NON-KÄHLER COMPACT COMPLEX MANIFOLDS ASSOCIATED TO NUMBER FIELDS

KARL OELJEKLAUS & MATEI TOMA

Abstract. For algebraic number fields K with s > 0 real and 2t > 0 complex embeddings and ”admissible” subgroups U of the group of integer units of K we construct and investigate (s + t)-dimensional compact complex manifolds X(K, U). We show among other things that such manifolds are non-Kähler but admit locally conformally Kähler metrics when t = 1. In particular we disprove a conjecture of I. Vaisman.

1. Notations, construction and first properties

Consider an algebraic number field K, that is a finite extension field of the field of rational numbers ℚ. Let n := (K : ℚ) be its degree. The field K admits precisely n = s + 2t distinct embeddings σ_{1}, ...σ_{n} into ℂ, where we suppose that σ_{1}, ...σ_{s} are the real embeddings, σ_{s+1}, ..., σ_{n} are the complex ones and that σ_{s+i} = σ̄_{s+i+t} for 1 ≤ i ≤ t. We shall suppose throughout the paper that both s and t are strictly positive. Furthermore, let O_{K} denote the ring of algebraic integers of K. This is a free ℤ-module of rank n. In fact our construction works also for arbitrary orders O of K, i.e. for subrings O of O_{K} which have rank n as ℤ-modules. Set now m := s + t and consider the ”geometric representation” of K:

σ : K → ℂ^{m}, σ(a) := (σ_{1}(a), ..., σ_{m}(a)).

It is known that the image σ(O_{K}) of O_{K} through σ is a lattice of rank n in ℂ^{m}, cf. [1], 2.3.1., p. 95ff. We thus get a properly discontinuous action of O_{K} on ℂ^{m} by translations. Consider furthermore the following multiplicative action of K on ℂ^{m}: for a ∈ K and z ∈ ℂ^{m} set

az := (σ_{1}(a)z_{1}, ..., σ_{m}(a)z_{m}).

For a ∈ O_{K}, aσ(O_{K}) is contained in σ(O_{K}). Let O_{K}^{∗} denote the group of units in O_{K} and

O_{K}^{∗,+} := {a ∈ O_{K}^{∗} | σ_{i}(a) > 0 for all 1 ≤ i ≤ s}.

2000 Mathematics Subjet Classification: 32J18, 32M17.

Key words and phrases. Compact complex manifolds, algebraic number fields, algebraic units, locally conformally Kähler metrics.

The authors thank the Deutsche Forschungsgemeinschaft for financial support within its Schwerpunkt ”Globale Methoden in der komplexen Geometrie”. The second author was also supported by the Max-Planck-Gesellschaft during the preparation of this paper.

Furthermore the authors thank L. Ornea for motivating them to look for examples of locally conformally Kähler manifolds and W. Bruns, Ph. Eyssidieux and A. Zaharescu for discussions on algebraic number theory.

2 OELJEKLAUS & TOMA

Since for s > 0 the only torsion elements of O_{K}^{∗} are 1 and −1, Dirichlet’s Units Theorem allows us to write O_{K}^{∗} = G ∪ (−G), where G is a free abelian (multiplicative) group of rank m − 1. One may choose G so that it contains O_{K}^{∗,+}, automatically with finite index. We denote by ℍ the upper complex half-plane, ℍ := {z ∈ ℂ | ℑmz > 0}. Combining the additive action of O_{K} with the induced multiplicative action of O_{K}^{∗,+} we get an action of O_{K}^{∗,+} ⋉ O_{K} on ℂ^{m} which is free and leaves ℍ^{s} × ℂ^{t} invariant. We shall now choose a subgroup U of rank s of O_{K}^{∗,+} such that the action of U ⋉O_{K} on ℍ^{s} ×ℂ^{t} becomes properly discontinuous, thus yielding a smooth quotient which will be shown to be compact. In order to do this we consider the logarithmic representation of units

l : O_{K}^{∗} → ℝ^{m}, l(u) := (ln|σ_{1}(u)|, ..., ln|σ_{s}(u)|, 2ln|σ_{s+1}(u)|, ..., 2ln|σ_{m}(u)|),

cf. [1] 2.3.3.. Dirichlet’s Units Theorem implies that l(O_{K}^{∗,+}) is a full lattice in the subspace L := {x ∈ ℝ^{m}| ∑_{i=1}^{m} x_{i} = 0} of ℝ^{m}. Since t > 0, the projection pr : L ↦ ℝ^{s} given by the first s coordinate functions is surjective. Thus there exist subgroups U of rank s of O_{K}^{∗,+} such that pr(l(U)) is a full lattice in ℝ^{s}. Such a subgroup will be called admissible for K. Take now U admissible for K. The quotient ℍ^{s} ×ℂ^{t}/σ(O_{K}) is clearly diffeomorphic to a trivial torus bundle (ℝ_{>0})^{s} × (S^{1})^{n} and U operates properly discontinuously on it since it induces a properly discontinuous action on the base (ℝ_{>0})^{s} by our choice. Differentiably the quotient of this action is a fiber bundle over (S^{1})^{s} with (S^{1})^{n} as fiber. We thus get an m-dimensional compact complex affine manifold

X = X(K, U) := (ℍ^{s} × ℂ^{t})/(U ⋉ O_{K}).

This paper is devoted to the description of these complex manifolds.

Remark 1. 1. For every choice of natural numbers s and t, algebraic number fields with precisely s real and 2t complex embeddings exist.

Since we don’t know of any source for this observation we include here an argument we owe to Ph. Eyssidieux. Proof. Consider the non-empty open set D of points a = (a_{1}, ..., a_{n}) ∈ ℚ^{n} such that the polynomials P = X^{n} +a_{1}X^{n−1} +...+a_{n} admit exactly s real distinct roots and 2t complex non-real roots. The open set D will contain arbitrarily large open balls since the map (a_{1}, ..., a_{n}) ↦ (ba_{1}, b^{2}a_{2}, ..., b^{n}a^{n}) leaves it invariant for any choice of rational numbers b. Choose now a prime number p and P̃ = X^{n} + ã_{1}X^{n−1} + ... + ã_{n} ∈ ℤ[X] an Eisenstein polynomial with respect to p, that is p|ã_{i} for all i but p^{2} ∤ ã_{n}. Then the set ã + p^{2}ℤ^{n} intersects D and consists only of Eisenstein hence irreducible polynomials. □

Remark 1. 2. For s = 1, t = 1 and U = O_{K}^{∗,+}, X(K, U) is an Inoue-Bombieri surface S_{M}; cf. [3].

COMPACT COMPLEX MANIFOLDS 3

Remark 1.3. When s = 1 or t = 1 all subgroups U of rank s of O_{K}^{∗,+} are admissible for K. But this need not be the case in general as the following example shows. Take two field extensions K′ and K′′ of ℚ with corresponding numbers of real and complex embeddings s′ = 1, t′ = 2, s′′ = 2, t′′ = 1 and K the composite of K′ and K′′. Then s = 2 but O_{K′}^{∗,+} is not admissible for K.

Lemma 1. 4. Let U be a subgroup of O_{K}^{∗} not contained in ℤ. Then the following are equivalent:

• The action of U on O_{K} admits a proper non-trivial invariant submodule of lower rank.

• There exists some proper intermediate field extension ℚ ⊂ K′ ⊂ K with U ⊂ O_{K′}^{∗}.

Proof. Suppose M is a proper ℤ-submodule of O_{K} which is invariant under U and with 0 < rank M = r < n. We consider the coefficient ring of M, O_{M} := {a ∈ K|aM ⊂ M}. We have U ⊂ O_{M}, hence O_{M} is not contained in ℚ. Let now K′ be the field of fractions of O_{M}. We have to show that K′ ≠ K. Let x ∈ K′ be a primitive element for K′/ℚ with x = a/b, a, b ∈ O_{M}. Then the action of x on M is described by an r × r matrix with rational coefficients in terms of a basis of M. If K′ and K coincided, then the characteristic polynomial of x would allow a factor of degree r over ℚ. This proves the lemma in one direction. The converse is clear. □

Definition 1.5. We shall call the manifold X(K, U) of simple type if U does not satisfy the equivalent conditions of the previous lemma.

Lemma 1. 6. Let ℚ ⊂ K′ ⊂ K be a proper intermediate extension and U ⊂ O_{K′}^{∗} an admissible subgroup for K. Let s′, 2t′ be the numbers of distinct real and respectively complex embeddings of K′. Then s = s′, t′ > 0 and U is admissible for K′.

Proof. The restrictions to K′ of two different real embeddings of K cannot coincide since U ⊂ K′ and U is admissible for K. Thus s′ ≥ s. We show now that s ≥ s′ as well. Let k := (K : K′). The restriction to K′ of a real embedding of K will have to coincide with the restrictions of exactly k − 1 complex embeddings of K. In particular since these restrictions are real these k − 1 complex embeddings occur in complex conjugate pairs. So k − 1 is even. Suppose now that there is a real embedding of K′ which is not the restriction of any real embedding of K. Such an embedding has then to be the restriction of exactly k complex embeddings of K and k would then be even by the same reason as above. Thus s = s′. By Dirichlet’s Units Theorem and since U has rank s, t has to be positive. It is clear now that U is admissible for K′. □

Remark 1. 7. If X(K, U) is not of simple type with ℚ ⊂ K′ ⊂ K as intermediate extension and U ⊂ O_{K′}, then there exists a holomorphic foliation of X(K, U) with a leaf

4 OELJEKLAUS & TOMA

isomorphic to X(K′, U). Just look at the foliation of ℂ^{m} defined by the translates V_{K′} +v, v ∈ ℂ^{m} of the complex vector subspace V_{K′} of ℂ^{m} spanned by σ(O_{K′}). Its restriction to ℍ^{s} × ℂ^{t} is invariant under the action of U ⋉ O_{K} and thus descends to X(K, U). It is clear that (V_{K′} ∩ (ℍ^{s} × ℂ^{t}))/U ⋉ O_{K′} is a leaf of this foliation which is compact since U is admissible for K′.

Remark 1. 8. Not all manifolds X(K, U) are of simple type. Keeping the notations of Remark 1.3 take for instance K′, K′′ with s′ = s′′ = t′ = t′′ = 1 but non isomorphic and K their composite. Then s = 1, t = 4 and this time O_{K′}^{∗,+} is admissible for K. Note however that for any choice of K there are infinitely many X(K, U) of simple type, since the number of intermediate extensions of K is finite.

2. Invariants and metrics

We investigate here some properties of the varieties X(K, U), where K is a number field as before and U is admissible for K. We start with some preparations for the computation of the first Betti numbers of X(K, U).

Remark 2. 1. Let a ∈ O_{K}^{∗} and consider its action on Hom_{ℤ}(O_{K}, ℂ) by (af)(x) := f(a^{−1}x) for f ∈ Hom_{ℤ}(O_{K}, ℂ) and x ∈ O_{K}. Then the restrictions to O_{K} of the embeddings σ_{1}, ..., σ_{n} of K give a basis of Hom_{ℤ}(O_{K}, ℂ) of eigenvectors for this action with associated eigenvalues σ_{1}(a^{−1}), ..., σ_{n}(a^{−1}).

Lemma 2.2. Let θ = ∑_{i=1}^{n} a_{i}σ_{i}, a_{i} ∈ ℂ be a ℚ-valued ℚ-linear form on O_{K}. Then either all coefficients a_{i} are non-zero or they all vanish.

Proof. It is easy to see that there exists a primitive element α for K/ℚ in O_{K}. Then θ(α^{k}) ∈ ℚ for 0 ≤ k ≤ n−1. Let α_{i} := σ_{i}(α) be the roots of the characteristic polynomial of α. The rationality condition for θ can be written as a linear system of equations for the coefficients a_{i}:

∑_{i=1}^{n} a_{i}α_{i}^{k} = b_{k}, 0 ≤ k ≤ n − 1,

for some rational numbers b_{0}, ..., b_{n−1}. Let A be the coefficient matrix (α_{i}^{k})_{1≤i≤n, 0≤k≤n−1} of this system. By the choice of α we have det A ≠ 0. We suppose now that θ ≠ 0, so not all b_{i}-s vanish, and that one of the a_{i}-s is zero, say a_{n} = 0. This means that the determinant of the matrix obtained from A by replacing the last column with the free vector b = (b_{0}, ..., b_{n−1}) shall vanish. Hence expanding this determinant after its last column gives us the following linear dependency relation over ℚ among the coefficients of the polynomial Π_{1≤i≤n−1}(X − α_{i}):

b_{n−1} + b_{n−2}s_{1} + ... + b_{0}s_{n−1} = 0.

COMPACT COMPLEX MANIFOLDS 5

Here we denoted by s_{i} the i-th elementary symmetric function in a_{1}, ..., a_{n−1}. Now we express inductively the elementary symmetric functions in a_{1}, ..., a_{n−1} in terms of those in a_{1}, ..., a_{n} and the powers of α_{n}:

s_{1}(a_{1}, ..., a_{n−1}) = s_{1}(a_{1}, ..., a_{n}) − α_{n},
s_{2}(a_{1}, ..., a_{n−1}) = s_{2}(a_{1}, ..., a_{n}) − α_{n}s_{1}(a_{1}, ..., a_{n−1}) =
= s_{2}(a_{1}, ..., a_{n}) − α_{n}s_{1}(a_{1}, ..., a_{n}) + α_{n}^{2}, ...

This leads to a non-trivial relation over ℚ among 1, α_{n}, ..., α_{n}^{n−1} which contradicts the choice of α. □

Proposition 2. 3. For all X = X(K, U) the first Betti number is b_{1}(X) = s. When X is of simple type one also has b_{2} = \binom{s}{2}.

Proof. The cohomology groups of X with coefficients in ℚ are isomorphic to those of its fundamental group. We thus have to compute H^{1}(U ⋉ O_{K}; ℚ) and H^{2}(U ⋉ O_{K}; ℚ). The Lyndon-Hochschild-Serre spectral sequence associated to the short exact sequence

0 → O_{K} → U ⋉ O_{K} → U → 0

gives

E_{2}^{pq} = H^{p}(U; H^{q}(O_{K}; ℚ)) ⟹ H^{p+q}(U ⋉ O_{K}; ℚ)

and an exact sequence of low degree terms:

0 → H^{1}(U; ℚ^{O_{K}}) → H^{1}(U ⋉ O_{K}; ℚ) → H^{1}(O_{K}; ℚ)^{U} → H^{2}(U; ℚ^{O_{K}}) → H^{2}(U ⋉ O_{K}; ℚ);

cf. [4], 6.8. Here ℚ is seen as a trivial U ⋉ O_{K}-module. Then H^{1}(O_{K}; ℚ) ≅ Hom(O_{K}; ℚ) is a non-trivial U-module via:

(uf)(x) := f(u^{−1}x), for all u ∈ U, f ∈ Hom(O_{K}, ℚ), x ∈ O_{K};

cf. [4] 6.8.1. Thus H^{1}(O_{K}; ℚ)^{U} := {f ∈ Hom(O_{K}, ℚ) | uf = f, for all u ∈ U} and this last space is trivial by Remark 2.1. Thus H^{1}(U ⋉ O_{K}; ℚ) ≅ H^{1}(U; ℚ^{O_{K}}) ≅ H^{1}(U; ℚ) ≅ H^{1}(ℤ^{s}; ℚ) ≅ ℚ^{s}. Moreover, the map H^{2}(U; ℚ^{O_{K}}) → H^{2}(U ⋉ O_{K}; ℚ) is injective. We only need to prove that it is surjective as well when X is of simple type. To see this it is enough to check that in this case the terms E_{2}^{0,2} and E_{2}^{1,1} of the spectral sequence vanish. Consider first E_{2}^{0,2} = H^{0}(U; H^{2}(O_{K}; ℚ)) = H^{2}(O_{K}; ℚ)^{U} ≅ Alt^{2}(O_{K}; ℚ)^{U}. This is the space of alternating 2-forms on O_{K} which are fixed by U. Let γ = ∑_{1≤i<j≤n} a_{ij}σ_{i} ∧ σ_{j} ∈ Alt^{2}(O_{K}; ℚ)^{U} with a_{ij} ∈ ℂ. The fact that γ is invariant under the action of some u ∈ U means that σ_{i}(u)σ_{j}(u) = 1 whenever a_{ij} ≠ 0; cf. Remark 2.1. From this we get a_{ij} = 0 for all 1 ≤ i < j ≤ s since U is admissible for K. The relation σ_{i}(u)σ_{j}(u) = 1 for u ∈ U and the fact that X is of simple type imply moreover that a_{ij} = 0 whenever 1 ≤ i ≤ s and that for each i > s there exists at most one j = j(i) > i with a_{ij} ≠ 0. (Otherwise we would get two equal embeddings σ_{j} = σ_{j′}.) Thus γ = ∑_{s<i<n} a_{ij(i)}σ_{i}∧σ_{j(i)}.

6 OELJEKLAUS & TOMA

Let α ∈ O_{K} be a primitive element for K. Then γ(α^{k}, 1) ∈ ℚ for all k ∈ ℤ, that is ∑_{s<i<n} a_{ij(i)}(σ_{i}(α^{k}) − σ_{j(i)}(α^{k})) ∈ ℚ for all k ∈ ℤ. But then we get a rational 1-form ∑_{s<i<n} a_{ij(i)}(σ_{i} − σ_{j(i)}) which by Lemma 2.2 has to vanish. We now check that E_{2}^{1,1} = H^{1}(U; H^{1}(O_{K}; ℚ)) is trivial. Since U is free abelian we reduce ourselves by the Lyndon-Hochschild-Serre spectral sequence for 0 → ℤ → ℤ^{s} → ℤ^{s−1} → 0 to the computation of H^{1}(ℤ; H^{1}(O_{K}; ℚ)) where ℤ here is the subgroup generated by some u ∈ U. Now H^{1}(ℤ; H^{1}(O_{K}; ℚ)) ≅ H^{1}(O_{K}; ℚ)_{ℤ} ≅ H^{1}(O_{K}; ℚ)/ < uf − f | f ∈ H^{1}(O_{K}; ℚ) >; cf. [4] 6.1.4. But the action of u − id is invertible by Remark 2.1 hence H^{1}(ℤ; H^{1}(O_{K}; ℚ)) vanishes. □

Lemma 2.4. Every holomorphic function on ℍ^{s} × ℂ^{t}/σ(O_{K}) is constant.

Proof. Take any element v ∈ ℍ^{s}. We shall first prove the following Claim The image of {v} × ℂ^{t} in (v + ℝ^{s}) × ℂ^{t}/σ(O_{K}) is dense in this space. We shall just check that 0 × ℂ^{t} has a dense image in ℝ^{s} × ℂ^{t}/σ(O_{K}). For this it is enough to prove that the image of O_{K} through σ′ = (σ_{1}, ...σ_{s}) : O_{K} → ℝ^{s} is dense in ℝ^{s}. Consider the connected component V of 0 of the topological closure of σ′(O_{K}) in ℝ^{s} and the ℤ-submodule M := σ^{′−1}(V) of O_{K}. If V ≠ ℝ^{s} we would have rank M < n. Take now α ∈ O_{K} a primitive element for K. On O_{K} we have a multiplicative action of α. The submodule αO_{K} of O_{K} has finite index so the induced linear action of α on ℝ^{s} will leave V invariant. Thus M also remains invariant under the action of α. But this would imply that the characteristic polynomial of α admits a factor of degree rank M over ℚ, which is absurd. Take now a holomorphic function f on ℍ^{s} × ℂ^{t}/σ(O_{K}) and v ∈ ℍ^{s}. Since f is bounded on (v +ℝ^{s})×ℂ^{t}/σ(O_{K}) ≃ (S^{1})^{n} its lift f̃ to ℍ^{s} ×ℂ^{t} will be bounded on each (v +ℝ^{s})×ℂ^{t} hence constant on {v}×ℂ^{t}. By our Claim it follows now that f̃is constant on (v+ℝ^{s})×ℂ^{t}. But then f̃ must be constant on ℍ^{s} × ℂ^{t} by the identity principle. □

Proposition 2. 5. The following vector bundles on X = X(K, U) are flat and admit no non-trivial global holomorphic sections:

Ω_{X}^{1}, Θ_{X}, K_{X}^{⊗k}, for all k ≠ 0.

Moreover dim H^{1}(X, O_{X}) ≥ s. In particular κ(X) = −∞ and X is non-Kähler.

Proof. Let z_{1}, ..., z_{m} be the standard complex coordinate functions on ℍ^{s}×ℂ^{t}. A section ω of K_{X}^{⊗k} lifted to ℍ^{s} × ℂ^{t} will have the form ω̃ = f(dz_{1} ∧ ... ∧ dz_{m})^{⊗k}. Since this section descends to ℍ^{s} × ℂ^{t}/σ(O_{K}) it follows from Lemma 2.4 that f is constant on ℍ^{s} × ℂ^{t}. Moreover if f ≠ 0, the invariance of ω̃ with respect to U gives (Π_{i=1}^{m}σ_{i}(u))^{k} = 1 for all u ∈ U. Multiplying this by (Π_{i=1}^{m}σ̄_{i}(u))^{k} = 1 and using the fact that (Π_{i=1}^{n}σ_{i}(u))^{k} = 1 we get (Π_{i=1}^{s}σ_{i}(u))^{k} = 1 which contradicts the admissibility of U.

COMPACT COMPLEX MANIFOLDS 7

In the case of Ω_{X}^{1} the automorphy factors are σ_{i}(u), i = 1, ..., m and it is clear that none of them equals 1. An analogous argument works for Θ_{X} using the vector fields ∂/∂z_{i}. The flatness of these bundles is evident. The statement on dim H^{1}(O_{X}) follows now from Proposition 2.3 and the exact sequence of sheaves on X:

0 → ℂ → O → dO → 0.
□

Remark 2. 6. The above proof also shows that the group U is uniquely determined by the complex structure of X(K, U) since its elements are automorphy factors for Ω_{X}^{1}. In particular when X is of simple type its complex structure determines both K and U.

Corollary 2. 7. The group of holomorphic automorphisms of X(K, U) is discrete. It is infinite when t > 1 since the elements of O_{K}^{∗}/U induce automorphisms of X(K, U).

It is known that the Inoue-Bombieri surfaces S_{M} admit locally conformally Kähler metrics. This means that there is a representation ρ : π_{1}(S_{M}) → ℝ_{>0} and a closed strongly positive (1, 1)-form ω on the universal cover of S_{M} such that g^{∗}ω = ρ(g)ω for all g ∈ π_{1}(S_{M}); cf. [2]. We now investigate the existence of locally conformally Kähler metrics more generally on the manifolds X(K, U). Example. When t = 1 all manifolds X(K, U) admit locally conformally Kähler metrics. Consider indeed the following potential

F : ℍ^{s} × ℂ → ℝ, F (z) := \frac{1}{Π_{j=1}^{s}(i(z_{j} − z̄_{j}))} + |z_{m}|^{2}.

Then ω := i∂ ¯∂F gives the desired Kähler metric on ℍ^{s} × ℂ.

Remark 2. 8. The manifolds X(K, U) with s = 2 and t = 1 give counterexamples to a conjecture of I. Vaisman, according to which a compact locally conformally Kähler manifold admitting even Betti numbers with odd index and non-zero Betti numbers with even index should already be Kähler; (see [2], p. 8).

Proof. We have the following Betti numbers for X(K, U): b_{0} = b_{6} = 1, b_{1} = b_{5} = 2, b_{2} = b_{4} = 1 and b_{3} = 0. In fact, here X(K, U) is of simple type and therefore we can apply Proposition 2.3 to get b_{1} and b_{2}. For b_{3} note that the Euler characteristic equals c_{3}(X(K, U)) = 0, since Θ is flat. □

Proposition 2. 9. When s = 1 and t > 1 there exists no locally conformally Kähler metric on X(K, U).

Proof. Let s = 1, ω = ∑_{1≤i, j≤m} g_{ij}dz_{i} ∧ dz̄_{j} a closed strictly positive (1, 1)-form on ℍ×ℂ^{t} and ρ : U ⋉O_{K} → ℝ_{>0} a representation such that g^{∗}ω = ρ(g)ω for all g ∈ U ⋉O_{K}. We shall show that t = 1.

8 OELJEKLAUS & TOMA

It is clear that ρ factorizes through a representation of U which we denote again by ρ. Since ω descends to (ℍ × ℂ^{t})/σ(O_{K}) ≃ ℝ_{>0} × (S^{1})^{n}, we may assume by averaging over (S^{1})^{n} that the coefficients g_{ij} are constant in the directions of σ(O_{K}). In particular they are constant on the subspaces {v} × ℂ^{t} for each v ∈ ℍ. Since dω = 0, this implies that for i, j > 1 the coefficients g_{ij} are constant on the whole of ℍ × ℂ^{t}. By the compatibility of ω with ρ we thus get

ρ(u) = |σ_{2}(u)|^{2} = |σ_{3}(u)|^{2}... = |σ_{m}(u)|^{2}, ∀u ∈ U.

Consider now a non-trivial element u of U and its characteristic polynomial X^{n} − a_{1}X^{n−1} + ... + a_{2t}X − 1. This polynomial must be irreducible, otherwise there would exist some i > 1 such that σ_{1}(u) = σ_{i}(u) ∀u ∈ U. But this would imply σ_{1}(u) = 1 which is impossible. We have

σ_{1}(u) = \frac{1}{ρ(u)^{t}} ,

a_{1} = \frac{1}{ρ(u)^{t}} + ∑_{j=2}^{m}(σ_{j}(u) + σ̄_{j}(u)),

a_{2t} = ∑_{j=1}^{m} \frac{1}{σ_{j}(u)} = ρ(u)^{t} + \frac{∑_{j=2}^{m}(σ_{j}(u) + σ̄_{j}(u))}{ρ(u)} = ρ(u)^{t} + \frac{a_{1}}{ρ(u)} − \frac{1}{ρ(u)^{t+1}} .

Thus ρ(u) satisfies the following equation:

ρ(u)^{n} − a_{2t}ρ(u)^{t+1} + a_{1}ρ(u)^{t} − 1 = 0.

Since ℚ[σ_{1}(u)] ⊂ ℚ[ρ(u)] these field extensions must be equal, hence ρ(u) is a non-torsion unit in O_{K} having the same property as u, namely that its images through the complex embeddings of K have the same absolute value: ρ(u)^{−1/t} = σ_{1}(u)^{1/t^{2}}. But the same argument as before yields a new non-torsion unit ρ(u)^{−1/t} which for t > 1 satisfies the equation X^{n} − 1 = 0. This is a contradiction! □

References

[1] Borevich, A. I., Shafarevich, I. R.: Number theory, Academic Press, New York-London 1966.

[2] Dragomir S., Ornea L.: Locally conformal Kähler geometry, Progress in Mathematics, Boston, Birkhäuser 1998.

[3] Inoue, M.: On surfaces of class VII_{0}, Invent. Math. 24 (1974), 269-310.

[4] Weibel, C. A.: An introduction to homological algebra, Cambridge University Press, Cambridge 1994

Karl Oeljeklaus: LATP-UMR(CNRS) 6632, CMI-Université d’Aix-Marseille I, 39, rue Joliot-Curie, F-13453 Marseille Cedex 13, France.

  E-mail address: karloelj@cmi.univ-mrs.fr

Matei Toma: Institute of Mathematics of the Romanian Academy and Max-Planck-Institut für Mathematik, Postfach 7280, 53072 Bonn, Germany.

E-mail address: matei@mpim-bonn.mpg.de