L Battisti

K Oeljeklaus

HOLOMORPHIC LINE BUNDLES OVER DOMAINS IN COUSIN GROUPS AND THE ALGEBRAIC DIMENSION OF OT-MANIFOLDS

Keywords: AMS CLASSIFICATION 2010: 32J18, 32Q57, 32J99

In this paper we extend results due to Vogt on line bundles over Cousin groups to the case of domains stable by the maximal compact subgroup. This is used in the sequel to show that the algebraic dimension of OT-manifolds is zero. In the last part we establish that certain Cousin groups, in particular those arising from the construction of OT-manifolds, have finite-dimensional irregularity.

Introduction

A connected complex Lie group which admits no non-constant holomorphic functions is called a Cousin group, or also a toroidal group in the older literature. These groups are named after P. Cousin (see [START_REF] Cousin | Sur les fonctions triplement périodiques de deux variables[END_REF]). If C is an ndimensional Cousin group, it is abelian and can be realized as the quotient of C n by a lattice Λ of C n of rank n + m, with 1 ≤ m ≤ n.

It is a fact that the line bundle associated to a positive divisor on a compact Kähler manifold has a non-trivial real Chern class. In [10], C. Vogt proved the following in the non-compact context: Let L be a topologically trivial line bundle on a Cousin group. Then L is holomorphically trivial if and only if it admits a non-trivial holomorphic section. This result had already been proved by Cousin [START_REF] Cousin | Sur les fonctions triplement périodiques de deux variables[END_REF], in the special case m = 1. His original proof was rewritten by A.T. Huckleberry and G.A. Margulis in [START_REF] Huckleberry | Invariant analytic hypersurfaces[END_REF], who used it to establish a theorem on hypersurfaces in quotients of semi-simple complex Lie groups, see also [START_REF] Berteloot | Invariant plurisubharmonic functions and hypersurfaces on semisimple complex Lie groups[END_REF], [START_REF] Gilligan | Homogeneous Kähler and Hamiltonian manifolds[END_REF].

In the first section of the present paper we generalize Vogt's result to the case of a domain U in a Cousin group C = C n /Λ whose inverse image in C n is convex. This assumption implies in particular that U is invariant under the action of the maximal compact subgroup of C.

In the next section we use the previous result to confirm the non-existence of complex hypersurfaces in OT-manifolds. In particular, the algebraic dimension of these manifolds is always zero. This result was proved in a special case by L. Ornea and M. Verbitsky in [8]. OT-manifolds were introduced in [START_REF] Oeljeklaus | Non-Kähler compact complex manifolds associated to number fields[END_REF] and we will recall their construction later.

The last section is devoted to the observation that if the coordinates of the lattice points of a Cousin C group are all algebraic integers then the irregularity dim C H 1 (C, O) is finite. This article is organized as follows. In the first section we recall the notations of [9] and [10], and prove the following: Theorem 2.9. Let U be an open subset of a Cousin group C ∼ = C n /Λ whose inverse image ‹ U in C n is a convex domain. Let L be a topologically trivial holomorphic line bundle over U . One has H 0 (U, L) = 0 if and only if L is holomorphically trivial.

In the second section, we first recall the construction of OT-manifolds and then prove: Theorem 3.5. Let X be an OT-manifold. Then there are no complexanalytic hypersurfaces on X, in particular the algebraic dimension of X is zero.

Before stating the result of the last section we recall: 1. The space

H 1 (C, O) is finite-dimensional.
2. Let P = (I n S) be a period basis of Λ. Then there exist constants C > 0 and a 0 such that t σS + t τ C exp(-a|σ|) for all σ ∈ Z n \ {0} and all τ ∈ Z m , where n + m is the rank of Λ. 3. Every line bundle over C comes from a theta factor.

Applying a generalization of Liouville's theorem we obtain the following which applies in particular to Cousin groups arising in the construction of OT-manifolds: Theorem 4.3. Let Λ ⊂ C n be a lattice such that C = C n /Λ is a Cousin group with a period basis whose coefficients are all algebraic numbers, then C satisfies the equivalent conditions of theorem 4.1.

Preliminaries

Let us consider a domain U of a Cousin group

C ∼ = C n /Λ whose inverse image ‹ U = π -1 (U ) in C n is a convex domain, where π : C n → C n /Λ is the quotient map.
In particular, U is invariant under the action of RΛ/Λ, admits no non-constant holomorphic functions and ‹ U is invariant under the action of RΛ and is evidently Stein. We recall notations, definitions and results from [9] which adapt directly to our situation. Definition 2.1. A map α : Λ× ‹ U → C * is called a factor of automorphy if it satisfies the following properties:

a) α λ : ‹ U → C * , α λ (z) := α(λ, z) is holomorphic for all λ ∈ Λ, b) α(0, z) = 1 for all z ∈ ‹ U , c) α(λ + λ ′ , z) = α(λ, z + λ ′ )α(λ ′ , z) for all λ, λ ′ ∈ Λ and all z ∈ ‹ U .
If α is a factor of automorphy, for every λ ∈ Λ there exists a holomorphic function a λ : ‹ U → C unique up to an additive constant 2iπk λ with k λ ∈ Z, such that α(λ, z) = exp(a λ (z)). Definition 2.2. A map a : Λ × ‹ U → C is called a summand of automorphy if the following three conditions are satisfied: a) a(λ, where T = (I m S) is the period basis of a complex torus of dimension m and R is a real matrix; a lattice Λ defined by such a matrix gives a Cousin group if and only if t σR ∈ Z 2m for all σ ∈ Z n-m \ {0}.

• ) : ‹ U → C is holomorphic for all λ ∈ Λ, b) a(0, z) = 0 for all z ∈ ‹ U , c) a(λ + λ ′ , z) = a(λ, z + λ ′ ) + a(λ ′ , z) for all λ, λ ′ ∈ Λ

2.1.

Vogt's results generalized. Given a factor of automorphy on ‹ U , one gets a line bundle L on U by proposition 2.4. The following proposition characterizes the topologically trivial bundles obtained that way. Its proof is the same as in the case of a Cousin group, see for instance [START_REF] Abe | Line bundles, cohomology and quasi-abelian varieties[END_REF], Lemma 2.1.9. Proposition 2.6. Let L p → U be a holomorphic line bundle given by a factor of automorphy α : Λ × ‹ U → C * . Then the following two conditions are equivalent: a) the bundle L is topologically trivial, b) there is a summand of automorphy a : Λ× ‹ U → C such that α(λ, z) = exp(a(λ, z)).

The previous proposition now gives a first normal form in the class of a summand of automorphy: Proposition 2.7. Let Λ ⊂ C n be a lattice whose period basis is given in the form (I n S). Then every summand of automorphy b :

Λ × ‹ U → C is equivalent to a summand of automorphy a : Λ × ‹ U → C with the following properties: a) a(σ, z) = 0 for all σ ∈ Z n and all z ∈ ‹ U , b) for all λ ∈ Λ, the holomorphic function a λ : ‹ U -→ C z -→ a(λ, z) is Z n -periodic.
Proof: The projection π : One also has a second normal form for summands of automorphy: Proposition 2.8. Let Λ ⊂ C n be a lattice whose basis is given under the form

C n → C factors through C n /Z n : C n π 2 -→ C n /Z n π 1 -→ C, with π = π 1 • π 2 . The image of ‹ U by π 2 in C n /Z n ∼ = (C * ) n is
P = Ç 0 T I n-m R å , such that C = C n /Λ is a Cousin group. Then every summand of automor- phy b : Λ× ‹ U → C is equivalent to a summand of automorphy a : Λ× ‹ U → C with the following properties: a) a(λ, z) = a(λ, z m+1 , ..., z n ) for all λ ∈ Λ and all z ∈ ‹ U , b) a(λ, z) = 0 for all λ ∈ Ç 0 Z n-m å and all z ∈ ‹ U , c) for all λ ∈ Λ, the holomorphic function a λ : ‹ U -→ C z -→ a(λ, z) is Ç 0 Z n-m å -periodic.
The proof given in [9] readily adapts to our case so we will not repeat it here. Now, we state the main result of this section:

Theorem 2.9. Let U be an open subset of a Cousin group

C ∼ = C n /Λ whose inverse image ‹ U in C n is a convex domain.
Let L be a topologically trivial holomorphic line bundle over U . One has H 0 (U, L) = 0 if and only if L is holomorphically trivial.

Proof: The sufficient condition is clear so we prove the necessary condition. We choose the second normal form of the lattice of proposition 2.5. Since L is topologically trivial, proposition 2.6 implies that it is given by a summand of automorphy a : Λ × ‹ U → C. By using the normal form of proposition 2.8 we may assume that:

• a(λ, z) = a(λ, z m+1 , ..., z n ) for all λ ∈ Λ and

• a λ : ‹ U → C, z → a(λ, z) is Ç 0 Z n-m å -periodic for all λ.
Now consider the Fourier series of a λ :

a λ (z) = σ∈Z n-m a λ,σ e 2iπ t σ(z m+1 ,...,zn) .
Since a is a summand of automorphy, we have:

a λ (z + λ ′ ) + a λ ′ (z) = a λ ′ (z + λ) + a λ (z) hence: a λ,σ (1 -e 2iπ t σλ ′ 2 ) = a λ ′ ,σ (1 -e 2iπ t σλ 2 ) for λ, λ ′ ∈ Λ (where λ 2 [resp. λ ′ 2 ]
is the (n-m)-tuple consisting of the n-m last coordinates of λ [resp. λ ′ ]) and z ∈ ‹ U . We want to extend the map a to RΛ × ‹ U such that

a(x + x ′ , z) = a(x, z + x ′ ) + a(x ′ , z) for all x, x ′ ∈ RΛ, z ∈ C n . (1)
First, we define a(x, z) for an element x of RΛ of the form x = rλ by:

a(x, z) := σ∈Z n-m a λ,σ C(σ, r, λ)e 2iπ t σ(z m+1 ,...,zn) ,
where

C(σ, r, λ) =              1 -e 2iπr t σλ 2 1 -e 2iπ t σλ 2 if t σλ 2 ∈ Z lim u→0 1 -e iru 1 -e iu = r if t σλ 2 ∈ Z.
The Fourier series defining a(x, z) is convergent since for any fixed real r, the factors C(σ, r, λ) are uniformly bounded by r. Now for an element of RΛ of the form x = n+m j=1 r j λ j we define the differentiable function a(x, z) everywhere on RΛ × ‹ U using equation [START_REF] Abe | Line bundles, cohomology and quasi-abelian varieties[END_REF].

Consider a section s of L. We shall prove that if s vanishes at one point, then it is constant equal to zero. This section corresponds to a holomorphic function f

: ‹ U → C, satisfying f (z + λ) = e a(λ,z) f (z) (2)
for all λ ∈ Λ and all z ∈ ‹ U . Let z 0 ∈ ‹ U , consider the function:

t z 0 : RΛ -→ C x -→ f (z 0 + x)e -a(x,z 0 ) .
Then, for all λ ∈ Λ and all x ∈ RΛ, one has:

t z 0 (x + λ) = f (z 0 + x + λ)e -a(x+λ,z 0 ) = f (z 0 + x)e a(λ,z 0 +x)-a(x+λ,z 0 ) = f (z 0 + x)e -a(x,z 0 ) = t z 0 (x).
Hence t z 0 is a Λ-periodic function and therefore bounded by a constant C z 0 . This implies that for all x ∈ RΛ,

|f (z 0 + x)| C z 0 |e a(x,z 0 ) | = C z 0 e ℜa(x,z 0 ) . (3)
Let C m Λ be the maximal complex subspace of RΛ. We need that the function x → ℜ(a(x, z 0 )) is bounded. In order to satisfy this, we will slightly modify the map a. There exists a unique linear map g z 0 : C m Λ → C such that ℜ(g z 0 (λ 1 )) = ℜ(a(λ 1 , z 0 )) for all λ ∈ Λ.

We then consider the summand of automorphy a z 0 : Λ × ‹ U → C defined by: a z 0 (λ, z) = a(λ, z) -g z 0 (z + λ) + g z 0 (z) = a(λ, z) -g z 0 (λ), it defines the same line bundle as a. Hence, up to exchanging a with a z 0 , one can now assume that ℜ(a(λ 1 , z 0 )) = 0 for all λ ∈ Λ. For x ∈ C m Λ and λ ∈ P , one has:

a(x + λ 1 , z 0 ) = a(x, z 0 + λ 1 ) + a(λ 1 , z 0 ) = a(x, z 0 ) + a(λ 1 , z 0 )
(since by assumption on a, in its second argument it only depends on the last n -m coordinates), so the restriction of ℜa to

C m Λ × {z 0 } is ‹ T .Z 2m - periodic therefore bounded. Consequently, the restriction of f (z 0 + • ) to C m
Λ is a bounded holomorphic function, so it is constant and this holds for all z 0 ∈ ‹ U . This lets us see that f does not depend on the first m coordinates; we can see f as a holomorphic function over

‹ U 2 = {z 2 | z ∈ ‹ U }. The zeros of f are (I n-m R)-invariant, because of condition (2). Since C n /Λ is a Cousin group, there is no σ ∈ Z n-m \ {0} with t σR ∈ Z 2m so the group gen- erated by (I n-m R) is dense in R n-m . It is a consequence that if f (w) = 0 for some w ∈ ‹ U 2
, then f vanishes on w + R n-m and also on the intersection of ‹ U 2 and the complex linear subspace C n-m generated by w + R n-m , i.e.

‹ U 2 ∩ C n-m = ‹ U 2 .
Finally, if s is a non-trivial section of L, it never vanishes and L is holomorphically trivial. Now we give a reformulation of the previous proposition which will be used in the next section: Proof: Assume that L is topologically trivial. By theorem 2.9, since it admits a (non trivial) section s 1 given by D, it is holomorphically trivial. This means that it has a holomorphic section s 2 which never vanishes. The quotient s 1 /s 2 is a non-constant holomorphic function on U (since s 1 vanishes on D and not on the complement of D), a contradiction. One can also notice that s 1 vanishes on D so it is identically zero by the proof of the previous theorem, which leads to a contradiction too.

Hypersurfaces on OT-manifolds

The goal of this section is to prove that the algebraic dimension of an OTmanifold is zero. We first briefly recall the construction of OT-manifolds.

3.1. Construction of OT-manifolds. Let K be a number field of degree n over Q; call s (resp. 2t) the number of real (resp. complex) embeddings of K, so that n = s + 2t. For the construction, one needs to assume that the integers s and t are non-zero.

Then σ(O K ) is a lattice of rank n in C m (where we set m := s + t) hence we have a properly discontinuous action of σ(O K ) on C m . We also define By Dirichlet's units theorem, l(O * ,+ K ) is a lattice in the vector space

l : O * ,+ K → R m
L := {x ∈ R m | m i=1 x i = 0}.
The projection pr : L → R s given by the s first coordinates is surjective so there are subgroups A of rank s of O * ,+ K such that pr(l(A)) is a lattice of rank s of R s . Such an A is called admissible.

We then look at the quotient X := X(K, A) = (H s × C t )/(A ⋉ O K ); it is a complex compact manifold of dimension m. This manifold is called an Oeljeklaus-Toma manifold, or OT-manifold.

We know ([7], lemma 2.4) that C m /σ(O K ) admits no non-constant holomorphic function. In other words, the complex Lie group C := C m /σ(O K ) is a Cousin group.

We recall two lemmas and a definition (see [START_REF] Oeljeklaus | Non-Kähler compact complex manifolds associated to number fields[END_REF]) for further use: Lemma 3.1. Let A be a subgroup of O * ,+ K which is not contained in Z. Then the two following conditions are equivalent:

1. The action of A on O K admits a (non-trivial) proper invariant submodule of lower rank.

There exists a proper intermediate field extension

Q ⊂ K ′ ⊂ K with A ⊂ O * ,+ K ′ . Definition 3.2.
We say that X(K, A) is of simple type if A does not satisfy one of the equivalent conditions of the previous lemma.

Lemma 3.3. Let Q ⊂ K ′ ⊂ K an intermediate field extension with A ⊂ O * ,+
K ′ an admissible subgroup for K. Let s ′ , 2t ′ be the number of real and complex embeddings of K ′ respectively. Then s = s ′ , t ′ > 0 and A is admissible for K ′ .

For more details, see [START_REF] Oeljeklaus | Non-Kähler compact complex manifolds associated to number fields[END_REF].

Algebraic dimension.

We shall first consider the case of an OTmanifold of simple type. Proposition 3.4. Let X be an OT-manifold of simple type. Then it admits no irreducible divisor; in particular, the algebraic dimension of X is zero.

Proof: Assume that D is an irreducible divisor of X of multiplicity 1. We consider X = X(K, A) as the quotient of U := (H s × C t )/σ(O K ) (which is diffeomorphic to (R >0 ) s × (S 1 ) n ) by A ∼ = Z s . One has the following commutative diagram:

U A ∼ = Z s p / / (S 1 ) s+2t q X (S 1 ) s+2t q ′ (R >0 ) s Z s p ′ / / (S 1 ) s (4) Because X is of simple type, one has dim H 2 (X) = s 2 = dim H 2 ((S 1 ) s )
by proposition 2.3 of [START_REF] Oeljeklaus | Non-Kähler compact complex manifolds associated to number fields[END_REF]. Thus the first Chern class c 1 (L(D)) of the line bundle L(D) over X associated to D is the pull-back by q ′ of an element ω ∈ H 2 ((S 1 ) s , Z). The fact that the map (q ′ ) * : H 2 ((S 1 ) s , Z) → H 2 (X, Z) is injective is established in the proof of proposition 2.3 of [START_REF] Oeljeklaus | Non-Kähler compact complex manifolds associated to number fields[END_REF]. The commutativity of diagram (4) implies that the pull-back ω over U of ω by q ′ • p is the same as the pull-back by p ′ • q of ω. Furthermore, one has (p ′ ) * (ω) = 0 since H 2 ((R >0 ) s , Z) = 0. Hence ω = 0. This is a contradiction to lemma 2.10, since ω = c 1 ( ‹ L(D)) where ‹ L(D) is the bundle associated to the divisor p -1 (D) of U . Now we can omit the hypothesis that X is of simple type: Theorem 3.5. Let X be an OT-manifold. Then there are no complexanalytic hypersurfaces on X, in particular the algebraic dimension of X is zero.

Proof: By proposition 3.4, we can assume that X is not of simple type. Consequently there exists a field extension Q ⊂ K ′ ⊂ K with A ⊂ O * ,+ K ′ and we call s ′ (respectively, 2t ′ ) the number of real (respectively, complex) embeddings of K ′ . Since A is admissible for K, we have s = s ′ and A is admissible for K ′ (see lemma 3.3). Without loss of generality, we can suppose that K ′ is the smallest subfield of K such that A ⊂ O * ,+ K ′ , i.e. X(K ′ , A) is of simple type. Call σ 1 , ..., σ s , σ s+1 , ..., σ s+t , σs+1 , ..., σs+t the s + 2t embeddings of K and σ

′ 1 , ..., σ ′ s , σ ′ s+1 , ..., σ ′ s+t ′ , σ′ s+1 , ..., σ′ s+t ′ the s + 2t ′ embeddings of K ′ .
As before, we note

σ : O K -→ C s+t a -→ (σ 1 (a), ..., σ s+t (a)).
We look at the complex linear subspace

V K ′ of C s+t spanned by σ(O K ′ ). We denote by C (respectively, C ′ ) the Cousin group C s+t /σ(O K ) (respec- tively, V K ′ /σ(O K ′ )). The group C ′ is a closed complex Lie subgroup of C. We study the two open sets U := (H s × C t )/σ(O K ) ⊂ C and U ′ := ((H s × C t ) ∩ V K ′ )/σ(O K ′ ) ⊂ C ′ .
The quotients U/A and U ′ /A are OTmanifolds that we call X and X ′ respectively; moreover, X ′ is a compact complex submanifold of X.

We consider the following commutative diagram:

0 / / C ′ / / C q / / / / C/C ′ / / 0 U ′ / / ? O O p ′ U ? O O p / / q(U ) ? O O X ′ i / / X (5)
Let D be a irreducible divisor of multiplicity 1 of X and let L(D) be the associated holomorphic line bundle. By restriction we get a bundle

L ′ := i * L(D) over X ′ .
Since X ′ is of simple type, the line bundle (p ′ ) * (L ′ ) over U ′ is topologically trivial, see the proof of proposition 3.4. The maximal complex subgroup The inverse image D := p -1 (D) is a divisor of U . We shall now prove that the divisor D is saturated by the leaves of the U ′ -foliation of U . We have

H ′ ∼ = C t ′ of the maximal compact torus T ′ ∼ = (S 1 ) s+2t ′ of C ′ is a subgroup of the maximal complex subgroup H ∼ = C t of the maximal compact torus T ∼ = (S 1 ) s+2t of C. We denote by B ∼ = C t-t ′ a connected complex subgroup of T such that B × H ′ ∼ = H. Since B is
c 1 (L( D)) = p * (c 1 (L(D))) ∈ H 2 (U, Z). (6) Since the diagram (5) commutes, we have c 1 (L( D)| U ′ ) = c 1 (L( D))| U ′ = c 1 (p * L ′ ) = 0 = c 1 (b * L( D))| U ′ .
There are three possible cases for the intersection b( D) ∩ U ′ . Either this intersection is U ′ , or it is empty (these two cases fit our purpose), or it is a divisor of U ′ . The last case can not occur since c 1 (b * L( D))| U ′ would be non-zero by lemma 2.10, which is a contradiction to [START_REF] Huckleberry | Invariant analytic hypersurfaces[END_REF]. Hence D is saturated by the leaves of the U ′ -foliation of U . Now, D induces a divisor q( D) of q(U ). One has q(U ) = C/C ′ because the leaves of the U ′ -foliation of U are in bijection with those of the C ′foliation of C. Hence, q( D) lifts as a divisor in C, which is stable under the action of the group A ⋉ C ′ . We still denote by D this divisor.

Since the group O * ,+ K is abelian, the action of O * ,+ K on U induces an action of O * ,+ K on X. Let η ∈ O * ,+ K , we have ηD = η D. We use the C ′ -invariance of ηD to write η D+C ′ = η D hence D is both C ′ -invariant and η -1 C ′ -invariant.

Let J be the smallest connected complex subgroup of C containing C ′ and ηC ′ for all η ∈ O * ,+ K . If J is a proper subgroup of C, its maximal compact subgroup is defined by a sublattice of O K which is stable under the action of O * ,+ K . This is impossible because there is no sublattice of O K stable by O * ,+ K . Indeed, if it were the case, since O * ,+ K contains a primitive element of K, this element should have a minimal polynomial with degree strictly smaller than the one of K and this is impossible. Finally, J = C and D is invariant by C, which is a contradiction. Hence, X admits no divisor and its algebraic dimension is zero. 1. The space H 1 (C, O) is finite-dimensional.

2. Let P = (I n S) be a period basis of Λ. Then there exist constants C > 0 and a 0 such that t σS + t τ C exp(-a|σ|) for all σ ∈ Z n \ {0} and all τ ∈ Z m , where n + m is the rank of Λ. 3. Every line bundle over C comes from a theta factor.

Let α be a real irrational algebraic number. In his paper, Vogt gives the following example of a lattice in C 2 given by P = Ç 0 1 i

1
P (z 1 , ..., z m ) = N 1 k 1 =0 • • • Nm km=0 a k 1 ,...,km z k 1 1 • • • z km m ∈ Z[z 1 , ..., z m ].
If P (α 1 , ..., α m ) is non-zero, then one has the inequality

|P (α 1 , ..., α m )| L(P ) 1-δn m k=1 L(α k ) -δN k n/n k ,
with δ = 1 if all the α i are real, δ = 1/2 otherwise and where L(P ) is the sum of the absolute values of the coefficients of P (and L(α) is the quantity L(µ), with µ being the minimal polynomial of α).

4.2.

A class of Cousin groups satisfying condition n • 2. We shall show:

Theorem 4.3. Let Λ ⊂ C n be a lattice such that C = C n /Λ is a Cousin group with a period basis whose coefficients are all algebraic numbers, then C satisfies the equivalent conditions of theorem 4.1.

Proof: Let us write a period basis of Λ as a matrix

(I n S)
where I n is the identity matrix of size n and S = (s i,j ) is a matrix with n rows and m columns whose coefficients are algebraic numbers over Q.

In order the check that the second condition of theorem 4.1 is satisfied, we have to verify that there exist constants C > 0 and a 0 such that for all µ ∈ Z n \ {0} and all ν ∈ Z m , one has t µS + t ν C exp(-a|µ|).

It is enough to prove this inequality for any non-zero coordinate of the vector t µS + t ν and then consider the infinity norm of this vector. Denote t µ = (µ 1 , ..., µ n ) and t ν = (ν 1 , ..., ν m ). Then the k-th coordinate of the vector t µS + t ν (for k ∈ {1, ..., m}) is

µ 1 s 1,k + • • • + µ n s n,k + ν k .
We now distinguish the two following cases: either we have the inequality

|µ 1 s 1,k + • • • + µ n s n,k | C ′′ (|µ 1 | + • • • + |µ n |) q C ′′ exp(-|q|(|µ 1 | + • • • |µ n |))
where C ′′ > 0 and q < 0 are constants which do not depend on µ and ν.

Application

We recall that in the construction of an OT-manifold, the group C := C m /σ(O K ) is a Cousin group. As a corollary of theorem 4.

Theorem 4 . 1 (

 41 [10]). Let C = C n /Λ be a Cousin group. Then the following conditions are equivalent:

  Stein by Grauert-Docquier's theorem. Let L be the line bundle defined by the summand of automorphy b. By the previous proposition, L is topologically trivial, so is π * 1 L. Because every topologically trivial line bundle over a Stein manifold is holomorphically trivial, one obtains a trivialization of π * L by taking the pull-back of a trivialization of π * 1 L. The summand of automorphy a : Λ × ‹ U → C given by this trivialization is equivalent to b in the sense of definition 2.3 and it satisfies condition a). Condition b) is a consequence of both a) and the definition of a summand of automorphy.

Lemma 2 .

 2 10. Let U be an open set of a Cousin group C ∼ = C n /Λ whose inverse image ‹U in C n is a convex domain and let D be a positive divisor of U . Let L be the line bundle over U associated to D. Then L is not topologically trivial, i.e. its first Chern class c 1 (L) ∈ H 2 (U, Z) is not zero.

  by l(a) = (log |σ 1 (a)|, ..., log |σ s (a)|, 2 log |σ s+1 (a)|, ..., 2 log |σ m (a)|).

  a subgroup of T it acts on U and this action is transitive on the leaves of the U ′ -foliation of U induced by the C ′ -foliation of C. The induced action of B on H 2 (U, Z) is trivial since H 2 (U, Z) is discrete and B is connected, i.e. we have b * c 1 (L) = c 1 (L) for every line bundle L above U and all b ∈ B.

4 . 1 : 4 . 1 .

 4141 A special class of Cousin groups 4.1. Introduction. In [9], Vogt exhibits a special class of Cousin groups by prooving Theorem Let C = C n /Λ be a Cousin group. Then the following conditions are equivalent:

  |ν k | 2|µ 1 s 1,k + • • • + µ n s n,k |, or its converse holds. Suppose first |ν k | 2|µ 1 s 1,k + • • • + µ n s n,k |. By theorem 4.2, we have |µ 1 s 1,k + • • • + µ n s n,k + ν k | C(|µ 1 | + • • • + |µ n | + |ν k |) p ,where C > 0 and p < 0 are constants independent of µ and ν. Recall here that we assumed thatµ 1 s 1,k + • • • + µ n s n,k + ν k is non-zero.By assumption on ν k , we have(|µ 1 | + • • • + |µ n | + |ν k |) p (|µ 1 | + • • • + |µ n | + 2|µ 1 s 1,k + • • • + µ n s n,k |) p C ′ (|µ 1 | + • • • + |µ n |) p C ′ exp(-|p|(|µ 1 | + • • • + |µ n |)),with C ′ independent of µ and ν.Now suppose that |ν k | > 2|µ 1 s 1,k + • • • + µ n s n,k |.We write the reverse triangle inequality:|µ 1 s 1,k + • • • + µ n s n,k + ν k | |ν k | -|µ 1 s 1,k + • • • + µ n s n,k | = |ν k | -|µ 1 s 1,k + • • • + µ n s n,k | |µ 1 s 1,k + • • • + µ n s n,k |. If µ 1 s 1,k + • • • + µ n s n,k vanishes, we have |ν k | 1 and the result is obtained; otherwise, we can again use theorem 4.2:

  and all z ∈ ‹ U .

	Definition 2.3. Two factors of automorphy α and β are called equivalent if there is a holomorphic function h : ‹ U → C * satisfying
	β(λ, z) = h(z + λ)α(λ, z)h -1 (z).
	Two summands of automorphy are called equivalent if the induced factors
	of automorphy are equivalent.
	Since ‹ U is Stein, one has the following proposition (see [9]):
	Proposition 2.4. There is a bijection between equivalence classes of factors
	of automorphy and equivalence classes of line bundles over U .
	We recall now two normal forms for the lattice Λ defining a Cousin group
	(see [9], propositions 1 and 2):
	Proposition 2.5. Let C = C n /Λ be a Cousin group. Then the real rank of
	Λ is r = n + m with 1 m n and:
	1. The Cousin group C has a period basis, i.e. a basis of Λ, of the form
	P = (I n S); a lattice Λ defined by such a matrix gives a Cousin group if and only if t σS ∈ Z m for all σ ∈ Z n \ {0}. Ç 0 T 2. The Cousin group C has a period basis of the form P = I n-m R

å

  α 0 Theorem 4.2. Let α 1 , ..., α m be algebraic numbers, of respective degrees n k , with deg Q(α 1 , ..., α m ) = n, and let

	å
	which defines
	a Cousin group satisfying condition n • 2. Here we give a general theorem
	in this setting which in particular applies to all Cousin groups appearing in
	the construction of OT-manifolds.
	We shall use the following generalization of Liouville's theorem which can
	be found in [4] (theorem 1.5, page 27):

  3, we have: Corollary 4.4. The Cousin group C satisfies condition n • 2 of theorem 4.1. 8. Liviu Ornea and Misha Verbitsky, Oeljeklaus-Toma manifolds admitting no complex subvarieties, Math. Res. Lett. 18 (2011), no. 4, 747-754. 9. C. Vogt, Line bundles on toroidal groups, J. Reine Angew. Math. 335 (1982), 197-215. 10., Two remarks concerning toroidal groups, Manuscripta Mathematica 41 (1983), no. 1-3, 217-232. .univ-mrs.fr, karloelj@cmi.univ-mrs.fr

	Laurent BATTISTI & Karl OELJEKLAUS
	AIX-MARSEILLE UNIVERSIT	É
	CNRS -LATP -UMR 7353
	F-13453 MARSEILLE
	France
	battisti@cmi

The theorem given in [9], p.

has 8 equivalent assertions, here we only recall three of them.


Laurent Battisti
email: laurent.battisti@yahoo.com

Karl Oeljeklaus

Karl Oeljeklaus Ruhr-Universit

Ä T Bochum

A GENERALIZATION OF SANKARAN AND LVMB MANIFOLDS

In this paper we describe the construction of a new class of non-Kähler compact complex manifolds. They can be seen as a generalization of Sankaran, OT and LVMB manifolds. Moreover, we give properties of these new spaces. Their Kodaira dimension is -∞ and under a mild condition they have algebraic dimension equal to zero.

Introduction

In this article we construct a new family of non-Kähler complex compact manifolds by the combination of a method due to Bosio ([5]) and another due to Sankaran ([12]). The class of manifolds constructed in this paper appears as a generalization of already known examples of non-Kähler manifolds, namely LVMB and Sankaran manifolds (ibid) along with OT manifolds ( [START_REF] Oeljeklaus | Non-Kähler compact complex manifolds associated to number fields[END_REF]).

Although the field of non-Kähler geometry remains relatively unexplored, there is nevertheless constant progress. New classes of non-Kähler compact complex manifolds have been constructed and studied recently.

The first example is given by the class of LVMB manifolds. Their construction is due to Bosio (see [START_REF] Bosio | Variétés complexes compactes: une généralisation de la construction de Meersseman et López de Medrano-Verjovsky[END_REF]) and can be summarized as follows. Given a family of subsets of {0, ..., n} all having 2m + 1 elements (where n and m are integers such that 2m n) and a family of n + 1 linear forms on C m satisfying technical conditions, one can find an open subset U of P n (C) and an action of a complex Lie group G ∼ = C m on P n (C), such that the quotient U/C m is a compact complex manifold. These manifolds generalize Hopf and Calabi-Eckmann manifolds and they also generalize a class of manifolds due to Meersseman ([8]), called LVM manifolds.

OT manifolds were constructed in [START_REF] Oeljeklaus | Non-Kähler compact complex manifolds associated to number fields[END_REF] by the second author and M. Toma. One starts by choosing an algebraic number field K having s > 0 (resp. 2t > 0) real (resp. complex) embeddings. Then, for a nice choice of a subgroup A of the groups of units O * K of K, the quotient X(K, A) of H s ×C t under the action of A ⋉ O K is a complex compact manifold. The required

The first author would like to thank Prof. A. T. Huckleberry and Prof. P. Heinzner for invitation as postdoc to Bochum's Ruhr-Universität. The second author is partially supported by the ANR project MNGNK, decision #ANR-10-BLAN-0118.

AMS CLASSIFICATION 2010: 32J18, 32Q57, 32J99 condition on A is that the projection on the s first coordinates of its image through the logarithmic representation of units

ℓ : O * K -→ R
is a full lattice in R s . Such an A is called admissible. When there exists no proper intermediate field extension Q ⊂ K ′ ⊂ K with A ⊂ O * K ′ , one says that the manifold X(K, A) is of simple type.

In [START_REF] Sankaran | A class of non-Kähler complex manifolds[END_REF], G. Sankaran studies the action of a discrete group W isomorphic to Z b over the open subset U of a toric manifold, chosen to obtain a compact quotient U/W . The group W is a subgroup of the group of units of a number field K and the construction of the infinite fan defining the toric manifold narrowly depends on W . This construction generalizes previous ones due to Inoue, Kato, Sankaran himself and Tsuchihashi. In the following, we will refer to the manifolds constructed in [START_REF] Sankaran | A class of non-Kähler complex manifolds[END_REF] as Sankaran manifolds.

Our construction combines the methods of Bosio and Sankaran. First, we take the quotient of a well-chosen n-dimensional toric manifold X ∆ under the action of a complex Lie subgroup G ∼ = C t of (C * ) n such that the quotient X of X ∆ by G is a (non-necessarily compact) manifold, then we choose a suitable open subset of X on which a discrete group W acts and gives a compact quotient. The group W is a subgroup of the group of units of a number field K having s > 0 real and 2t > 0 complex embeddings, with s + 2t = n. Of course, the choices of W and ∆ are strongly related, as in Sankaran's case. We obtain an (s + t)-dimensional non-Kähler complex compact manifold.

The paper is organized as follows: In section 1 we introduce all notations and results that will be needed in the sequel.

The next section is devoted to the description of the construction itself, which is subdivided into two steps. At the end of section 2, some remarks are made on the structure of the manifolds and we show how they generalize other known classes of manifolds.

In section 3 we provide results on invariants and geometric properties of the new manifolds. We will prove that their Kodaira dimension is -∞, that they are not Kähler and finally show that their algebraic dimension is zero under a mild technical assumption. Moreover, we compute the second Betti number of OT manifolds under the same assumption, a fact which was already known for OT manifolds of simple type.

We conclude the paper by giving a concrete example of a 3-dimensional manifold.

Preliminaries

Before starting the construction, we need to introduce or recall notations and results. Most of them can be found in Sankaran's paper ( [START_REF] Sankaran | A class of non-Kähler complex manifolds[END_REF]), others come from [START_REF] Oeljeklaus | Non-Kähler compact complex manifolds associated to number fields[END_REF] and [START_REF] Battisti | LVMB manifolds and quotients of toric varieties[END_REF]. Detailed proofs are only provided when modifications are needed to fit in with the current context.

The section is divided into four parts, as follows: First we settle the number-theoretic ground of our construction. Then, we study complex Lie groups which will arise later, in particular a certain Cousin group, i.e. a complex Lie group without non-constant holomorphic functions. Furthemore, we state some facts about manifolds with corners associated to a fan and finally construct a special fan.

1.1. Number-theoretic notations and results. Let K be a number field having n = s + 2t distinct embeddings in C and let σ 1 , ..., σ s be the s real and σ s+1 , ..., σ n the 2t complex (non real) ones. Up to re-ordering these embeddings, we can assume that the relation σ s+i = σs+t+i holds for 1 i t. We also require that both s and t strictly positive.

Define the map

σ K : K -→ R s × C 2t k -→ (σ 1 (k), ..., σ s (k), σ s+1 (k), ..., σ n (k)). Note O K the set of algebraic integers of K. The image σ K (O K ) is a lattice of rank n in R s × C 2t ⊂ C n . Let O * K be the group of units of O K . Since s 1, Dirichlet's Unit Theorem tells us that O * K is isomorphic to {±1} × Z s+t-1 . The group O * K acts on C n by componentwise multiplication: η • (z 1 , ..., z n ) := (σ 1 (η)z 1 , ..., σ n (η)z n ). For all η ∈ O * K , set η i := σ i (η) if i = 1, ..., s and η i := |σ i (η)| if i = s + 1, ..., s + t. Call O * ,+
K the set of units η ∈ O * K such that η i > 0 for all i = 1, ..., s. The following theorem is due to Sankaran. For the convenience of the reader, we give the proof. Theorem 1.1 ([12], Theorem 2.1). For all b ∈ {1, ..., s}, there exists a subgroup W < O * ,+ K of rank b satisfying the following condition, called "Assumption C"1 : for all η = 1 ∈ W , either there is an i b such that η i > η j for every j > b, or there is an i b such that η i < η j for every j > b.

Proof. Set g := s + tb and to every η ∈ O * ,+ K associate an element of R bg via the map:

ϕ b : O * ,+ K -→ R bg η -→         ln(η 1 /η b+1 ) • • • ln(η 1 /η s+t ) . . . ln(η i /η b+j ) . . . ln(η b /η b+1 ) • • • ln(η b /η s+t )        
.

For i = 1, ..., b, let π i,b be the projection

π i,b : R bg -→ R g (t i,j ) -→ (t i,j ) j=1,...,g
i.e. π i,b maps an element of R bg written in matrix form to its i-th row.

The fact that W satisfies Assumption C means that for all η = 1 ∈ W , there exists i ∈ {1, ..., b} such that

π i,b (η) ∈ Q b := {(x 1 , ..., x g ) ∈ R g | the x j 's are non-zero and of same sign}. Call U b the vector space ϕ b (O * ,+ K ) ⊗ R ∼ = R s+t-1 ⊂ R bg .
In order to prove the existence of a group W of rank b verifying Assumption C, it is enough to find a b-dimensional linear subspace A ⊂ U b that is generated by elements of ϕ b (O * ,+ K ) and such that for every x = 0 ∈ A there exists an i ∈ {1, ..., b} with π i,b (x) ∈ Q b . We proceed by induction.

For b = 1 the result is clear, as it is enough to take A to be any line passing through the origin and any point of Q 1 .

Assume the result is proved at rank b-1: there exists a (b-1)-dimensional linear subspace A ′ of U b-1 having the following property:

for all x ∈ A ′ , there exists i ∈ {1, ..., b -1} such that π i,b-1 (x) ∈ Q b-1 .
In this case, for all x ∈ A ′ , the integer i above satisfies:

π i,b (x) ∈ Q b . Now, fix a projection along A ′ , say π A ′ : U b → R g . There is a map L ∈ GL g (R) such that the following diagram commutes: U b π b,b Ð Ð π A ′ 0 0 R g ∼ = π b,b (U b ) L G G U b /A ′ ∼ = R g .

Now we consider

Q A ′ := L(Q b ). Take a line ℓ ⊂ Q A ′ ∪{0}, then the subspace A := π -1
A ′ (ℓ) fits our purpose. To see this, pick an x ∈ A. Either we have x ∈ A ′ and in this case there is an i ∈ {1, ..., b} such that

π i,b (x) ∈ Q b , or π A ′ (x) ∈ ℓ \ {0} ⊂ Q A ′ and π A ′ (x) = L(π b,b (x)), which means that π b,b (x) ∈ Q b .
1.2. A Cousin group. Consider now the linear subspace H := {(0, ..., 0, z s+t+1 , ..., z s+t+t ) | z s+t+1 , ..., z s+t+t ∈ C} ∼ = C t of C n and call π H the projection from C n to C s+t with respect to H, given by the s + t first coordinates.

Lemma 1.2. The restriction of π H to σ K (O K ) is injective. Proof. It is sufficient to check that H ∩ σ K (O K ) = {0}
, which is straightforward because the σ i 's are embeddings. Now let W be as in the previous theorem for some b ∈ {1, ..., s}. Notice that the action of W on C n (resp.

C n /H ∼ = C s+t ) induces an action on σ K (O K ) (resp. π H (σ K (O K ))).
Take an integral basis of K and call B K its image by σ K . This is a basis of C n over the complex numbers. Denote by E the vector space generated over R by the vectors of σ K (O K ), note B := (e 1 , ..., e n ) the canonical basis of C n and B ′ the basis (e 1 , ..., e s , e s+1 + e s+t+1 , -i(e s+1e s+t+1 ), ..., e s+t + e s+2t , -i(e s+te s+2t )).

In the latter basis, the matrix of an element η ∈ W is written as:

                  σ 1 (η) 0 . . . σ s (η) ℜ(σ s+t+1 (η)) -ℑ(σ s+t+1 (η)) ℑ(σ s+t+1 (η)) ℜ(σ s+t+1 (η)) . . . ℜ(σ n (η)) -ℑ(σ n (η)) 0 ℑ(σ n (η)) ℜ(σ n (η))                   .
Moreover, B ′ is an R-basis for the vector space E. To see this, it is enough to prove the following:

Lemma 1.3. Let P B K ,B ′ be the change-of-basis matrix from B K to B ′ . Then, all entries of P B K ,B ′ are real, in other words one has P B K ,B ′ ∈ GL n (R).
Proof. Notice that P B K ,B ′ = P B K ,B P B,B ′ and that the last 2t columns of P B K ,B are pairwise conjugated, meanwhile the s first columns are real. Hence, all the coefficients of the vectors P B K ,B e j for j ∈ {1, ..., s} are real. Now, denote by h 1 , ..., h t , h 1 , ..., h t the last 2t columns of P B K ,B and observe that P B K ,B (e s+j + e s+t+j ) = 2ℜ(h j ) and P B K ,B (-i(e s+je s+t+j )) = -2ℑ(h j ) for j = 1, ..., t. The lemma is thus proven.

Denote by H the vector space generated over R by the last 2t vectors of B ′ . This is a 2t-dimensional linear subspace of E and we note π H : E → E/ H the quotient map.

Remark 1.4. The shape of the matrix of an element η ∈ W in the basis B ′ shows that such an element induces a linear map from the real vector space E ∼ = R n to itself and that this linear map descends to the quotient E/ H ∼ = R s , where it is given by η

: (x 1 , ..., x s ) → (σ 1 (η)x 1 , ..., σ s (η)x s ). Furthermore, one has H ∩ σ K (O K ) = {0}.
Now we describe the linear subspace H in the basis B K ; it is generated by the t vectors (h 1 , ..., h t ), where h i is the vector e s+t+i of the canonical basis written in the basis B K (according to notations of lemma 1.3).

The group

H is a closed Lie subgroup of C n /σ K (O K ) ∼ = (C * ) n , via the following map ι: H -→ (C * ) n             0 . . . 0 z s+t+1 . . . z s+t+t             -→             exp(2iπ t i=1 h i,1 z s+t+i ) . . . exp(2iπ t i=1 h i,n z s+t+i )             , (2)
where h i,j is the j-th coordinate of the vector h i . We still denote by H the image of H under this map. We shall see that the group (C n /σ K (O K ))/H only has trivial holomorphic functions.

A connected complex Lie group admitting no non-constant holomorphic functions is called a Cousin group, or a toroidal group. One has:

Corollary 1.5. The quotient of the complex Lie group C n /σ K (O K ) ∼ = (C * ) n by H is a Cousin group, which we call C 0 . We denote by p the quotient map p : C n /σ K (O K ) → C 0 . Proof. By lemma 1.2 above, the quotient of C n /σ K (O K ) by H is isomorphic to the quotient of C s+t = C n /H by π H (σ K (O K )), which is a Cousin group by lemma 2.4 of [9]. Lemma 1.6. The subgroups H and (S 1 ) n of C n /σ K (O K ) intersect trivially.
Proof. An element of this intersection corresponds to an element of the linear subspace H satisfying the following equation:

P B,B K             x 1 . . . x n             =             0 . . . 0 z s+t+1 . . . z s+t+t            
, where P B,B K is the change-of-basis matrix from the canonical basis to B K and the x i 's are real. The i-th and (i+t)-th components of every vector of the basis B K are conjugated (for i = s+1, ..., s+t). Since the x i 's are real numbers, the i-th and (i + t)-th components of the vector (0, ..., 0, z s+t+1 , ..., z s+t+t ) are also conjugated, hence z s+t+1 = ... = z s+t+t = 0.

Remark 1.7. The quotient of (C * ) n by (S 1 ) n is given by the map ord :

(C * ) n -→ R n (z 1 , ..., z n ) -→ (-ln |z 1 |, ..., -ln |z n |).
With equation ( 2), we see that the space ord(H) ∼ = R 2t is generated by the 2t vectors ℜ(h j ) and ℑ(h j ) (for j ∈ {1, ..., t}), in other words, ord(H) = H. 1.3. Manifolds with corners. In [START_REF] Battisti | LVMB manifolds and quotients of toric varieties[END_REF], one can find the definition of the manifold with corners of a (non-necessarily rational) fan ∆ of R n , denoted by Mc(∆). Heuristically, this is a partial compactification of R n obtained the following way: for every cone σ in ∆ one sends a complementary subspace of span(σ) "at infinity" in the direction of σ. One also defines a topology on this space and when ∆ is rational, this space is (homeomorphic to) the manifold with corners of ∆ in the usual toric-geometrical sense, i.e. X ∆ /(S 1 ) n . This construction was inspired by the first chapter of [START_REF] Ash | Smooth compactifications of locally symmetric varieties[END_REF].

We need the following two lemmas:

Lemma 1.8 ([2]
, lemma 1.17). Let σ be a cone of a fan ∆. Then the closure

S of σ ⊂ Mc(∆) in Mc(∆) is compact. Lemma 1.9 ([2], lemma 2.3). Let ∆ be a fan in R n and E ∼ = R k be a linear subspace of R n .

The action of E on Mc(∆) is proper if and only if the restriction of the quotient map

π : R n → R n /E to the support |∆| of ∆ is injective.
We state the following definition and proposition, the proof of which is straightforward: Definition 1.10. Let ∆ be a (non-necessarily rational) fan of R n and G a discrete group of GL n (R). We say that G acts on ∆ if for every cone σ ∈ ∆ and every g ∈ G, one has g(σ) ∈ ∆. This action is called free if g(σ) = σ when g is not the identity of G and σ = {0} is a cone of ∆. This action is called properly discontinuous if for every σ ∈ ∆, the set 1.4. A suitable fan. In the vector space E, we construct a rational fan ∆ with respect to the lattice O K ⊂ E. First this fan has to be W -invariant. Furthermore, the image of its support under the projecting map π H must be included in an open degenerate proper cone Ω ⊂ π H (E) ∼ = R s invariant under the action of W . Degenerate means that the closure of the cone Ω contains a non-trivial linear subspace of π H (E). If these conditions are satisfied, then there is an action of W on the toric manifold associated to this fan.

{g ∈ G | (g(σ) ∩ σ) \ {0} = ∅} is finite.
The projection π H is injective on σ K (O K ) and W -equivariant. Assume there is a fan ∆ ′ in π H (E) that is generated by elements of π H (σ K (O K )) and whose support is such a cone Ω. Take its pullback via the map π H . This is the desired fan ∆.

To construct the fan ∆ ′ in π H (E), we follow the steps of Sankaran in [START_REF] Sankaran | A class of non-Kähler complex manifolds[END_REF] (theorem 2.5). Again, we only give proofs of results when they need modifications, all the others adapt readily.

Let Ω be an open degenerate cone in π H (E) invariant under W . We start by describing the maximal vector subspace contained in its closure: Lemma 1.13 ([12], lemma 1.2). The vector space N of maximal dimension contained in the closure of Ω is of the form

N = {(x 1 , ..., x s ) ∈ π H (E) | x i 1 = • • • = x i s-h = 0}, with h > 0.
Remark 1.14. Up to renumbering the coordinates of π H (E) and the σ i 's simultaneously, we may assume that N is the set

{(x 1 , ..., x s ) ∈ π H (E) | x 1 = • • • = x s-h = 0}.
This allows us to describe Ω explicitly:

Lemma 1.15 ([12], lemma 1.3). One has Ω = N × L + where L = {x ∈ π H (E) | x s-h+1 = ... = x s = 0} and L + = {x ∈ L | ± x i > 0, i ∈ {1, ..., s -h}}.
Up to composition with a suitable element of O * K , we may assume that L + is the set {x ∈ L | x i > 0, i ∈ {1, ..., s -h}}. Now, we take an integer h ∈ {1, ..., s -1} and consider the expression of Ω given by lemma 1.15. To ensure that L + /W is a compact manifold, we choose the rank b of W to be equal to the dimension s-h of L. Remark 1.18. Let's recall the idea of Sankaran's proof: there is a real number δ > 0 such that

C ⊂ C δ := {v ∈ Ω | b i=1 v 2 i δ j>b v 2 j }. (3)
It is hence enough to establish the result for C δ . This can be done by observing that there exist N > 1 and a ∈ N such that η k C δ ⊂ C N k δ for k > a. A crucial remark for the next part of the construction is that the constants a and N can be chosen independently of η ∈ W . For further details, see the proof of theorem 2.4 in [START_REF] Sankaran | A class of non-Kähler complex manifolds[END_REF]. Assumption C and the fact that b < s are used in this proof.

Theorem 1.19 ([12], theorem 2.5). There exists an infinite fan ∆ ′ of R s stable under the action of W , whose support

|∆ ′ | is (Ω \ L + ) ∪ {0}
, such that all its cones are generated by elements of π H (O K ) and ∆ ′ /W is a finite set of cones.

Remark 1.20. In particular, ∆ ′ is constructed in such a way that the action of W on it is free and properly discontinuous in the sense of definition 1.10.

Construction of the new manifolds

Now we are able to start the construction. It is divided into two steps. First, we choose a group W as in theorem 1.1 and get an infinite fan on which W acts thanks to theorem 1.19. We take the quotient X of its associated toric manifold by the complex Lie group H and check that we have an action of W on the complex manifold X.

The second step is to find a suitable open subset U of X so that its quotient under the action of W becomes a compact manifold.

2.1.

Step 1. Let K be a number field, with s > 0 real and 2t > 0 complex embeddings. Choose an integer b ∈ {1, ..., s -1}. Theorem 1.1 shows the existence of a subgroup W of O * ,+ K of rank b satisfying Assumption C. Take a fan ∆ ′ in R s as in theorem 1.19. Note ∆ its preimage π -1 H (∆ ′ ), which is a rational fan of E with respect to O K and consider its associated toric manifold denoted by X ∆ . The group H acts as a closed subgroup of (C * ) n (see equation ( 2)) on X ∆ and we have the following: Lemma 2.1. The action of H on X ∆ is free and proper.

Proof. To check that this action is proper, we first observe that the group H = ord(H) ∼ = H acts properly on Mc(∆). This is a consequence of lemma 1.9 and the fact that the map π H : E → E/ H is injective on |∆| by construction. Finally, it is clear that the action of H on X ∆ is proper if and only if the action of H on Mc(∆) is proper because (S 1 ) n is compact.

The isotropy group of a point x ∈ X ∆ is a compact subgroup of H ∼ = C t and hence trivial.

The preceding lemma tells us that the quotient X := X ∆ /H is a complex manifold and since (C * ) n is abelian, the following diagram is commutative:

X ∆ (S 1 ) n G G H ∼ = C t Mc(∆) ord(H) = H ∼ = R 2t X q p((S 1 ) n ) ∼ = (S 1 ) n G G Mc(π H (∆)) (4)
The action of (S 1 ) n on X ∆ descends to X via the group p((S 1 ) n ) ∼ = S 1 ) n . We note q : X → Mc(π H (∆)) the quotient map for this action. A detailed proof of the fact that Mc(∆)/ H and Mc(π H (∆)) are homeomorphic can be found in section 2.2.2, pp. 558 & 559 of [START_REF] Battisti | LVMB manifolds and quotients of toric varieties[END_REF].

Lemma 2.2. The action of W on X ∆ descends to an action on X wich we note (η, x) → η • x action.

Proof. We shall prove that the action of W normalizes the subgroup H of (C * ) n . Let η ∈ W , x ∈ X ∆ and z = (z 1 , ..., z t ) ∈ H. First, η is equivariant with respect to η :

(C * ) n -→ (C * ) n (t 1 , ..., t n ) -→ (t a 1,1 1 • • • t a 1,n n , ..., t a i,1 1 • • • t a i,n n , ..., t a n,1 1 • • • t an,n
n ) where (a i,j ) is the matrix of the linear map η in the basis B K , see for instance [START_REF] Ewald | Combinatorial convexity and algebraic geometry[END_REF], theorem 6.4 p. 244. This means that one has η(ι(z)x) = η(ι(z))η(x). The map ι is defined in equation [START_REF] Battisti | LVMB manifolds and quotients of toric varieties[END_REF]. Moreover, η(ι(z)) = ι(σ s+t+1 (η)z 1 , ..., σ s+t+t (η)z t ), hence we have:

η(ι(z)x) = ι(σ s+t+1 (η)z 1 , ..., σ s+t+t (η)z t )η(x).

Lemma 4), we have: q(η • H.x) = q(H.η(x)) = H. ord(η(x)). We first prove that ord(η(x)) = η(ord(x)) on the dense open set (C * ) n of X ∆ .

One has:

ord(x) = - 1 2π (ln |x 1 |, ..., ln |x n |) , and
ord(η(x)) = - 1 2π (ln(|x a 1,1 1 • • • x a 1,n n |), ..., ln(|x a n,1 1 • • • x an,n n |)) = - 1 2π n i=1 a 1,i ln |x i |, ..., n i=1 a n,i ln |x i | = η(ord(x)).
By the density of (C * ) n in X ∆ , this equality holds on X ∆ . Hence

q(η • H.x) = H. ord(η(x)) = H.η(ord(x)) = η • ( H. ord(x)).

2.2.

Step 2. The last step consists of finding an open subset U of X on which the action of W is free and properly discontinuous. This shows that the quotient U/W is a complex manifold. It turns out that the action of W on the manifold with corners Mc(π H (∆)) carries enough informations for our purpose. In this space, we find an open subset on which the action of W is free and properly discontinuous, with a compact fundamental domain. This proves the compactness of the quotient. Then, we pull everything back to X via the map q.

Let η 1 , ..., η b be generators of W . Taking their inverses if necessary, we may assume they all satisfy the following Assumption C + : for every i ∈ {1, ..., b} and every nondegenerate closed cone C ⊂ Ω ∪ {0}, one has Proof. For the first statement, notice that W is a lattice of R b via the map w → (ln η 1 , ..., ln η b ). The second one comes from the fact that every element η of W \ W >1 satisfies η i < 1 for all i ∈ {1, ..., b}.

Lemma 2.5. The action of W on U := Mc(π H (∆)) \ |Ω| * c is free, properly discontinuous and admits a compact fundamental domain. The same holds for the action of W on the preimage U := q -1 (U ) in X.

Proof. The fact that the action is free and properly discontinuous is a consequence of proposition 1.11. By theorem 1.19, there is a finite set of cones Σ := {σ 1 , ..., σ d } such that W • Σ = ∆ ′ . We still assume that η 1 , ..., η b is a family of generators of W satisfying assumption C + and B is the set defined directly before lemma 2.4. Let us consider the closure D in Mc(π H (∆)) of the set

D := Ñ η∈W + η(|Σ|) ∩ B é D 1 ∪ Ñ |Σ| ∩ η∈W >1 η(B) é D 2 .
Here, |Σ| is the union of the cones of Σ and W + is the set of elements of W satisfying assumption C + , along with the identity. Notice in particular that W + ⊂ W >1 . Figure 1 gives a picture of D for s = 2 and b = 1.

To prove the lemma, it is enough to show that D is a compact fundamental domain for the action of W .

For compactness, first notice that D 1 is bounded. This is a consequence of remark 1.18. Indeed, there is an upper-bound for the b first coordinates of points in D 1 because of the definition of B, while for the sb last coordinates, we observe that the inclusion η a C δ ⊂ C N a δ holds for a ∈ N Since Σ is a finite set, the closure of D 2 is the union of the closures of σ ∩ η∈W >1 η(B) for σ ∈ Σ. Such a set is contained in σ \ B(0, C) for some constant C > 0. Indeed, let x = (x 1 , ..., x s ) ∈ B and η ∈ W >1 . For all η ∈ W >1 , there exists an integer i b such that η i 1, so without loss of generality we may assume η 1 1. The following inequalities hold:

η(x) 2 = s i=1 η 2 i x 2 i η 2 1 x 2 1 x 2 1 min y∈B y 2 1 > 0.
The last inequality is a consequence of the fact that the projection of B on the b first coordinates is a compact set in R b having no point with a zero coordinate. All the constants C i := min y∈B y 2 i for i = 1, ..., b are strictly positive. Let C be the square root of the smallest of these constants. One has η(x) C. Finally, the closure of D 2 is compact by lemma 1.8.

In order to finish the proof, we check that D is a fundamental domain for the action of W .

First observe that for all x ∈ Ω \ L + , there are two elements η and

η ′ ∈ W such that x ∈ η(|Σ|) ∩ η ′ (B). One has either η -1 η ′ ∈ W + ⊂ W >1 or ηη ′-1 ∈ W + . This respectively implies that either η -1 (x) ∈ |Σ| ∩ η -1 η ′ (B) or η ′-1 (x) ∈ ηη ′-1 (|Σ|) ∩ B. If x ∈ L + , lemma 2.4 gives the conclusion since B ∩ L + ⊂ D.
For the remaining case, take a point x = y + ∞ • τ in D on a component at infinity of Mc(π H (∆)). The second conclusion of lemma 2.4 implies then that there is a constant C ′ > C such that for every cone σ of π H (∆), one has

σ \ B(0, C ′ ) = Ñ σ ∩ η∈W >1 η(B) é \ B(0, C ′ ).
Now let π be the projection from R s to (R s ) τ with respect to L(τ ), where (R s ) τ ⊕ L(τ ) = R s . Let also σ be the s-dimensional cone having τ as a face satisfying π(y) ∈ π(σ) and let g be an element of the group W such that g(σ) ∈ Σ. Such a g always exists because of the definition of Σ. One has:

g(x) = g(y) + ∞ • g(τ ).
Up to replacing y by y + w with w ∈ L(τ ), one may assume that y ∈ σ, hence g(y) ∈ g(σ) and the conclusion follows. We only need to prove the existence of σ. For this, take a generating set {v 1 , ..., v k } of τ , so that τ is written as:

τ = R 0 v 1 + • • • + R 0 v k .
Consider the open ball of radius ε > 0 centered at

v 0 := v 1 + • • • + v k ∈ τ . Since the support of π H (∆) is (Ω \ L + ) ∪ {0}
, there exists an ε > 0 small enough such that B(v 0 , ε) ⊂ Ω \ L + . Then, by shrinking ε if necessary, we may assume that B(v 0 , ε) only intersects the cones containing τ as a face, in view of the freeness of the W -action. Write

B(v 0 , ε) = B(v 0 , ε) ∩ σ>τ dim σ=s σ.
The image of B(v 0 , ε) under the projection π is an open neighborhood of 0 ∈ π(R s ) and the set π( σ>τ dim σ=s σ) is a union of cones, so it is invariant under homothetic transforms of positive ratio. Hence one has π(

σ>τ dim σ=s σ) = π(R s ),
there is a cone σ ∈ π H (∆) admitting τ as a face and π(y) ∈ π(σ).

Remark 2.6. In a similar fashion one can describe a compact fundamental domain showing that Sankaran manifolds are compact. Actually, in [START_REF] Sankaran | A class of non-Kähler complex manifolds[END_REF], p. 47 the author mentions this possibility to obtain compactness, although it did not seem feasible in general.

Finally, the previous lemma implies: Proposition 2.7. The quotient Y := U/W is a complex compact manifold of dimension s + t.

Remarks.

a) The open subset U ⊂ X intersects the Cousin group C 0 defined in corollary 1.5. Note that U 0 := U ∩ C 0 is a dense open subset of U . The quotient of U 0 by W is not compact and this is the reason for adding divisors at infinity to U 0 . In fact the irreducible components of these divisors are generalized LVMB manifolds in the sense of theorem 5.2 of [START_REF] Battisti | LVMB manifolds and quotients of toric varieties[END_REF]. To see this, first observe that the preimage of U \ U 0 in X ∆ is the set σ∈∆\{0} orb(σ). Let σ = R 0 v be a 1-dimensional cone of ∆, with v ∈ σ K (O K ). The fan of the toric manifold orb(σ), denoted by ∆/σ, is obtained by taking the image of every cone of ∆ having σ as a face under the projection map with respect to span(σ), see section VI.4 in [START_REF] Ewald | Combinatorial convexity and algebraic geometry[END_REF]. Note π σ : E ∼ = R n → E/ span(σ) ∼ = R n-1 this projection. One gets a finite fan in E/ span(σ) and since H trivially intersects σ K (O K ), the images of H and σ K (O K ) under π σ have a trivial intersection in E/ span(σ). Note π H the projection from E/ span(σ) with respect to π σ (H); this is an injective map on |∆/σ|. The fan π H (∆/σ) is complete because the closure of σ in Mc(∆) is compact by lemma 1.8. All the hypotheses of theorem 5.2 of [START_REF] Battisti | LVMB manifolds and quotients of toric varieties[END_REF] are satisfied. b) Our construction generalizes other known classes of manifolds.

i) As for the case b = 0, which corresponds to the case that W is trivial, one recovers the LVMB manifolds.

ii) It is also possible to extend the construction to the case b = s. In this case the fan ∆ then has to be the trivial fan and this leads to the description of OT manifolds, see [START_REF] Oeljeklaus | Non-Kähler compact complex manifolds associated to number fields[END_REF] and Y is the compact quotient of C 0 by W .

iii) When replacing the linear subspace H by the trivial space {0}, the rank b of W varies between 1 and s + t -1 and one gets the construction of Sankaran manifolds as in [START_REF] Sankaran | A class of non-Kähler complex manifolds[END_REF].

Invariants and geometric properties

(U/W, Q) ∼ = H 2 (W ⋉ O K , Q).
As in [START_REF] Oeljeklaus | Non-Kähler compact complex manifolds associated to number fields[END_REF], we use the Lyndon-Hochschild-Serre spectral sequence associated to the short exact sequence

0 → O K → W ⋉ O K → W → 0. One has E p,q 2 = H p (W, H q (O K , Q)) ⇒ H p+q (W ⋉ O K , Q
) and the following exact sequence:

0 → H 1 (W, Q O K ) → H 1 (W ⋉ O K , Q) → E 0,1 2 H 1 (O K , Q) W → H 2 (W, Q O K ) → H 2 (W ⋉ O K , Q) 1 → H 1 (W, H 1 (O K , Q)) E 1,1 2 where H 2 (W ⋉ O K , Q) 1 is defined by the exact sequence 0 → H 2 (W ⋉ O K , Q) 1 → H 2 (W ⋉ O K , Q) → H 2 (O K , Q) W E 0,2 2 .
See for instance [START_REF] Sah | Cohomology of split group extensions[END_REF].

If one proves that E 0,1 2 = E 0,2 2 = E 1,1 2 = 0, the result follows. For the fact that E 0,1 2 = E 1,1 2 = 0, the proof adapts readily from [START_REF] Oeljeklaus | Non-Kähler compact complex manifolds associated to number fields[END_REF], proposition 2.3, so we will not repeat it here.

As for the group

E 0,2 2 = H 2 (O K , Q) W ∼ = Alt 2 (O K , Q) W , recall that an element of Alt 2 (O K , Q) W is of the form γ = i<j a i,j σ i ∧ σ j with a i,j ∈ C.
Moreover, the W -invariance of γ means that for every pair (i, j) such that a i,j = 0, one has σ i (η)σ j (η) = 1 for all η ∈ W . Now W contains a nonreciprocal unit η 0 and therefore the relation σ i (η 0 )σ j (η 0 ) = 1 can never hold for any choice of i < j. Hence γ is trivial and so is the group E 0,2 2 . Proposition 3.8. If the group W contains at least one non-reciprocal unit, then U 0 /W contains no complex hypersurface; in particular the algebraic dimension of Y is zero.

Proof. One has the following commutative diagram:

U 0 W ∼ = Z b G G R s-b × (S 1 ) n q U 0 /W R s-b × (S 1 ) n q ′ (R >0 ) b W G G (S 1 ) b (5)
The open set U 0 is diffeomorphic to (R >0 ) b × R s-b × (S 1 ) n , W acts properly discontinuously on (R >0 ) b and U 0 /W is an R s-b × (S 1 ) n -bundle over (S 1 ) b .

Because W contains at least one non-reciprocal unit, the previous proposition gives that the map (q ′ ) * : H 2 ((S 1 ) b , Z) → H 2 (U 0 /W, Z) is injective and the proof of proposition 3.4 of [START_REF] Battisti | Holomorphic line bundles over domains in Cousin groups and the algebraic dimension of OT-manifolds[END_REF] adapts readily. Finally, Y only has a finite number of divisors, namely those added at infinity to compactify U 0 /W , hence its algebraic dimension is equal to zero.

Remark 3.9. The second Betti number of an OT manifold of simple type was already computed in proposition 2.3 of [START_REF] Oeljeklaus | Non-Kähler compact complex manifolds associated to number fields[END_REF]. In fact the proof of proposition 3.7 above shows that we can replace the simplicity condition with the assumption that s is odd.

Corollary 3.10. Let X = X(K, A) be an OT manifold. Let s > 0 and 2t > 0 be the number of real and complex embeddings of K respectively.

Then, if s is odd, one has dim R H 2 (X, R) = Ä s 2 ä .
Proof. Notice that this result holds with the same proof as above, under the assumption that the group A ∼ = Z s contains at least one non-reciprocal unit. If s is odd, K can not contain any reciprocal unit. Indeed if K contained such a unit, say α, the degree of Q(α) over Q would be even and divide the degree of K.

An example

To conclude this paper, we describe a concrete example of a manifold obtained by the above construction. In what follows we continue to use the same notations.

Definition 4.1. A Salem number is a real algebraic integer γ > 1 such that all its conjugates are of modulus smaller or equal to 1, with equality for a least one of them.

Remark 4.2. By using the fact that it admits a root of modulus 1, one easily proves that the minimal polynomial P of a Salem number γ is palindromic, i.e. it satisfies X deg P P (1/X) = P (X). This implies that the roots of P are γ, 1/γ and complex numbers of absolute value 1. In particular, the minimal polynomial of a Salem number has degree at least 4 and a Salem number is necessarily a unit of even degree.

Example 4.3. (See [START_REF] Bertin | Pisot and Salem numbers[END_REF], p. 85) It is possible to describe all polynomials of degree 4 which are minimal polynomials of a Salem number. These are the polynomials with integer coefficients of the form X 4 +q 1 X 3 +q 2 X 2 +q 1 X +1 with 2(q 1 -1) < q 2 < -2(q 1 + 1). The minimal polynomial of the smallest Salem number of degree 4 is X 4 -X 3 -X 2 -X + 1. Now we consider the polynomial P (X) := X 4 -X 3 -X 2 -X + 1. Its roots are α ≈ 1,722 (truncated value), α -1 , β and β where β is a complex number of modulus 1 and ℑ(β) > 0. Note σ 1 , σ 2 , τ 1 and τ 1 the associated embeddings of the field K := Q[X]/ P in R (for the first two ones) and C (for the last two ones).

One can check by computation that the family (1, α, α 2 , α 3 ) forms an integral basis of K, see section 2.6 of [START_REF] Stewart | Algebraic number theory and Fermat's last theorem[END_REF] for a method. The image under the map σ K of this family is the following basis of C 4 : Here, H is the subgroup {(0, 0, 0, z) ∈ C 4 | z ∈ C} of C 4 . In the basis B K , H is the set {(-βz, β(1β)z, (β -1)z, z) | z ∈ C} and the embedding ι : H → (C * ) 4 is given by ι(z) = (e -2iπβz , e 2iπβ(1-β)z , e 2iπ(β-1)z , e 2iπz ) .

B K = áá 1 1 1 1 ë , á α α -1 β β ë , á α 2 α -2 β 2 β
The cone Ω ⊂ R 2 is the open half-plane R >0 × R. We define a fan of R 2 whose cones are generated by elements of π H (O K ) and which is invariant under the action of W the following way: Let σ 1 be the cone positevely generated by the vectors (1, 1), (α, α -1 ) and σ 2 the cone generated by (1, -1), (α, -α -1 ) and call ∆ ′ the fan generated by W.{σ 1 , σ 2 }. It is clear that (Ω \ (R >0 × {0})) ∪ {0} is the support of ∆ ′ . For a picture of the situation in R 2 , we refer to figure 1.

In this example the divisors that we add to U 0 so that its quotient by W is compact are chains of Hopf surfaces. Indeed, let σ ∈ ∆ ⊂ E ∼ = R 4 be a 1-dimensional cone, say σ is generated by (α, α -1 , β, β) for simplicity. By the first remark of section 2.3, the quotient of orb(σ) by H is a generalized LVMB manifold. The fan ∆/σ of R 3 is generated by the images of the two 2-dimensional cones containing σ under π σ : R 4 → R 4 / span(σ). Up to a linear isomorphism we can assume that this is a subfan of the fan of P 3 (C). In other words, orb(σ)/H is an LVMB manifold. If we adopt the notations of [START_REF] López De Medrano | A new family of complex, compact, nonsymplectic manifolds[END_REF], this situation corresponds to the case n = 4, k = 3, n 1 = n 2 = 1 and n 3 = 2 hence orb(σ)/H is a Hopf surface (section 4.(b), p. 260, ibid.).

Or, one also can notice that orb(σ)/H is necessarily a Hopf surface by Potters' theorem on the classification of almost homogeneous complex compact surfaces ( [START_REF] Potters | On almost homogeneous compact complex analytic surfaces[END_REF]).

Hence we have built a complex manifold of dimension 3 which consists of a dense open set that is the quotient of a Cousin group by a discrete group, compactified by two cycles of Hopf surfaces. A computation shows that each of these Hopf surfaces contains the two elliptic curves C/ 1, β and C/ 1, 1β . Now since 1β = (β -1)/β, these two elliptic curves are isomorphic. However, because W only contains reciprocal units, one can not apply proposition 3.8 and hence we don't know yet the algebraic dimension of this manifold. On the other hand, if one carries the construction with the group W instead of W , then we know that the resulting manifold has its algebraic dimension equal to zero.

  s+t a -→ (ln |σ 1 (a)|, ..., ln |σ s (a)|, 2 ln |σ s+1 (a)|, ..., 2 ln |σ s+t (a)|)

  For a fan ∆, we set |∆| * := |∆| \ {0} and |∆| * c is the complement of this set in R n . One has: Proposition 1.11. Let ∆ be a (non-necessarily rational) fan and G a discrete group acting freely and properly discontinuously on ∆. Then this action induces an action of G on Mc(∆)\|∆| * c which is also free and properly discontinuous. Remark 1.12. Notice that Mc(∆) \ |∆| * c = (Mc(∆) \ R n ) ∪ |∆| * , i.e. this set is the support of ∆ (except 0) to which we add the components "at infinity" of Mc(∆).

  In particular, we have 1 b < s. Remember that given an integer b ∈ {1, ..., s -1}, theorem 1.1 guarantees the existence of a subgroup W of rank b of O * ,+ K satisfying Assumption C. Proposition 1.16 ([12], proposition 1.4). The action of W on Ω is free and properly discontinuous. Proposition 1.17 ([12], proposition 2.2). If C ⊂ Ω∪{0} is a nondegenerate closed cone, for all η ∈ W one has: a∈Z η a C ⊂ L + .

Lemma 2 . 4 .

 24 a∈N η a i C ⊂ L + . (Remark 1.18.) Define the affine subspace H 0 := {(1, ..., 1 b times , x 1 , ..., x s-b ) | x 1 , ..., x s-b ∈ R} ⊂ E/ H and set H i := η i (H 0 ) for i = 1, ..., b. Note B the (unbounded) convex envelope of H 0 , H 1 , ..., H b . Let B b be the image of B by the projection on the b first coordinates of E/ H. Then one has η∈W η(B b ) = (R + * ) b and the union η∈W \W >1 η(B b ) is bounded, where W >1 is the set of elements of W having at least one of their b first coordinates greater or equal to 1.

2 D 2 Figure 1 .

 221 Figure 1. The fundamental domain D.

Lemma 3 . 1 .

 31 The open set U admits no non-constant holomorphic functions.Proof. The manifold X contains the complex Lie group C 0 as an open dense subset, see corollary 1.5. As U 0 = U ∩ C 0 is an open set of C 0 stable under the action of the maximal compact subgroup (S 1 ) n of C 0 , it has no nonconstant holomorphic functions.The following lemma will helps us to compute the Kodaira dimension of our manifolds: Proposition 3.7. If the group W contains at least one non-reciprocal unit, then dim H 2 (U 0 /W, Z) = The open set U 0 /W of Y is the quotient of its contractible universal covering H b × C s+t-b by the group W ⋉ O K , hence one has H 2

Moreover, α and 1 -.

 1 α are two fundamental units of O * K . Since s = 2 one has Ω = N × L + with h = dim N = 1 = dim L = b (see lemma 1.13 for notations). It is clear that the two subgroups W := α Z and W := 1α Z of O * ,+ K satisfy Assumption C and they both are of rank b = 1.The action of W on C 4 is given by the following diagonal matrix:In the basis B K , the matrix of the linear map associated to M is the companion matrix of P :

  2.3. The group W acts on Mc(∆) and this action descends toMc(π H (∆)). Moreover, the map q is equivariant with respect to the action of W .Proof. For an element x ∈ X ∆ (resp. y ∈ Mc(∆)), we note H.x, respectively H.y, the H-orbit of x, respectively the H-orbit of y. The proof of the previous lemma shows that the action of W descends to Mc(∆). The pro-

jection map of E ⊂ Mc(∆) with respect to H commutes with the action of W , see remark 1.4, hence there is an action of W on Mc(π H (∆)). Note this action again by (η, H.y) → η • H.y. Now we prove that q is W -equivariant. Let H.x ∈ X and η ∈ W . By lemma 2.2 and the commutativity of diagram (

In[START_REF] Sankaran | A class of non-Kähler complex manifolds[END_REF], Sankaran formulates two others assumptions (A and B) which we do not use here.

Lemma 3.2. Let G ∼ = C t be a closed subgroup of (C * ) n and Ω be an open subset of an n-dimensional toric manifold Z. Assume that the action of G on Ω is proper and that the quotient Ω/G admits no non-constant holomorphic functions. Then the Kodaira dimension of Ω/G is -∞.

Proof. Let π : Ω → Ω/G be the quotient map. Choose a basis v 1 , ..., v n of the Lie algebra of (C * ) n such that v 1 , ..., v t is a basis of the Lie algebra of G. Define the vector fields v * 1 , ..., v * n on Ω/G by v * i := dπ y (v i (y)) for y ∈ π -1 (x) and consider the wedge product τ

Proof. Use the previous lemma with G = H, Z = X ∆ and Ω = π -1 (U ) where π : X ∆ → X is the quotient map under the action of H.

A consequence of the previous lemma is also: Proof. Let ρ : W → C be a group homomorphism. We will associate to this homomorphism a principal C-bundle above Y and show that if this bundle is trivial then ρ is also trivial, which will lead to the desired inequality. Recall that an element α of O * K is called a reciprocal unit if α -1 is a conjugate of α over Q. In the following we shall prove that the algebraic dimension of Y is zero if there is such a unit in W .