^{∗}). Since evidently Hom(SL_{2}(ℂ), ℂ^{∗}) = 1, proposition 11 now implies that there exists flat line bundles on X = SL_{2}(ℂ)/Γ which can not be defined by a unitary representation ρ : Γ → S^{1} .

10. Some complex analytic preparation

In order to prove the ”equivalence criterion” for representations of the fundamental group with relatively compact image, we will need some complexanalytic preparation.

Lemma 10. Let X be a compact complex manifold, Γ = π_{1}(X), X̃ its universal covering and X̃ × Γ → X̃ the usual action of deck transformations. Let Y be a Stein complex manifold and K a compact topological group acting from the right on Y by biholomorphic transformation. Assume that there exists a group homomorphism ρ : π_{1}(X) → K and a holomorphic map F : X̃ → Y such that

F (xγ) = F (x) · ρ(γ)

for all γ ∈ π_{1}(X).

Then F is constant.

Proof. Since Y is a Stein manifold, it admits a strictly plurisubharmonic exhaustion function ϕ. By integration over K with respect to Haar measure we may assume that ϕ is K -invariant. Then g = ϕ∘F is π_{1}(X)-invariant and induces a plurisubharmonic function g_{0} : X → ℝ which is constant, because X is compact. It follows that g = ϕ ∘ F is constant. Now assume that F is not constant. Then there exists a holomorphic map j : D → X̃ such that F ∘ j is non-constant, where D denotes the unit disc {z ∈ ℂ : |z| < 1}. If F ∘ j : D → Y is not constant, then D contains a dense open subset Ω where F ∘ j is immersive, i.e. D(F ∘ j) : T_{x}D → T Y is injective for x ∈ Ω. But this implies that ϕ ∘ F ∘ j : Ω → ℝ^{+} is strictly plurisubharmonic. This contradicts ϕ ∘ F being constant. Hence F must be constant.

11. Bounded representations of the fundamental group

We now study representations of the fundamental group with relatively compact image. Here the assumption ”parallelizable” is not used.

Theorem 8. Let X be a compact complex manifold. Let H be a Stein complex Lie group. Let ρ_{1}, ρ_{2} : Γ = π_{1}(X) → H be group homomorphisms such that the images ρ_{i}(Γ) are relatively compact in H.

Then the induced bundles E_{1} , E_{2} are holomorphically equivalent if and only if there exists an element g ∈ H such that

ρ_{1}(γ) = g · ρ_{2}(γ) · g^{−1}

for all γ ∈ Γ, i.e. if and only if ρ_{1} and ρ_{2} are conjugate.

Note that GL_{n}(ℂ) and more generally every complex Lie subgroup of GL_{n}(ℂ) are Stein.

Proof. Assume that the bundles E_{1} and E_{2} defined by ρ_{i} are holomorphically equivalent. Then there exists a holomorphic map f : X̃ → H such that

f(xγ) = ρ_{1}(γ^{−1}) · f(x) · ρ_{2}(γ).  (∗)

The images ρ_{i}(Γ) are subgroups of H which constitute relatively-compact subsets. Recall that for every subgroup in a topological group the closure is again a subgroup. Hence the closures of the ρ_{i}(Γ) form compact subgroups K_{i} of H. Now K = K_{1}×K_{2} is a compact group with an action on the Stein manifold H defined by

μ(k_{1}, k_{2}) : g ↦ k_{1}^{−1} · g · k_{2}.

In this form (∗) reads as

f(xγ) = μ(ρ(γ))(f(x))

with ρ = (ρ_{1}, ρ_{2}). Thanks to the proposition of the preceding section it follows that such a map f must be constant. If f is constant with value g ∈ H, then (∗) translates to

ρ_{1}(γ) = g · ρ_{2}(γ) · g^{−1}.

For the special case H = ℂ^{∗} the theorem implies the well-known fact that two line bundles on a compact complex manifold X given by unitary representations ρ, ρ′ : π_{1}(X) → S^{1} = {z ∈ ℂ^{∗} : |z| = 1} are holomorphically equivalent if and only if ρ = ρ′.

Corollary. Let X be a compact complex parallelizable manifold, H a Stein complex Lie group, ρ : π_{1}(X) → H a group homomorphism with relatively compact (but not trivial) image.

Then ρ defines a bundle which is not holomorphically trivial.

In particular, non-trivial representations with finite image define non-trivial bundles. We will use this fact later to prove that every compact complex manifold admits a non-trivial holomorphic vector bundle.

12. Some algebraic preparation

The purpose of this section is to collect several auxiliary results on algebraic group actions. These results will be needed in the subsequent section on antiholomorphic representations.

Let G be an algebraic group acting algebraically on an algebraic variety Z (everything being defined over an algebraically closed field k). Then every G-orbit in Z is a constructible set which is open in its closure (with respect to the Zariski-topology). This implies: Either G acts transitively on Z or there exists a non-trivial G-invariant closed subvariety W ⊂ Z (Non-trivial means that W is neither empty nor the whole variety Z.) In the second case the ideal I(W) is a G-stable subvectorspace of the algebra of regular functions k[Z]. Now every regular function f ∈ k[Z] is contained in a finite-dimensional G-stable subvectorspace of k[Z], hence every f ∈ I(W) is contained in a finitedimensional G-stable subvectorspace of I(W).

Recall that for a unipotent group G acting on a finite-dimensional vector space V ≠ {0} the subvectorspace V^{G} of G-fixed points is always positivedimensional. Recall furthermore that for an affine variety Z with non-trivial closed subvariety W there always exists a regular function f ∈ k[Z] with f ∈ I(W) but f ≢ 0.

Together these facts imply the following well-known result.

Proposition 12. Let G be a unipotent group acting on an affine variety Z, both defined over an algebraically closed field k.

Then either G acts transitively or there exists a non-constant invariant regular function f ∈ k[Z].

Proof. If G does not act transitively, there is a non-trivial G-stable subvariety W and a non-constant regular function f_{0} ∈ I(W). Furthermore f_{0} is contained in a G-stable finite-dimensional subvectorspace V ⊂ I(W). Finally choose f ∈ V^{G} \ {0}.

Corollary. Let G be a unipotent group acting on an affine variety W, both defined over an algebraically closed field k. Let S ⊂ W be a subset.

Then either S is contained in a G-orbit, or there exists a G-stable closed subvariety X ⊂ W with S ⊂ X and f ∈ k[X]^{G} such that f|_{S} is non-constant.

Proof. Let GS denote the union of all G-orbits intersecting S and apply the proposition to the Zariski-closure Z = \overline{GS}. Finally note that a G-invariant regular function can not be constant on S unless it is constant on \overline{GS}.

The above results apply to any algebraically closed field. Now we return to the complex situation.

Lemma 11. Let X be a complex manifold, U a unipotent group acting on an affine variety Z (both U and Z being defined over ℂ). Let S ⊂ Z be a subset which is relatively compact with respect to the euclidean topology, and ϕ : X → Z be a holomorphic map with ϕ(X) ⊂ US. Furthermore assume that every bounded holomorphic function on X is constant.

Then ϕ(X) is contained in one U -orbit.

Proof. Note that the assumptions of the lemma imply that f ∘ϕ is a bounded holomorphic function on X for every f ∈ ℂ[Z]^{U} . Therefore this statement is a consequence of the preceding algebraic result.

Lemma 12. Let f, g be functions on ℂ^{n} (n ∈ ℕ), f holomorphic, g antiholomorphic. Assume that h defined by h(z) = f(z) + g(z) is a bounded function.

Then both f and g are constant.

Proof. By construction h is a bounded harmonic function on ℂ^{n}, hence constant. Hence ∂h = 0 = ¯∂h. Now df = ∂h and dg = ¯∂h. Thus g and f are constant, too.

Proposition 13. Let U be a complex unipotent group with complex conjugation ξ. Assume that U acts on an affine variety Z and that there are a holomorphic map ϕ : U → Z and a map α : U → Z with relatively compact image. Moreover assume that α(u) = ξ(u) · ϕ(u) for all u ∈ U.

Then both α and ϕ are constant.

Proof. First we apply lemma 12 (with X = U, S = α(U)). This yields that ϕ(U) is contained in a single U -orbit. We have to show that this U -orbit is a fixed point. Assume the contrary. Then there exists a surjective U -equivariant morphism to π : U(z_{0}) → ℂ (with z_{0} ∈ Z such that ϕ(U) ⊂ U(z_{0})). A transitive action of a unipotent group on ℂ is necessarily given by translations. Therefore the relation α(u) = ξ(u) · ϕ(u) yields holomorphic functions f, g ∈ O(U) such that h(u) = π ∘ α(u) = ḡ(u) + f(u). The preceding lemma implies that g is constant. But this is in contradiction to U acting transitively on ℂ. Therefore the U -orbit U(z_{0}) must have been a fixed point, hence α = ϕ. Now boundedness of α(U) in Z implies that F ∘ ϕ is a bounded holomorphic function on U for every ℂ[Z]. Since Z is affine and every bounded holomorphic function on U is constant, it follows that ϕ is a constant map.

Proposition 14. Let G be a linear-algebraic group with G = G′ (defined over ℂ or any algebraically closed field k).

Then there exists a Zariski-open subset Ω_{0} ⊂ G such that for every g ∈ Ω_{0} there exist unipotent elements u, v ∈ G with g = uv.

99 C.12

Proof. Let U denote the unipotent radical of G, π : G → G/U the natural projection. Then g ∈ G is unipotent if and only if π(g) is unipotent. Therefore it suffices to consider the case where U = {e}, i.e. we may assume that G is reductive. Then G = G′ implies that G is semisimple.

Let G_{u} denote the set of all unipotent elements and Ω = G_{u} · G_{u}. Clearly Ω is a constructible subset of G invariant under conjugation. Due to constructibility it is enough to prove that Ω is Zariski-dense. First let us consider the case G ≃ SL_{2}(k). Note that any 2×2-matrix A with det(A) = 1 and T race(A) = λ+λ^{−1} is conjugate to (^{λ}_{λ^{−1}}) unless T race(A) ∈ {−2, 2}. Observe that the trace of the matrix product

\binom{1}{s 1} · \binom{1 t}{1} = ( \begin{matrix}1 & t \\ s & 1 + st \end{matrix} )

equals 2+st. It follows that every A ∈ SL_{2}(k) is contained in Ω except possibly those with T race(A) ∈ {−2, 2}. In particular Ω is Zariski-dense in G for G = SL_{2}(k).

In the case of an arbitrary semisimple algebraic group G, the union of all maximal tori constitutes a Zariski-dense subset. Furthermore, for every torus, the union of all one-dimensional subtori is Zariski-dense. It follows that the union of all onedimensional subtori forms a Zariski-dense subset in G. Now every one-dimensional torus A in a semisimple algebraic group S may be embedded in a subgroup isomorphic to SL_{2}(k) or P SL_{2}(k). Hence the result for G ≃ SL_{2}(k) (which immediately implies the result for P SL_{2}(k)) yields the result for the general case.

Lemma 13. Let G be a connected algebraic group, Ω a Zariski-open subset, Γ a Zariski-dense subgroup.

Then Ω ∩ Γ generates Γ as group.

Proof. Let γ ∈ Γ. Now S = {g ∈ G : γg^{−1} ∈ Ω} ∩ Ω is Zariski-open, hence has non-trivial intersection with Γ. Let ω ∈ S ∩ Γ. Then γ = (γω^{−1})ω with γω^{−1} , ω ∈ Ω ∩ Γ.

13. Antiholomorphic representations

We continue our preparations for the desired classification theorem of flat vector bundles over X = G/Γ with G semisimple without SL_{2}(ℂ)-factors. We need some basic information on essentially antiholomorphic representations.

Definition. Let G be a complex Lie group, Γ a discrete subgroup, H a complex Lie group.

A group homomorphism ρ : Γ → H is called essentially antiholomorphic if there exists an antiholomorphic Lie group homomorphism ρ_{0} : G → H and a group homomorphism ζ : Γ → H such that

100 C.13 (i) the image set {ζ(γ) : γ ∈ Γ} is relatively compact in H; (ii) ρ(γ) = ρ_{0}(γ) · ζ(γ) for γ ∈ Γ;

Remark. Let G, Γ, H as above, Γ_{0} ⊂ Γ a subgroup of finite index, ρ : Γ → H a group homomorphism. Assume that there exists an antiholomorphic Lie group homomorphism ρ_{0} : G → H such that ρ|_{Γ_{0}} = ρ_{0}|_{Γ_{0}} . Then ρ is essentially antiholomorphic (see Lemma I.1).

Our next goal is to classify essentially antiholomorphic representations. The following lemma is the first step.

Lemma 14. Let G be a connected complex Lie group, Γ a discrete cocompact subgroup, H a linear complex Lie group, ρ : Γ → H an essentially antiholomorphic group homomorphism.

Then the antiholomorphic group homomorphism ρ_{0} : G → H and the group homomorphism ζ : Γ → H required by the definition of essentially antiholomorphic are unique.

Furthermore ρ_{0}(g) and ζ(γ) commute for all g ∈ G, γ ∈ Γ.

Proof. The second assertion follows directly from the functional equation ζ(γ) = ρ(γ)ρ_{0}(γ)^{−1} and the assumption that all the maps ζ, ρ and ρ_{0} are group homomorphisms.

To check uniqueness, assume the converse, i.e. let ρ̃_{0} and ζ̃ be group homomorphisms which also fulfill the conditions of the definition. Fix an embedding H ↪ GL(V) and let τ denote complex conjugation on GL(V). Define a holomorphic map f : G → GL(V) by f(g) = τ (ρ_{0}(g)ρ̃_{0}(g)^{−1}). Then

f(gγ) = τ (ρ_{0}(gγ)ρ̃_{0}(gγ)^{−1})
= τ (ρ_{0}(g)ρ_{0}(γ)ρ̃_{0}(γ)^{−1}ρ̃_{0}(g)^{−1})
= τ (ρ_{0}(g)ζ(γ)^{−1}ρ(γ)ρ(γ)^{−1}ζ̃(γ)ρ̃_{0}(g)^{−1})
= τ (ζ(γ)^{−1}ρ_{0}(g)ρ̃_{0}(g)^{−1}ζ̃(γ))
= τ(ζ(γ)^{−1}f(g)τ(ζ(γ))

Since G/Γ is compact, the last equality implies that the image of f is relatively compact in GL(V). But every bounded holomorphic function on a connected complex Lie group is constant. Hence f is constant, i.e. ρ_{0} = ρ̃_{0} and therefore ζ = ζ̃.

Corollary. Let G be a complex Lie group, Γ a discrete cocompact subgroup, ρ : Γ → GL(V) an irreducible representation which is essentially antiholomorphic.

Then there exist vector spaces V_{i} (i = 1, 2), an antiholomorphic representation ρ_{0} : G → GL(V_{1}) and a relatively compact representation ζ : Γ → GL(V_{2}) such that ρ = ρ_{0}|_{Γ} ⊗ ζ.

Thus for a given essentially antiholomorphic representation ρ we have a unique antiholomorphic representation ρ_{0} of G and a unique relatively compact representation ζ of Γ. These are called the associated antiholomorphic resp. bounded representations of ρ. Now we can classify essentially antiholomorphic representations in the following way.

Let R_{ea}(Γ) denote the set of equivalence classes of irreducible representations of Γ which are essentially antiholomorphic. Let R_{bd}(Γ) denote the set of all irreducible representations ρ of Γ with ρ(Γ) relatively compact. Let R_{ℂ}(G) denote the set of equivalence classes of irreducible holomorphic representations of G.

Proposition 15. There is a bijective map

μ : R_{bd}(Γ) × R_{ℂ}(G) → R_{ea}(Γ)

given by μ(ζ, ρ_{0}) = ζ ⊗ (ρ̄_{0}|_{Γ}).

Proof. The only assertion not yet proved is the following: Let ρ̄_{0} ∈ R_{ℂ}(G) and ζ ∈ R_{bd}(Γ). Then ρ = ρ_{0} ⊗ ζ is irreducible. To prove this let W and V denote the respective vector spaces, i.e. ρ_{0} : G → GL(W) and ζ : Γ → GL(V). Let Γ_{0} denote the kernel of ζ : Γ → GL(V) and 1 denote the trivial representation in GL(V). Then Γ_{0} is still cocompact in G and ρ|_{Γ_{0}} = ρ_{0} ⊗ 1|_{Γ_{0}} . It follows that every ρ(Γ_{0})-invariant subspace of W ⊗ V must also be ρ_{0}(G)-invariant. Hence every ρ(Γ)- invariant subspace of W ⊗ V takes the form W ⊗ V_{1} with V_{1} ⊂ V . But irreducibility of ζ implies that there is no such non-trivial invariant subspace.

The following theorems underline the importance of essentially antiholomorphic representations.

Theorem 9. Let G be a simply-connected complex Lie group, Γ a discrete cocompact subgroup, H a complex linear-algebraic group, ρ, ρ̃ : Γ → H two essentially antiholomorphic group homomorphisms. Assume that G = G′.

Then the H -principal bundles over X = G/Γ defined by ρ and ρ̃ are isomorphic if and only if ρ and ρ̃ are conjugate, i.e. if and only if there is an element h ∈ H such that ρ(γ) = hρ̃(γ)h^{−1} for all γ ∈ Γ.

Proof. There are group homomorphisms with bounded image ζ, ζ̃ : Γ → H such that ρ(γ) = ζ(γ)ρ_{0}(γ) and ρ̃(γ) = ζ̃(γ)ρ̃_{0}(γ). Furthermore ζ(Γ) and ζ̃(Γ) commute with ρ_{0}(G) resp. ρ̃_{0}(G).

The induced principal bundles are equivalent if and only if there exists a holomorphic map ϕ : G → H such that

ϕ(gγ) = ρ̃(γ^{−1})ϕ(g)ρ(γ)  (1)

for all γ ∈ Γ, g ∈ G. Define

α(g) := ρ̃_{0}(g)ϕ(g)ρ_{0}(g)^{−1}  (2)

It follows that

α(gγ) = ρ̃_{0}(gγ)ϕ(gγ)ρ_{0}(gγ)^{−1}
= ρ̃_{0}(g)ρ̃_{0}(γ)ρ̃(γ)^{−1}ϕ(g)ρ(γ)ρ_{0}(γ)^{−1}ρ_{0}(g)^{−1}
= ρ̃_{0}(g)ζ̃(γ)^{−1}ϕ(g)ζ(γ)ρ_{0}(g)^{−1}
= ζ̃(γ)^{−1}ρ̃_{0}(g)ϕ(g)ρ_{0}(g)^{−1}ζ(γ)
= ζ̃(γ)^{−1}α(g)ζ(γ)  (3)

Now let F denote a compact subset of G such that ΓF = G. Since ζ and ζ̃ have finite image, it follows that ζ̃(Γ) · α(F) · ζ(Γ) is compact. Hence the above calculations imply that α(G) is relatively-compact in H.

There is an antiholomorphic G-action on H given by μ(g)(h) = ρ̃_{0}(g)hρ_{0}(g)^{−1} . In this notation α(g) = μ(g)(ϕ(g)). Let U denote a unipotent subgroup of G with complex conjugation. Then ϕ|_{U} and α|_{U} are constant (proposition 14). Thus there exists an element c ∈ H such that ϕ(u) = c and α(u) = c for every unipotent element u ∈ G. Now equation (2) implies that ρ̃_{0}(u) = cρ_{0}(u)c^{−1} for every unipotent element u ∈ G. Since G is generated by its unipotent elements, we obtain

ρ̃_{0}(g) = cρ_{0}(g)c^{−1}

for all g ∈ G.

By proposition 15 we know that G contains a Zariski-open subset Ω such that for all g ∈ Ω there exist unipotent elements u, v ∈ G such that g = uv. Since Γ is Zariski-dense in G, it is clear that Ω ∩ Γ generates Γ as group. Let γ ∈ Ω ∩ Γ, γ = uv with u, v unipotent. Then

c = α(v) = α(u^{−1}γ) = ζ̃(γ)^{−1}α(u^{−1})ζ(γ) = ζ̃(γ)^{−1}cζγ,

where the second equality follows from equation (3). Since

η(γ) : x ↦ ζ̃(γ)cζ(γ)^{−1}

is a group action, it follows that c = ζ̃(γ)cζ(γ)^{−1} for all γ ∈ Γ. Hence

ζ̃(γ) = cζ(γ)c^{−1}

and consequently

ρ̃(γ) = cρ(γ)c^{−1}

for all γ ∈ Γ.

103 C.14 14. Classification of homogeneous vector bundles for G semisimple without SL_{2}(ℂ)-factor

Given the above results on essentially antiholomorphic representations we are now in a position to prove the following classification theorem.

Theorem 10. Let G be a simply-connected semisimple complex Lie group, Γ a discrete cocompact subgroup, X = G/Γ. Assume that no direct factor of G is isomorphic to SL_{2}(ℂ).

Then there is a one-to-one correspondence between indecomposable homogeneous vector bundles over X and pairs (ϕ, ζ), where ϕ : G → GL(V_{1}) is an irreducible holomorphic representation and ζ : Γ → GL(V_{2}) is an irreducible representation of Γ such that ζ(Γ) forms a relatively compact subgroup of GL(V_{2}).

The correspondence is given in the following way: The vector bundle associated to (ϕ, ζ) is the flat bundle given by the representation ρ = (ϕ̄|_{Γ}) ⊗ ζ of the fundamental group π_{1}(X) ≃ Γ. (ϕ̄ is the antiholomorphic representation obtained from ϕ by complex conjugation.)

An arbitrary homogeneous vector bundle E is in a unique way a direct sum E = ⊕_{i}E_{i} of indecomposable homogeneous vector bundles E_{i}.

(A vector bundle E over a compact complex manifold X is called indecomposable if and only if there does not exist any non-trivial decomposition E = E_{1} ⊕ E_{2} .)

Proof. Every holomorphic vector bundle E over a compact complex manifold X admits a decomposition E = ⊕_{i}E_{i}. The sequence of factors E_{i} is unique up to permutation [2]. E is homogeneous if and only if every summand E_{i} is homogeneous. Thus it is enough to consider indecomposable vector bundles.

We have already seen that (for G semisimple) every homogeneous vector bundle is flat. In view of the result on essentially antiholomorphic representations obtained in the preceding section the only fact yet to prove is that every flat vector bundle can be defined by an essentially antiholomorphic representation.

For this we need the superrigidity theorem. (This is the point where it is necessary to exclude SL_{2}(ℂ)-factors.) Let ρ : Γ → GL(V) be an irreducible representation. By superrigidity there exists a continuous representation ρ_{0} : G → GL(V) such that ρ_{0} and ρ coincide on a subgroup of finite index Γ_{0} ⊂ Γ. Now Γ_{0} is still cocompact. Hence the density theorem applies and we may conclude that ρ_{0} is irreducible. It follows that for each simple factor S_{i} of G the restriction ρ_{0}|_{S_{i}} is either holomorphic or antiholomorphic. This implies that there exists a holomorphic representation ϕ : G → GL(V) and an essentially antiholomorphic representation τ of Γ such that

104 C.14

ρ(γ) = ϕ(γ)τ(γ) and ϕ(g)τ(γ) = τ(γ)ϕ(g) for all g ∈ G, γ ∈ Γ. Thus ϕ : G → GL(V) is a holomorphic map satisfying

ϕ(gγ) = τ(γ^{−1})ϕ(g)ρ(γ).

Hence the bundles defined by ρ and τ are isomorphic. Thus every flat vector bundle is isomorphic to a vector bundle defined by an essentially antiholomorphic representation.

15. Finite representations of the fundamental group

Here we will prove that compact complex parallelizable manifolds have many non-trivial holomorphic vector bundles which are induced by representations of the fundamental group with finite image.

We have seen already that given a connected complex Lie group H and a representation of the fundamental group ρ : π_{1}(X) → H with finite, but nontrivial image the induced H -principal bundle over X is not holomorphically trivial.

Now we encounter the following question: Given a manifold X which is not simply-connected, does there always exist a non-trivial finite representation of the fundamental group? This is indeed the case for homogeneous complex manifolds, but not for arbitrary complex manifolds.

Theorem (Malcev). Let K be a field, Γ a finitely generated subgroup of GL_{n}(K).

Then Γ is residually finite, i.e. for every element g ∈ Γ there exists a group homomorphism ρ : Γ → S to a finite group S such that ρ(g) ≠ e.

We apply this to parallelizable manifolds.

Proposition 16. Let X be a compact complex parallelizable manifold.

Then π_{1}(X) is residually finite, i.e. for every element g ∈ π_{1}(X) there exists a group homomorphism ρ : π_{1}(X) → S to a finite group S such that ρ(g) ≠ e.

Proof. The fundamental group π_{1}(X) is finitely generated, because X is a compact manifold, see e.g. [32,Th.6.16]. Now X may be realized as quotient of simplyconnected complex Lie group G by a discrete subgroup Γ with Γ ≃ π_{1}(X). Since G is simply-connected, it may be embedded into a complex-linear group GL_{m}(ℂ). Hence we are entitled to apply Malcevs theorem.

In fact there exists a non-trivial finite representation for arbitrary homogeneous manifolds (not necessarily parallelizable).

Proposition 17. Let G be a complex Lie group, H a closed complex Lie subgroup. Assume that X = G/H is not simply-connected.

Then there exists a non-trivial group homomorphism from π_{1}(X) to a finite group.

Proof. There is no loss in generality in assuming G to be simply-connected. Then π_{1}(G/H) ≃ H/H^{0} . (As usual H^{0} denotes the connected component of e in H.) Furthermore G may be embedded into a complex-linear group GL_{n}(ℂ). Let N denote the normalizer N_{GL_{n}(ℂ)} (H^{0}) of H^{0} in GL_{n}(ℂ). The normalizer of a connected Lie subgroup of a linear-algebraic group may be realized as isotropy group of a certain action on a Grassmann manifold induced by the adjoint representation. This implies that the normalizer of a connected Lie subgroup of linear-algebraic group is always a linear-algebraic subgroup. Now let H̄^{0} denote the closure of H^{0} in GL_{n}(ℂ) with respect to the algebraic Zariski-topology. Then N/H̄^{0} is a linear-algebraic group, hence admits a faithful linear representation. It follows that either H/H^{0} admits a nontrivial finite representation fibering through H/(H ∩H̄^{0}) or H ⊂ H̄^{0} . But H̄^{0}/H^{0} is commutative. Hence in the second case H/H^{0} is a finitely generated abelian group. Every finitely generated abelian group is isomorphic to some direct product ℤ^{k} × A with A finite and therefore clearly admits a non-trivial finite representation.

For non-homogeneous manifolds there is no such result.

Example. There exists a three-dimensional compact complex manifold with infinite fundamental group such that every finite representation of the fundamental group is trivial.

Proof. By a result of Higman [15] there exists a finitely presentable group without any non-trivial subgroup of finite index. Furthermore there is a recent result of Taubes [34] that for every finitely presentable group there exists a three-dimensional compact complex manifold with this group as fundamental group.

16. On the number of finite representations

In the preceding section we proved that for a discrete cocompact subgroup Γ in a complex Lie group G there always exist some group homomorphisms to finite groups.

For the case where G is semisimple without SL_{2}(ℂ)-factors we will now demonstrate that there are only finitely many non-equivalent group homomorphisms with finite image from Γ to a fixed Lie group. This implies that for a

106 C.16 given number r ∈ ℕ there are only finitely many non-equivalent homogeneous vector bundles of rank r over X = G/Γ.

This implies that under some assumptions for each r ∈ ℕ there are only finitely many non-equivalent flat vector bundles of rank r over a compact parallelizable manifold.

Theorem 11. Let S be a semisimple, simply-connected complex Lie group without SL_{2}(ℂ)-factor and Γ a lattice.

Then for every Lie group H with finitely many connected components there exists only finitely many non-conjugate group homomorphisms ρ : Γ → H. (Two group homomorphisms ρ, τ are conjugate if there exists an element h ∈ H such that ρ(γ) = hτ(γ)h^{−1} for all γ ∈ Γ.)

Moreover, for every such H there exists a normal subgroup of finite index Γ_{H} ⊂ Γ such that Γ_{H} ⊂ ker ρ for every ρ ∈ Hom(Γ, H).

Corollary. Let S be a semisimple, simply-connected complex Lie group without SL_{2}(ℂ)-factor and Γ a lattice. Let H be a complex Lie group with finitely many connected components.

Then there exists only finitely man non-isomorphic H -principal bundles over S/Γ admitting a flat connection.

For the proof of the theorem we need some preparation. First we recall that by Margulis superrigidity it suffices to consider representations with finite image.

Lemma 15. Let S be a connected linear Lie group, Γ a finitely generated infinite subgroup. Let Λ_{k} = ⋂_{ρ} ker ρ where ρ runs over all group homomorphisms from Γ to finite groups of order ≤ k.

Then Λ_{k} is an infinite normal subgroup of Γ.

Proof. Normality is obvious. By Selberg’s theorem we know that Γ contains a torsionfree subgroup of finite index. It follows that Γ contains a subgroup H ≃ (ℤ, +). Let S denote the set of all elements γ ∈ Γ which are k^{!} -divisible, i.e. for which there exists an element λ ∈ Γ with λ^{(k^{!})} = γ. With H ≃ ℤ we have k^{!}ℤ ⊂ H ∩ S, hence S is an infinite set. Now S ⊂ Λ_{k}, hence Λ_{k} is infinite.

We furthermore need Jordan’s theorem on subgroups of finite linear groups.

Theorem (Jordan [17],[7]). There exists a function k : ℕ → ℕ such that for every finite subgroup F ⊂ GL_{n}(ℂ) there exists a normal abelian subgroup A◁F such that F/A has at most k(n) elements.

Corollary. Let H be a (real) Lie group with finitely many connected components. Then there exists a number k(H) such that for every finite subgroup F ⊂ H there exists an normal abelian subgroup A ◁ F such that F/A has at most k(H) elements.

Proof of the corollary. Let K be a maximal compact subgroup of H. Then every finite subgroup of H is conjugate to a finite subgroup of K. Since K is linear, there exists a faithful representation ρ : K → GL_{n}(ℂ) for some n ∈ ℕ. Therefore this is a corollary of Jordan’s theorem.

Finally we need the result that for a given finite group and a given Lie group with only finitely many connected components there exist only finitely many group homomorphisms up to conjugacy.

Proposition 18. Let G be a finite group, H a (real) Lie group with finitely many connected components.

Then up to conjugacy there exist only finitely many group homomorphisms from G to H.

Proof. Clearly the image of such a group homomorphism f : G → H is finite, hence compact. Since all the maximal compact Lie subgroups of H are conjugate, there is no loss of generality in assuming that H is compact.

Let us assume that there exists an infinite sequence of pairwise non-conjugate group homomorphisms f_{n} : G → H. Since H is compact, we may choose a subsequence such that f_{n}(x) converges for every x ∈ G. Evidently the limit map f : G → H is again a group homomorphism. Theorem 5.3. of [24] now implies that for all n sufficiently large the image group f_{n}(G) is conjugate to a subgroup of f(G). This contradicts the assumption that all the f_{n} are pairwise non-conjugate, because f(G) has only finitely many subgroups and every such subgroup has only finitely many automorphisms.

Proposition 19. Let S be a simply-connected semisimple complex Lie group without SL_{2}(ℂ)-factors, Γ a lattice, H a (real) Lie group with finitely many connected components.

Then there exists a normal subgroup Γ_{H} of finite index in Γ such that Γ_{H} ⊂ ker ρ for every group homomorphism ρ : Γ → H with finite image.

Proof. Let k = k(H) as in the corollary to Jordan’s theorem above. Then ρ(Λ_{k}) must be abelian for every group homomorphism ρ : Γ → H with finite image. By Margulis’ theory every infinite normal subgroup of Γ is of finite index. Hence Λ_{k} is of finite index in Γ. Moreover the commutator group Λ′_{k} is of finite index in Λ_{k}, hence of finite index in Γ. Thus Γ_{n} := Λ′_{k} has the desired property.

108 C.16

Now the proof of the theorem is an immediate consequence of the fact that a finite group (such as Γ/Γ_{n}) has only finitely many non-equivalent irreducible representations.

By use of superrigidity we obtain in particular.

Corollary. Let S be a simply-connected semisimple complex Lie group without SL_{2}(ℂ)-factors, Γ a lattice and H a Lie group with finitely many connected components.

Then there exist only finitely many non-conjugate group homomorphisms from Γ to H.

Corollary. Let S be a simply-connected semisimple complex Lie group without SL_{2}(ℂ)-factors and Γ a discrete cocompact subgroup in S.

Then for each r ∈ ℕ there exist only finitely many non-equivalent homogeneous vector bundles of rank r over X = S/Γ.

Superrigidity results for lattices Γ ⊂ S are often stated in the form that for certain groups H every group homomorphism from a lattice to H either has finite (or relatively compact) image or extends to a group homomorphism from S to H.

Our theorem implies such a result for torsion groups.

Corollary. Let S be a simply-connected semisimple complex Lie group, without SL_{2}(ℂ)-factors, Γ a lattice, T a n-torsion group for some n ∈ ℕ, i.e. g^{n} = e for all g ∈ T.

Then for every group homomorphism ρ : Γ → T the image ρ(Γ) is finite.

17. Existence of vector bundles

Bănică posed the question, whether every compact complex manifold admits a non-trivial holomorphic vector bundle. We are now in a position to give an affirmative answer.

Theorem 12. Let X be a positive-dimensional compact complex manifold.

Then X admits a non-trivial holomorphic vector bundle E with rank(E) ≤ dim X.

Proof. If the holomorphic tangent bundle of X is non-trivial, there is nothing to prove. Thus we may assume that X is a parallelizable, i.e. X ≃ G/Γ where G is a simply-connected complex Lie group and Γ a discrete cocompact subgroup. If G ≠ G′, then the Albanese torus Alb(X) is non-trivial, and there are non-trivial line bundles on X obtained by pull-back from Alb(X). Finally, if G = G′, let ρ denote the representation of the fundamental group π_{1}(X) ≃ Γ which is the restriction of the complex conjugate of the adjoint representation of G. Theorem 9 implies that in this case the vector bundle E induced by ρ is non-trivial. Furthermore, the construction of ρ implies rank(E) = dim(X).

Given the fact that every compact complex manifold admits a non-trivial holomorphic vector bundle, there are two natural questions. First: Is the statement on the rank optimal? Second: Does similar results hold in other categories?

Concerning the first question we note that already for surfaces it is possible that a compact complex manifold does not admit any non-trivial line bundle, namely this happens for K3-surfaces of algebraic dimension zero.

Concerning the second question we remark that clearly every projective manifold admits a line bundle L with c_{1}(L) ≠ 0.

However, in the differentiably category there is no general existence theorem for non-trivial vector bundles. In fact, the three-dimensional sphere S^{3} is an example for a compact differentiable manifold X such that every principal bundle (with finite-dimensional structure group) over X is trivial. To see this, note that S^{3} is the suspension of S^{2} . This implies that (topologically) H -principal bundles over S^{3} are classified by homotopy classes of continuous maps from S^{2} to H. But π_{2}(H) = {1} for every Lie group H ([9]). Thus S^{3} does not admits any non-trivial vector bundle.

18. Topological structure of flat bundles

Our goal here is to prove that for a compact complex parallelizable manifold every homogeneous vector bundle is almost topologically trivial. In particular we will see that for a homogeneous vector bundle over a compact complex parallelizable manifold all the Chern classes vanish.

This result is specific to parallelizable manifolds, e.g. for homogeneous rational manifolds like ℙ_{n}(ℂ) every line bundle is homogeneous. In contrast, for parallelizable manifolds, homogeneous bundles are close to flat bundles. In fact for many special cases we proved that a homogeneous bundle is automatically flat.

Flat vector bundles are always close to be topologically trivial. This is true, because the presence of a flat connection implies that all Chern classes are zero. Now every (continuous) complex vector bundle of rank k over a compact differentiable manifold X is obtained as a pull-back of the universal vector bundle over the classifying space BU(k) via a continuous map f : X → BU(k). Topological triviality of the vector bundle is equivalent to the assumption that f is homotopic to a constant map. Vanishing of all Chern classes is equivalent to the property that the induced homomorphism of cohomology rings

f^{∗} : H^{•}(BU(k), ℂ) → H^{•}(X, ℂ)

is trivial. In this sense flat vector bundles are always almost topologically trivial.

First we prove a general statement on flat bundles with solvable structure group (which is possibly well-known). Namely, if X is a compact differentiable manifold and E a flat bundle with solvable structure group, then there exists a finite covering τ : X_{1} → X such that τ^{∗}E is topologically trivial.

We then proceed to prove the main theorems of this section. We will prove the following: Let H be a complex linear Lie group, E a homogeneous H - principal bundle over a compact complex parallelizable manifold X. Then there exists a finite covering τ : X_{1} → X and a flat H -principal bundle E_{1} → X_{1} such that there is a topological isomorphism τ^{∗}E ≃ E_{1} . This implies that for every homogeneous vector bundle over a compact complex parallelizable manifold the Chern classes vanish.

For compact complex tori every flat vector bundle is topologically trivial. However already for nilmanifolds it is quite possible that H_{1}(X, ℤ) contains torsion. In this case it follows that there exist flat complex line bundles which are not topologically trivial.

We begin with some considerations about line bundles. As usual let P ic(X) denote the group of holomorphic line bundles over a complex manifold X.

Lemma 16. Let X be a compact complex manifold, A the torsion part of H^{2}(X, ℤ) and δ : P ic(X) → H^{2}(X, ℤ) the natural group homomorphism.

Then for every flat line bundle L we have δ(L) ∈ A. Conversely for every element α ∈ A there exists a flat line bundle L over X with δ(L) = α.

There is an isomorphism Ext(H_{1}(X, ℤ), ℤ) ≃ A and a non-functorial isomorphism between A and the torsion part of H_{1}(X, ℤ) ≃ π_{1}(X)^{ab}.

Proof. Since X is compact, the homology groups are finitely generated. It follows that H_{1}(X, ℤ) ≃ ℤ^{r} × T for some finite abelian group T . The short exact sequence of abelian groups

0 → ℤ → ℂ → ℂ^{∗} → 0

yields a long exact Ext-sequence

… → Hom(T, ℂ) → Hom(T, ℂ^{∗}) → Ext(T, ℤ) → Ext(T, ℂ) → …

Now Hom(T, ℂ) = 0 and Ext(T, ℂ) = 0, hence there is an isomorphism between Hom(T, ℂ^{∗}) and Ext(T, ℤ). Furthermore by the universal coefficient theorem for cohomology

0 → Ext(H_{1}(X, ℤ), ℤ) → H^{2}(X, ℤ) → Hom(H_{2}(X, ℤ), ℤ) → 0

The group Hom(H_{2}(X, ℤ), ℤ) is free. Hence the finite group Ext(H_{1}(X, ℤ), ℤ) = Ext(T, ℤ) ≃ Hom(T, ℂ^{∗}) is isomorphic to the torsion part of H^{2}(X, ℤ).

This lemma implies in particular that every torsion element of H^{2}(X, ℤ) is contained in the image of δ : P ic(X) → H^{2}(X, ℤ). It follows that there exists a topologically non-trivial flat line bundle as soon as H_{1}(X, ℤ) contains torsion. This happens already for nilmanifolds. For instance, there is a threedimensional nilmanifold X with Ext(H_{1}(X), ℤ) ≃ ℤ_{2} given by X = G/Γ with

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

and

Γ = \{( \begin{matrix}1 & x & z \\ & 1 & y \\ & & 1 \end{matrix} ) : x, y ∈ ℤ ⊕ iℤ; z ∈ ℤ ⊕ \frac{1}{2}iℤ\} .

Proposition 20. Let M be a compact differentiable manifold and E → M an H -principal bundle for some connected Lie group H. Assume that E is given by a representation of the fundamental group ρ : π_{1}(M) → H and that H is solvable.

Then there exists a finite covering π : M_{1} → M such that π^{∗}E → M is (differentiably) trivial.

If H_{1}(M, ℤ) is torsion-free, then E itself is trivial.

Proof. First we want to remark that a differentiable bundle is differentiably trivial if and only if it is topologically trivial.

Let K be a maximal compact Lie subgroup of H. Since H is solvable, K is commutative, hence K ≃ (S^{1})^{m}. Now the structure group of E can be reduced to K, hence it suffices to prove the statement for H = K. Considering one factor after the other, it is enough to deal with the case H ≃ S^{1} .

Now we consider the exact sequences of sheaves 1 → ℤ → ℝ → S^{1} → 1 and 1 → ℤ → E → S → 1 where as usual ℤ, ℝ and S^{1} denote the sheaves of locally

112 C.18

constant functions with values in that group. We obtain a commutative diagramm of exact sequences

0 ⟶ ℤ ⟶ ℝ ⟶ S^{1} ⟶ 0
∥ ↓ ↓
0 ⟶ ℤ ⟶ E ⟶ S ⟶ 0

which induce a commutative cohomology diagram

H^{1}(M, ℝ) ⟶^{β} H^{1}(M, S^{1}) ⟶^{δ} H^{2}(M, ℤ) ⟶ H^{2}(M, ℝ)
↓  ↓^{p}  ∥  ↓
0 = H^{1}(M, E) ⟶ H^{1}(M, S) ⟶^{∼} H^{2}(M, ℤ) ⟶ H^{2}(M, E) = 0

Next recall that H^{1}(M, A) ≃ Hom(H_{1}(M), A) for A an abelian group (resp. the sheaf of locally constant functions with values in that group). Thus H^{1}(M, S^{1}) classifies homomorphisms ρ : π_{1}(M) → S^{1} (observe that H_{1}(M) is the abelianization of π_{1}(M)). On the other hand H^{1}(M, S) classifies the S^{1} –principal bundles. But now Ext(H_{1}(M), ℤ) = 0 implies that β : H^{1}(M, ℝ) → H^{1}(M, S^{1}) is surjective, i.e. δ : H^{1}(M, S^{1}) → H^{2}(M, ℤ) is the zero map. Since H^{k}(M, E) = 0 (k > 0) implies H^{1}(M, S) ≃ H^{2}(M, ℤ), it follows that the map p : H^{1}(M, S^{1}) → H^{1}(M, S) is the zero map. Thus any S^{1} –principal bundle is trivial if it is induced by a representation of the fundamental group.

Remarks.

(1) In the statement of the proposition, the choice of the covering depended on E. Actually it is possible to choose a finite covering τ : M_{1} → M such that τ^{∗}E is (differentiably) trivial for all flat H –principal bundles for all connected solvable Lie groups H. It suffices to choose M_{1} in the following way: Let F denote a torsion-free subgroup of finite index in H_{1}(M, ℤ) (which exists, because H_{1}(M, ℤ) is finitely generated for M compact). Then let Γ = ϕ^{∗}(F) with ϕ : π_{1}(M) → H_{1}(M, ℤ) and let M_{1} → M be the covering corresponding to Γ ⊂ π_{1}(M).

(2) The assumption of compactness of the manifold M is essential for the statement. For instance, let Γ = ℚ/ℤ, M a manifold with π_{1}(M) ≃ Γ and L → M the line bundle defined by the natural embedding of the fundamental group π_{1}(M) = ℚ/ℤ ↪ ℂ/ℤ ≃ ℂ^{∗}. Then π^{∗}L is topologically non-trivial for any covering π : M_{1} → M except the universal covering.

Theorem 13. Let X be a compact complex parallelizable manifold, E → X a homogeneous vector bundle.

Then all the Chern classes c_{k}(E) vanish.

Remarks.

(1) For flat vector bundles it is obvious that all the Chern classes do vanish, because the Chern classes of a vector bundle may be calculated in terms of the curvature of a connection and by definition a flat vector bundle has a connection with vanishing curvature.

(2) For an arbitrary holomorphic vector bundle E over a compact complex manifold X, the vanishing of all Chern classes does not imply that there exists a finite covering τ : X_{1} → X with τ^{∗}E → X_{1} topologically trivial. For example, there exists a vector bundle over ℙ_{2} such that all Chern classes vanish, but the bundle is not topologically trivial (see [5]).

(3) Parallelizability of X implies that homogeneous bundles are closely related to flat bundles. For arbitrary homogeneous vector bundles over arbitrary compact homogeneous manifolds the corresponding assertion is clearly wrong, e.g. every line bundle over ℙ_{n} is homogeneous.

(4) It is absolutely essential to require that the structure group is linear. For instance, the Iwasawa-manifold is a torus-principal bundle over a torus. This is a homogeneous bundle with non-linear structure group, and it is certainly not topologically isomorphic to a flat bundle.

The proof of the theorem results from the lemmata below. We recall the standard fact from fiber bundle theory that every principal

bundle with contractible structure group is (topologically) trivial. This can be generalized to the following statement:

Let G = S × U be a semi-direct product of Lie groups with U contractible. Let π : G → S ≃ G/U denote the natural projection.

Then every G-principal bundle E is topologically isomorphic to the S -principal bundle E′ associated via π : G → S. This is a particularly useful formulation for our purposes, because if E

is homogeneous with respect to some group action on the base space, then evidently E′ is homogeneous as well.

Lemma 17. Let X = G/Γ be a compact complex parallelizable manifold, H a connected linear complex Lie group and S a maximal connected semisimple Lie subgroup of H.

Let E → X be a homogeneous H -principal bundle. Then there exists a finite covering τ : X_{1} → X and a homogeneous S -principal bundle E′ → X_{1} such that τ^{∗}E and E′ are topologically isomorphic (as H -principal bundles).

Proof. Fix an embedding H ↪ GL_{N} (ℂ) and let H̄ denote the Zariski-closure of H in GL_{N} (ℂ). The commutator group H′ of H is linear-algebraic [8]. Therefore H̄/H′ is a commutative linear-algebraic group. Since H/H′ ⊂ H̄/H′, it follows that H/H′ ≃ ℂ^{k} × (ℂ^{∗})^{l} for some k, l ∈ ℕ. Thus the H/H′ -principal bundle over X associated to E is isomorphic to a direct product of ℂ- and ℂ^{∗} -principal bundles. We have already seen that every homogeneous ℂ^{∗} -principal bundle over X is flat and therefore becomes trivial after passing through a finite covering. Moreover any ℂ-principal bundle is topologically trivial. Hence there exists a finite covering τ : X_{1} → X such that the structure group of τ^{∗}E can be reduced to H′. Using the Levi-decomposition of H′ and the observation made before the lemma, we may further reduce the structure group to S.

We are now in a position to prove that every homogeneous bundle over a parallelizable manifold is topologically equivalent to a flat one.

Proposition 21. Let X = G/Γ be a compact complex parallelizable manifold and H a connected linear complex Lie group.

Let E → X be a homogeneous H -principal bundle. Then there exists a finite covering τ : X_{1} → X and a flat H -principal bundle E_{1} over X_{1} such that τ^{∗}E and E_{1} are topologically isomorphic.

Proof. By the above lemma we may assume that H is semisimple and thereby in particular that H is linear-algebraic. Since E is homogeneous, there exists an essential extension of connected complex Lie groups

1 → L → G̃ →^{σ} G → 1

such that the G-action on X = G/Γ lifts to a G̃-action on E. Let I = σ^{−1}(Γ). Then E is isomorphic to a fiber product E = G̃ ×_{I} H → X = G̃/I given by some group homomorphism ρ : I → H. Now we may replace H by the Zariski-closure of ρ(I) in H. By passing to a finite cover of X we may assume that this is connected. A new application of the above lemma yields that we may further reduce the structure group to a maximal connected semisimple Lie subgroup of \overline{ρ(I)}. Iterating this process, it is clear that we may simultanously assume that H is semisimple and connected and that ρ(I) is Zariski-dense in H.

Let R̃ denote the radical of R. Since ρ(I) is Zariski-dense, it is clear that the Zariski-closure of ρ(I ∩ R̃) in H is a normal solvable subgroup, hence finite. Once again replacing X by a finite cover, we may assume that I ∩ R̃ ⊂ ker ρ. But L ⊂ R̃ (because the extension (∗) is essential, cf. proposition 5), hence ρ : I → H actually fibers through Γ ≃ I/L. Thus we obtain a flat bundle.

This proposition implies immediately that for a homogeneous vector bundle over a compact complex parallelizable manifold the Chern classes vanish. (Note that for a finite covering τ : X_{1} → X the induced homomorphism of cohomology

115 C.18 rings τ^{∗} : H^{•}(X, ℂ) → H^{•}(X_{1}, ℂ) is injective (see e.g. [12,Prop. VIII.10.10]). Hence c_{k}(τ^{∗}E) = 0 implies c_{k}(E) = 0.)

Line Bundles

19. Powers of line bundles

Here we confirm that certain properties of line bundles are not affected by passing to a finite tensor power. In particular this is true for the existence of (flat) holomorphic connections as well as for the existence of a holomorphic structure on a topological complex line bundle.

Lemma 18. Let X be a complex manifold, L a holomorphic line bundle, k ∈ ℕ Then L admits a holomorphic connection resp. flat holomorphic connection if and only if L^{k} admits such a connection.

Proof. A connection D on L induces a connection D̃ on L^{k} via

D̃(σ) = kσ̃ ⊗ ^{k} \sqrt{σ̃}^{−1} ⊗ D (^{k} \sqrt{σ̃})  (∗)

The k -th root of a section in L^{k} is a section in L which is well-defined only up to multiplication with a k -th root of unity. Taking the same branch of the k -th root at both occurances of the k -th root in (∗) makes the formula well-defined.

In the converse direction a connection D̃ on L^{k} yields a connection D on L via

D(σ) = \frac{1}{k}σ^{1−k} ⊗ D̃(σ^{k}).  (∗∗)

This is exactly the reverse procedure of (∗).

In a joint local trivialiazation D and D̃ are given by D(σ) = ω ⊗ σ + dσ resp. D̃(σ) = ω̃ ⊗ σ + dσ for some one-forms ω, ω̃. From (∗) resp. (∗∗) one can deduce ω̃ = kω. Since dω resp. dω̃ yields the curvature form, it follows that D is flat if and only if D̃ is flat.

Corollary. Let X be a compact complex manifold with P ic(X) = H^{1}(X, O^{∗}) finite.

Then every holomorphic line bundle over X admits a flat holomorphic connection and is therefore induced by a representation ρ : π_{1}(X) → S^{1} .

Lemma 19. Let X be a complex manifold, L → X a topological complex line bundle (i.e. a complex line bundle with continuous transition functions), k ∈ ℕ.

Then L admits a structure of a holomorphic line bundle if and only if L^{k} does.

Proof. Topological line bundles are parametrized by H^{2}(X, ℤ). An element α ∈ H^{2}(X, ℤ) corresponds to a line bundle admitting a holomorphic structure if and only if it is mapped to zero by the natural group homomorphism H^{2}(X, ℤ) → H^{2}(X, O). This implies the assertion, because H^{2}(X, O) is a complex vector space and therefore a torsion-free abelian group.

20. Line bundles: Comparison with Kähler manifolds

Now we want to compare the above results with the situation for compact Kähler manifolds.

For compact complex manifolds X we consider the following properties for line bundles:

(α) There is a line bundle L̃ ∈ P ic^{0}(Alb(X)) such that Alb^{∗} L̃ ≃ L^{k} for some

k ∈ ℕ; (β) L admits a flat holomorphic connection; (β′) L is given by a representation of the fundamental group; (γ) L is admits a holomorphic connection and L^{k} is topologically trivial for

some k > 0, (δ) A tensor power L^{k} is topologically trivial for some k ∈ ℕ; (δ′) A tensor power L′ = L^{k} for some k ∈ ℕ is divisible as an element in the

group P ic(X), i.e. for every m ∈ ℕ there exists a line bundle H ∈ P ic(X) such that H^{m} ≃ L′. For arbitrary compact complex manifold there are the following implica-

tions:

α ⟺ β′ ⟹ β ⟹ γ ⟹ δ ⟺ δ′

Proof. By the theorem of Appell-Humbert (see e.g. [26]) every topologically trivial line bundle on a torus is induced by a representation of the fundamental group. Hence α ⇒ β′. The equivalence β ⇔ β′ is a classical result of Atiyah [3]. β ⇒ γ ⇒ δ is obvious. Finally note that the group of topologically trivial line bundles P ic^{0}(X) admits a surjection H^{1}(X, O) → P ic^{0}(X). Hence P ic^{0}(X) is a divisible group. On the other hand topologically line bundles are parametrized by H^{2}(X, ℤ) which is a finitely generated abelian group. This suffices to prove δ ⇔ δ′.

The converse implications hold only under special assumptions.

Proposition 22. α ⇐ β holds if and only if the projection on the Albanese X → A = Alb(X) induces an isomorphism H^{1}(X, ℂ) ≃ H^{1}(A, ℂ).

β ⇐ γ holds if and only if the natural group homomorphism H^{1}(X, O) → H^{1}(X, dO) induced by f ↦ df is the zero homomorphism. This happens if and only if E_{2}^{0,1} ≃ E_{3}^{0,1} for the Hodge-Frölicher spectral sequence.

γ ⇐ δ holds if and only if the natural group homomorphism H^{1}(X, O) → H^{1}(X, Ω^{1}) (induced by the ℂ-module sheaf homomorphism given by f ↦ df) is zero. This is equivalent to E_{1}^{0,1} ≃ E_{2}^{0,1} for the Hodge-Frölicher spectral sequence.

α ⇐ δ holds if and only if the projection map to the Albanese induces an isomorphism H^{1}(X, O) ≃ H^{1}(Alb(X), O).

Proof. Evidently α ⇔ β′ holds if and only if the groups Hom(π_{1}(Alb(X)), ℂ^{∗}) and Hom(π_{1}(X), ℂ^{∗}) coincide up to torsion. This is equivalent to H^{1}(X, ℂ) ≃ H^{1}(Alb(X), ℂ), because H^{1}(M, ℂ) ≃ Hom(π_{1}(M), ℂ) for any topological space M.

If a line bundle L is topologically trivial, then there is a cocycle f_{ij} ∈ O(U_{ij}) (for some Leray cover) such that L is defined by g_{ij} = exp(f_{ij}). Explicit calculations show that there exists a holomorphic connection if and only if the cohomology class of d log g_{ij} = dg_{ij}/g_{ij} = df_{ij} in H^{1}(X, Ω^{1}) vanishes. (Here Ω^{1} denotes the sheaf of holomorphic one-forms.) A flat holomorphic connection exists if and only if d log g_{ij} defines the zero cohomology class in H^{1}(X, dO), where dO denotes the sheaf of d-closed holomorphic one-forms. For the Hodge-Frölicher spectral sequence it is well-known that E_{1}^{p,q} ≃ H^{q}(X, Ω^{p}) with d_{1} : E_{1}^{p,q} → E_{1}^{p+1,q} induced by ordinary differentiation d : Ω^{p} → Ω^{p+1} . Hence E_{1}^{0,1} ≃ E_{2}^{0,1} if and only if the natural group homomorphism H^{1}(X, O) → H^{1}(X, Ω^{1}) is zero. Now we consider the long exact cohomology sequence induced by the short exact sequence of sheaves 0 → ℂ → O → dO → 0. We obtain

0 → H^{0}(X, dO) → H^{1}(X, ℂ) → H^{1}(X, O) → H^{1}(X, dO) → …

Let V denote the subvectorspace of H^{1}(X, O) for which there exists a flat holomorphic connection on the associated line bundle. Then (non-functorially) H^{1}(X, ℂ) ≃ H^{0}(X, dO) ⊕ V . On the other hand the Hodge-Frölicher spectral sequence yields E_{∞}^{1} ≃ E_{3}^{1,0} ⊕ E_{3}^{0,1} with E_{∞}^{1} ≃ H^{1}(X, ℂ) and E_{3}^{1,0} ≃ E_{2}^{1,0} ≃ H^{0}(X, dO). In this way V ≃ E_{3}^{0,1} . Hence β ⇐ γ if and only if E_{2}^{0,1} = E_{3}^{0,1} .

Finally α ⇐ δ is clearly equivalent to P ic^{0}(X) ≃ P ic^{0}(Alb(X)). Using the exponential sequence it is easy to verify that this holds if and only if H^{1}(X, O) ≃ H^{1}(Alb(X), O).

Thus all the terms E_{r}^{0,1} of the Hodge-Frölicher spectral sequence are significant for line bundles and the existence of (flat) holomorphic connections. (E_{r}^{0,1} ≃ E_{3}^{0,1} for r ≥ 3).

Corollary. For a compact Kähler manifold X all the properties α, β, γ, δ are equivalent.

119 C.20

For compact complex parallelizable manifolds the picture is more complicated.

Proposition 23. Let X = G/Γ be a compact complex parallelizable manifold. Let S = G/R where R denotes the radical of G. Then one has the following implications

α ⇐ β holds if and only if G′Γ is closed and (Γ ∩ G′)/Γ′ is finite. In particular this is true, if S contains no SL_{2}(ℂ)-factor.

β ⇐ γ holds for every compact complex parallelizable manifold.

The properties α, β, γ and δ are equivalent if G is nilpotent or semisimple without SL_{2}(ℂ)-factor or G ≃ SL_{2}(ℂ) with Γ/Γ′ finite.

There are parallelizable examples with α ⇍ β and γ ⇍ δ. Namely, if X = G/Γ with G = SL_{2}(ℂ) and Γ/Γ′ is infinite, then dim H^{1}(X, ℂ) is larger than H^{1}(Alb(X), ℂ) (Alb(X) is trivial, because G = G′). Hence α ⇍ β in this case.

Nakamura has calculated H^{1}(X, O) explicitly for certain solvmanifolds (see [28]). This yields an example with γ ⇍ δ.

As mentioned above β ⇐ γ for compact Kähler manifolds as well as for all compact complex parallelizable manifolds. However, this does not hold for arbitrary compact complex manifolds. Cordero, Fernandez and Gray produced an example for which E_{2}^{0,1} ≠ E_{3}^{0,1} . It is a four-dimensional compact complex manifold, diffeomorphic to a real nilmanifold, but not complex parallelizable. The above considerations show that on this manifold there exists a topologically trivial line bundle which does admit a holomorphic connection, but no flat holomorphic connection.

Property γ stated that the line bundle has a holomorphic connection and a finite tensor power of the bundle is topologically trivial. One may ask whether the presence of a holomorphic connection implies that a finite power is topologically trivial. For compact Kähler manifolds as well as for compact complex parallelizable manifolds this is indeed the case.

Lemma 20. Let X be a compact manifold, which is Kähler or complex parallelizable. Let L be a line bundle which admits a holomorphic connection. Then there exists a k ∈ ℕ such that L^{k} is topologically trivial.

Proof. If X is complex-parallelizable, then the existence of a holomorphic connection implies that there exists a flat holomorphic connection by theorem 6.

Now assume that X is Kähler. The presence of a holomorphic connection implies that there exists a d-closed (2, 0)-form representing c_{1}(L). On the other hand every line bundle admits a hermitian connection, hence there is also a d-closed (1, 1)-form representing c_{1}(L). Thus Hodge-decomposition forces c_{1}(L) = 0.

Thus c_{1}(L) = 0 in both cases. For a compact manifold this implies that a finite power of L is topologically trivial.

However, we do not know whether there is such an implication for an arbitrary compact complex manifold. (There is no such implication for a noncompact manifold. For instance, if X is a Stein manifold, then every line bundle admits a holomorphic connection, because H^{1}(X, Ω^{1}) = 0.)

21. The first Chern class

Using the results on homogeneous vector bundles, we will prove the following:

Theorem 14. Let S be a semisimple complex Lie group, Γ a discrete cocompact subgroup, E a holomorphic vector bundle over S/Γ.

Then c_{1}(E) = 0.

The proof will be obtained through an analysis of the S -action on various cohomology spaces.

Group actions on H^{1}(X, O^{∗})

Lemma 21. Let G be a complex Lie group acting on a complex space X by biholomorphic transformations. Consider

… → H^{1}(X, ℤ) → H^{1}(X, O) → H^{1}(X, O^{∗}) → H^{2}(X, ℤ) → …  (∗)

There is an induced G-action on these cohomology groups (by group automorphisms) such that (∗) is equivariant. Moreover, the G-action on the complex vector space H^{1}(X, O) is holomorphic.

Proof. Functoriality properties imply the existence of the induced actions. Holomorphicity of the G-action on H^{1}(X, O) has been discussed in B.§6.

Before continuing our argumentation we recall some language. Given a complex space X the group H^{1}(X, O^{∗}) is denoted by P ic(X), the image of the natural group homomorphism H^{1}(X, O^{∗}) → H^{2}(X, ℤ) is denoted by NS(X), and the kernel by P ic^{0}(X). Thus there is a short exact sequence

0 → P ic^{0}(X) → P ic(X) → NS(X) → 0  (∗)

Lemma 22. For every complex space X the short exact sequence (∗) is split.

Proof. The obstruction to such a splitting is contained in Ext(NS(X), P ic^{0}(X)). As a complex vector space the additive group H^{1}(X, O) is divisible, i.e. for every element g and every natural number n there exists an element h such that nh = g. There is a surjective group homomorphism H^{1}(X, O) → P ic^{0}(X). Hence P ic^{0}(X) is divisible, too. This implies Ext(NS(X), P ic^{0}(X)) = 0.

A similar arguments yields the following.

Lemma 23. Let A be a commutative Lie group and A^{0} the connected component of e. Then A is isomorphic (as a Lie group) to A^{0} × A/A^{0} .

Proof. Since A/A^{0} is discrete, it suffices to prove that 0 → A^{0} → A → A/A^{0} → 0 splits as a sequence of abstract groups. But connected commutative Lie groups are divisible, hence Ext(A/A^{0}, A^{0}) = 0.

Lemma 24. Let A be a commutative complex Lie group and A^{0} the connected component of e ∈ A. Let S be a connected semisimple complex Lie group acting on A by holomorphic group automorphisms.

Then the natural projection π : A^{S} → A/A^{0} is surjective (where A^{S} denotes the fixed-point set).

Proof. The projection A → A/A^{0} is equivariant. Let G denote the group of all holomorphic group automorphisms of A inducing the identity map on A/A^{0} . Evidently S ⊂ G, since A/A^{0} is discrete and S connected. There is an exact sequence

1 → C → G → Aut(A^{0}) → 1.

The group C consists of all holomorphic group automorphisms of A = A^{0} × Λ which may be written in the form (a, λ) ↦ (a + f(λ), λ) for some f ∈ Hom(Λ, A^{0}). Clearly C is a commutative normal Lie subgroup of G. Furthermore G is a semidirect product Aut(A^{0})× C. The Levi-Malcev-decomposition for the connected component G^{0} of G implies that all maximal connected semisimple Lie subgroups of G are conjugate. Since C is normal and commutative, it follows that S is conjugate to a subgroup of Aut(A^{0}) by an element ϕ ∈ G ⊂ Aut(A). Thus ϕ(Λ) ⊂ A^{S} which implies surjectivity of A^{S} → A/A^{0} .

Proposition 24. Let X be a complex space, S a connected semisimple complex Lie group acting on X. Let NS(X) denote the image of H^{1}(X, O^{∗}) → H^{2}(X, ℤ). Assume that H^{1}(X, O) is finite-dimensional.

Then the induced group homomorphism H^{1}(X, O^{∗})^{S} → NS(X) is surjective.

122 C.21

Proof. Since S is connected, every s ∈ S induces an automorphism L_{s} : X → X which is homotopic to the identity map. Therefore the S -action on H^{•}(X, ℤ) is trivial. Let V denote the complex subvectorspace of H^{1}(X, O) spanned by H^{1}(X, ℤ). Then S fixes V point-wise. Let W denote the image of V in H^{1}(X, O^{∗}) = P ic(X). Then P ic^{0}(X)/W ≃ H^{1}(X, O)/V is a finite-dimensional complex vector space on which S acts holomorphically. Now we recall that 0 → P ic^{0}(X) → P ic(X) → NS(X) → 0 splits. Endowing NS(X) with the discrete topology, we obtain a commutative complex Lie group A ≃ P ic(X)/W with A^{0} ≃ P ic^{0}(X)/W on which S acts holomorphically. Now the preceding lemma implies surjectivity of P ic(X)^{S} → NS(X).

Corollary. Let S be a connected complex semisimple Lie group acting holomorphically on a complex space X with dim_{ℂ} H^{1}(X, O) < ∞.

Then every holomorphic line bundle is topologically equivalent to a homogeneous line bundle.

Corollary. Let S be a simply-connected complex semisimple Lie group, Γ a discrete cocompact subgroup, X = S/Γ.

Then NS(X) is finite. Furthermore there is a subgroup Γ_{0} ⊂ Γ of finite index such that π^{∗}L is topologically trivial for every line bundle L ∈ P ic(X), where π denotes the finite covering S/Γ_{0} → X.

Remark. Despite the finiteness of NS(X) the group H^{2}(X, ℤ) is not necessarily finite. For instance there is a discrete cocompact subgroup Γ ⊂ SL_{2}(ℂ) such that b_{1}(SL_{2}(ℂ)/Γ) = 1 (see A.§3). By the Künneth formula this implies that H^{2}(X, ℤ) is infinite for X = (SL_{2}(ℂ)/Γ) × (SL_{2}(ℂ)/Γ).

Proof of the corollary. Since S is semisimple, every homogeneous vector bundle is flat. For a flat line bundle, the corresponding element in H^{2}(X, ℤ) is a torsion element. Thus the above corollary implies that NS(X) is a torsion group. Compactness of S/Γ implies that H^{2}(X, ℤ) and therefore NS(X) is finitely generated. A finitely generated abelian torsion group is finite. Thus NS(X) is finite. By the universal coefficient theorem there is a (non-functorial) isomorphism between the torsion part of H^{2}(X, ℤ) and the torsion part of H_{1}(X, ℤ) for compact X. Recall that a flat line bundle is induced by a representation ρ : H_{1}(X, ℤ) → ℂ^{∗} and is topologically trivial if and only if ρ vanishes on the torsion part of H_{1}(X, ℤ). Let τ : Γ = π_{1}(X) → H_{1}(X, ℤ) denote the natural projection, Λ a torsion-free subgroup of finite index in H_{1}(X, ℤ) (which exists, because H_{1}(X, ℤ) is finitely generated) and Γ_{0} = τ^{−1}(Λ). Then Γ_{0} does the job.

The theorem now follows using the fact that c_{1}(E) = c_{1}(det E) for any vector bundle E.

123 C.21

For later application we note some further consequences.

Proposition 25. Let S be a complex semisimple Lie group, Γ discrete cocompact, X = S/Γ, Z a compact Kähler manifold, f : Z → X a holomorphic map and L ∈ P ic(X).

Then either f^{∗}L is holomorphically trivial or H^{0}(Z, f^{∗}L) = {0}.

Proof. Since Z is Kähler, every non-trivial line bundle with non-trivial section has non-vanishing first Chern class contrary to f^{∗}c_{1}(L) = 0.

One may pose the following question: Given a topological complex line bundle L on a complex manifold X does there exists a holomorphic structure on L? This is equivalent to the question whether the group homomorphism

P ic(X) = H^{1}(X, O^{∗}) → H^{2}(X, ℤ)

is surjective. We can answer this for a special case.

Lemma 25. Let S = SL_{2}(ℂ), Γ a discrete cocompact subgroup with Γ/Γ′ finite, X = S/Γ.

Then P ic(X) → H^{2}(X, ℤ) is surjective.

Proof. This group homomorphism is part of a long exact sequence

… → P ic(X) →^{c} H^{2}(X, ℤ) → H^{2}(X, O) → …

Now H^{2}(X, O) is a complex vectorspace, hence torsion-free. Thus surjectivity of c is implied by finiteness of H^{2}(X, ℤ). Since X is compact, finiteness of H^{2}(X, ℤ) is equivalent to H^{2}(X, ℂ) = {0} which follows from proposition B.7.

Existence of non-homogeneous Vector Bundles

22. The Serre-construction

Here we want to describe a method of construction which yields the existence of non-homogeneous vector bundles of rank two over X = SL_{2}(ℂ)/Γ for Γ discrete, cocompact with Γ/Γ′ finite.

Our goal is to prove the following

Theorem 15. Let S = SL_{2}(ℂ), Γ a torsion-free discrete cocompact subgroup with Γ/Γ′ finite and C an elliptic curve in X = S/Γ.

Then there exists a unique non-homogeneous holomorphic vector bundle E of rank two with as section σ ∈ H^{0}(X, E) such that C = {x : σ(x) = 0}.

Furthermore E|_{C} is trivial.

It is well-known that SL_{2}(ℂ) admits many such discrete subgroups. Furthermore the results of Chapter A imply that any compact quotient manifold of SL_{2}(ℂ) by a discrete subgroup contains many elliptic curves. More precisely: If Γ is a torsion-free discrete cocompact subgroup of S = SL_{2}(ℂ), then for every γ ∈ Γ the centralizer C(γ) has a closed orbit in S/Γ. Evidently such a closed orbit must be an elliptic curve.

A key element in the proof of the theorem is the vanishing theorem on H^{1}(X, O) derived earlier.

Finally we investigate deformations of the vector bundles obtained in this way. It turns out that there are only the obvious deformations. Every small deformation is obtained by translation. Thus it is not possible to obtain further examples of non-homogeneous vector bundles by deformation.

Proposition 26. Let S,Γ, X, E be as above, P a complex space U → P × X a holomorphic vector bundle, p ∈ P such that U_{p} ≃ E where U_{p} → X is the vector bundle obtained from U by embedding X into P × X as {p} × X. Furthermore let Ẽ → S × X denote the vector-bundle Ẽ = μ^{∗}E obtained by pull-back via the multiplication map μ : S × X → X.

Then there exists an open neighbourhood W of p in P and a holomorphic map f : W → S such that F^{∗}Ẽ = U with F = (f, id_{X}).

A key step for this result on deformations is the calculation of certain cohomology groups.

Lemma 26. Let E be as above. Then dim_{ℂ} H^{k}(X, E^{∗} ⊗ E) = 2 for 0 ≤ k ≤ 3 and

dim_{ℂ} H^{k}(X, E) = \{ \begin{matrix}1 & for k = 0, 3 \\ 0 & for k = 1, 2 \end{matrix}

We also prove a result in the converse direction.

Theorem 16. Let S = SL_{2}(ℂ), Γ be a torsion-free discrete cocompact subgroup with Γ/Γ′ finite and E a non-homogeneous vector bundle on X = S/Γ of rank 2. Assume that there exists a non-trivial section σ ∈ H^{0}(X, E). Assume in addition that the zero-set {σ = 0} is irreducible and of multiplicity one. Then {x : σ(x) = 0} is an elliptic curve and the bundle E may be reconstructed by the Serre-construction.

Proof. First recall that there are no hypersurfaces in X, hence C = {σ = 0} is a curve. Observe that, given a vector field X ∈ Γ(X, T X) and a section σ ∈ Γ(X, E) we may calculate X(σ) in a local trivialization. Of course the result depends on the trivialization. But, (this is easy to check explicitly) the restriction of X(σ) to the zero-set V (σ) = {σ = 0} does not! Hence for any vector field X on X we obtain a section Xσ ∈ Γ(C, E|_{C}). Moreover, if two vector fields X and Y span the normal bundle N_{C,X} at some point p ∈ C, then Xσ and Yσ span E|_{C} at p. Thus Xσ∧Yσ defines a non-trivial section ζ in det E|_{C}. This section ζ must vanish at the points where X and Y fail to span the normal bundle N_{C,X}. Using this, one can show that either N_{C,X} is trivial or det E|_{C} admits a section which vanishes at some point q ∈ C, but not everywhere. However, the second case is impossible. To see this, let π : Ĉ → C denote the normalization of C. Then Ĉ is a Kähler manifold and we may apply proposition 25. Thus the normal bundle has to be trivial. This implies that C is parallelizable, i.e. an elliptic curve.

We will apply the so-called Serre-construction in order to obtain nonhomogeneous vector bundles over certain quotients SL_{2}(ℂ)/Γ (Γ discrete).

In the context of complex-projective spaces the Serre-construction is wellknown, see e.g. [14],[29],[33]. Nevertheless we will briefly recall the main arguments in order to underline that they are still valid in a non-projective setup.

We formulate now the general result which we will use.

Theorem (Serre-Construction). Let X be a complex manifold, Y a complex submanifold^{∗} of codimension two with Γ(Y, O_{Y}) = ℂ and L a line bundle over X such that L ≃ det I_{Y} /I_{Y}^{2} where L denote the sheaf of holomorphic sections in L and I_{Y} the ideal sheaf of Y . Assume that H^{2}(X, L) = {0}.

Then there exists a vector bundle E of rank two over X with a section σ ∈ Γ(X, E) such that Y = {x : σ(x) = 0}. (If moreover H^{1}(X, L) = {0}, then E is unique.)

Assume in addition that there exists no section in L^{∗} vanishing on Y but not vanishing everywhere. Then Γ(X, E) is generated by σ as Γ(X, O_{X})- module.

For parallelizable manifolds this has the following consequence.

Corollary. Let X = SL_{2}(ℂ)/Γ with Γ discrete and H a one-dimensional subgroup with compact orbit Y = H/(H ∩ Γ). Assume H^{1}(X, O_{X}) = {0}.

Then there exists a holomorphic vector bundle E of rank two on X with a section σ ∈ Γ(X, E) such that Y = {σ = 0}. Moreover Γ(X, E) = ℂσ, if Γ is Zariski-dense in SL_{2}(ℂ).

Proof. Any subvectorspace V transversal to Lie(H) in Lie(G) yields a trivialization of the normal bundle N_{Y,X}. Hence we may choose L as the trivial line bundle. Since X is parallelizable, K_{X} is trivial. Hence Serre-duality implies that H^{2}(X, L) = H^{2}(X, O_{X}) is dual to H^{1}(X, O_{X}), thus zero by assumption. Therefore we can apply the theorem. Finally recall that the density assumption implies that there are no hypersurfaces on X [16]. Thus any section in the line bundle L^{∗} vanishing on Y must vanish everywhere.

This implies the first theorem, because we proved already that H^{1}(X, O) = 0 under the assumptions of the theorem.

The statement on the sections Γ(X, E) = ℂσ has interesting consequences. It implies that the curve C used to construct E may be recovered from E simply as the zero-set of any non-trivial section. This clearly implies that E is not a homogeneous vector bundle. Moreover it implies that there are infinitely many non-isomorphic non-homogeneous vector bundles of rank two on X, because there are infinitely many elliptic curves on X.

Proof of the theorem. Instead of E itself we construct E^{∗}, the coherent sheaf of sections in the dual bundle E^{∗}. The sheaf E^{∗} is constructed as an extension of coherent sheafs

0 → L → E^{∗} → I_{Y} → 0.

∗ Actually the condition of smoothness of Y may be relaxed to the requirement that Y should be a local complete intersection.

Such extensions are classified by Ext^{1}(I_{Y} , L). In our situation (L locally free, I_{Y} is a coherent sheaf which locally admits a free resolution of length one) a lemma of Serre implies that the coherent sheaf associated to an element e ∈ Ext^{1}(I_{Y} , L) is locally free if and only if e_{y} generates Ext^{1}(I_{Y} , L)_{y} as a O_{X,y} -module sheaf at every point y ∈ Y . By Nakayama’s lemma this is equivalent to the assumption that e_{y} generates Ext^{1}(I_{Y} , L)(y) as complex vectorspace. (Here, as usual, S(x) = S_{x}/m_{x}S_{x} for any coherent sheaf S, where m_{x} is the maximal ideal of the local ring O_{X,x}.)

In order to find such an e ∈ Ext^{1}(I_{Y} , L), we will prove Ext^{1}(I_{Y} , L) ≃ O_{Y} and that the natural map

Ext^{1}(I_{Y} , L) → Γ(Ext^{1}(I_{Y} , L)) ≃ Γ(O_{Y}) ≃ ℂ

is surjective.

There is a spectral sequence

E_{2}^{p,q} = H^{p}(X, Ext^{q}(I_{Y} , L)) ⇒ E_{∞}^{p+q} = Ext^{p+q}(I_{Y} , L)

yielding the following exact sequence (”lower term sequence”)

0 → H^{1}(X, Hom(I_{Y} , L)) → Ext^{1}(I_{Y} , L) → → H^{0}(X, Ext^{1}(I_{Y} , L)) → H^{2}(X, Hom(I_{Y} , L)) → …

Since Y is a smooth submanifold of codimension two (actually needed: locally complete intersection of codimension two), we know that

Ext^{k}(O_{Y} , S) ≃ \{ \begin{matrix}0 & for k = 0, 1 \\ Hom(det I_{Y} /I_{Y}^{2} , S) & for k = 2 \end{matrix}

for every locally free sheaf S, thus in particular for S = L. Now we deduce the long exact Ext^{k}(·, L) sequence associated to

0 → I_{Y} → O_{X} → O_{Y} → 0

obtaining

0 → Hom(O_{Y} , L) → Hom(O_{X}, L) → Hom(I_{Y} , L) → Ext^{1}(O_{Y} , L) …

Thus Ext^{k}(O_{X}, L) = 0 for k > 0 implies Ext^{k}(I_{Y} , L) ≃ Ext^{k+1}(O_{Y} , L) for all k > 0. In particular Ext^{1}(I_{Y} , L) ≃ Hom(det I_{Y} /I_{Y}^{2} , L) ≃ O_{Y} where the second equivalence follows from our assumption L|_{Y} ≃ det I_{Y} /I_{Y}^{2} . Furthermore Hom(O_{Y} , L) is the zero sheaf, since L is locally free and O_{Y} is a torsion O_{X} -module sheaf. Hence

Hom(I_{Y} , L) ≃ Hom(O_{X}, L) ≃ L.

128 C.22

This means that locally every O_{X} -module sheaf homomorphism from I_{Y} to L is obtained via embedding I_{Y} ↪ O_{X} and multiplying f_{x} ∈ (I_{Y})_{x} ⊂ O_{X,x} by a section s_{x} ∈ L_{x}. Summarizing we obtain

0 → H^{1}(X, L) → Ext^{1}(I_{Y} , L) → H^{0}(Ext^{1}(I_{Y} , L)) → H^{2}(X, L)

with Ext^{1}(I_{Y} , L) ≃ O_{Y} .

Finally we have to study sections in E. For this purpose we consider the long exact Ext(·, O_{X})-sequence associated to 0 → L → E^{∗} → I_{Y} . Since Ext^{1}(I_{Y} , O_{X}) ≃ O_{Y} we obtain

0 → Hom(I_{Y} , O_{X}) ≃ O_{X} → E → L^{∗} → O_{Y} → 0

Now for any surjective sheaf homomorphism L^{∗} → O_{Y} the kernel sheaf must be L^{∗} ⊗ I_{Y} . Therefore

0 → Γ(X, O_{X}) → Γ(X, E) → Γ(X, L^{∗} ⊗ I_{Y}) → …

Since by assumption there are no sections in L^{∗} vanishing in Y but not vanishing everywhere, it follows that Γ(X, O_{X}) ⟶^{α} Γ(X, E) is an isomorphism. Let σ = α(1). By construction σ is a section in E inducing a O_{X} -sheaf homomorphism from E^{∗} to O_{X} which fibers E^{∗} → I_{Y} → O_{X} where the first map is surjective and the second map is ordinary inclusion. Hence {x : σ(x) = 0} = Y .

23. Deformations of vector bundles

For a holomorphic vector bundle E over a fixed compact complex space X there exists a versal deformation space and the Zariski-tangent space is exactly H^{1}(X, Hom(E, E)) [13]. Thus we are led to calculate the dimension of this cohomology group.

Lemma 27. Let S = SL_{2}(ℂ), Γ a discrete cocompact subgroup such that Γ/Γ′ is finite, X = S/Γ, Y ⊂ X an elliptic curve and E → X a holomorphic vector bundle of rank two with a section σ ∈ Γ(X, E) such that Y = {x : σ(x) = 0}.

Then dim_{ℂ} H^{1}(X, Hom(E, E)) = 2.

Proof. Exploitation of the long exact cohomology sequence associated to

0 → I_{Y} → O_{X} → O_{Y} → 0

yields

dim_{ℂ} H^{k}(X, I_{Y}) = \{ \begin{matrix}0 & for k = 0, 1 \\ 1 & for k = 2, 3 \end{matrix} .

129 C.23

Using duality between H^{k}(X, E) and H^{3−k}(X, E^{∗}), an evaluation of

0 → O_{X} → E → I_{Y} → 0

yields

dim_{ℂ} H^{k}(X, E) = \{ \begin{matrix}1 & for k = 0, 3 \\ 0 & for k = 1, 2 \end{matrix}

Now we apply Hom(E^{∗}, ·) to

0 → O_{X} → E → I_{Y} → 0

and obtain

0 → E → Hom(E, E) → E ⊗ I_{Y} → 0  (∗)

(The higher Ext-sheaves Ext^{k}(E, S) with k ≥ 1 vanish for every coherent sheaf S because E is locally free.)

0 → E ⊗ I_{Y} → E → E ⊗ O_{Y} → 0  (∗∗)

Using the triviality E|_{Y} the last sequence yields

dim_{ℂ} H^{k}(X, E ⊗ I_{Y}) = \{ \begin{matrix}1 & for k = 0, 3 \\ 2 & for k = 1, 2 \end{matrix}

Finally exploitation of (∗) yields

dim_{ℂ} H^{k}(X, Hom(E, E)) = 2 for all k = 0, 1, 2, 3.

A family of vector bundles over a complex space X is given by a vector bundle U → S × X. For every point s ∈ S this gives a vector bundle U_{s} → X as pull-back via the natural embedding X ↪ G × X given by x ↦ (s, x). Now any vector bundle E over a G-space X yields a family Ẽ → G×X by Ẽ = μ^{∗}E where μ : G × X → X is the map defining the group action. Given a section σ ∈ H^{0}(X, E) with C = {x : σ(x) = 0} one obtains sections σ_{g} ∈ H^{0}(X, E_{g}) with g(C) = {x : σ(x) = 0}.

In our situation it follows that E_{g} is not isomorphic to Eg̃ unless g(C) = g̃(C). Hence there is already a two-dimensional effective deformation induced by translations. Since the Zariski tangent space of the versal deformation space is two-dimensional, too, it follows that every small deformation is obtained by a translation.

References

1. Altman, A.B.; Kleiman, S.L.: Introduction to Grothendieck Duality Theory. LN 146 Springer 1970 2. Atiyah, M.: On the Krull-Schmidt theorem with applications to sheaves.

Bull. Soc. Math. France 84 , 307–317 (1956)

3. Atiyah, M.: Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc. 85, 181–207 (1957)

4. Atiyah, M.: K-Theory. Benjamin (1967)

5. Atiyah, M.; Rees, F.: Vector Bundles on Projective 3-space. Invent. math. 36, 131–153 (1976)

6. Barth, W.; Peters, C.; van de Ven, A.: Compact complex surfaces. Springer 1984

7. Boothby, W.; Wang, H.: On the finite Subgroups of Connected Lie Groups. Comm. Math. Helv. 39, 281–294 (1964)

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

9. Cartan, E.: La topologie des groupes de Lie. Enseign. math. 35, 177–200 (1936)

10. Cordero, L.; Fernandez, M.; Gray, A.: La suite spectrale de Frölicher et les nilvariétés complexes compactes. C.R. Acad. Sci. Paris 305 , 753–756 (1987)

11. Cordero, L.; Fernandez, M.; Gray, A.: The Froehlicher spectral sequence for compact nilmanifolds. Ill. J. Math. 35, no.1 , 56–67 (1991)

12. Dold, A.: Lectures on Algebraic Topology. Springer Berlin Heidelberg New York 1972.

13. Forster, O.; Knorr, K.: Über die Deformationen von Vektorraumbündeln auf kompakten komplexen Räumen. Math. Ann. 209 , 291–346 (1974)

14. Hartshorne, R.: Varieties of small codimension in projective space. Bull. A.M.S. 80 , 1017-1032 (1974)

15. Higman, G.: A finitely generated infinite simple group. J. London Math. Soc. 26, 61–64 (1951)

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

17. Jordan, C.: Mémoire sur les équations différent linéaires à intégrale algébriques. J. Reine Angew. Math. 84, 89–215 (1878)

18. Magnus, W.: On residually finite groups. Bull. A.M.S. 75 , 305–325 (1969)

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

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

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

22. Matsushima, Y.: Fibrés holomorphes sur un tore complexe. Nagoya Math. J. 14, 1–24 (1959)

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

24. Montgomery, D.; Zippin, L.: Topological Transformation Groups. Interscience Publishers 1955

25. Morimoto, A.: Sur le groupe d’Automorphismes d’un espace fibré principal analytique complexe. Nagoya Math. J.13, 157–168 (1959)

26. Mumford, D.: Abelian varieties. Oxford University Press 1970

27. Murakami, S.: Sur certains espaces fibrés principaux holomorphes admettant des connexions holomorphes. Osaka Math. J. 11, 43-62 (1959)

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

29. Okonek, Ch.; Schneider, M.; and Spindler, H.: Vector Bundles on Complex Projective Spaces. Progress in Math. 3, Birkhäuser Boston, 1980

30. Raghunathan, M.S.: On the first cohomology of discrete subgroups of semisimple Lie groups. Amer. J. Math. 87 , 103–139 (1965)

31. Raghunathan, M.S.: Vanishing Theorems for Cohomology Groups Associated To Discrete Subgroups of Semisimple Lie Groups. Osaka J. Math. 3 , 243-256 (1966)

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

33. Serre, J.P.: Sur les modules projectifs. Sem. Dubreil-Pisot exposé 2. (1960/61)

34. Taubes, C.H.: The existence of self-dual conformal structures. J. Diff. Geom. 36, 163–253 (1992)

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

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

37. Winkelmann, J.: Every compact complex manifold admits a holomorphic vector bundle. Revue Roum. Math. Pures et Appl. 38, 743-744 (1993)