Structure of nilmanifolds

D. On the structure of complex nilmanifolds

1. The results

A nilmanifold X is a compact complex manifold with holomorphically trivial tangent bundle and nilpotent fundamental group. It is well-known that such a manifold is biholomorphic to a quotient of a nilpotent complex Lie group G by a discrete subgroup Γ. Furthermore, such a nilmanifold is a tower of torus principal bundles. Here we are interested in the structure of those tori occuring in this tower. We will prove that (up to isogeny) they are all products of simple tori with complex multiplication, unless the nilmanifold is somewhat degenerate. Conversely, every torus with complex multiplication occurs in the tower of a nilmanifold in a non-trivial way. Here a torus T is defined as a torus with complex multiplication if End_{ℚ}(T) contains a numberfield K such that [K : ℚ] = 2 dim_{ℂ}(T). This implies that K is totally complex. Tori with complex multiplicated are rather special tori. For instance, up to isomorphism there exist only countably many tori with complex multiplication. A one-dimensional torus ℂ/ ⟨1, τ⟩ has complex multiplication if and only if τ^{2} ∈ ℚ + τℚ. However, a torus with complex multiplication is not necessarily an abelian variety. In fact, for every totally complex number field K which is not a CM-field there exists a torus T with complex multiplication by K such that T admits no non-constant meromorphic functions at all. (A totally complex number field K is called a CM-field, if it contains a totally real subfield K^{+} with [K : K^{+}] = 2.) We use this to give an example of a non-trivial nilmanifold without non-constant meromorphic functions. Now we present the main result.

Theorem 1. Let G be a connected, simply-connected complex nilpotent Lie group and Γ a discrete subgroup such that X = G/Γ is compact. Let G^{k} denote the descending central series (i.e. G^{0} = G, G^{k+1} = [G, G^{k}]) and C_{k} the ascending central series (i.e. C_{0} = {e}, C_{k+1} = {c : cgc^{−1}g^{−1} ∈ C_{k} ∀g ∈ G}.)

Let

X = X^{n} ⟶^{T_{n}} X^{n−1} ⟶^{T_{n−1}} … ⟶^{T_{2}} X^{1} = T_{1} = Alb(X)

denote the tower of torus principal bundles associated to the descending central series, (i.e. X^{n} = G/G^{n}Γ, T_{n} = G^{n−1}/G^{n}Γ ∩ G^{n−1} and

X = X_{0} ⟶^{S_{1}} X_{1} ⟶^{S_{2}} … ⟶^{S_{n−1}} X_{n−1} = S_{n}

Compact complex parallelizable manifolds

the tower of torus principal bundles associated to the ascending central series. (It is well-known [3][20] that all the G^{k}Γ and C_{k}Γ are closed in G.)

Then for k > 1 both S_{k} and T_{k} are isogenous to a product of simple tori with complex multiplication.

If we impose a non-degeneracy condition on the nilmanifold, we can deduce some conditions for Alb(X).

For the statement of this result we need to explain the notion of a decomposition series for a torus T. This is a series of subtori {e} = T_{0} ⊂ T_{1} ⊂ … ⊂ T_{m} = T such that all the quotients T_{k+1}/T_{k} are positive-dimensional and simple. These quotients T_{k+1}/T_{k} are called simple factors for this composition series. Up to ordering and isogenies, they are independent of the particular choice of a composition series, i.e. depend only on T. If T is algebraic, then it is isogenous to a direct product of these simple factors.

Proposition 1. Assume the notations of the above theorem.

Assume that Z = C_{1} ⊂ G′ = G^{1} . Then the composition series both for Alb(X) and for any torus embedded in X as complex submanifold has the property that every simple factor admits complex multiplication.

If in addition C ⊂ G′ with C = {c ∈ G : cgc^{−1}g^{−1} ∈ G^{2}}, then Alb(X) is isogenous to a product of simple tori with complex multiplication.

The condition C ⊂ G′ holds in particular in the following two cases:

(i) G is a generalized Heisenberg group, i.e. Z = G′. (ii) G is a maximal unipotent subgroup in a complex semisimple Lie group.

(In the second case C ⊂ G′ can be verified easily using the theory of root systems for semisimple Lie groups.)

Example. Let G be a complex Heisenberg group, i.e. a simply-connected nilpotent complex Lie group G with one-dimensional center Z. (This implies G′ = Z.) Let Γ be a discrete cocompact subgroup. Then X = G/Γ ⟶^{C} G/G′Γ = Alb(X) is the Albanese, and our theorem implies that C = G′/(G′∩Γ) is an elliptic curve with complex multiplication and Alb(X) is isogenous to a direct product of copies of C. ^{∗}

∗ The article On the Picard Group of a compact complex nilmanifold in the Rocky Mt. J. Math. 17, 65–76 (1987) contains some alleged examples of nilmanifolds which would provide counterexamples to this conclusion. These examples are based on the assumption that certain subgroups (given in terms of their generators) are discrete.

But actually they fail to be discrete.

We have also a result in the converse direction, confirming that every torus with complex multiplication occurs in such a way.

Theorem 2. Let g be a nilpotent ℚ-Lie algebra, K a totally complex number field, [K : ℚ] = 2g.

Then there exists a simply-connected nilpotent complex Lie group G, and a discrete cocompact subgroup Γ such that Lie(G) is isomorphic to a direct product of g copies of g ⊗_{ℚ} ℂ and every torus G^{k}/(G^{k+1}Γ) is a direct product of tori with complex multiplication by K.

If K is not a CM-field, then G and Γ may be choosen in such a way that none of the tori G^{k}/(G^{k+1}Γ) admits a non-constant meromorphic function.

Considering number fields which are neither totally real nor totally complex, one can generalize this method of construction in order to obtain interesting non-compact nilmanifolds X such that every holomorphic function on X is constant, but every compact analytic subset is finite.

We have another type of converse to theorem 1 which characterizes nilmanifolds among solvmanifolds.

Theorem 3. Let G be a simply-connected solvable complex Lie group, Γ a discrete cocompact subgroup. Assume that the center Z is contained in the commutator group G′.

Then G is nilpotent if and only if every simple factor in a decomposition series of Alb(X) ≃ G/G′Γ has complex multiplication.

(By a result of Barth and Otte [3] G′Γ is closed in G, hence Alb(X) ≃ G/G′Γ.) For investigating the structure of tori occuring in the tower of a nilmanifold, it is essential to use the structure for tori. For abelian varieties, there is a very satisfying structure theory. However, we need a structure theory for arbitrary tori, not only the algebraic ones. Many results for abelian varities do not hold for arbitrary tori. Most important, an arbitrary torus is not necessarily isogenous to a product of simple tori. Thus we first have to develope some structure theory for non-algebraic tori.

One tool is the already mentioned notion of a composition series.

Proposition 2. Every torus T admits a composition series, i.e. an ascending sequence of subtori {e} = T_{0} ⊂ … T_{m} = T such that all the quotient S_{k} = T_{k+1}/T_{k} are positive-dimensional and simple. The sequence S_{k} is called the sequence of simple factors for this composition series.

If there exists another composition series, the sequence of simple factors differs only by ordering and isogenies.

If there is a surjective morphism of tori T → T′ or an injective morphism of tori T′ → T, then the sequence of simple factors for T′ is contained in the sequence of simple factors for T (up to ordering and isogenies).

An algebraic torus always equals the sum of all its simple subtori. In contrast, an arbitrary torus is generated by all its irreducible subtori. Here a torus T is called irreducible, if it is not generated by its proper subtori. We prove that irreducible non-simple tori can not occur in our context. The reason is the following:

Proposition 3. Let T = ℂ^{g}/Λ be an irreducible torus, E a commutative subalgebra of End_{ℚ}(T). Assume that Λ ⊗ ℚ is a principal E -module, i.e. generated as E -module by one element.

Then T is simple and E is a number field.

Commutativity of E is essential, cf. the example in section 6.

For a nilpotent group, the commutator map ζ(g, h) = ghg^{−1}h^{−1} is especially useful. In particular it yields a ℤ-bilinear map G/G′ × G/G′ → G′/G^{2} . This enables us to conclude that the tori we encounter will have large endomorphism algebras. Furthermore these algebras must contain large commutative subalgebras. This follows from an auxiliary, purely algebraic result on bilinear forms on modules. Then the above stated result implies that all tori occuring in the tower of a nilmanifold must be semisimple, i.e. isogenous to a direct product if simple tori. Moreover these simple tori must have complex multiplication.

For solvable, non-nilpotent groups it is quite possible that G′ = G^{2} . Hence the ℤ-bilinear map from G/G′ × G/G′ → G′/G^{2} is not useful in this context. Thus the proof of theorem 1 does not hold for solvable non-nilpotent groups.

In order to prove the converse statement, theorem 3, we note that for a non-nilpotent Lie group the adjoint representation is never nilpotent. From this we deduce the following structure theorem on solvmanifolds.

Proposition 4. Let G be a simply-connected solvable complex Lie group, N its nilradical, Γ a discrete cocompact subgroup in G, T = G/NΓ (NΓ is closed in G by a result of Mostow). Let T = ℂ^{g}/Λ.

Then there exists a homomorphism of complex Lie groups ϕ : ℂ^{g} → (ℂ^{∗})^{k} with discrete kernel such that ϕ(Λ) is contained in the \overline{ℚ}-rational points, i.e. ϕ(Λ) ⊂ (\overline{ℚ}^{∗})^{k}.

It should be noted that such a Lie group homomorphism ϕ is very much transcendental and thus does not carry algebraic numbers to algebraic numbers.

Using Baker’s theorem from transcental number theory we prove the following

Proposition 5. Let Λ be a lattice in ℂ^{g} such that Λ ⊂ \overline{ℚ}^{g}. Let ϕ : ℂ^{g} → (ℂ^{∗})^{k} be a complex Lie group homomorphism such that ϕ(Λ) ⊂ (\overline{ℚ}^{∗})^{k}.

Then ϕ is constant.

Now any torus with complex multiplication admits a lattice, such that all elements in the lattice are \overline{ℚ}-rational. Therefore this result implies that G/NΓ is never a torus with complex multiplication. From this we deduce theorem 3.

2. Nilpotent groups

For a group G and subsets A, B ⊂ G, let [A, B] denote the subgroup of G generated by all elements of the form aba^{−1}b^{−1} with a ∈ A, b ∈ B. Then the descending central series G^{k} is defined recursively by G^{0} = G and [G, G^{k−1}] = G^{k}. G is nilpotent, if G^{k} = {e} for some k. As usual, G′ = G^{1} is called the commutator group.

It is also possible to define nilpotent groups via the ascending central series. This series is defined as follows: C_{0} = {e} and C_{k} = {c ∈ G : cgc^{−1}g^{−1} ∈ C_{k−1}∀g ∈ G} for k ≥ 1. G is nilpotent if and only if C_{k} = G for some k ∈ ℕ. C_{1} is simply the center of G and C_{k}/C_{k−1} is the center of G/C_{k−1} .

Under special circumstances the commutator map ζ : (g, h) ↦ ghg^{−1}h^{−1} is a group homomorphism in both variables.

Lemma 1. Let G be a group, H a subgroup and assume that [G, H] is central. Then ζ : G × H → G defined by

ζ : (g, h) ↦ ghg^{−1}h^{−1}

is a group homomorphism in both variables.

Note that the assumption implies in particular that ζ(g, h) = e if g ∈ G′ or h ∈ H′, because [G, H] is abelian.

Proof. This follows by explicit calculation.

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

A similar calculation yields ζ(g, h_{1}h_{2}) = ζ(g, h_{1})ζ(g, h_{2}).

Due to this lemma, the commutator map yields ℤ-bilinear maps

G/G′ × G^{k−1}/G^{k} → G^{k}/G^{k+1}

and

G/G′ × C_{k−1}/C_{k} → C_{k}/C_{k+1}.

In the first case, the image of the map generates G^{k}/G^{k+1} as a ℤ-module (i.e. as an abelian group). In the second case the map is non-degenerate in the second variable in the following sense: For each c ∈ C_{k−1} \ C_{k} there exists an element g ∈ G such that gcg^{−1}c^{−1} ∈ C_{k} \ C_{k+1} . (This follows immediately from the definition of the C_{k}.)

3. Nilmanifolds

We recall the fundamental fact that for nilmanifolds many natural subgroups have closed orbits.

Theorem 4. Let G be a connected (real or complex) nilpotent Lie group, Γ a discrete cocompact subgroup.

Then all the subgroups G^{(k)} , G^{k}, C_{k} have closed orbits in G/Γ.

As usual, G^{(0)} = G, G^{k+1} = [G, G^{k}] (derived series) G^{(k+1)} = [G^{(k)}, G^{(k)}] C_{0} = {e} (descending central series) and C_{k+1} = {c ∈ G : [c, G] ⊂ C_{k}} (ascending central series).

From this one may deduce the following

Theorem 5. Let X = G/Γ be a complex nilmanifold. Then there exists a tower of torus principal bundles

X = X_{n} → X_{n−1} → … → X_{0},

where X_{0} is a torus, given by X_{i} = G/G^{i+1}Γ.

For the ascending central series the assumption of cocompactness of Γ can be weakened.

Theorem (Barth-Otte [3]). Let G be a complex nilpotent Lie group, Γ a closed Lie subgroup such that every holomorphic function on G/Γ is constant.

Then all the C_{k} have closed orbits in G/Γ.

There is an example due to K. Oeljeklaus of a quotient G/Γ of a complex nilpotent Lie group G by a discrete subgroup Γ such that every holomorphic function on G/Γ is constant, but the G′ -orbits are not closed. [19,p.64]. Hence this theorem applies only to the ascending central series and not to the descending central series. The central goal of this chapter is to answer the question: Which tori can actually occur in this tower, if X = G/Γ is non-trivial, e.g. not a direct product with a torus?

4. Compact complex tori

In order to study complex nilmanifolds, we need some basic facts on compact complex tori. For abelian varieties, all the results in this section are trivial or well-known. However, complex nilmanifolds are never projective manifolds unless they are already abelian. Therefore it would be quite artificial to restrict our attention to nilmanifolds with projective Albanese.

A major difficulty in dealing with non-algebraic tori arises from the fact that the Poincaré Reducibility Theorem does not hold for arbitrary non-alge-braic torus.

In what follows a torus is always a compact complex torus, i.e. a quotient of a complex vector space ℂ^{n} by a lattice Λ of rank 2n. We always fix a basepoint e in a torus. Then there is a unique structure as a commutative complex Lie group with e as neutral element. A morphism between tori is holomorphic map, which we require to take the neutral element to the neutral element. It is well-known that every such morphism is a Lie group homomorphism and linear, if pull-backed to the universal covering.

A very important notion for tori is that of an isogeny. An isogeny between two tori S, T is a holomorphic correspondence, given by a graph Φ ⊂ S × T such that both projections Φ → S and Φ → T are finite unramified coverings. Isogenies define an equivalence relation among tori and it is often useful to think of two tori being essentially equal if there is an isogeny between them. Isogenies between two given tori S and T are not necessarily maps, hence not contained in Hom(S, T). However, there is a natural 1-1 correspondence between isogenies and invertible elements in Hom(S, T) ⊗_{ℤ} ℚ.

We call a torus T algebraic if it is a projective manifold. This is equivalent to assume that T is Moishezon. It is also equivalent to assume that T is the complex space associated to an abelian variety defined over ℂ. Furthermore T = ℂ^{g}/Λ is algebraic if and only if there exists a Riemann form H. A Riemann form H is a positive definite Hermitian form with Im H(Λ × Λ) ⊂ ℤ.

Definition. A torus T is simple if it does not contain any subtorus S with 0 < dim(S) < dim(T).

A torus T is semisimple if it is isogenous to a direct product of simple tori.

By a theorem of Poincaré every algebraic torus is semisimple. However, there are non-algebraic tori which are not semisimple.

Many results on abelian varieties generalize easily to semisimple tori. For instance, it is a direct consequence of the definition that every subtorus of a semisimple torus is again semisimple.

Lemma 2. Let

0 → T_{1} → T_{2} →^{α} T_{3} → 0  (∗)

be an exact sequence of tori. Assume that T_{2} is semisimple.

Then all of the T_{i} are semisimple. Moreover there exists an element s ∈ Hom(T_{3}, T_{2}) ⊗_{ℤ} ℚ such that α ∘ s = id_{T_{3}} , i.e. (∗) splits on the isogeny level.

Thus quotients and subtori of semisimple tori are semisimple.

This enables us to make the following definition.

Definition /Proposition. Let T be a torus. Let T_{0} be the smallest subtorus for which T/T_{0} is semisimple. Then the projection π_{ss} : T → T_{ss} = T/T_{0} is called semisimple reduction of T.

Proof. Let T_{1} , T_{2} be subtori of T with T/T_{i} (i = 1, 2) semisimple. Then T/(T_{1}∩T_{2}) may be embedded in T/T_{1} × T/T_{2} . Therefore T/(T_{1} ∩ T_{2}) is again semisimple. This implies that T_{0} is uniquely determined.

By the preceding lemma it is clear that dim T_{ss} > 0 for every positivedimensional torus.

Lemma 3. Every torus T admits a non-constant morphism to a simple torus.

Proof. By induction on dim(T). If dim(T) = 1, then T is simple. If T is not simple (hence dim(T) > 1), there is a subtorus S and by induction hypothesis T/S admits a non-constant morphism to a simple torus.

The semisimple reduction fulfills a universality property.

Lemma 4. Let T, S be tori, f : T → S a morphism, S semisimple. Then there exists a morphism f_{0} : T_{ss} → S such that f = f_{0} ∘ π_{ss}.

Proof. Let T′ denote the kernel of f. Then T/T′ is contained in the semisimple torus S, hence semisimple itself. It follows that ker π_{ss} ⊂ T′ by the construction of the semisimple reduction.

Similarily, it follows that T → T_{ss} defines a covariant functor from the category of complex tori to the category of semisimple complex tori.

Another variation on this theme yields the following.

Lemma 5. Let T be a torus, S a simple torus. Then there exists a surjective morphism π form T to some torus T′ such that

1) T′ is isogenous to a direct product of copies of S,

2) Every morphism from T to S fibers through π : T → T′.

We mentioned already that every torus admits a non-constant morphism onto a simple torus. By induction this fact allows the definition of a composition series.

Definition. A composition series for a torus T is a sequence of subtori

{e} = T_{0} ⊂ T_{1} ⊂ … ⊂ T_{d} = T

such that T_{i}/T_{i−1} is simple.

The sequence of simple tori S_{i} = T_{i}/T_{i−1} is called the sequence of simple factors of the composition series T_{i}.

Owing to the preceding lemma, it is clear that every torus has a composition series. However, we need some kind of uniqueness.

Lemma 6. Let T be torus, T_{i} and T′_{i} two composition series. Then the two sequences of simple factors coincide up to ordering and isogenies.

Proof. We will prove this by induction on dim(T). Note that the lemma is trivially true for dim(T) = 1.

Let Z = T_{1} . This is a simple subtorus of T . Consider the morphisms τ_{i} : Z → T′_{i}/T′_{i−1} = S′_{i}. Since Z and S′_{i} are simple, every τ_{i} is either constant or an isogeny. It is clear that exactly τ_{i} is an isogeny for exactly one number i = i_{0} . Now we have two composition series Z_{i} and Z′_{i} for T/S as follows: Z_{i} = T_{i+1}/S for all i, Z′_{i} = T′_{i}/(S ∩ T′_{i}) for i < i_{0} and Z′_{i} = T′_{i+1}/(S ∩ T′_{i+1}) for i ≥ i_{0} . The statement now follows from the induction hypothesis.

Thus the sequence of simple factors does not depend on the particular choice of a composition series, if we ignore ordering and isogenies. This legitimizes the following definition.

Definition. A simple torus S is called a simple factor for a torus T, if it is isogenous to a simple factor of a composition series of T.

In a similar way to the proof of the above lemma one can prove the following:

Lemma 7. Let S, T be tori. Assume that there is a surjective morphism τ : S → T or an injective morphism i : T → S.

Then (up to ordering and isogenies) the sequence of simple factors for T is a subsequence of the sequence of simple factors for S.

Definition. A torus T is irreducible if T is not the sum of all its proper subtori.

A semisimple torus is irreducible if and only if it is simple.

Lemma 8. A torus T is irreducible if and only if there is a subtorus T_{0} with {e} ⊆ T_{0} ⊊ T such that T/T_{0} is simple and T_{0} contains every non-trivial subtorus of T.

Proof. Obviously T is irreducible if there exists such a T_{0} . Conversely, let T be an irreducible torus. Then T_{0} defined as the sum of all proper subtori does the job.

Corollary. Let T be a torus which is not irreducible.

Then there exists a surjective morphism from T to a semisimple torus S which is not simple.

Lemma 9. Every torus T equals the sum of all its irreducible subtori.

Proof. By induction on dim(T) as follows. Let T be a torus. If there exist subtori T_{i} ⊊ T with T = ⊕_{i}T_{i}, the statement follows by induction. If there exists no such family of subtori, then T is already irreducible.

Caveat: These irreducible subtori may have large intersection.

Definition. For an irreducible torus T with T_{0} as above the quotient T/T_{0} is called the dominant simple factor for T.

Lemma 10. Let f : S → T be a surjective morphism of tori, S irreducible. Then T is irreducible, too. Furthermore the dominant simple factors for S and T are isogenous.

The relevance of dominant simple factors relies on the following fact.

Lemma 11. Let T be a torus, S a simple factor.

Then T contains an irreducible subtorus U with a dominant simple factor isogenous to S.

Proof. Consider all subtori Z_{i} of T for which S is a simple factor. Choose U as one of the Z_{i} with the minimal possible dimension. Then S does not occur as simple factor for any subtorus of U. This implies that S is not a simple factor for the sum of all proper subtori of U. Since S is a simple factor for U, it follows that U is irreducible.

Lemma 12. Let T be a torus, T_{i} a family of irreducible subtori with T = ∑_{i} T_{i}. Let T → T/S = T_{ss} be the semisimple reduction. Then every simple factor of T_{ss} is a dominant simple factor for some T_{i}.

Proof. Let A be a simple factor of T_{ss}. Then there exists a surjective morphism τ from T to a torus S′ isogenous to S. If T_{i} is irreducible with a dominant simple factor not isogenous to S, then the restriction of τ to T_{i} must be constant. Now T = ∑_{i} T_{i} implies that there is a T_{i} such that τ|_{T_{i}} is not constant.

For an arbitrary compact complex manifold X, the theorem of Siegel states that the field of meromorphic functions M(X) is a finitely generated field extension of ℂ with transcendence degree bounded by dim_{ℂ} X. This is the main ingredient for the proof of the existence of an algebraic reduction.

Theorem (Algebraic Reduction, see e.g. [24]). Let X be a compact complex manifold. Then there exists a projective variety V and a surjective meromorphic map f : X → V with connected fibers such that f^{∗}ℂ(V) = f^{∗}M(V) = M(X).

The number dim_{ℂ}(V) = trdeg_{ℂ}M(X) is called algebraic dimension of X and denoted by a(X). X is called Moishezon if a(X) = dim_{ℂ}(X). For parallelizable manifolds (thus in particular for tori) the theorem on the algebraic reduction specializes to the following:

Theorem (Algebraic reduction for parallelizable manifolds). Let X be a compact complex manifold for which the holomorphic tangent bundle T X is holomorphically trivial.

Then there exists an abelian variety V and a surjective connected holomorphic map f : X → V such that f^{∗}ℂ(V) = M(X).

This follows easily from the corresponding result for arbitrary compact complex-homogeneous manifolds, which is due to Grauert and Remmert [5].

This has an immediate consequence for simple tori.

Corollary. Let T be a simple torus. Then either T is algebraic or every meromorphic function on T is constant.

Some tori admit closed complex subspace which are not subtori. This has implications for the decomposition series.

Proposition 6. Let T be a torus, S_{i} the sequence of simple factors for a decomposition series {0} = T_{0} ⊂ … ⊂ T_{n} = T.

Assume that T contains a closed complex subspace Z which is not a subtorus. Then at least one of the S_{i} is algebraic, i.e. isomorphic to a complex abelian variety.

Proof. There exists a subtorus A ⊂ T such that Z is A-invariant and Z/A is a complex space of general type contained in the quotient torus T/A ([24]). Let ⟨Z/A⟩ denote the subtorus of T/A generated by Z/A. For some n ∈ ℕ there is a surjective morphism (Z/A)^{n} → ⟨Z/A⟩. Hence ⟨Z/A⟩ is Moishezon and therefore an abelian variety. Now every simple factor for T/A is also simple factor for T . Hence there must be a simple factor for T which is isomorphic to an abelian variety.

5. Endomorphism algebras

In this section we summarize basic facts on the endomorphism algebra of a torus. Most of this is well-known, see e.g. [8],[23].

For each torus T the set End(T) = Hom(T, T) is a ring with composition as multiplication and pointwise addition in T as addition in End(T). Thus End(T) is a ℤ-algebra. However, often it is easier to work with the ℚ-algebra End_{ℚ}(T) = End(T)⊗_{ℤ} ℚ. For instance, the ℤ-algebra End(T) is not invariant under isogenies.

Furthermore, an endomorphism ϕ of a torus T yields an invertible element in End_{ℚ}(T) if and only if ker ϕ is finite.

Consider a torus T as quotient of complex vectorspace V = ℂ^{g} by a lattice Λ. Given an endomorphism ϕ : T → T, this canonically induces endomorphisms on the tangent space at e of T, which we may identify with V , as well as on the fundamental group of T, which we may identify with Λ. Thus we obtain natural representations of ℚ-algebras

ρ_{r} : End_{ℚ}(T) → End_{ℚ}(Λ ⊗_{ℤ} ℚ)

(often called the rational representation) and

ρ_{a} : End_{ℚ}(T) → End_{ℂ}(V)

(sometimes called the analytic representation). These representations are related, because Λ ⊗_{ℤ} ℝ is canonically isomorphic to the 2g -dimensional real vectorspace obtained from V ≃ ℂ^{g} by restriction of scalars. It follows that ρ_{r} ≃ ρ_{a} ⊕ ρ̄_{a} considered as representations in End_{ℝ}(ℝ^{2g}).

Theorem A. Let T = ℂ^{n}/Λ be a simple torus. Then E = End_{ℚ}(T) is a skewfield with dim_{ℚ}(E) ≤ dim_{ℝ}(T).

Sketch of Proof. Simplicity of T implies that E is an integral domain. Furthermore, E is a finite-dimensional over ℚ, because ρ_{r} : E → End(Λ ⊗ ℚ) is injective. Now a finite-dimensional algebra over a field is necessarily a skewfield, provided it is an integral domain. Finally, the dimension bound follows because Λ ⊗ ℚ is a E -left vector space.

For our applications the most important case involves tori with complex multiplication.

Definition. A torus T (not necessarily simple) has complex multiplication (by a number field K) if End_{ℚ}(T) contains a number field K with [K : ℚ] = 2 dim_{ℂ}(T) as subring.

Caveat: This notion is often reserved to algebraic tori, a restriction which would be unnatural for us.

Example. A one-dimensional torus T = ℂ/ ⟨1, τ⟩_{ℤ} is a torus with complex multiplication if and only if τ^{2} = pτ + q for some p, q ∈ ℚ.

Complex multiplication is a rather rare phenomenon among tori.

Lemma 13. Let T be a torus with complex multiplication. Then T is isogenous to some S × … × S, S simple.

Proof. For dimension reasons, every torus T contains a simple subtorus S. Now let A denote the subtorus of T which is the sum of all subtori isogenous to S. Recall that a morphism from a simple torus to any torus is either constant or an isogeny to the image. Therefore A is stabilized by all endomorphisms of T . However, the assumption of complex multiplication implies that K ⊂ End_{ℚ}(T) does not stabilize any proper subtorus. Hence A = T .

Corollary. Let T be a torus with complex multiplication.

Then either T is algebraic or a(T) = 0, i.e. every meromorphic function is constant.

Let T = V/Λ be a torus with complex multiplication by a complex number field K. It is important to study the natural representations of K ⊂ End_{ℚ}(T).

Lemma 14. Let T be a torus with complex multiplication by a number field K. Let 2g = [K : ℚ] and let τ_{1}, …, τ_{2g} : K → ℂ denote the 2g distinct embeddings of K into ℂ.

Then ρ_{r} : K → End_{ℚ}(Λ ⊗ ℚ) is isomorphic to the representation ⊕_{i=1}^{2g}τ_{i}.

Proof. This is immediate, since Λ ⊗ ℚ is a one-dimensional K -vectorspace.

Corollary. Let T be a torus with complex multiplication by a number field K, 2g = [K : ℚ]. Then there exist g distinct, pairwise non-conjugate embeddings σ_{1}, … σ_{g} → ℂ such that ρ_{a} ≃ ⊕_{i=1}^{g}σ_{i}. In particular K is totally complex, i.e. admits no real embedding.

Proof. Recall ρ_{r} ≃ ρ_{a} ⊕ ρ̄_{a}. It follows that ρ_{a} is isomorphic to the direct sum of a choice of g embeddings σ_{i} : K → ℂ such that every embedding of K in ℂ equals one of the σ_{i} or σ̄_{i}. The existence of such a choice implies that K is totally complex.

Thus for each torus with complex multiplication by a (totally complex) number field K one obtains a type in the sense of the definition below.

Definition. Let K be a totally complex number field, [K : ℚ] = 2g. A type σ for K is a choice of g complex embeddings which are mutually distinct and non-conjugate.

Since there are g pairs of conjugate embeddings, there are exactly 2^{g} types for a given totally complex number field K of degree 2g.

Definition. Let K be a totally complex number field and σ a type. A torus T has complex multiplication of type (K, σ) if the complex linear representation ρ_{a} : K → End(ℂ^{g}) induced by the representation on the tangent space at e is equivalent to the sum of the complex embeddings of K contained in σ.

Theorem 6. Let K be a totally complex number field, O_{K} the ring of algebraic integers and σ a type.

Then Λ = σ(O_{K}) = {(σ_{1}(p), …, σ_{g}(p)) : p ∈ O_{K}} is a lattice in ℂ^{g} such that V/Λ is a torus with complex multiplication type (K, σ).

Conversely, every torus T with complex multiplication type (K, σ) is isogenous to V/Λ.

The following consequence will be needed later on.

Corollary. Let T by a g -dimensional torus with complex multiplication. Then there exists a lattice Γ ⊂ ℂ^{g} such that T ≃ ℂ^{g}/Γ and Γ ⊂ \overline{ℚ}^{g}, where \overline{ℚ} denotes the algebraic closure of ℚ in ℂ.

Given a torus with complex multiplication, there are criteria how to determine whether the torus is simple and whether it is algebraic.

Definition. Let k, K be a totally complex number fields, k ⊂ K. A type σ for K is lifted from k, if there is type σ_{k} for k such that σ consists of all complex embeddings of K whose restriction to k is contained in σ_{k}.

Proposition 7. Let T be a torus with complex multiplication type (K, σ). Then T is simple if and only if σ is not lifted from a proper totally complex subfield k ⊊ K.

There are totally complex number fields such that every type is lifted from a proper subfield. For instance, let p be a prime number, K = ℚ(\sqrt{p}, i). Then every type of K is lifted either from ℚ(i) or from ℚ(\sqrt{p}i). Thus, given a totally complex number field K, it is not always possible to find a simple torus with complex multiplication by K.

Now we recall the characterization of the algebraic tori with complex multiplication. For this we need the notion of a CM-field. (CM stands for ”complex multiplication”).

Definition. A totally complex number field K is called a CM-field if and only if it is the quadratic extension of a totally real number field.

It is easy to see that K is a CM-field, if it admits a totally real subfield K^{+} and a totally imaginary element α such that K = K^{+}(α).

Theorem (Lang, Th.4.4,p.19). Let T be a torus with complex multiplication type (K, σ). Then T is algebraic if and only if σ is lifted from a CM-field.

Thus for the construction of non-algebraic tori with complex multiplication one needs a totally complex number field K with a type not lifted from a CMsubfield. We will show that such a type always exists unless K itself is a CM-subfield.

Lemma 15. Let K be a number field which contains a CM-subfield K_{0} .

Then K contains a CM-subfield K′ such that every CM-subfield of K is contained in K′.

Proof. Let K^{+} denote the subfield of K which consists of all totally real elements of K. Let α be a totally imaginary element in K_{0} such that K_{0} = (K^{+}∩K_{0})(α). Define K′ = K^{+}(α). Let K_{1} be an arbitrary CM-subfield of K. Then K_{1} = (K^{+} ∩ K_{1})(β) for a totally imaginary element β. Now αβ is totally real, hence contained in K^{+} . It follows that K_{1} ⊂ K^{+}(β) = K^{+}(α) = K′.

Corollary. Let K be a totally complex number field which is not a CM-field. Then K has a type which is not lifted from a CM-subfield.

Proof of the corollary. If K has no CM-subfields, there is nothing to prove. Otherwise let K′ be the maximal CM-subfield provided by the lemma. Then every type which lifts from a subfield of K′ may be lifted through K′. Since 2h = [K′ : ℚ] < 2g = [K : ℚ], it follows that there are 2^{g} −2^{h} > 0 types not lifted from a CM-subfield.

Corollary. Let K be a totally complex number field which is not a CM-field.

Then there exists a torus T with complex multiplication by K such that every meromorphic function on T is constant.

6. Endomorphism algebras of irreducible tori

For an abelian variety, the endomorphism algebra is always semisimple. The proof (see any book on abelian varieties) carries over to arbitrary semisimple tori. However, for non-semisimple tori the structure of the endomorphism algebra can be more intricate. In this section we study the structure of the endomorphism algebra of irreducible non-semisimple tori.

We will prove in particular that given a large commutative subalgebra in the endomorphism algebra of a torus the very existence of this subalgebra implies that the torus is semisimple. This is a key ingredient for our later proof that the Albanese of a nilmanifold is always semisimple unless the nilmanifold is degenerate in a certain sense.

To illustrate the phenomena occuring for non-semisimple tori, we first give an example.

Example. Let C = ℂ/ ⟨1, τ⟩_{ℤ} be an elliptic curve with complex multiplication, i.e. τ^{2} = pτ + q for some p, q ∈ ℚ. Let α ∈ ℂ, but α ∉ ℚ(τ). Consider the torus T with period matrix

Π = ( \begin{matrix}1 & τ & 0 & α \\ 0 & 0 & 1 & τ \end{matrix} )

Then T is irreducible with a unique subtorus T′. Both T′ and T/T′ are isomorphic to C. By explicit calculations, one can check that

End_{ℚ}(T) = \{( \begin{matrix}x̄ & t(x)α + y \\ 0 & x \end{matrix} ) : x, y ∈ ℚ(τ)\}

with t : ℚ(τ) → ℚ defined by t(aτ + b) = a for a, b ∈ ℚ. There is a two-sided ideal

N = \{( \begin{matrix}0 & y \\ 0 & 0 \end{matrix} ) : y ∈ ℚ(τ)\} ,

which consists of all nilpotent elements in E. The algebra E is not commutative and not semisimple.

Now we start with the general theory for non-semisimple tori.

Let T be an irreducible torus. By definition, this means that there is a subtorus {e} ⊆ T′ ⊊ T such that T′ contains every proper subtorus of T. The quotient T/T′ is simple. We represent T as a quotient of a complex vectorspace V by a lattice Λ. Then T′ = V′/Λ′ with Λ′ = V′ ∩ Λ for some subvectorspace V′ ⊂ V . Let E = End_{ℚ}(T). Then we have the following basic facts.

Lemma 16. The algebra E admits a two-sided ideal N such that the following are equivalent:

i) ϕ ∈ N, ii) ϕ(T) ⊂ T′, iii) ϕ is not an isogeny, i.e. not invertible. iv) ϕ is nilpotent.

Proof. We may define N by N = {ϕ ∈ E : ϕ(T) ⊂ T′}. Since T′ contains every proper subtorus of T , it follows that ϕ ∈ N if and only if ϕ is not an isogeny. This implies that N is a two-sided ideal. To check nilpotency, choose ϕ ∈ N and let A = ∩_{i∈ℕ}ϕ^{i}(T). For dimension reasons, A = ϕ^{n}(T) for some n ∈ ℕ. Now A = ϕ^{2n}(T) implies that ϕ^{n}|_{A} is an isogeny. Hence T = A + ker (ϕ^{n}). Since T is irreducible, and A ≠ T , it follows that T = ker (ϕ^{n}). Thus every ϕ ∈ N is nilpotent.

Lemma 17. There is an injective algebra homomorphism E/N → End_{ℚ}(T/T′) and an injective homomorphism of additive groups N → Hom(T, T′).

Proof. Both assertions follow immediately form the definition of N.

Lemma 18. Let T = ℂ^{g}/Λ be an irreducible torus, T′ the sum of all proper subtori, Λ′ ⊂ Λ the corresponding subgroup of the lattice.

Let E = End_{ℚ}(T) and γ ∈ Λ, but γ ∉ Λ′. Then Eγ is a free E - submodule of Λ ⊗ ℚ.

Proof. Assume that there is a ϕ ∈ E with ϕγ = 0. With γ ∉ Λ′ this implies that ker ϕ is not contained in T′. Hence T = ker ϕ, i.e. ϕ = 0.

We are now approaching the main result of this section. It shows that the non-commutativity in our above example is not a coincidence.

Proposition 8. Let T = V/Γ be an irreducible torus and E = End_{ℚ}(T). Assume that E contains a commutative subalgebra F such that Λ ⊗ ℚ is a principal F -module.

Then T is simple, F = E and F is a number field, i.e. T admits complex multiplication.

The previous example shows that commutativity of F is essential for the proposition.

The proof requires several steps. For brevity we make the following definition.

Definition. A torus T = V/Λ has weak CM (weak complex multiplication) if the endomorphism algebra End_{ℚ}(T) contains a commutative subalgebra F for which Λ ⊗ ℚ is a principal F -module.

Thus our goal is to prove that a torus has weak complex multiplication (weak CM) if and only if it has complex multiplication.

Lemma 19. Let S be a simple torus with weak CM.

Then S has complex multiplication.

Proof. Since S is simple, E = End_{ℚ}(S) is a skewfield. By the assumption of weak CM, E admits a commutative ℚ-subalgebra F such that Λ ⊗ ℚ is a principal F - module. Thus a forteriori Λ⊗ℚ is a one-dimensional E -left vectorspace. Now E = F for dimension reasons, i.e. E is commutative.

Lemma 20. Let T be an irreducible torus with weak CM. Then there exists a subtorus A ⊊ T′ such that T/A likewise has weak CM and T′/A is the only (non-trivial) subtorus of T/A.

Furthermore T/T′ is isogenous to T′/A.

Proof. The problem is to construct quotients of T in such a way that enough endomorphisms of T descend to endomorphisms of the quotient.

Let τ : T′ → T′_{ss} denote the semisimple reduction of T′, and C = ker τ. Since T′ and therefore C are canonically defined, it is clear that ϕ(C) ⊂ C for all ϕ ∈ End_{ℚ}(T). Hence all endomorphisms of T can be pushed forward to endomorphisms of T/C. This implies that T/C has weak CM and we may assume that T′ is semisimple.

Now let ϕ ∈ N. Then ϕ(T) is an irreducible subtorus of T′. Since T′ is semisimple, every irreducible subtorus is simple. It follows that ϕ(T) is a simple subtorus of T′ and ϕ(T′) = {e} for all ϕ ∈ N.

Recall that E/N injects into End_{ℚ}(T/T′). Therefore the assumption that Λ⊗ℚ is a principal F -module implies that T′ = ∑_{ϕ∈N} ϕ(T). It follows that every simple factor of T′ is isogenous to T/T′. Furthermore this implies that for every simple subtorus S ⊂ T′ there is a ϕ ∈ N with ϕ(T) = S. Dimension reasons imply E = F . Now αϕ = ϕα for ϕ ∈ N, α ∈ F , clearly implies that every α ∈ F must stabilize the image of ϕ for all ϕ ∈ N. It follows that every simple subtorus (hence: every subtorus) of T′ is stabilized by all α ∈ E. Consequently every quotient of T has weak CM as well. We may therefore assume that T′ is simple and isogenous to T/T′.

Proposition 9. There is no irreducible torus T with weak CM such that the maximal non-trivial subtorus T′ is simple and isogenous to T/T′.

Proof. We maintain the above notation. Since E/N injects into End_{ℚ}(T/T′), it is clear that S = T/T′ is a simple torus with complex multiplication. Therefore there is a totally complex numberfield K and a type σ : K → ℂ^{g} such that S is isogenous to ℂ^{g}/Γ with Γ = σ(O_{K}). There is an induced representation of K in End_{ℂ}(ℂ^{g}) given by x : (z_{1}, …, z_{g}) ↦ (σ_{1}(x)z_{1}, …, σ_{g}(x)z_{g}). By abuse of language, this algebra homomorphism from K to End_{ℂ}(ℂ^{g}) is again denoted by σ.

Replacing T be a suitable isogenous torus, we may represent T as a quotient of the vector space V ⊕ V with V ≃ ℂ^{g} by a discrete subgroup Λ with (v_{1}, v_{2}) ∈ Λ if and only if v_{1}, v_{2} − ζ(v_{1}) ∈ Γ for some ℚ-linear map ζ : V → V . (Note that the ”coordinates” v_{i} are vectors in V and not numbers). Since only ζ|_{Γ} is relevant and V ≃_{ℝ} Γ ⊗ ℝ, we may assume that ζ is ℝ-linear. However, we can not assume that ζ is ℂ-linear. Now for every A ∈ End_{ℂ}(V) we have a ℂ-linear change of coordinates given by v′_{1} = v_{1} , v′_{2} = v_{2} + A(v_{1}) which transforms ζ into A + ζ with addition taking place in End_{ℝ}(V). We will keep this in mind.

All endomorphisms of T lift to ℂ-linear endomorphisms of V ⊕ V and therefore are representable in the algebra M(2, End_{ℂ}(V)) of 2 × 2-matrices with entries in End_{ℂ}(V). Clearly E is a subalgebra of

\{( \begin{matrix}σ(x) & A \\ 0 & σ(y) \end{matrix} ) : x, y ∈ K; A ∈ End_{ℂ}(V)\} .

151 D.6

Furthermore

N = \{( \begin{matrix}0 & σ(z) \\ 0 & 0 \end{matrix} ) : z ∈ K\} .

Consider an element ϕ = ({A \atop 0}{B \atop C}). Commutativity of E implies that ϕη = ηϕ for all η = ({0 \atop 0}{Z \atop 0}) with Z ∈ σ(K). This reduces to AZ = ZC. Since A, Z, C ∈ σ(K), which is commutative, this implies that A = C for all ϕ ∈ E.

The Zariski-closure of σ(K^{∗}) in GL_{ℂ}(V) is evidently the Cartan subgroup H consisting of all endomorphisms

(z_{1}, …, z_{g}) ↦ (λ_{1}z_{1}, …, λ_{g}z_{g})

with λ_{i} ∈ ℂ^{∗}. Now let G denote the complex linear-algebraic subgroup of GL_{ℂ}(V ⊕ V) defined by

G = \{( \begin{matrix}A & B \\ 0 & A \end{matrix} ) : A ∈ H, B ∈ End_{ℂ}(V)\} .

Let E^{∗} denote the multiplicative group of invertible elements of E. Then E^{∗} is a subgroup of G. Let A denote the closure of E^{∗} in G with respect to the (complexalgebraic) Zariski-topology. A is commutative, because E^{∗} is commutative. It follows that A = A_{s} × A_{u} with A_{s} reductive and A_{u} unipotent. Now

U = \{( \begin{matrix}1 & B \\ 0 & 1 \end{matrix} ) : B ∈ End_{ℂ}(V)\}

is the maximal unipotent subgroup of the solvable algebraic group G. Hence A_{u} ⊂ U. Consider the projection π : G → G/U. The quotient G/U is reductive and σ(K^{∗}) being dense in H implies that π(E^{∗}) (hence a forteriori π(A)) is dense in G/U. Therefore π(A_{s}) = G/U. This in turn implies that A_{s} is a Cartan subgroup of G. Now every two Cartan subgroups are conjugate. It follows that (by a suitable change of coordinates) we may assume that

A_{s} = \{( \begin{matrix}A & 0 \\ 0 & A \end{matrix} ) : A ∈ H\} .

Now the centralizer of A_{s} in G is just

C_{G}(A_{s}) = \{( \begin{matrix}A & B \\ 0 & A \end{matrix} ) : A ∈ H, B ∈ H̄\}

where H̄ denotes the closure of H in End_{ℂ}(V), i.e. H̄ consists of all endomorphisms given by

(z_{1}, …, z_{g}) ↦ (λ_{1}z_{1}, …, λ_{g}z_{g})

with λ_{i} ∈ ℂ. Since A is commutative and contains A_{s} as well as

(1 + N) = \{( \begin{matrix}1 & B \\ 0 & 1 \end{matrix} ) : B ∈ σ(K)\} ,

152 D.6

it follows that A = C_{G}(A_{s}).

Now let x ∈ O_{K}, x ≠ 0. Then γ_{0} = σ(x) ∈ Γ. Since

γ_{0} = (σ_{1}(x), …, σ_{g}(x))

and σ_{i}(x) ≠ 0 for all i, there is a B ∈ H̄ ⊂ End_{ℂ}(V) such that Bγ_{0} = −ζ(γ_{0}) (with ζ ∈ End_{ℝ}(V) defined as above). A change of coordinates v′_{1} = v_{1} , v′_{2} = v_{2} + Bv_{1} does not affect A, because ({1 \atop 0}{B \atop 1}) ∈ A. Thus we may assume that ζ(γ_{0}) = 0. i.e. (0, γ_{0}) ∈ Λ.

The number field K is a finite extension of ℚ. Hence there exists primitive element τ ∈ K such that K = ℚ(τ). Then τ^{2g} = ∑_{k=0}^{2g−1} p_{k}τ^{k} with p_{k} ∈ ℚ. Let ϕ ∈ E be such that the natural projection p : E → E/N ≃ K maps ϕ to τ. Then E is generated (as a ℚ-algebra) by N and ϕ. Furthermore ϕ ∈ A. Hence ϕ = ({θ \atop 0}{ξ \atop θ}) with θ = σ(τ) and ξ ∈ H̄. The ℚ-vector space Λ ⊗ ℚ is now generated by the elements (γ, 0) with γ ∈ Γ together with the elements ϕ^{n}(0, γ_{0}) with n ∈ ℕ. Recall that τ^{2g} = ∑_{k=0}^{2g−1} p_{k}τ^{k}. This implies that ϕ^{2g} − ∑_{k} p_{k}ϕ^{k} ∈ N. Hence

[ϕ^{2g} − ∑_{k} p_{k}ϕ^{k}] ( \begin{matrix}0 \\ γ_{0} \end{matrix} ) = ( γ̃ ; 0 )

for some γ̃ ∈ Γ. Observe that

ϕ^{m} = ( \begin{matrix}θ & ξ \\ 0 & θ \end{matrix} )^{m} = ( \begin{matrix}θ^{m} & mθ^{m−1}ξ \\ 0 & θ^{m} \end{matrix} ) .

(Here as well as in the following calculations it is essential that θ and ξ commute.) Thereby we obtain

ξ [2gθ^{2g−1} − ∑_{k} p_{k}kθ^{k−1}] γ_{0} ∈ Γ ⊗ ℚ.

Now recall that τ is a primitive element for [K : ℚ]. Therefore θ^{0}, …, θ^{2g−1} are ℚlinearly independent, which implies that the expression [2gθ^{2g−1} + …] in the formula above can not be zero. It follows that there is a number y ∈ K^{∗} such that ξσ(y) ∈ Γ ⊗ ℚ = σ(K). Thus ξσ(y) = σ(z) with y ∈ K^{∗}, z ∈ K. Recall that ξ ∈ H̄. Hence there are complex numbers λ_{1}, …, λ_{g} such that

ξ : (z_{1}, …, z_{g}) ↦ (λ_{1}z_{1}, …, λ_{g}z_{g}).

It follows that λ_{i} = σ_{i}(z)/σ_{i}(y) = σ_{i}(z/y) for all i. Thus ξ ∈ σ(K). It follows that ξ(γ) ∈ Γ ⊗ ℚ for all γ ∈ Γ. This furthermore implies that ϕ^{n}(0, γ_{0}) ∈ Γ ⊗ ℚ for all n ∈ ℕ. Finally this implies that Λ ⊗ ℚ = {(v_{1}, v_{2}) : v_{i} ∈ Γ ⊗ ℚ}. But this contradicts the assumption that T is irreducible, because it implies that T is isogenous to V/Γ × V/Γ.

153 D.7 7. Primitive elements

Let T = ℂ^{g}/Λ be a torus. To each element λ in the lattice Λ we may associate a subtorus Θ(λ) in the following way. We consider the complex line ⟨λ⟩_{ℂ} in ℂ^{g} and let Θ(λ) be the smallest subtorus of T containing π (⟨λ⟩_{ℂ}), where π : ℂ^{g} → T denotes the natural projection. Equivalently, Θ(λ) may be defined as the smallest subtorus S of T with λ ∈ Image (i_{∗} : π_{1}(S) → π_{1}(T)), identifying Λ with π_{1}(T).

Definition. Let T = ℂ^{g}/Λ be a torus. An element λ ∈ Λ is called primitive if the associated subtorus Θ(λ) (defined as above) is irreducible.

The set of all primitive elements in Λ is denoted by Λ^{prim}.

Lemma 21. Let f : T → S denote a morphism of tori, f_{∗} : Λ → Γ the associated group homomorphism between the lattices of T and S.

Let λ ∈ Λ. Then f (Θ(λ)) = Θ(f_{∗}λ).

Furthermore f_{∗}(Λ^{prim}) ⊂ Γ^{prim}.

Proof. The inclusion f (Θ(λ)) ⊃ Θ(f_{∗}λ) is obvious as well as Θ(λ) ⊂ f^{−1} (Θ(f_{∗}λ)). This yields the first assertion. The second is an immediate consequence, since a quotient of an irreducible torus is again irreducible.

Lemma 22. Let T = ℂ^{g}/Λ be an irreducible torus, T_{0} a maximal proper subtorus, Λ_{0} the corresponding sublattice.

Then Θ(λ) = T for all λ ∈ Λ \ Λ_{0} .

Proof. Obvious.

Corollary. Let T = ℂ^{g}/Λ be a torus.

Then Λ ⊗ ℚ is generated (as ℚ-vectorspace) by Λ^{prim}.

Proof. Thanks to the lemma, this is clearly true for irreducible tori. This implies the statement for arbitrary tori, because every torus equals the sum of it irreducible subtori.

The ℤ-module Λ is not necessarily generated by Λ^{prim}, e.g. let S be a simple torus, τ_{0} ∈ End(S) an involution, τ = (τ_{0}, τ_{0}) and T = S × S/τ.

Lemma 23. Let T = ℂ^{g}/Λ be an irreducible torus, λ ∈ Λ^{prim} with Θ(λ) = T and E = End_{ℚ}(T).

Then Eλ is a free E -submodule of the E -module Λ ⊗ ℚ.

Proof. We have to show that ϕλ = 0 implies ϕ = 0 for all ϕ ∈ E. But this is evident, since ϕλ = 0 implies Θλ ⊂ ker ϕ.

8. Bilinear forms on modules

Here we deduce an auxiliary algebraic result on bilinear forms on modules over possibly non-commutative rings. We will need this later on in order to deduce that all the tori in the tower of a non-degenerate nilmanifold admit weak complex multiplication.

Proposition 10. Let F be a finite-dimensional associative algebra over a field, M a F -left module, λ, ρ : M → F, B : M × M → M maps such that

B(x, y) = λ(x)y = ρ(y)x

for all x, y ∈ M. Furthermore assume that there are elements m, n, p ∈ M such that F p is a free F -module and B(m, n) = p.

Then F is commutative and M is a principal free F -module generated by each of the elements m, n and p, i.e. M = F m = F n = F p.

Proof. We start by evaluating

B(m, n) = λ(m)n = ρ(n)m = p.

Since F p is free, this implies that right multiplication by λ(m), ρ(n) in F is injective. Now F is a finite-dimensional vector space over a field, thus injective linear maps are bijective. It follows that right multiplication by λ(m) resp. ρ(n) is a bijective selfmap of F . Hence λ(m), ρ(n) admit left inverses. It follows that F m = F n = F p. In particular F m is a free F -module. Therefore B(m, m) = λ(m)m = ρ(m)m implies λ(m) = ρ(m). For the sake of brevity, let α = λ(m) = ρ(m). For arbitrary x ∈ M, consider

B(m, x) = λ(m)x = αx = ρ(x)m.

Since α has an left-inverse, x ∈ F m. Thus M = F m, i.e. M is a principal F - module. Now let x = am with a ∈ F . Then B(am, m) = λ(am)m = αam, hence λ(am) = αa. Similarily one obtains ρ(am) = αa. Therefore

B(am, bm) = αabm = αbam

for all a, b ∈ F . Since α has a left inverse and F m is free, it follows that ab = ba for all a, b ∈ F .

9. Bilinear forms for tori

Definition. Let T_{i} (i = 1, 2, 3) be tori with lattices Λ_{i} and universal coverings T̃_{i}.

A bilinear form for the T_{i} is a ℂ-bilinear map B : T̃_{1} × T̃_{2} → T̃_{3} such that B(Λ_{1} × Λ_{2}) ⊂ Λ_{3} .

155 D.9

Caveat: Such a bilinear form does not yield a map from T_{1} × T_{2} → T_{3} . However for λ ∈ Λ_{1} one obtains a morphism of tori B′(λ) : T_{2} → T_{3} induced by B(λ, ·) : T̃_{2} → T̃_{3} . The similar statements hold for λ ∈ Λ_{2} .

Using our results on primitive elements, it follows that B(Λ_{1}^{prim} × Λ_{2}^{prim}) ⊂ Λ_{3}^{prim}.

Now, given such a bilinear form we have the left kernel

K̃_{1} = {v ∈ T̃_{1} : B(v, w) = 0 ∀w ∈ T̃_{2}},

the right kernel K̃_{2} ⊂ T̃_{2} and the image Image(B) ⊂ T̃_{3} . Observe that v ∈ K̃_{1} if and only if B(v, λ) = 0 for every λ ∈ Λ_{2} . This implies that K̃_{1}/Λ_{1} equals the intersection of the kernels of the morphisms of tori T_{1} → T_{3} induced by the B(·, λ) with λ ∈ Λ_{2} . Therefore K_{1} = K̃_{1}/(K̃_{1} ∩ Λ_{1}) is a subtorus of T_{1} . Similarily for K_{2} = K̃_{2}(K̃_{2}∩Λ_{2}). Furthermore, if S̃ denotes the ℂ-vector space spanned by the image of B, similar arguments yields that S = S̃/(S̃ ∩ Λ_{3}) is a subtorus of T_{3} . We call these subtori K_{i}, S the subtori associated to the kernel resp. image of B.

We are interested in these bilinear forms, because they arise in a canonical way from nilmanifolds.

Lemma 24. Let X = G/Γ be a nilmanifold, G^{k} the descending and C_{k} the ascending central series.

Then the commutator map ζ : (g, h) ↦ ghg^{−1}h^{−1} induces bilinear forms for each of the triples G/G′Γ, G^{k}/(G^{k+1}Γ ∩ G^{k}), G^{k+1}/(G^{k+2}Γ ∩ G^{k+1}) and G/G′, C_{k+2}/(C_{k+1}Γ ∩ C_{k+2}), C_{k+1}/(C_{k}Γ ∩ C_{k+1} for all k ≥ 0. Let B^{k} resp. B_{k} denote these bilinear forms.

Then for B_{k} the right kernel is trivial, while for B^{k} the image generates everything.

Proof. This is an immediate consequence of the results in the section on nilmanifolds.

Furthermore such bilinear forms are related to the bilinear forms on modules discussed in the preceding section.

Lemma 25. Let T = ℂ^{g}/Λ be an irreducible torus, γ_{i} ∈ Λ^{prim} with T = Θ(γ_{i}), B a bilinear form for T_{1} , T_{2} , T_{3} with T ≃ T_{i} for i = 1, 2, 3 such that B(γ_{1}, γ_{2}) = γ_{3} . Furthermore let E = End_{ℚ}(T) and M = Λ ⊗ ℚ.

Then B induces a map B_{0} : M × M → M, such that there exist maps λ : M → E, ρ : M → E with

B(x, y) = λ(x)y = ρ(y)x.

Moreover Eλ_{3} is a free E -submodule of M.

Proof. Evidently B induces a map Λ ⊗ ℚ × Λ ⊗ ℚ → Λ ⊗ ℚ simply by restriction.

Since B induces a morphism of tori B(γ, ·) : T → T , it is clear that for each γ_{1} ∈ Λ there exists an element λ(γ) ∈ E such that B(γ, y) = λ(γ)y.

Finally, Eγ_{3} is a free E -module due to Lemma 23.

Now we will use these bilinear forms to deduce some properties of the tori which are involved.

Lemma 26. Let B be a bilinear form for tori T_{1}, T_{2}, T_{3} with lattices Λ_{i}, λ_{i} ∈ Λ_{i}^{prim} with B(λ_{1}, λ_{2}) = λ_{3} and let Θ_{i} = Θ(λ_{i}) denote the irreducible subtori of T_{i} associated to λ_{i}.

Then

B(Θ̃_{1} × Θ̃_{2}) ⊂ Θ̃_{3}

Moreover, Θ_{3} is a simple torus with complex multiplication and there exist subtori K_{i} ⊂ Θ_{i} (i = 1, 2) such that Θ_{3} ≃ Θ_{i}/K_{i}.

Proof. Let

C_{1} := ker[B(·, λ_{2}) : Θ_{1} → Θ_{3}]

and

C_{2} := ker[B(λ_{1}, ·) : Θ_{2} → Θ_{3}].

Then Θ_{i}/C_{i} ≃ Θ_{3} (i = 1, 2). Now consider λ ∈ Λ_{1} with B(λ, λ_{2}) = 0. Since Θ_{3} is irreducible and equals the subtorus Θ(λ_{2}) associated to λ_{2} (as defined in the section on primitive elements), it follows that B(λ, v) = 0 for all v ∈ Θ̃_{2} as soon as B(λ, λ_{2}) = 0. Thus B induces a bilinear form for the triplet of tori Θ_{1}/C_{1} , Θ_{2}/C_{2} , Θ_{3} . Furthermore these tori are pairwise isomorphic. Now lemma 25, Prop. 9 and Prop. 7 imply that Θ_{3} is a simple torus with complex multiplication.

Corollary. Let B be a bilinear form for tori T_{1} , T_{2} , T_{3} , and let S denote the subtorus of T_{3} associated to the image and K_{i} ⊂ T_{i} (i = 1, 2) the subtori associated to the kernel.

Then S and both T_{i}/K_{i} are isogenous to products of simple tori with complex multiplication.

Proof. Recall that for any torus T = ℂ^{g}/Λ, the set of primitive elements Λ^{prim} generates Λ ⊗ ℚ as ℚ-vector space. Therefore S is generated by its simple subtori with complex multiplication. It follows that S is isogenous to a direct product of simple tori with complex multiplication. Now K_{i} (for i = 1, 2) is the intersection of kernels of morphisms of tori to S. Thereby T_{i}/K_{i} is isomorphic to a subtorus of a S^{N_{i}} for N_{i} ∈ ℕ sufficiently large. This implies that T_{i}/K_{i} is likewise isogenous to a product of simple tori with complex multiplication.

157 D.9

This completes the proof of theorem 1.

10. A nilmanifold without non-constant meromorphic functions

Here we construct a non-trivial complex nilmanifold of algebraic dimension zero.

Proposition 11. Let G_{0} be the usual complex Heisenberg group, i.e.

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

Let G = G_{0} × G_{0} , n ∈ ℤ \ {0}, ξ ∈ ℂ such that ξ^{2} = n + i and Γ the set of all elements of the form

(( \begin{matrix}1 & a_{1} + b_{1}ξ & a_{3} + b_{3}ξ \\ & 1 & a_{2} + b_{2}ξ \\ & & 1 \end{matrix} ) , ( \begin{matrix}1 & ā_{1} − b̄_{1}ξ̄ & ā_{3} − b̄_{3}ξ̄ \\ & 1 & ā_{2} − b̄_{2}ξ̄ \\ & & 1 \end{matrix}))

with a_{j}, b_{j} ∈ ℤ + iℤ. Then Γ is a discrete subgroup of G and the quotient G/Γ is a compact nilmanifold without non-constant meromorphic functions.

We start with the construction of a certain number field. Fix n ∈ ℤ \ {0}. Choose ξ ∈ ℂ with ξ^{2} = n + i. Let K = ℚ(i)(ξ). A general element of K is given by a + bξ with a, b ∈ ℚ(i). The four embeddings of K into ℂ map a + bξ to a + bξ, a − bξ, ā + b̄ξ̄ and ā − b̄ξ̄ respectively. Thus an element a + bξ is totally real, if both a and bξ are real. However bξ ∈ ℝ implies (bξ)^{2} ∈ ℝ ∩ ℚ(i) = ℚ, since b ∈ ℚ(i) and ξ^{2} = n + i. Now (bξ)^{2} = p ∈ ℚ implies N_{ℚ(i)|ℚ}(b^{2}) · (n^{2} + 1) = p^{2} , which is impossible for b ≠ 0, since n^{2} + 1 is not a square in ℚ. Thus K contains no totally real elements except the rational ones. In particular, K is not a CM-field. Furthermore, using Lemma 15 it follows that ℚ(i) is the only CM-subfield of K.

Lemma 27. Let ξ ∈ ℂ with ξ^{2} = n + i, n ∈ ℤ \ {0}. Let K = ℚ(i)(ξ) with type σ given by σ_{1}(z) = a+bξ and σ_{2}(z) = ā−b̄ξ̄ for z = a+bξ (a, b ∈ ℚ(i)). Let R = {a + bξ : a, b ∈ ℤ + iℤ}

Then T = ℂ^{2}/σ(R) is a simple two-dimensional complex torus with complex multiplication by K such that every meromorphic function on T is constant.

Proof. Clearly R = {a + bξ : a, b ∈ ℤ + iℤ} ⊂ O_{K}. Thus R is a subring of O_{K} with O_{K}/R finite (because rank_{ℤ}(R) = 4 = [K : ℚ]). In particular ℂ^{2}/σ(O_{K}) is isogenous to ℂ^{2}/σ(R) for every type σ. As we have seen above, K contains no CMsubfields except ℚ(i) and the type σ is clearly not lifted from ℚ(i). Therefore T is not algebraic. This implies the statement, since a torus with complex multiplication is either algebraic or does not admit any non-constant meromorphic function at all.

Proof of the proposition. First note that Γ is indeed a discrete subgroup. This is verified easily using the fact that R = {a + bξ : a, b ∈ ℤ + iℤ} is a ring and σ_{i}|_{R} : R → ℂ are ring-homomorphisms. Recall that for any nilmanifold X every meromorphic function is a lift from Alb(X). In our case this implies that G/Γ has no non-constant meromorphic function.

11. A general method of construction

Let K be a totally complex numberfield, [K : ℚ] = 2g. Let σ = (σ_{1}, …, σ_{g}) a type for K, i.e. a choice of g mutually distinct and non-conjugate embeddings K → ℂ.

Any field homomorphism ϕ : K → K′ yields a covariant functor from K - varieties to K′ -varieties. For an affine variety V ⊂ 𝔸^{n} defined by polynomials P_{i} ∈ K[X_{1}, …, X_{n}] this functor simply takes V to the variety ^{ϕ}V defined by the polynomials ϕ_{∗}P_{i} ∈ K′[X_{1}, …, X_{n}].

Thus a type σ = (σ_{1}, …, σ_{g}) on a totally complex number field K yields a covariant functor

σ : V ↦ ^{σ}V = _{1}^{σ}V × … × _{g}^{σ}V.

This functor takes n-dimensional K -varieties to gn-dimensional ℂ-varieties. It takes K -groups to algebraic groups defined over ℂ.

For an affine space 𝔸^{n}, the map σ maps 𝔸^{n}(O_{K}) onto a lattice in 𝔸^{ng}(ℂ) = ℂ^{ng}.

We will apply these methods to construct nilmanifolds.

Definition. A nilpotent algebraic group defined over a field k is called a nilpotent flag group, if it is k -isomorphic to the affine space 𝔸^{n} endowed with a group structure defined by a k -morphism m : 𝔸^{n} × 𝔸^{n} → 𝔸^{n} with m = (m_{1}, …, m_{n}) such that

m_{i}(x_{1}, …, x_{n}, y_{1} …, y_{n}) = x_{i} + y_{i} + P_{i}(x_{1}, …, x_{i−1}, y_{1}, …, y_{i−1})

for some polynomials P_{i}.

Note: If k is algebraically closed, then every unipotent k -group is a nilpotent flag group. For arbitrary fields this is not true.

Lemma 28. Let g be a nilpotent Lie algebra over a field k of characteristic zero. Then there exists a nilpotent flag k -group G with g = Lie(G).

Proof. By the Ado-Iwasawa theorem, there is an embedding ρ : g → Lie GL_{n}(k) such that all the ρ(v) (v ∈ g) are nilpotent. Thus the exponential power series exp(A) = ∑_{k} 1/k^{!}A^{k} restricts to a polynomial isomorphism of varieties from ρ(g) to a unipotent subgroup U of GL_{n}(k). Now suitable linear coordinates on g induce the structure of a nilpotent flag group on U.

Lemma 29. Let G be a nilpotent flag group defined over a field k of characteristic zero. Let O be an additive subgroup of k, which generates k as a ℚ-vector space.

Then G is k -isomorphic to 𝔸^{n} (as k -variety) with the group structure defined by a polynomial map m : 𝔸^{n} × 𝔸^{n} → 𝔸^{n} such that all coefficients of m are in O.

Proof. Note that for each x ∈ k there is a natural number q ∈ ℕ such that qx ∈ O. Using this, one can prove easily by induction on dim(G) that by an application of a linear change of coordinates of the type

(z_{1}, …, z_{n}) ↦ (p_{1}z_{1}, …, p_{n}z_{n}) (p_{i} ∈ ℕ)

one can achieve that all the coefficients of the polynomial map defining group multiplication are in O.

These auxiliary result enable us to construct interesting nilmanifolds.

Theorem 7. Let K be a totally complex numberfield, σ : K → ℂ^{g} a type and g a nilpotent K -Lie algebra.

Then there exists a torus T with complex multiplication type (K, σ), a connected complex nilpotent Lie group G and a discrete cocompact subgroup Γ ⊂ G such that

1) Lie(G) ≃ g ⊗_{σ_{1}} ℂ × … g ⊗_{σ_{g}} ℂ.

2) Each of the tori G^{k}/(G^{k+1} ∩ Γ) is isogenous to a direct product of copies of T.

Proof. Let G_{K} be a nilpotent flag group defined over K such that Lie G_{K} = g. Now G_{K} is isomorphic to the affine space 𝔸^{n} as a K -variety and due to the preceding lemma we may assume that the multiplication map is given by a polynomial map such that all the coefficients are contained in the ring of algebraic integers O_{K}. This implies that 𝔸^{n}(O_{K}) = {(z_{1}, …, z_{n}) : z_{i} ∈ O_{K}} is a subgroup of G_{K}. Now the type σ yields a complex unipotent group G given by G = ^{σ}G_{K} with a group homomorphism ζ : G_{K} → G given by x ↦ (σ_{1}(x), …, σ_{g}(x)). Finally recall that σ : K → ℂ^{g} embedds O_{K} as a discrete cocompact subgroup in (ℂ^{g}, +). Using induction, this implies that σ (G_{K}(O_{K})) is a discrete cocompact subgroup of G. Moreover T = ℂ^{g}/σ(O_{K}) is torus with complex multiplication type (K, σ) which yields the second assertion, again by induction.

12. Examples of non-compact nilmanifolds

A variation of this procedure described above may be used to construct interesting non-compact nilmanifolds composed out of Cousin groups. (A Cousin group is a commutative complex Lie group without non-constant holomorphic functions.) The key point is to consider number fields which are not totally complex.

Proposition 12. Let K be a number field with r real embeddings K → ℝ and s pairs of conjugate complex embeddings. Let σ = (σ_{1}, …, σ_{r+s}) a choice of r + s mutually distinct and non-conjugate embeddings such that σ_{i} is real for i ≤ r. Let g be a nilpotent K -Lie algebra.

Then there exists a Cousin group H ≃ ℂ^{r+s}/σ(O_{K}), a connected complex nilpotent Lie group G and a discrete subgroup Γ ⊂ G such that

1) Lie(G) ≃ g ⊗_{σ_{1}} ℂ × … g ⊗_{σ_{g}} ℂ.

2) G^{k}Γ is closed in G for all k and G^{k}/(G^{k+1} ∩ Γ) is isogenous to a direct product of copies of H.

3) If r > 0, then every compact analytic subset of X = G/Γ is finite. If s > 0, then every holomorphic function on X is constant.

Once one has derived the technical lemma below, the proof is an easy generalization of the proof of theorem 2. Therefore we only prove the subsequent lemma.

Lemma 30. Let K be a number field with r real embeddings and s pairs of conjugate complex embeddings. Let g = r + s. Let O_{K} denote the ring of algebraic integers.

Let σ_{1}, …, σ_{g} be a choice of distinct and pairwise non-conjugate embeddings K → ℂ such that σ_{1}, …, σ_{r} are real.

Then σ(O_{K}) = {(σ_{1}(x), …, σ_{g}(x)) : x ∈ O_{K}} is a discrete subgroup of ℂ^{g}.

If r > 0, then G = ℂ^{g}/σ(O_{K}) has no non-finite compact analytic subset. If s > 0, then G admits no non-constant holomorphic functions.

Proof. Every embedding of a number field K into ℂ induces an absolute value on K by pulling-back the ordinary absolute values on ℂ. Moreover, two embeddings σ, τ : K → ℂ induce independent absolute values unless σ = τ or σ = τ̄. Thus the Artin-Whaples Approximation theorem implies that for all x = (x_{1}, …, x_{g}) ∈ K^{g} and ϵ > 0 there exists an element z ∈ K such that ∥ζ(x) − σ(z)∥ < ϵ where ζ : K^{g} → ℂ^{g} denotes the map given by ζ(x) = (σ_{1}(x_{1}), …, σ_{g}(x_{g})). Now the image of ζ(K^{g}) is easily seen to be dense in ℝ^{r} ⊕ ℂ^{s}. Thus σ(K) is dense in V = ℝ^{r} ⊕ ℂ^{s}. Since K ≃ O_{K}⊗_{ℤ}ℚ as ℤ-module and dim_{ℝ}(V) = [K : ℚ], it follows that σ(O_{K}) is a lattice in V . Now K acts by complex-linear endomorphisms on ℂ^{g} stabilizing the discrete subgroup σ(O_{k}). Since K ≃ O_{K} ⊗ℚ, it follows that no subgroup of Λ = σ(O_{K}) can be stable under all endomorphisms of G = ℂ^{g}/Λ. It follows that for r > 0 the group G cannot contain a compact complex Lie subgroup. Since any connected compact analytic subset (containing 0) would generate a compact Lie subgroup, this implies that every compact analytic subset is finite (for r > 0).

We still have to show that there are no non-constant holomorphic functions on G for s > 0. Recall that for any complex Lie group G there exists a holomorphic reduction, i.e. a holomorphic map π : G → H onto a Stein Lie group H such that all holomorphic functions on G are lifted from H [15]. Observe furthermore that a complex Stein manifold of complex dimension n is homotopy-equivalent to a CW - complex of real dimension n. Therefore a commutative complex Lie group ℂ^{g}/Γ can not be Stein unless rank_{ℤ}(Γ) ≤ dim_{ℂ}(ℂ^{g}) = g. It follows that for s > 0 the group G is not Stein. Furthermore it is clear that, for any complex-linear projection π : ℂ^{g} → ℂ^{d} with ℂ^{d}/π(Λ) Stein, there must be a non-trivial intersection ker π ∩ Λ. Since Λ has no non-trivial subgroups stable under all endomorphisms of G, this implies that there are no non-constant holomorphic functions on G.

13. Solvmanifolds

Our results obtained so far are specific to nilmanifolds. They do not apply to solvmanifolds.

Example. Let E be an elliptic curve (i.e. a one-dimensional torus) without complex multiplication, A ∈ SL_{2}(ℤ) with eigenvalues α, α^{−1} such that |α| < 1. Let X′ = ℂ^{∗} × E × E and X = X′/ ∼ with (x, y, z) ∼ (αx, A \binom{y}{z}).

Then X is a three-dimensional compact complex solvmanifold, which is a E × E -bundle over Alb(X) ≃ ℂ/ ⟨2πi, τ⟩_{ℤ} with e^{2πiτ} = α.

Neither E nor Alb(X) have complex multiplication. To check that Alb(X) has no complex multiplication, note that Alb(X) = ℂ/ ⟨1, τ⟩ with e^{2πiτ} = α.

162 D.13 The theorem of Lindemann-Weierstraß implies that τ is not algebraic. But ℂ/ ⟨1, τ⟩ has complex multiplication if and only if [ℚ(τ) : ℚ] = 2.

We will now generalize this special result. First we need an auxiliary result in transcendental number theory. For this we recall Baker’s theorem which is a generalization of the Lindemann-Weierstraß theorem.

Theorem (Baker [2]). Let α_{1}, …, α_{n} be non-zero algebraic numbers. Fix a branch of the logarithm and assume that 1, log α_{1}, …, log α_{n} are linearly dependent over \overline{ℚ} (the algebraic closure of ℚ in ℂ).

Then log α_{1}, …, log α_{n} are already linearly dependent over ℚ.

This result on linear forms of logarithms constitutes the base for the following auxiliary result in transcental number theory.

Proposition 13. Let V = (ℂ^{n}, +), S = (ℂ^{∗})^{d}. Let Λ be a discrete subgroup of V such that every holomorphic function on V/Λ is constant (e.g. a lattice). Let ϕ : V → S be a complex Lie group homomorphism. Assume Λ ⊂ \overline{ℚ}^{n} and

ϕ(Λ) ⊂ (\overline{ℚ}^{∗})^{d}.

Then ϕ is constant.

Proof. First note that there is no loss in generality in assuming d = 1.

The statement is trivially true for d = 0. Thus we may argue by induction. Now let Σ denote the kernel of ϕ|_{Λ} : Λ → S. Furthermore let W = ⟨Σ⟩_{ℂ} denote the complex subvectorspace of V spanned by Σ. We claim: dim_{ℂ}(W) = rank_{ℤ} Σ. To prove the claim, let U = ⟨Σ⟩_{ℝ} and K = U ∩ iU. We have to show that K = {0}. Now K is a complex subvector space of V which is defined over \overline{ℚ}. It follows that V′ = V/K ≃ ℂ^{m} may be endowed with complex coordinates such that π(Λ) ⊂ V′(\overline{ℚ}) = \overline{ℚ}^{m}. Moreover there is a subgroup Λ′ ⊂ π(Λ) which is a lattice in V′. (A lattice has to be cocompact and discrete and π(Λ) itself is possibly not discrete). Thus we may invoke induction, if we can show that ϕ : V → S fibers through V/K. This is equivalent to ϕ|K ≡ 1. Now ϕ|Σ ≡ 1 and U/Σ is compact, hence ϕ(K) ⊂ ϕ(U) is a compact subset of S ≃ ℂ^{∗}. By the theorem of Liouville it follows that ϕ_{K} ≡ 1. Thus invoking induction we may assume that K = {0} which implies dim_{ℂ}(W) = rank_{ℤ} Σ as claimed.

There is a vector a ∈ ℂ^{n} such that ϕ(z) = e^{a·z} with a · z = ∑_{i} a_{i}z_{i}. We may fix a basis λ_{1}, …, λ_{n} of V ≃ ℂ^{n} in such a way that λ_{j} ∈ Λ for all j and λ_{j} ∈ Σ for j ≤ k = rank_{ℤ} Σ. Then a_{i} ∈ iπℤ for i ≤ k. Moreover iπ, a_{k+1}, …, a_{n} are ℚ-linearly independent because ker ϕ ∩ Λ is contained in the ℚ-vector space spanned by a_{1}, …, a_{k}. Choose an element λ = ∑_{j} β_{j}a_{j} of Λ. If k < n, we require β_{n} ∉ ℚ.

163 D.13

(This is possible, because otherwise

∑_{j} β_{j}λ_{j} ↦ e^{Nβ_{n}}

will be a non-constant Λ-invariant holomorphic function on V for N >> 0.) Now ϕ(λ) = e^{∑_{j} β_{j}a_{j}} and all the numbers ϕ(λ_{j}) = e^{a_{j}} are algebraic. Furthermore Λ ⊂ \overline{ℚ}^{n} implies that β_{j} ∈ \overline{ℚ}, i.e. log ϕ(λ) is \overline{ℚ}-linearly dependent on iπ, a_{1}, …, a_{n}. Thus Baker’s theorem implies that ∑_{j} β_{j}a_{j} is ℚ-linearly dependent on iπ, a_{1}, …, a_{n}. If k = rank Σ < n, then a_{n} is ℚ-linearly dependent on iπ, a_{1}, …, a_{n} and we arrive at a contradiction, since β_{n} ∉ ℚ. Thus we may assume k = n. In this case ∑_{j} β_{j}a_{j} = log(ϕ(λ)) is a ℚ-multiple of iπ. But this implies that ϕ(λ) is a root of unity. Recall that for k = n the element λ ∈ Λ was choosen arbitrarily. Hence ϕ(λ) is a root of unity for all λ ∈ Λ. Since Λ is a finitely generated group, it follows that ϕ(Λ) = E is a finite subgroup of S ≃ ℂ^{∗}. Thus we obtain an induced Lie group homomorphism ϕ_{0} : V/Λ → (ℂ^{∗}/E) ≃ ℂ^{∗}. Since V/Λ does not admit any non-constant holomorphic function, it follows that ϕ_{0} and therefore ϕ are constant.

Lemma 31. Let G be a solvable complex Lie group, N its nilradical, i.e. the maximal connected nilpotent normal Lie subgroup.

Then there exists a complex Lie group homomorphism ρ → (ℂ^{∗})^{d} (d = dim(G)), such that

(i) The connected component of the kernel of ρ equals N. (ii) For each g ∈ G, ρ_{1}(g), …, ρ_{d}(g) ∈ ℂ^{∗} are the eigen-values for Ad(g) ∈

Lie(G).

Proof. Let Ad : G → GL(Lie G) denote the adjoint representation. Since this is a complex linear representation of a connected solvable Lie group, there is a Borel group B of GL(Lie G) with Ad(G) ⊂ B. Now B′ = [B, B] is nilpotent and conjugate to the group of all unipotent upper triangular matrices. Moreover B′ is the set of all unipotent elements in B. In particular Ad(N) ⊂ B′. Conversely B′ is nilpotent and Ad^{−1}(B′) is a central extension of B′, hence likewise nilpotent. It follows that N equals the connected component of Ad^{−1}(B′). Finally, for each b ∈ B ∈ GL(Lie G) the eigenvalues of b depend only on the projection π(b) ∈ B/B′, since B is conjugate to the group of upper triangular matrices. Thus ρ = π ∘ Ad : G → B/B′ does the job.

We want to apply this to groups with lattices. For this purpose we need the following fact.

Lemma 32. Let V be a complex vectorspace, Λ a lattice, ϕ ∈ GL(V) with ϕ(Λ) ⊂ Λ.

Then all the eigen-values of ϕ are algebraic.

164 D.13

Proof. This follows immediately from V ≃ Λ ⊗_{ℤ} ℝ.

Proposition 14. Let G be a solvable complex connected Lie group, Γ a lattice, N the nilradical.

Then G/NΓ is a torus, hence isomorphic to a quotient V/Λ of a complex vector space V by a lattice Λ.

Furthermore there exists a complex Lie group homomorphism ϕ : V → (ℂ^{∗})^{n} (n = dim_{ℂ}(G)) such that ker ϕ is discrete and for all λ ∈ Λ all the ϕ_{i}(λ) are algebraic numbers.

Proof. It is a result of Mostow [16] that NΓ is closed in G, which implies the first assertion. Moreover, a result of Malcev [10] implies that all the N^{k}Γ are closed in NΓ, hence closed in G. Now the adjoint action clearly stabilizes all the Lie(N^{k}), furthermore induces the trivial action on Lie(G)/ Lie(N), because G′ ⊂ N. It follows that all the eigenvalues of Ad(G) are eigenvalues of some action on N^{k}/N^{k+1} with Ad(Λ) stabilizing the lattice Λ_{k} = (N^{k} ∩Λ)/(N^{k+1} ∩Λ). Hence all the eigen-values of Ad(Λ) are algebraic by the preceding lemma. With the help of lemma 31 this implies the last assertion of the proposition.

Corollary. Let X = G/Γ be a complex solvmanifold. Assume that Z ⊂ G′.

Then G is nilpotent if and only if Alb(X) is isogenous to a product of simple tori with complex multiplication.

Remark. Due to a result of Barth and Otte [3] the commutator group G′ has closed orbits in G/Γ. Therefore Alb(X) = G/G′Γ.

Proof. The nilradical N of G contains the commutator group G′, hence G/NΓ is a torus. It follows that G/NΓ is a quotient of the Albanese Alb(X). Thus G/NΓ is likewise isogenous to a product of simple tori with complex multiplication. It follows that G/NΓ ≃ ℂ^{g}/Λ for a lattice Λ with Λ ⊂ \overline{ℚ}^{g}. Now prop. 9 and prop. 10 together imply that G/NΓ is a point, i.e. G = N.

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