Mappings and cohomology

B. Mappings and Cohomology

1. Survey In this chapter we study holomorphic mappings and cohomology groups in

the context of compact complex parallelizable manifolds. Many of the results obtained in this chapter are of a preparatory character.

We prove that many holomorphic mappings are automatically equivariant. In particular, if X = G/Γ is a compact complex parallelizable manifold, Y a normal complex space and f : X → Y a surjective connected holomorphic map, then necessarily Y is parallelizable too. In fact Y ≃ G/HΓ for some normal complex Lie subgroup H ⊂ G such that F : G/Γ → G/HΓ is simply the natural projection.

If G is a simply-connected simple complex Lie group, Γ a discrete cocompact subgroup, then self-maps of X = G/Γ have a very special form: Every holomorphic self-map of X is either constant or biholomorphic, i.e. an element in Aut_{O}(X). Moreover Aut_{O}(X)^{0} is a quotient of S by a finite subgroup and Aut_{O}(X)/ Aut_{O}(X)^{0} is finite.

Meromorphic maps to compact complex parallelizable manifolds are automatically holomorphic.

Extending vanishing results of Raghunathan we prove that H^{1}(G/Γ, O) = 0 for G semisimple and Γ/Γ′ finite. This implies in particular that the complex structure on X = G/Γ is rigid.

Finally we study the Hodge-Frölicher spectral sequence and the Albanese. For a compact complex manifold X the projection map to the Albanese Alb : X → A = Alb(X) induces homomorphisms of the cohomology groups Γ(·, dO), H^{1}(·, ℂ) and H^{1}(·, O). These group homomorphisms are always injective. For Kähler manifolds they are bijective. We prove that they are likewise bijective for many though not all parallelizable manifolds. On the other hand for each of this three cohomology groups there exists a compact complex parallelizable manifold for which this homomorphism is not surjective.

Let Γ denote a discrete cocompact subgroup in SL_{2}(ℂ). We note that the Betti-numbers of X = SL_{2}(ℂ)/Γ can be determined completely and that they depend only on r = dim_{ℚ}(Γ/Γ′) ⊗ ℚ.

2. Holomorphic mappings In I.§6 we mentioned already that, by universality of Tits-fibration, a com-

Compact complex parallelizable manifolds

pact complex parallelizable manifold X = G/Γ never admits a non-constant G-equivariant map to a projective space ℙ_{n}(ℂ). This is particularly useful, because proper connected surjective holomorphic maps are automatically equivariant. This is a result which basically goes back to Blanchard [3]. In the formulation cited below it is due to Remmert and van de Ven [20].

Theorem. Let f : X → Y be a surjective proper holomorphic map with connected fibers between normal complex spaces. Assume that there exists a holomorphic action of a connected complex Lie group G on X.

Then there exists a holomophic G-action on Y for which f is equivariant.

This has the following consequence for holomorphic maps from parallelizable manifolds.

Corollary. Let X be a compact complex parallelizable manifold, Y a complex space and f : X → Y a surjective holomorphic map. Then there exists a compact complex parallelizable manifold Y_{1} , a connected surjective holomorphic map f_{1} : X → Y_{1} and a surjective finite holomorphic map g : Y_{1} → Y such that g ∘ f_{1} = f.

In particular, if Y is normal and the fibers of f are connected, then Y is a compact parallelizable manifold.

Proof. First note that f fibers through the normalization of Y . Then Stein factorization implies that there is a normal complex space Y_{1} and holomorphic maps f_{1} : X → Y_{1} and g : Y_{1} → Y such that f_{1} is connected, g is finite and g ∘ f_{1} = f. The above theorem implies that f_{1} : X → Y_{1} is equivariant for any action of a connected complex Lie group on X. Hence Y_{1} is homogeneous, i.e. Y_{1} ≃ G/H where G is a complex Lie group and H a closed complex Lie subgroup. Now we have the Tits-fibration ζ : G/H → G/N_{G}(H^{0}) where the image G/N_{G}(H^{0}) can be embedded equivariantly into a projective space. It follows that ζ is constant, hence H^{0} is normal in G and the G-action reduces to an action of the quotient Lie group G/H^{0} . Therefore Y_{1} is parallelizable.

Corollary. Let G be a connected complex Lie group, Γ a discrete cocompact subgroup and H a connected complex Lie subgroup such that HΓ is closed in G.

Then H is normal in G.

Proof. By the proposition G/HΓ is parallelizable.

3. Holomophic mappings to parallelizable manifolds

As mentioned above, it is well-known that every surjective proper connected holomorphic map is equivariant for every connected complex Lie group acting on the source manifold. In general one can not drop the assumption that the map is surjective and connected. However, if the target space is parallelizable, then this is permissable.

Proposition 1. Let X be a connected compact complex manifold, Y a connected compact complex parallelizable manifold, f : X → Y a holomorphic map and G a connected complex Lie group acting on X.

Then there exists a G-action on Y such that f is G-equivariant.

Proof. Fix a point x ∈ X and a holomorphic vector field V ∈ Γ(X, T X). Now consider the pushed-forward tangent vector f_{∗}(V_{x}) ∈ T_{f(x)}Y . Since Y is compact parallelizable there exists a unique holomorphic vector field Z ∈ Γ(Y, T Y) such that f_{∗}(V_{x}) = Z_{f(x)} . Now for every p ∈ X and every holomorphic one-form ω ∈ Γ(Y, T^{∗}Y) we obtain

ω_{f(p)}(f_{∗}V_{p}) = f^{∗}ω(V)(p) = f^{∗}ω(V)(x) =
= ω_{f(x)}(f_{∗}V_{x}) = ω_{f(x)}(Z_{f(x)}) = ω(Z)(f(x)) = ω(Z)(f(p))

(note that f^{∗}ω(V) resp. ω(Z) are holomorphic functions on X resp. Y , hence constant). It follows that f_{∗}V_{p} = Z_{f(p)} for all p ∈ X. Thus f is equivariant for the one-parameter group corresponding to V resp. Z. Since V was choosen arbitrarily, it follows that f is equivariant for any connected complex Lie group acting on X.

Note: In fact the assumption of compactness of X and Y may be weakened to the assumption that neither X nor Y admit non-constant holomorphic functions.

The automorphism group of a parallelizable manifold

It is relatively easy to describe the connected component of the automorphism group of a compact complex parallelizable manifold.

Lemma 1. Let G be a connected complex Lie group, Γ a discrete cocompact subgroup, X = G/Γ.

Then Aut_{O}(X) is a complex Lie group acting holomorphically on X and Aut_{O}(X)^{0} ≃ G/(Z ∩ Γ) where Z denotes the center of G.

Proof. It is a result of Bochner and Montgomery [4] that for every connected compact complex manifold the automorphism group can be endowed with the structure of a complex Lie group (finite-dimensional, with countably many connected components) such that the action on the manifold is holomorphic. Moreover the Lie algebra of holomorphic vector fields is isomorphic to the Lie algebra associated to this complex Lie group in a natural way.

Parallelizability of X = G/Γ implies that there are no other holomorphic vector fields besides those induced by the G-action. Hence Aut_{O}(X)^{0} ≃ G/L where L denotes the set of all elements g ∈ G acting trivially on G/Γ. Evidently h ∈ L if and only if ghg^{−1} ∈ Γ for all g ∈ G. Since G is connected and Γ discrete, it follows that h ∈ L if and only if h ∈ Z ∩ Γ.

In general Aut_{O}(X)^{0}/ Aut_{O}(X) can be infinite. For instance, consider X = ℂ^{n}/(ℤ^{n} + iℤ^{n}) with n ≥ 2. Here Aut_{O}(X)/ Aut_{O}(X)^{0} is isomorphic isomorphic to GL_{n}(ℤ + iℤ).

However, we will show now that in the semisimple case Aut_{O}(X) has only finitely many connected components.

Theorem 1. Let S be a simply-connected semisimple complex Lie group, Γ a discrete cocompact subgroup, X = S/Γ.

Then Aut_{O}(X)/ Aut_{O}(X)^{0} is finite.

Proof. Note that S is the universal covering of Aut_{O}(X)^{0} , hence Aut_{O}(X) acts naturally on S. Let Λ denote the kernel of ϕ : Aut_{O}(X) → Aut(S). This means that Λ is the group of all automorphisms of X which commute with every s ∈ S. It follows that every λ ∈ Λ stabilizes the set Z = {x ∈ X : S_{x} = Γ} ≃ N/Γ with N = N_{S}(Γ). Evidently N is a closed Lie subgroup of S. Now N^{0} acts by conjugation on Γ. Since N^{0} is connected and Γ discrete, it follows that N^{0} centralizes Γ. The density results now imply that N^{0} is central in S. Thus semisimplicity of S implies N^{0} = {e}, i.e. N is discrete. Now Γ ⊂ N and cocompactness of Γ imply that N/Γ is finite. Thus the action of Λ on X stabilizes a finite subset, viz. N/Γ. Hence Λ admits a subgroup of finite index Λ_{1} such that every λ ∈ Λ_{1} admits a fixed point in X. But Λ_{1} commutes with S and the S -action on X is transitive. Hence the equation λs(x) = sλ(x) = s(x) for s ∈ S, λ ∈ Λ_{1} , x ∈ X with λ(x) = x implies that every λ ∈ Λ_{1} acts trivially. But Λ_{1} ⊂ Aut_{O}(X). Hence Λ_{1} = {e} and consequently ϕ : Aut_{O}(X) → Aut(S) has finite kernel. Furthermore, since the outer automorphism group is finite for every connected complex semisimple Lie group, it follows that Aut_{O}(X)/ Aut_{O}(X)^{0} is finite.

4. Holomorphic self-maps

Lemma 2. Let G be a semisimple (complex or real) Lie group, Γ a discrete, cocompact subgroup, ϕ ∈ Aut(G). Assume ϕ(Γ) ⊂ Γ.

Then ϕ(Γ) = Γ.

60 B.4

Proof. If ϕ(Γ) ≠ Γ, then Γ_{n} = ϕ^{−n}(Γ) form an infinite ascending sequence of discrete cocompact subgroups in G. By a result of H.C. Wang [23] there is no such sequence.

Lemma 3. Let G be a simply-connected Lie group, ϕ : G → G a Lie group homomorphism with discrete kernel.

Then ϕ is a Lie group automorphism.

Proof. Since ϕ has discrete kernel, the associated Lie algebra homomorphism Lie(ϕ) : Lie(G) → Lie(G) is injective. It follows that Lie(ϕ) is bijective, hence invertible. Now G being simply-connected implies that ϕ is invertible, too. Thus ϕ is an automorphism.

Lemma 4. Let G, H be simply-connected complex Lie groups, Γ, Λ discrete cocompact subgroups of G resp. H, X = G/Γ, Y = H/Λ. Let f : X → Y be a holomorphic map with f(eΓ) = eΛ, f̃ : G → H its lift to the universal coverings of X resp. Y which can be identified with G resp. H.

Then f̃ is a homomorphism of complex Lie groups.

Proof. This is a direct consequence of f being G-equivariant.

Proposition 2. Let G be a simply-connected semisimple complex Lie group, Γ a discrete cocompact subgroup, X = G/Γ and f : X → X a holomorphic map. Assume that at least one of the fibers of f is discrete.

Then f is an automorphism of X.

Proof. By proposition 1 the map f is G-equivariant. The induced map between the universal coverings f_{∗} : G → G is a homomorphism of complex Lie groups Evidently the kernel of f_{∗} is discrete, since the fibers of f are discrete. Now lemma 3 above implies that f_{∗} is an automorphism of G. By lemma 2 f_{∗}(Γ) ⊂ Γ implies f_{∗}(Γ) = Γ, i.e. f is biholomorphic.

Corollary. Let G be a simply-connected simple complex Lie group, Γ a discrete cocompact subgroup, X = G/Γ and f : X → X a holomorphic map.

Then f is either constant or an automorphism.

Proof. By the preceding results, f is G-equivariant and the image is biholomorphic to some quotient G/H with H^{0} normal in G. Since G is simple, either G = H or H is discrete. Thus either f is constant or f has discrete fibers. In the latter case the proposition implies f ∈ Aut_{O}(X).

Note: Instead of requiring G to be simple, it is enough to demand that Γ is an irreducible discrete cocompact subgroup in a semisimple complex Lie group G.

61 B.5 5. Meromorphic maps to parallelizable manifolds

It is well-known that every meromorphic map from a manifold to a torus is holomorphic. This can be extended to maps with values in parallelizable manifolds.

Proposition 3. Let Y be a compact complex parallelizable manifold, X an arbitrary complex manifold, f : X → Y a meromorphic map.

Then f is holomorphic.

Proof. Let E(f) denote the exceptional set. For every x ∈ E(f) there is an open neighbourhood U(x) such that V = U(x) \ E(f) is simply-connected (because X is smooth and codim(E(f)) ≥ 2). Now f|V : V → Y lifts to the universal covering of Y , which is a simply-connected complex Lie group, hence Stein. Again using codim(E(f)) ≥ 2 it follows that f extends as holomorphic map from V to U. Thus for every x ∈ E(f) there exists an open neighbourhood U(x) such that f extends as a holomorphic map from U(x) \ E(f) to U(x). By identity principle these local extensions are compatible. Thus f is holomorphic.

Corollary. Let X be a connected compact complex parallelizable manifold, Bim(X) the group of bimeromorphic self-maps. Then Bim(X) = Aut_{O}(X).

6. Induced Actions on Cohomology Groups

Let G be a complex Lie group acting holomorphically on a complex space X. Every g ∈ G induces a biholomorphic map μ(g) : X → X which in turn induces a ℂ-linear automorphism on the cohomology groups H^{k}(X, ℂ) and H^{k}(X, O). Thus we obtain induced actions of G on the cohomology groups H^{k}(X, ℂ) and H^{k}(X, O). If G is connected, then every μ(g) is homotopic to id_{X}. Hence the induced action on H^{•}(X, ℂ) is trivial for every connected Lie group G.

Let us now consider compact complex spaces X. Then H^{k}(X, O) is finitedimensional. We want to show that for every complex Lie group acting holomorphically on X the induced action on H^{•}(X, O) is holomorphic. For this purpose we fix an element g ∈ G and choose an open neighbourhood V (g) ⊂ G which is biholomorphic to a ball. Then V (g) is Stein and a Leray spectral sequence argument for the projection map π : V ×X → V yields that H^{k}(V ×X, O) is naturally isomorphic to the space of holomorphic mappings from V to H^{k}(X, O). The group action gives us a holomorphic map ϕ : V × X → X defined by ϕ(h, x) = h · x inducing a ℂ-linear map

ϕ^{∗} : H^{k}(X, O) → H^{k}(V × X, O).

Now for any α ∈ H^{k}(X, O), g ∈ G be the pulled-back element f^{∗}α which corresponds to a holomorphic map f : V → H^{k}(X, O) such that f(g) = g(α). Therefore the induced action of G on H^{k}(X, O) is holomorphic. Of course the same argumentation yields that G induces a holomorphic action on the cohomology groups of other canonically defined coherent sheaves like Ω^{k}, the sheaf of holomorphic k -forms.

Lescure gave a different proof for the holomorphicity of the induced actions on H^{q}(X, Ω^{p}) based on the Dolbeault-isomorphism H^{q}(X, Ω^{p}) ≃ H_{Dolb}^{p,q}(X) (see [12]).

It should be emphasized that, for a compact Kähler manifold, Hodge decomposition yields a canonical injection of the Dolbeault groups H^{q}(X, Ω^{p}) into H^{p+q}(X, ℂ). This implies that for any connected Lie group acting holomorphically on a compact Kähler manifold the induced action on the Dolbeault groups H^{q}(X, Ω^{p}) is trivial.

On the other hand, Lie group actions on non-Kähler manifolds need not be trivial. There is an example of Lescure (see [11]) of a ℂ^{∗} -action on a non-Kähler compact complex 4-manifold such that the induced action on H^{1}(X, O) is non-trivial.

7. Some topological preparations for finite coverings

In the next paragraph we will prove a vanishing theorem for H^{1}(X, O). For this purpose we need some topological results for finite coverings. Here we deduce these auxiliary resuls.

First we cite the following result from algebraic topology.

Proposition (see [6,p.314]). Let τ : X → Y be a finite covering of complex manifolds.

Then the induced ring homomorphism H^{•}(Y, ℂ) → H^{•}(X, ℂ) is injective.

We would like to have a similar result for local systems. Here we regard a local system L as a locally constant sheaf of complex vector spaces, i.e. a local system is a sheaf of locally constant sections in a complex vector bundle with locally constant transition functions.

The following should be well-known, but we provide a proof for convenience.

Lemma 5. Let Y be a complex manifold (or just a topological space), L → Y a local system, π : X → Y a finite (unramified) covering.

Then there is a split short exact sequence of local systems

0 → L → π_{∗}π^{∗}L → Q → 0

for some local system Q on X.

Proof. For x ∈ X the stalk L_{x} is a complex vector space and (π_{∗}π^{∗}L)_{x} is the vector space of mappings from the finite set π^{−1}(x) to L_{x}. There is an obvious embedding i : L_{x} → (π_{∗}π^{∗}L)_{x} which associates to each v ∈ L_{x} the constant map from π^{−1}(x) to L_{x} with value v. On the other hand there is a projection ρ : (π_{∗}π^{∗}L)_{x} → L_{x} given by

ρ(f) = \frac{1}{n} ∑_{y∈π^{−1}(x)} f(y)

with n = #π^{−1}(x). Now π_{∗}π^{∗}L_{x} ≃ i(L_{x}) ⊕ Q_{x} with Q_{x} = ker ρ.

Proposition 4. Let π : X → Y a finite (unramified) covering of complex manifolds (or topological spaces), L a local system on Y .

Then the induced homomorphism π^{∗} : H^{•}(Y, L) → H^{•}(X, π^{∗}L) is injective.

Proof. A Leray spectral sequence argument yields H^{•}(Y, π_{∗}π^{∗}L) ≃ H^{•}(X, π^{∗}L). The injectivity of H^{•}(Y, L) → H^{•}(Y, π_{∗}π^{∗}L) follows from the above lemma.

8. Vanishing Theorems for H^{1}(X, O)

We are now going to investigate under which circumstances H^{1}(X, O_{X}) = {0} for compact quotients X = S/Γ, S semisimple, Γ discrete.

In [18] Raghunathan proved

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

Then H^{1}(X, O_{X}) = {0} for X = S/Γ.

The assumption that no simple factor is isomorphic to SL_{2}(ℂ) is used in the proof only in order to ensure that H^{1}(X, L_{ρ}) = {0} for every local system induced by a holomorphic representation ρ : S → GL_{n}(ℂ).

However, using an other vanishing result of Raghunathan we will deduce the following result.

Proposition 5. Let S be a simply-connected semisimple complex Lie group, Γ a discrete cocompact subgroup such that Γ/Γ′ is finite, ρ : S → GL_{n}(ℂ) a holomorphic representation, L_{ρ} the induced local system.

Then H^{1}(S/Γ, L_{ρ}) = {0}.

Proof. For the trivial representation, vanishing of H^{1}(X, L_{ρ}) = H^{1}(X, ℝ) is equivalent to the finiteness of Γ/Γ′, since Γ is finitely generated and

H^{1}(X, ℝ) ≃ Hom(π_{1}(X), ℝ) ≃ Hom(Γ/Γ, ℝ).

For the general case we first assume that Γ is torsion-free. Let K be a maximal compact subgroup of S. Then K is a simply-connected compact real Lie group. Hence H^{q}(K, ℝ) = 0 for q = 1, 2. We consider the K -principal bundle ϕ : X = S/Γ → K\S/Γ = M which yields the following Leray spectral sequence:

H^{p+q}(X, L_{ρ}) ⇐ E_{2}^{p,q} = H^{p}(M, R^{q}ϕ_{∗}L_{ρ})

The bundle ϕ : X → M is a locally trivial K -principal bundle, since we required Γ to be torsion-free. Hence π_{0}(K) = π_{1}(K) = 0 imply that ϕ induces an isomorphism π_{1}(X) →^{∼} π_{1}(M). Via this isomorphism ρ induces a representation ρ̃ : π_{1}(M) → GL_{n}(ℂ) such that ϕ_{∗}L_{ρ} is isomorphic to the local system Lρ̃ associated to ρ̃. For any contractible open subset U ⊂ M the restriction of Lρ̃ to U is trivial and the cohomology groups H^{q}(ϕ^{−1}(U), ℂ) vanish for q = 1, 2, because ϕ^{−1}(U) is homotopyequivalent to K. It follows that the higher direct image sheaves R^{q}ϕ_{∗}L_{ρ} vanish for q = 1, 2. In particular E_{2}^{p,1} = 0. Now the lower term sequence yields

0 → E_{2}^{1,0} = H^{1}(M, Lρ̃) →^{∼} E_{∞}^{1} = H^{1}(X, L_{ρ}) → E_{2}^{0,1} = 0

The quotient K\S is a contractible topological space on which Γ acts freely from the right. Hence M = K\S/Γ is an Eilenberg-MacLane space K(Γ, 1). A well-known theorem of Eilenberg implies H^{1}(M, Lρ̃) ≃ H^{1}(Γ, ρ|_{Γ}). Thus we need a vanishing theorem for H^{1}(Γ, ρ|_{Γ}). Clearly it is enough to prove this under the additional hypothesis that ρ is irreducible.

Now we can make use of the results of Raghunathan. Taking into account that S is a complex Lie group, the main result of [17] states that H^{1}(Γ, ρ|_{Γ}) vanishes unless S contains a simple factor S_{0} with S_{0} ≃ SL_{2}(ℂ) such that the height weight of ρ|_{S_{0}} is an integral multiple of the highest weight associated to the ”natural” (real) representation SL_{2}(ℂ) → SL_{4}(ℝ) induced by the real isomorphism SL_{2}(ℂ) ≃ SO(3, 1). But ρ was assumed to be a holomorphic representation. Hence the heighest weight associated to ρ can not be a multiple associated to the real representation SL_{2}(ℂ) → SL_{4}(ℝ). Hence Raghunathan’s result implies H^{1}(Γ, ρ|_{Γ}) = 0. Therefore H^{1}(X, L_{ρ}) = {0} and the prove is complete for torsion-free Γ.

Due to a theorem of Selberg Γ admits a torsion-free subgroup Γ_{1} of finite index. Thus H^{1}(X_{1}, τ^{∗}L_{ρ}) = 0 for X_{1} = S/Γ_{1} . By proposition 4 this implies H^{1}(X, L_{ρ}) = 0.

As a consequence we obtain the following generalization of Raghunathan’s vanishing result.

65 B.8 Theorem 2. Let S be a simply-connected semisimple complex Lie group, Γ a discrete cocompact subgroup.

Assume that at least one of the following conditions are fulfilled:

(1) No direct factor of S is isomorphic to SL_{2}(ℂ);

(2) Γ/Γ′ is finite.

Then H^{1}(X, O) = {0} for X = S/Γ.

Proof. This follows using Raghunathan’s work in [18]. There he proved that for every representation ρ : S → GL(V) there is an isomorphism

H^{1}(X, L_{ρ}) ≃ (H^{1}(X, O) ⊗_{ℂ} V )^{S} .

As we have seen before in §6, S acts holomorphically on H^{1}(X, O). If H^{1}(X, L_{ρ}) vanishes for all holomorphic representations ρ, this implies that H^{1}(X, O) = 0, because we can choose V as the S -module dual to H^{1}(X, O). (For every S -module F the tensor product F ⊗F^{∗} contains a trivial S -submodule, because F ⊗F^{∗} ≃ Hom(F, F) and the identity homomorphism spans a one-dimensional trivial S -submodule of Hom(F, F).)

9. Deformations

Here we want to review some work on small deformations of the complex structure on compact complex parallelizable manifolds. For basic facts and notions on deformation theory see e.g. [10][7].

By a result of Andreotti and Stoll any small deformation of a compact complex torus is again a compact complex torus [1,Th.8 on p.339].

For solvmanifolds deformations were studied by Nakamura [15]. He showed that small deformations of solvmanifolds need not be parallelizable. This is demonstrated by a family of compact complex manifolds X_{t} whereas X_{0} is a three-dimensional solvmanifold while for t ≠ 0 the universal covering is not holomorphically convex. (This implies that for t ≠ 0 the manifold is not parallelizable, because for a compact complex parallelizable manifold the universal covering is a simply-connected complex Lie group and therefore a Stein manifold.) He used this example to demonstrate that the Kodaira-dimension κ(X) and the plurigenera p_{g}(X) = dim_{ℂ} Γ(X, K^{m}) are no deformation invariants, thereby giving a negative answer to a question of Iitaka.

As discussed in a preceding section, for many discrete cocompact subgroups Γ in semisimple complex Lie groups there is a vanishing result for H^{1}(S/Γ, O). By definition T_{X} ≃ O^{n} for a n-dimensional complex parallelizable manifold. Hence H^{1}(X, O) = {0} implies H^{1}(X, T) = {0} for X parallelizable. Now it is a basic fact from deformation theory that H^{1}(X, T) may be regarded as the Zariski tangent space for the versal family of infinitesimal deformations. (This versal family exists for all compact complex manifolds due to a result of Kuranishi.) Hence there are no infinitesimal deformations as soon as H^{1}(X, O) = {0}.

Proposition 6. Let G be a simply-connected semisimple complex Lie group, Γ a discrete cocompact subgroup. Assume in addition that one of the following conditions are fulfilled:

(i) No direct factor of G is isomorphic to SL_{2}(ℂ); (ii) G ≃ SL_{2}(ℂ) and Γ/Γ′ is finite.

Then X = G/Γ is rigid, i.e. there are no small deformations.

10. The Hodge-Frölicher spectral sequence

The Hodge-Frölicher spectral sequence relates Dolbeault- and DeRham-Cohomology on complex manifolds as follows:

H^{p+q}(X, ℂ) = E_{∞}^{p+q} ⇐ E_{•}^{p,q}

with

E_{1}^{p,q} = H_{Dolb}^{p,q}(X, ℂ) ≃ H^{q}(X, Ω^{p}).

It is a fundamental fact in Kähler geometry that this spectral sequence collapses in E_{1}^{p,q} for every compact Kähler manifold. More generally it collapses in E_{1}^{p,q} for all compact manifolds for which the Hodge-decomposition holds. This includes all Moishezon manifolds and moreover all manifold in Fujiki’s class C. It also collapses in E_{1}^{p,q} for all compact complex surfaces. Now E_{1}^{1,0} is isomorphic to the vector space of all holomorphic one-forms H^{0}(X, Ω^{1}), whereas E_{2}^{1,0} is isomorphic to the vector space H^{0}(X, dO) of all closed holomorphic one-forms. Hence E_{1}^{1,0} ≠ E_{2}^{1,0} for all compact complex parallelizable manifolds other than tori. Thus for compact complex parallelizable manifolds other than tori the Hodge-Frölicher spectral sequence cannot collapse before E_{2}^{p,q}. A result basically due to Sakane [5,Thm.9] states that for nilmanifolds it does collapse in E_{2}^{p,q}. The results of the next paragraph will yield the following:

Let X = SL_{2}(ℂ)/Γ with Γ discrete, cocompact and Γ/Γ′ finite. Then the Hodge-Frölicher spectral sequence collapses in E_{2}^{p,q}.

11. Betti-numbers of SL_{2}(ℂ)/Γ

Proposition 7. Let S = SL_{2}(ℂ), Γ a discrete cocompact subgroup with Γ/Γ′ finite, X = S/Γ.

Then the Betti-numbers b_{k}(X) = dim_{ℂ} H^{k}(X, ℂ) are given as follows

\begin{matrix}k & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ b_{k}(X) & 1 & 0 & 0 & 2 & 0 & 0 & 1 \end{matrix}

1^{st} Proof. We employ the Hodge-Frölicher spectral sequence

H^{k}(X, ℂ) = E_{∞}^{k} ⇐ E_{1}^{p,q} = H^{q}(X, Ω^{p})

Recall H^{1}(X, O) = {0}. By Serre-duality H^{2}(X, O) vanishes as well. Using parallelizability this implies E_{1}^{p,2} = E_{1}^{p,1} = 0 for all p. Therefore all the d_{r} vanish except possibly d_{1} : E_{1}^{p,0} → E_{1}^{p+1,0} and d_{1} : E_{1}^{p,3} → E_{1}^{p+1,3} . (In particular the spectral sequence collapses in E_{2}^{p,q}.) Now E^{p,0} = H^{0}(X, Ω^{p}) ≃ ⋀^{p} g^{∗} where d_{1} : E_{1}^{p,0} → E_{1}^{p+1,0} is simply differentiation. Thus an explicit calculation shows that E_{1}^{1,0} →^{d_{1}} E_{1}^{2,0} is surjective. Hence E_{2}^{2,0} = 0 and therefore E_{∞}^{2} = 0. Thus b_{2}(X) = 0. Furthermore b_{0}(X) = 0 for every connected manifold and b_{1}(X) = 0, because H^{1}(X, ℂ) ≃ Hom(Γ, ℂ). Poincaré-duality implies b_{k}(X) = b_{6−k}(X). Finally, due to Hopf index theorem, parallelizability implies χ_{top}(X) = ∑_{k}(−1)^{k}b_{k}(X) = 0. This yields b_{3}(X) = 2.

One may also use the Leray-spectral sequence.

2^{nd} Proof. We will first assume that Γ is torsion-free. The cohomology groups H^{k}(X, ℂ) may be interpreted as the sheaf cohomology for the sheaf \underline{ℂ} of locally constant complex-valued functions. Let K ≃ SU_{2}(ℂ) denote a maximal compact subgroup of S = SL_{2}(ℂ) and consider the projection ϕ : X → M = K\X = K\S/Γ. This is a locally trivial bundle, because Γ is torsion-free. Hence H^{1}(K, ℂ) = H^{2}(K, ℂ) = \{0\} imply the vanishing of the higher direct image sheaves R^{q}ϕ_{∗}\underline{ℂ} for q = 1, 2. For the Leray spectral sequence

H^{p+q}(X, ℂ) ⇐ E_{2}^{p,q} ≃ H^{p}(M, R^{q}ϕ_{∗}\underline{ℂ})

this implies E_{2}^{p,q} = 0 for q = 1, 2. We will next show that E_{2}^{2,0} = 0. By Poincaréduality for the manifold M we obtain H^{2}(M, ℂ) ≃ H^{1}(M, ℂ). Hence

E_{2}^{2,0} = H^{2}(M, ϕ_{∗}\underline{ℂ}) ≃ H^{2}(M, ℂ) ≃ H^{1}(M, ℂ) ≃ Hom(Γ, ℂ) = 0.

Thus E_{2}^{p,q} = 0 for all p, q with p + q = 2. Hence E_{∞}^{2} = H^{2}(X, ℂ) = 0. Moreover H^{1}(X, ℂ) = 0 (because Γ/Γ′ is finite) and H^{0}(X, ℂ) = ℂ. All other Betti-numbers now follow using Poincaré-duality for X and the fact that the Poincaré-Euler characteristic of X must vanish due to parallelizability of X. This completes the proof for the case where Γ is torsion-free.

For arbitrary Γ the theorem of Selberg [21] gives us a subgroup of finite index Γ_{1} ⊂ Γ and thereby a finite unramified covering G/Γ_{1} = X_{1} → X. Now H^{•}(X, ℂ) → H^{•}(X_{1}, ℂ) is injective, hence b_{1}(X_{1}) = b_{2}(X_{1}) = 0 imply b_{1}(X) = b_{2}(X) = 0. Evidently b_{0}(X) = 1. By Poincaré-duality b_{4}(X) = b_{5}(X) = 0 and b_{6}(X) = 1. Finally χ(X_{1}) = [Γ_{1} : Γ]χ(X) = 0 yields b_{3}(X) = 2.

68 B.11 Remark.

(1) Note that E_{2}^{2,0} → E_{∞}^{2} is always injective, because d_{r} : E_{r}^{p,q} → E_{r+1}^{p+r,q+1−r}. Hence the above argumentation yields a proof of the following more general statement. Let S = SL_{2}(ℂ), Γ a torsion-free discrete cocompact subgroup, r = rank_{ℤ}(Γ/Γ′), X = S/Γ. Then the Betti-numbers b_{k}(X) = dim_{ℂ} H^{k}(X, ℂ) are given as follows

\begin{matrix}k & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ b_{k}(X) & 1 & r & r & 2 & r & r & 1 \end{matrix}

(2) If G is a real nilpotent Lie group, Γ a discrete cocompact subgroup, then a result of Nomizu [16] implies that there is an isomorphism between Lie algebra cohomology and de Rham-Cohomology,

ϕ : H^{•}(Lie(G), ℝ) →^{∼} H^{•}(X, ℝ),

where Lie(G) is the associated real Lie algebra and X = G/Γ. The map ϕ is the natural map arising from the fact that the Lie algebra cohomology for the trivial module ℝ may be interpreted as the cohomology of the complex of G-right invariant differential forms on G. The above proposition implies that this homomorphism ϕ is likewise an isomorphism if G is SL_{2}(ℂ) regarded as real Lie group and Γ as discrete cocompact subgroup with Γ/Γ′ finite.

(3) For every discrete cocompact subgroup Γ ⊂ SL_{2}(ℂ) the quotient manifold contains curves (proposition A.6). Hence the proposition implies that for discrete cocompact subgroups Γ ⊂ SL_{2}(ℂ) with Γ/Γ′ finite the quotient manifold X = SL_{2}(ℂ)/Γ contains curves C ⊂ X whose fundamental class [C] ∈ H_{2}(X, ℂ) is zero.

12. The Albanese

Given a compact complex manifold X, there exists the Albanese torus Alb(X) together with a holomorphic map Alb : X → Alb(X) [3]. It has the following universality property:

Let T be a torus, f : X → T a holomorphic map. Then there exists a unique holomorphic map F : Alb(X) → T such that f = F ∘ Alb.

The Albanese may be constructed in the following way: Take V = Γ(X, dO) the vector space of closed holomorphic one-forms. Let V^{∗} denote the dual vectorspace. Define a map ψ : H_{1}(X, ℤ) → V^{∗} via

ψ(γ)(ω) = ∫_{γ} ω.

Let H denote the smallest closed complex Lie subgroup of (V^{∗}, +) containing ψ(H_{1}(X, ℤ)). Then Alb(X) = V^{∗}/H with Alb : X → Alb(X) induced by x ↦ ∫_{x_{0}}^{x} where x_{0} is a fixed base point in X.

For compact complex homogeneous manifolds this specializes to the following.

Proposition 8. Let G be a connected complex Lie group, Γ a closed complex Lie subgroup such that X = G/Γ is compact.

Let I denote the smallest closed complex Lie subgroup of G containing G′Γ. Then Alb(X) ≃ G/I with Alb : G/Γ → G/I = Alb(X) being the natural projection.

Proof. Universality yields that Alb : X → Alb(X) is equivariant for every automorphism of X. Thus the image is an orbit of a connected subgroup of Aut(Alb(X)), hence a subtorus. The uniqueness part of the universality property yields that this subtorus equals Alb(X). Hence Alb(X) = G/H for some closed complex Lie subgroup H of G. Evidently G/H being a torus implies G′Γ ⊂ H. Hence H ⊂ I with I defined as above. On the other hand G/I is a torus, hence I ⊂ H due to universality.

Corollary. Let G be a connected complex Lie group, Γ a closed complex Lie subgroup such that X = G/Γ is compact.

Then Alb : X → Alb(X) has connected fibers.

Proof of the corollary. Connectedness of G forces G′ and therefore G′Γ/Γ to be connected. Hence I must be connected.

Remark. It is clearly important whether or not I = G′Γ, i.e. whether or not G′Γ is closed in G. In A.§3 we proved the following: Let G be a simplyconnected complex Lie group, Γ a discrete cocompact subgroup, R the radical of G, S = G/R. Assume that no direct factor of S is isomorphic to SL(2, ℂ). Then G′Γ is closed in G.

We will now discuss how certain cohomology groups on X relate to those on Alb(X).

Theorem 3. Let X be a compact complex manifold and A = Alb(X). Consider the homomorphisms of cohomology groups

Γ(A, dO) →^{α} Γ(X, dO)

H^{1}(A, ℂ) →^{β} H^{1}(X, ℂ)

H^{1}(A, O) →^{γ} H^{1}(X, O)

induced by Alb : X → A.

Then the group homomorphisms α,β, γ are injective. For certain compact complex manifolds they are surjective as well.

(i) If X is Kähler then all three homomorphisms α, β, γ are isomorphisms; (ii) Assume that X is parallelizable, i.e. X = G/Γ with G simply-connected

complex Lie group, Γ discrete. Assume furthermore that G′Γ is closed in G and that (G′ ∩ Γ)/Γ′ is finite. Then both the homomorphisms α and β are isomorphisms.

(iii) Assume that X = G/Γ is parallelizable with G simply-connected complex Lie group, Γ discrete,cocompact. Assume in addition that one of the following conditions is fulfilled: α) G is nilpotent; β) G is semisimple and not factor of G is isomorphic to SL_{2}(ℂ); γ) G is semisimple and Γ/Γ′ is finite. Then γ is an isomorphism.

Before starting the proof we discuss the cohomology groups which are involved in the case of parallelizable manifolds.

Lemma 6. Let X = G/Γ be a compact complex manifold, where G is a simplyconnected complex Lie group and Γ a discrete cocompact subgroup.

Then Γ(X, dO) ≃ (g/g′)^{∗} and

H^{1}(X, ℂ) ≃ Hom(Γ, ℂ) ≃ (Γ/Γ′, ℂ).

Proof. Parallelizability implies that there is a one-to-one correspondence between H^{0}(X, Ω^{1}) and the vector space Lie(G)^{∗} of complex linear functionals on the Lie algebra of G. Now

dω(X, Y) = Xω(Y) − Y ω(X) − ω([X, Y])

for any holomorphic one-form ω and vector fields X, Y . Note that ω(X), ω(Y) are global holomorphic functions and therefore constant. It follows that dω(X, Y) = −ω([X, Y]). Hence dω = 0 if and only if the corresponding linear functional on Lie(G) vanishes on Lie(G′). Thus H^{0}(X, dO) ≃ Lie(G/G′)^{∗}. Furthermore H_{1}(X, ℤ) ≃ Γ/Γ′. Hence H^{1}(X, ℂ) ≃ Hom(Γ, ℂ).

Proof of the proposition. For Kähler manifolds it is well-known that all the group homomorphisms α, β, γ are isomorphisms. This is a consequence of the Hodgedecomposition.

Now let us prove that α, β, γ are injective for arbitrary compact complex manifolds X. Injectivity of α is an immediate consequence of the method of construction of Alb(X). The universality of Alb(X) implies that the map X → Alb(X) can not be lifted to any non-trivial covering of Alb(X). Therefore π_{1}(X) → π_{1}(Alb(X)) is surjective. Since H^{1}(X, ℂ) ≃ Hom(π_{1}(X), ℂ) this yields injectivity of β.

For the injectivity of γ we consider the image Z of X in A = Alb(X). Let Z̃ be a desingularization of Z such that X → A fibers through X → Z̃ → Z → A. Then X → A also fibers through X → Z̃ → Alb(Z̃) → A. It suffices to show that at each step the induced group homomorphism on H^{1}(·, O) is injective. Now Alb(Z̃) → A is a surjective morphism of tori, for which injectivity is easy to check. Z̃ is by construction a manifold in class C, hence bimeromorphic to a Kähler manifold. In particular Z̃ enjoys the Hodge-decomposition. This yields the injectivity of H^{1}(Alb(Z̃), O) → H^{1}(Z, O). Finally the projection X → Z̃ is a proper surjective holomorphic map. Such a map admits a Stein factorization X → Y → Z̃, with f : X → Y connected and g : Y → Z̃ finite (and both f and g proper and surjective). Here Y is an irreducible normal compact complex space.

Since f is proper and connected we obtain f_{∗}O_{X} ≃ O_{Y} , which implies the injectivity of f^{∗} : H^{1}(Y, O) → H^{1}(X, O). Finally define a subsheaf S ⊂ g_{∗}O_{Y} as follows: A function f ∈ g_{∗}O_{Y} (U) = O_{Y} (g^{−1}(U)) (with U open) in Z̃ belongs to S(U) if and only if ∑_{y∈g^{−1}(z)} f(y) = 0 for all z ∈ Z̃ not contained in the ramification locus. Then g_{∗}O_{Y} = S ⊕ OZ̃ where OZ̃ is embedded in g_{∗}O_{Y} in the obvious way. Thus OZ̃ is a direct summand of g_{∗}O_{Y} . This clearly implies injectivity of H^{1}(Z̃, O) → H^{1}(Y, O).

Next we deal with the assertions for parallelizable manifolds, first discussing statement (ii). For the above mentioned method for constructing Alb(X) we obtain V^{∗} ≃ Lie(G/G′) ≃ G/G′ (G/G′ is a simply-connected commutative Lie group!) and H_{1}(X, ℤ) ≃ Γ/Γ′ in a canonical way. Now closedness of G′Γ in G implies that ψ(H_{1}(X, ℤ) is discrete in V^{∗} and Alb(X) ≃ G/G′Γ. Furthermore we obtain H_{1}(X, ℤ) ≃ Γ/Γ′ and H_{1}(Alb(X), ℤ) ≃ Γ/(G′ ∩ Γ). Hence finiteness of (G′ ∩ Γ)/Γ′ implies H_{1}(X, ℂ) ≃ H_{1}(Alb(X), ℂ) and consequently H^{1}(X, ℂ) ≃ H^{1}(Alb(X), ℂ). With Γ(X, dO) ≃ (g/g′)^{∗} bijectivity of α is a direct consequence of Alb(X) ≃ G/G′Γ.

Bijectivity of γ for nilmanifolds is due to Kodaira [15,Thm.3].

For semisimple G the Albanese Alb(X) reduces to a point. Hence, for G semisimple bijectivity of γ simply means vanishing of H^{1}(X, O). Thus (iii) is a corollary to the vanishing theorem from section 8.

Nakamura gave an example of a three-dimensional solvmanifold for which H^{1}(Alb(X), O) → H^{1}(X, O) is not surjective [15].

Remark. The example of a compact parallelizable manifold X = G/Γ with non-closed G′ -orbits given in A.§3 yields a parallelizable example of a compact complex manifold with ψ(H_{1}(X, ℤ)) not closed in V^{∗}. The first example of

72 B.12 a compact complex manifold with such a property (a certain torus-principal bundle over ℙ_{1}(ℂ)) is due to Blanchard [3].

References

1. Andreotti, A.; Stoll,W.: Extension of holomorphic maps. Ann. of Math. 72, 312–349 (1960)

2. Barth, W.; Peters; van de Ven: Compact complex surfaces. Springer 1984

3. Blanchard, A.: Sur les variétés analytiques complexes. Ann. Sci. ecole norm. sup. 73, 157–202 (1956)

4. Bochner, S.; Montgomery, D.: Groups on analytic manifolds. Ann. of Math. 48 (2) , 659-669 (1947).

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

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

7. Douady, A.: Le probléme de modules pour les sous–espaces analytiques compacts d’un espace analytique donné. Ann. Inst. Fourier 16, 1–95 (1966) MR 43#2940

8. Frölicher: Relations between the cohomology groups of Dolbeault and topological invariants. Proc. Nat. Acad. Sci. U.S.A. 41, 641–644 (1955)

9. Griffiths, P.; Harris, J.: Principles of Algebraic Geometry. Pure and Applied Mathematics, Wiley-Interscience, New-York (1978)

10. Kodaira, K.: Complex Manifolds and Deformation of Complex Structures. Springer 1981

11. Lescure, F.: Action non triviale sur le premier groupe de comologie de Dolbeault. Comptes Rendus Acad. Sci. Paris 316, 823–825 (1993)

12. Lescure, F.: Cohomologie totale et courants de Dolbeault invariants. preprint (1994)

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

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

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

16. Nomizu, K.: On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. Math. (2) 59, 531–538 (1954)

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

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

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

20. Remmert, R.; van de Ven, T.: Zur Funktionentheorie homogener komplexer Mannigfaltigkeiten. Topology 2 , 137–157 (1963)

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

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

23. Wang, H.: On a maximality property of discrete subgroups with fundamental domain of finite measure. Amer. J. Math. 89 , 124–132 (1967)

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