On a generalization of Inoue and Oeljeklaus-Toma manifolds

TL;DR

This paper introduces a new family of complex manifolds generalizing Inoue and Oeljeklaus-Toma manifolds, allowing non-diagonalizable monodromies, and shows many do not admit Kähler structures or are homeomorphic to known types.

Contribution

It constructs a broad class of complex manifolds based on matrices in SL(N,Z) with non-diagonalizable monodromies, extending previous manifold constructions.

Findings

Many constructed manifolds do not admit Kähler structures.

Some manifolds are not homeomorphic to Oeljeklaus-Toma manifolds.

The construction includes non-diagonalizable monodromies.

Abstract

In this paper we construct a family of complex analytic manifolds that generalize Inoue surfaces and Oeljeklaus-Toma manifolds. To a matrix $M$ in $SL(N,\mathbb{Z})$ satisfying some mild conditions on its characteristic polynomial we associate a manifold $T(M,\mathbf{D})$ (depending on an auxiliary parameter $\mathbf{D}$). This manifold fibers over the $s$-dimensional torus $\mathbb{T}^s$, where $s$ is the number of real eigenvalues of $M$. The fiber is the $N$-dimensional torus $\mathbb{T}^{N}$, and the monodromy matrices are certain polynomials of the matrix $M$. The basic difference of our construction from the preceding ones is that we admit non-diagonalizable matrices $M$ and the monodromy of the above fibration can also be non-diagonalizable. We prove that for a large class of non-diagonalizable matrices $M$ the manifold $T(M,\mathbf{D})$ does not admit any K\"ahler structure and is not homeomorphic to any of Oeljeklaus-Toma manifolds.

Equations

$${\tau }_i(x_1,\dots ,x_s,v)=({\xi }_i^{(1)}{x}_1,\dots ,{\xi }_i^{(s)}{x}_s,A_i(v))$$

$$({\mathbb{R}}^{*}_{+})^{s}\to{\mathbb{R}}^{s},\ \ \ (x_{1},\ldots,x_{s})\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{}}$}}}}}}}\ignorespaces}}}}\ignorespaces(\log x_{1},\ldots,\log x_{s})$$

$$\psi:({\mathbb{R}}^{*}_{+})^{s}\times E\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 0.63889pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\lg\times\text{id}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{}}$}}}}}}}\ignorespaces}}}}\ignorespaces{\mathbb{R}}^{s}\times E$$

$${\tau }_i(a,b)=(a+\lg {\xi }_i,A_i(b)).$$

$${\tau }_i(x_1,\dots ,x_s)=({\xi }_i^{(1)}{x}_1,\dots ,{\xi }_i^{(s)}{x}_s)$$

$$({\mathbb{R}}_{+}^{\ast }{)}^s\approx ({\mathbb{R}}_{+}^{\ast }{)}^s\times \{0\}\subset ({\mathbb{R}}_{+}^{\ast }{)}^s\times E.$$

$${\mathbb{C}}^{s+2n}=V\oplus W\oplus \bar{W}$$

$$v_{i}=\Big{(}a^{(i)}_{1},\ldots,a^{(i)}_{s},b^{(i)}_{1},\ldots,b^{(i)}_{n}\Big{)}\in{\mathbb{R}}^{s}\times{\mathbb{C}}^{n}\approx{\mathbb{R}}^{s+2n}.$$

$$g_0:{\mathbb{C}}^s\times {\mathbb{C}}^n\to {\mathbb{C}}^s\times {\mathbb{C}}^n;(w_1,\dots ,w_s,z)\mapsto ({\alpha }_1{w}_1,\dots ,{\alpha }_s{w}_s,R^{\top }z)$$

$$AB=BR.$$

$${\overrightarrow{\alpha}}=(\alpha_{1},\ldots,\alpha_{s}),\ \ D({\overrightarrow{\alpha}})=\Big{(}D(\alpha_{1}),\ldots,D(\alpha_{s})\Big{)}.$$

$$g_i=D_i(g_0):{\mathbb{R}}^s\times {\mathbb{C}}^n\to {\mathbb{R}}^s\times {\mathbb{C}}^n;$$

$$\psi:({\mathbb{R}}^{*}_{+})^{s}\times\Big{(}{\mathbb{R}}^{s}\times{\mathbb{C}}^{n}\Big{)}\approx{\mathbb{H}}^{s}\times{\mathbb{C}}^{n}.$$

$$g_{i}(w,z)=\Big{(}D_{i}(\alpha_{1})w_{1},\ldots,D_{i}(\alpha_{s})w_{s},D_{i}(R^{\top})z\Big{)}$$

$$\sigma:K\to{\mathbb{R}}^{l}\times{\mathbb{C}}^{q};\ \ \ \sigma(x)=\big{(}\sigma_{1}(x),\ldots,\sigma_{l+q}(x)\big{)}$$

$$\mathcal{L}(x)=$$

$$\Big{(}\log|\sigma_{1}(x)|,\ldots,\log|\sigma_{l}(x)|,2\log|\sigma_{l+1}(x)|,\ldots,2\log|\sigma_{l+q}(x)|\Big{)}$$

$$H=\Big{\{}\lambda\in{\mathbb{R}}^{l+q}~{}|~{}\lambda_{1}+...+\lambda_{l+q}=0\Big{\}},$$

$${\sigma }_j(u_i)={\sigma }_j(P_i(t))=P_i({\gamma }_j).$$

$$\varphi ={\varphi }_0\times {\varphi }_1:\mathbb{Z}[t]/C\to \mathbb{Z}[t]/B_0\times \mathbb{Z}[t]/B_1$$

$$D_i\equiv Q_i\mathrm{mod}(B_0),\mathrm{and}{D}_i\equiv P_i\mathrm{mod}(B_1).$$

$$B_1(t)=t^4+3t^3+3t^2+3t+1,B_2(t)=t^4+4t^3+3t^2+4t+1.$$

$$R(B_1,A)=1,R(B_2,B_1)=1,R(B_2,A)=1;$$

$$\varphi :\mathbb{Z}[t]/C\to \mathbb{Z}[t]/B_0\times \mathbb{Z}[t]/B_1\times ...\times \mathbb{Z}[t]/B_k.$$

$$D_{j,i}\equiv E_{j,i}\mathrm{mod}(B_0),$$

$$D_{j,i}\equiv P_{j,i}\mathrm{mod}(B_j),$$

$$D_{j,i}\equiv 1\mathrm{mod}(B_{\mu })\mathrm{if}\mu \ne j\mathrm{and}\mu \ne 0.$$

$$H_1(B_l)\stackrel{(\varphi (t_{l+1}){)}_{\ast }-\text{Id}}{\to }{H}_1(B_l)\to H_1(B_{l+1})\to H_0(B_l)\to H_0(B_l).$$

$$h:G_{s,k}(\rho )\to H_1(G_{s,k}(\rho ))/Tors\approx {\mathbb{Z}}^s.$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

1. 1 Introduction
2. 2 Twisted diagonal actions
3. 3 Twisted diagonal actions associated with an integer matrix
4. 4 Constructing Dirichlet families
5. 5 The manifold $T(M,{\mathcal{D}})$
6. 6 Mapping multi-tori and their homological properties
7. 7 LCK structures on $T(M,{\mathcal{D}})$
8. 8 Relations with the Oeljeklaus-Toma construction
9. 9 Acknowledgements

1. Introduction

1.1. Background

In 1972 M. Inoue [8] constructed complex surfaces having remarkable properties: they have second Betti number equal to zero and contain no complex curves. Inoue surfaces attracted a lot of attention. It was proved by F. Bogomolov [2] (see also the works of J. Li, S.-T. Yau, and F. Zheng [14] and [15], and A. Teleman [20]) that each complex surface of class $VII_{0}$ with $b_{2}(X)=0$ and containing no complex curves is isomorphic to an Inoue surface. Inoue surfaces are not algebraic, and moreover they do not admit Kähler metric.

Let us say that a matrix $M\in\mathrm{SL}(2k+1,{\mathbb{Z}})$ is of type ${\mathcal{I}}$, if it has only one real eigenvalue which is irrational and strictly greater than $1$. Inoue’s construction associates to every matrix $M\in\mathrm{SL}(3,{\mathbb{Z}})$ of type ${\mathcal{I}}$ a complex surface $S_{M}$ obtained as a quotient of ${\mathbb{H}}\times{\mathbb{C}}$ by action of a discrete group (here ${\mathbb{H}}$ is the upper half-plane). This manifold fibers over $S^{1}$ with fiber ${\mathbb{T}}^{3}$

Inoue’s construction was generalized to higher dimensions in the work of K. Oeljeklaus and M. Toma [17]. The construction of Oeljeklaus and Toma uses algebraic number theory. It starts with an algebraic field $K$ of degree $n$. Such field admits $n$ embeddings to ${\mathbb{C}}$; denote by $s$ the number of real embeddings; then $n=s+2t$. Oeljeklaus and Toma  construct an action of a certain semidirect product ${\mathbb{Z}}^{s}\ltimes{\mathbb{Z}}^{2t+s}$ on ${\mathbb{H}}^{s}\times{\mathbb{C}}^{t}$, the quotient is a compact complex manifold of complex dimension $s+t$. It has interesting geometric properties, in particular, it does not admit a Kähler metric. The original Inoue surface corresponds to the algebraic number field generated by the eigenvalues of the matrix $M$.

The Oeljeklaus-Toma manifolds (*OT-manifolds * for short) have very interesting geometric properties, studied in [17]; in particular, they do not admit Kähler metric. These manifolds were recently studied by many authors. In the work of L. Ornea, M. Verbitsky, and V. Vuletescu [18] it is shown that in many cases the OT-manifolds do not contain proper analytic subvarieties. In the article [9] of N. Istrati and A. Otiman the De Rham cohomology of OT-manifolds is computed. The paper of D. Angella, M. Parton, and V. Vuletescu [1] is dedicated to the proof of the rigidity of the complex structure of the OT-manifolds. The non-existence of complex curves in OT-manifolds is proved in the paper [21] of S. Verbitsky.

In the paper [6] we introduced another generalization of the Inoue surfaces. Namely for every matrix $M$ of type ${\mathcal{I}}$ we constructed a complex non-Kähler manifold  $T_{M}$ that fibers over a circle and the monodromy of the fibration equals $M^{\top}$. If the matrix $M$ is non-diagonalizable, then the manifold  $T_{M}$ is not homeomorphic to any of the manifolds constructed by Oeljeklaus and Toma.

1.2. Outline of the paper

In the present paper we generalize our construction to the case of matrices $M$ with several real eigenvalues. Let $M\in SL(N,{\mathbb{Z}})$. Assuming some restrictions on the matrix $M$ (see Definition 4.11) we construct a manifold  $T(M,{\mathcal{D}})$ that fibers over ${\mathbb{T}}^{s}$ (where $s$ is the number of real eigenvalues of $M$) with fiber ${\mathbb{T}}^{s+2n}$ and the monodromy group isomorphic to ${\mathbb{Z}}^{s}$. The monodromy transformations are linear automorphisms of the fiber deduced from the matrix $M$ (see Theorem 5.1).

The contents of the paper section by section is as follows. In Section 2 we discuss some special properly discontinuous actions of semidirect product ${\mathbb{Z}}^{s}\ltimes{\mathbb{Z}}^{k}$ on $({\mathbb{R}}^{*}_{+})^{s}\times{\mathbb{R}}^{k}$. We call them twisted-diagonal actions. In Section 3 we show how to construct a twisted-diagonal actions from a matrix $M$ of special type. Section 4 is the central technical part of the work, here we study the Dirichlet families of integer polynomials associated to a matrix. The construction of the generalized Oeljeklaus-Toma  manifold is the subject of the main theorem of the paper (Theorem 5.1) which is stated and proved in Section 5. This theorem associates to each matrix $M$ of type ${\mathcal{J}}$ (see Definition 4.11) and a Dirichlet family ${\mathcal{D}}$ for $M$ a complex manifold $T(M,{\mathcal{D}})$. Sections 6 – 8 are about properties of the manifolds $T(M,{\mathcal{D}})$. We concentrate on the case when the matrix $M$ is non-diagonalizable, since this case is the most different from the manifolds construsted in [8], [17]. We prove in particular that if $M$ is non-diagonalizable, and the family ${\mathcal{D}}$ is primary (see Definition 4.14) then: – The manifold $T(M,{\mathcal{D}})$ does not admit any LCK-structure, in particular it does not admit a structure of a Kähler manifold. – The manifold $T(M,{\mathcal{D}})$ is not homeomorphic to any of OT-manifolds. In section 8 we show that some of OT-manifolds are biholomorphic to manifolds $T(M,{\mathcal{D}})$.

2. Twisted diagonal actions

In this subsection we consider some special groups of diffeomorphisms of the space $({\mathbb{R}}^{*}_{+})^{s}\times E$ where $E$ is a finite-dimensional real vector space.

Let $k=\dim E$. Let ${\mathcal{B}}$ be any basis in $E$ and denote by ${\mathcal{L}}$ the free abelian group generated by ${\mathcal{B}}$, so that ${\mathcal{L}}\approx{\mathbb{Z}}^{k}$. The group ${\mathcal{L}}$ acts on $E$ by translations. Let ${\mathcal{F}}=(\xi_{1},\ldots,\xi_{r})$ be any family of elements of $({\mathbb{R}}^{*}_{+})^{s}$. Each of vectors $\xi_{i}$ can be considered as a diagonal matrix in $GL(s,{\mathbb{R}})$. These matrices commute with each other and so we obtain an action of ${\mathbb{Z}}^{r}$ on ${\mathbb{R}}^{s}$. We can restrict it to $({\mathbb{R}}^{*}_{+})^{s}$. The direct product of the two actions above gives an action of the abelian group $G_{0}={\mathbb{Z}}^{r}\times{\mathcal{L}}$ on $({\mathbb{R}}^{*}_{+})^{s}\times E$. This action will be referred to as diagonal action of $G_{0}$.

Now we shall generalize this notion so as to define an action of a certain semidirect product of ${\mathbb{Z}}^{r}$ and ${\mathcal{L}}$ on $({\mathbb{R}}^{*}_{+})^{s}\times E$.

Let ${\mathcal{A}}=\{A_{1},\ldots,A_{r}\}\subset\text{\rm GL}(E)$ be a family of commuting linear automorphisms. Assume that every $A_{i}$ preserve the lattice ${\mathcal{L}}$, and moreover, determines an automorphism of ${\mathcal{L}}$. Define an action of ${\mathbb{Z}}^{r}$ on ${\mathcal{L}}$ by the following rule: the $i$-th free generator $\tau_{i}$ of ${\mathbb{Z}}^{r}$ acts as $A_{i}$. Consider the corresponding semidirect product $G={\mathbb{Z}}^{r}\ltimes{\mathcal{L}}.$

Definition 2.1**.**

Define an action of $G$ on $({\mathbb{R}}^{*}_{+})^{s}\times E$ as follows.

1. The subgroup ${\mathcal{L}}$ of $G$ acts by translations on $E$.

2. The generator $\tau_{i}$ of the group ${\mathbb{Z}}^{r}$ acts as follows

[TABLE]

This action will be called *twisted diagonal action * associated to ${\mathcal{F}},{\mathcal{L}}$ and ${\mathcal{A}}$. The quotient of $({\mathbb{R}}^{*}_{+})^{s}\times E$ by this action will be denoted by $Y({\mathcal{F}},{\mathcal{L}},{\mathcal{A}})$. The action will be called special if at least one of the linear maps $A_{i}:E\to E$ has no eigenvalue 1.

Definition 2.2**.**

The map

[TABLE]

will be denoted by $\lg$. For a family ${\mathcal{F}}=(\xi_{1},\ldots,\xi_{r})$ of elements of $({\mathbb{R}}^{*}_{+})^{s}$ we denote by $\lg{\mathcal{F}}$ the family $(\lg\xi_{1},\ldots,\lg\xi_{r})$ of elements of ${\mathbb{R}}^{s}$. A family ${\mathcal{F}}$ is called a log-basis if $s=r$ and $\lg{\mathcal{F}}$ is a basis in ${\mathbb{R}}^{s}$. $\hfill\triangle$

Definition 2.3**.**

The twisted diagonal action associated to ${\mathcal{F}},{\mathcal{L}}$ and ${\mathcal{A}}$ will be called non-degenerate if ${\mathcal{F}}$ is a log-basis of $({\mathbb{R}}^{*}_{+})^{s}$.

Proposition 2.4**.**

Any non-degenerate twisted diagonal action is properly discontinuous and the quotient is a compact manifold.

*Proof. *Consider the homeomorphism

[TABLE]

The action of $G$ on $({\mathbb{R}}^{*}_{+})^{s}\times E$ induces an action of $G$ on ${\mathbb{R}}^{s}\times E$ which satisfies the following:

1. The lattice ${\mathcal{L}}$ acts by translations on $E$.

2. The action of the generator $\tau_{i}\in{\mathbb{Z}}^{s}$ on ${\mathbb{R}}^{s}\times E$ is given by the formula

[TABLE]

The vectors $(\lg\xi_{1},\ldots,\lg\xi_{s})$ form a basis in ${\mathbb{R}}^{s}$ and the proposition follows. $\hfill\square$

Denote the matrix of $A_{i}$ in the basis ${\mathcal{B}}$ by $\mu_{i}$, then $\mu_{i}\in GL(k,{\mathbb{Z}})$.

Proposition 2.5**.**

There is a fibration $p:Y({\mathcal{F}},{\mathcal{L}},{\mathcal{A}})\to{\mathbb{T}}^{s}$ with fiber ${\mathbb{T}}^{k}$ and monodromy matrices equal to $\mu_{1},\ldots,\mu_{s}$. 2. 2)

This fibration admits a cross-section $\sigma$.

*Proof. *To prove 1) define a diagonal action of ${\mathbb{Z}}^{s}$ on $({\mathbb{R}}^{*}_{+})^{s}$ by the formula

[TABLE]

and observe that the first projection $({\mathbb{R}}^{*}_{+})^{s}\times E\to({\mathbb{R}}^{*}_{+})^{s}$ commutes with the actions of ${\mathbb{Z}}^{s}$ and determines a map $Y({\mathcal{F}},{\mathcal{L}},{\mathcal{A}})\to{\mathbb{T}}^{s}$ which is clearly a fibration with fiber $E/{\mathcal{L}}\approx{\mathbb{T}}^{k}$ and monodromy maps $\mu_{i}$ as required. As for the point 2) the cross-section is defined by the inclusion

[TABLE]

$\hfill\square$

3. Twisted diagonal actions associated with an integer

matrix

Definition 3.1**.**

We say that a polynomial $P\in{\mathbb{Z}}[t]$ is of type ${\mathcal{J}}_{0}$ if it has at least one real root and at least one imaginary root and all real roots are simple.

Let $M\in SL(N,{\mathbb{Z}})$. We say that $M$ is a matrix of type ${\mathcal{J}}_{0}$ if its characteristic polynomial if of type ${\mathcal{J}}_{0}$. $\hfill\triangle$

In this section we outline a procedure of constructing from a matrix $M$ of type ${\mathcal{J}}_{0}$ a twisted diagonal action. This procedure requires some further assumptions on $M$ which will be discussed in subsequent sections. Denote by $\alpha_{1},\ldots,\alpha_{s}$ the real eigenvalues of $M$ and by $\beta_{1},\ldots,\beta_{k},\bar{\beta}_{1},\ldots,\bar{\beta}_{k}$ the imaginary ones. We have then $0<s,0<k$. The size of the matrix equals $s+2n$ with $n\geqslant k$.

Denote by $f_{M}:{\mathbb{C}}^{s+2n}\to{\mathbb{C}}^{s+2n}$ the linear map corresponding to $M$. We have a decomposition of ${\mathbb{C}}^{s+2n}$ into a direct sum of complex $f_{M}$-invariant subspaces

[TABLE]

where $V$ is generated by the eigenvectors $a_{1},\ldots,a_{s}\in{\mathbb{R}}^{s+2n}$ of $M$, corresponding to the real eigenvalues $\alpha_{1},\ldots,\alpha_{s}$, and $W$ is the generalized eigenspace corresponding to all the imaginary eigenvalues. Let $b_{1},\ldots,b_{n}$ be a basis of $W$. Consider the vectors

[TABLE]

(here $1\leqslant i\leqslant s+2n$)

Lemma 3.2**.**

The family ${\mathcal{B}}=(v_{1},\ldots,v_{s+2n})$ is a basis of ${\mathbb{R}}^{s+2n}$.

*Proof. *It suffices to prove that the determinant of the matrix $Q$ formed by the columns of the coordinates of $(s+2n)$ real vectors $a_{1},\ldots,a_{s},\text{\rm Re }b_{1},\ \text{\rm Im }b_{1},\ldots,\text{\rm Re }b_{n},\ \text{\rm Im }b_{n}$ is non-zero. This determinant equals to a non-zero multiple of the determinant of the following family of complex vectors: $(a_{1},\ldots,a_{s},b_{1},\bar{b}_{1},\ldots b_{n},\bar{b}_{n})$, and this last family is a basis. $\hfill\square$

The next definition is basic for our purposes.

Definition 3.3**.**

Denote by $R$ the matrix of the restriction of the linear map $f_{M}$ to $W$. Denote by $\Delta$ the diagonal $(s\times s)$-matrix with diagonal entries $\alpha_{1},\ldots,\alpha_{s}$. Let $U\in GL(s+2n,{\mathbb{C}})$ be the direct sum $\Delta\oplus R$ of these matrices. Define a linear automorphism $g_{0}$ of ${\mathbb{C}}^{s}\times{\mathbb{C}}^{n}$ by the following formula

[TABLE]

(here $w_{i}\in{\mathbb{C}}$ for every $i$ and $z\in{\mathbb{C}}^{n}$). Its matrix equals $U^{\top}$. Observe that the above formula determines also an automorphism of ${\mathbb{R}}^{s}\times{\mathbb{C}}^{n}$, and a biholomorphism of ${\mathbb{H}}^{s}\times{\mathbb{C}}^{n}$ onto itself. These two isomorphisms will be denoted by the same symbol $g_{0}$ by a certain abuse of notation. $\hfill\triangle$

We will need a lemma from linear algebra.

Lemma 3.4**.**

Let $A$ be an $(m\times m)$-matrix. We will denote the corresponding mapping ${\mathbb{C}}^{m}\to{\mathbb{C}}^{m}$ by the same letter. Let $L$ be an $A$-invariant subspace, and ${\mathcal{B}}=(b_{1},\ldots b_{k})$ a basis in $L$. Denote by $B$ the $(m\times k)$-matrix formed by the columns of the vectors of ${\mathcal{B}}$, and by $R$ the matrix of $A~{}|~{}L:L\to L$ in the basis ${\mathcal{B}}$. Then

[TABLE]

*Proof. *Denote by ${\mathcal{E}}$ the canonical basis of ${\mathbb{C}}^{m}$. Observe that $AB$ is the matrix of the composition L\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}}}}}}}}}\ignorespaces}}}}\ignorespaces{\mathbb{C}}^{m}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 6.375pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{\scriptstyle{A}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}}}}}}}}}\ignorespaces}}}}\ignorespaces{\mathbb{C}}^{m} with respect to  the bases ${\mathcal{B}}$ and ${\mathcal{E}}$. Further, $BR$ is the matrix of the composition L\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 6.375pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.39166pt\hbox{\scriptstyle{A}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}}}}}}}}}\ignorespaces}}}}\ignorespaces L\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}}}}}}}}}\ignorespaces}}}}\ignorespaces{\mathbb{C}}^{m} with respect to  the same bases. The lemma follows. $\hfill\square$

Remark 3.5**.**

If $L={\mathbb{C}}^{m}$, the formula (3) is equivalent to the classical formula $R=B^{-1}AB$, relating the matrices of a linear map in two different bases.

We will now apply the preceding Lemma to the basis ${\mathcal{B}}$ from Lemma 3.2.

Proposition 3.6**.**

The lattice ${\mathcal{L}}$ generated by ${\mathcal{B}}$ is invariant with respect to  $g_{0}$, and the matrix of the map $g_{0}$ in ${\mathcal{B}}$ equals $M^{\top}$.

*Proof. *Put $u_{i}=(a^{(i)}_{1},\ldots,a^{(i)}_{s})\in{\mathbb{R}}^{s}$ and $w_{i}=(b^{(i)}_{1},\ldots,b^{(i)}_{n})\in{\mathbb{C}}^{n}$, so that $v_{i}=(u_{i},w_{i})$; here $1\leqslant i\leqslant s+2n$. We have $(\Delta u_{i})_{j}=\alpha_{j}a^{(i)}_{j}$; this equals $\sum_{k}M_{ik}a^{(k)}_{j}$ since $a_{i}$ is the eigenvector of $M$ with eigenvalue $\alpha_{i}$. Therefore $\Delta u_{i}=\sum_{k}M_{ik}u_{k}$.

Let $B$ be the matrix formed by the columns of coordinates of vectors $b_{i}$. Then the vector $R^{\top}w_{i}$ is the $i$-th column of the matrix $R^{\top}B^{\top}$. This matrix equals $(MB)^{\top}$ by Lemma 3.4 and $(MB)_{ij}=\sum_{k}M_{ik}b_{k}^{(j)}$. Therefore $g_{0}(v_{i})=\sum_{k}M_{ik}v_{k}=\sum_{k}(M^{\top})_{ki}v_{k}$ and the proposition is proved. $\hfill\square$ This proposition allows us to construct a twisted diagonal action of ${\mathbb{Z}}\ltimes{\mathcal{L}}\approx{\mathbb{Z}}\ltimes{\mathbb{Z}}^{s+2n}$ on ({\mathbb{R}}^{*}_{+})^{s}\times\Big{(}V\times W\Big{)}, following the procedure from Section 2. The input data is $({\mathcal{F}},{\mathcal{L}},{\mathcal{A}})$ where ${\mathcal{F}}$ consists of just one vector $(\alpha_{1},\ldots,\alpha_{s})$ and ${\mathcal{A}}$ consists of just one matrix $U^{\top}$. For the case $s=1$ this action is nothing else than the one constructed in our previous paper [6].

For the case $s>1$ this action is not non-degenerate. To construct a non-degenerate action we need a family of $s$ matrices, commuting with each other. We will choose these matrices to be of the form $D_{i}(U^{\top})$ where $D_{i}$ are integer polynomials of special type, which will be described now.

For any polynomial $D\in{\mathbb{Z}}[t]$ the vectors $a_{1},\ldots,a_{s}\in{\mathbb{R}}^{s+2n}$ are eigenvectors of $D(M)$ with eigenvalues $D(\alpha_{1}),\ldots,D(\alpha_{s}).$ Put

[TABLE]

Definition 3.7**.**

A family ${\mathcal{D}}=(D_{1},\ldots,D_{s})$ of integer polynomials will be called a Dirichlet family for $M$ or *a $D$-family * for short, if the following three conditions hold:

$\Delta 1)$ $D_{i}(M)\in SL(s+2n,{\mathbb{Z}})$ for every $i$;

$\Delta 2)$ $D_{i}(\alpha_{j})>0$ for every $i,j$;

$\Delta 3)$ The family {\mathcal{D}}(\overrightarrow{\alpha})=\Big{(}D_{1}(\overrightarrow{\alpha}),\ldots,D_{s}(\overrightarrow{\alpha})\Big{)} is a log-basis of $({\mathbb{R}}^{*}_{+})^{s}.$

Remark 3.8**.**

Observe that if a family ${\mathcal{D}}=(D_{1},\ldots,D_{s})$ satisfies only the conditions $\Delta 1)$ and $\Delta 3)$ it is easy to construct from it a Dirichlet family, namely, put $\tilde{\mathcal{D}}=(\tilde{D}_{1},\ldots,\tilde{D}_{s})$ where $\tilde{D}_{i}=D^{2}_{i}$.

We will construct such families later, now we are going to explain how to construct a twisted-diagonal action from a $D$-family. For every $i$ with $1\leqslant i\leqslant s$ the automorphism

[TABLE]

preserves the lattice ${\mathcal{L}}$; its matrix in the canonical basis equals $D_{i}(U^{\top})$. and in basis ${\mathcal{B}}$ it equals $D_{i}(M^{\top})$.

Applying Definition 2.1 to the input data $({\mathcal{F}},{\mathcal{L}},{\mathcal{D}})$ where ${\mathcal{F}}=(D_{1}(\overrightarrow{\alpha}),\ldots,D_{s}(\overrightarrow{\alpha}))$ and ${\mathcal{D}}=(D_{1}(U^{\top}),\ldots,D_{s}(U^{\top}))$ we obtain a twisted-diagonal action of $G={\mathbb{Z}}^{s}\ltimes{\mathbb{Z}}^{s+2n}$ on $({\mathbb{R}}^{*}_{+})^{s}\times E$ where $E={\mathbb{R}}^{s}\times{\mathbb{C}}^{n}.$ We have a natural diffeomorphism

[TABLE]

In the coordinates $\psi$ the action of $g_{i}$ is written as follows:

[TABLE]

(where $w\in{\mathbb{H}}^{s},\ z\in{\mathbb{C}}^{n}$). Thus every $g_{i}$ is a biholomorphism of ${\mathbb{H}}^{s}\times{\mathbb{C}}^{n}$ and we obtain the following.

Proposition 3.9**.**

Let ${\mathcal{D}}$ be a Dirichlet family of polynomials for a matrix $M$ of type ${\mathcal{J}}_{0}$. Then the formula (4) determines a twisted-diagonal action of the corresponding semidirect product $G={\mathbb{Z}}^{s}\ltimes{\mathbb{Z}}^{s+2n}$ on ${\mathbb{H}}^{s}\times{\mathbb{C}}^{n}$, and we have the following:

The quotient $Y$ has a natural structure of a complex manifold  of dimension $s+n$. 2. 2)

There is a fibration $p:Y\to{\mathbb{T}}^{s}$ with fiber ${\mathbb{T}}^{s+2n}$; the action of generators of the monodromy group ${\mathbb{Z}}^{s}$ is given by the matrices $D_{1}(M^{\top}),\ldots,D_{s}(M^{\top})$. 3. 3)

The fibration $p$ admits a cross-section.

*Proof. *The part 1) follows from the fact that $G$ acts by biholomorphisms, together with Proposition 2.4. The parts 2) and 3) follow from Proposition 2.5. $\hfill\square$

Remark 3.10**.**

For $s=1$ the family consisting of one polynomial $D_{1}(t)=t$ is obviously a Dirichlet family for any matrix $M$ of type ${\mathcal{J}}_{0}$. Thus we recover here the construction of generalized Inoue manifolds from [6], §2.

4. Constructing Dirichlet families

In this section we construct Dirichlet families associated to a given matrix $M\in SL(N,{\mathbb{Z}})$. The main result of this section is Theorem 4.13. The principal algebraic tool in this construction is Proposition 4.1, based on the Dirichlet’s unit theorem.

4.1. A corollary of the Dirchlet’s unit theorem

Proposition 4.1**.**

Let $P\in{\mathbb{Z}}[t]$ be an irreducible polynomial. Assume that $P$ has at least one real root, and at least one imaginary root. Denote the real roots of $P$ by $\gamma_{1},\ldots,\gamma_{l}$, and let $2q$ be the number of imaginary roots. Put $\overrightarrow{\gamma}=(\gamma_{1},\ldots,\gamma_{l})$.

Then there are polynomials $P_{1},\ldots,P_{l}\in{\mathbb{Z}}[t]$  such that :

1) $P_{i}$ are invertible in ${\mathbb{Z}}[t]/P$.

2) For every $i,j$ we have $P_{i}(\gamma_{j})>0$.

3) The vectors $P_{i}(\overrightarrow{\gamma})=(P_{i}(\gamma_{1}),\ldots,P_{i}(\gamma_{l}))$ with $1\leqslant i\leqslant l$ form a log-basis in $({\mathbb{R}}^{*}_{+})^{l}$.

4) The properties 1) – 3) hold if we replace $P_{i}$ by any other polynomial $\tilde{P}_{i}$  such that   $\tilde{P}_{i}\equiv P_{i}\mod P$.

*Proof. *Consider the algebraic extension $K$ of ${\mathbb{Q}}$ corresponding to $P$, that is, $K={\mathbb{Q}}[t]/P$. Let $m=\deg P$, then $m=l+2q=(K:{\mathbb{Q}})$. An embedding $K\hookrightarrow{\mathbb{C}}$ is called real if its image is in ${\mathbb{R}}$; an embedding which is not real is called complex. The field $K$ admits $l$ real embeddings and $2q$ complex embeddings. Denote the real embeddings by $\sigma_{1},\ldots,\sigma_{l}$ and the complex embeddings by $\sigma_{l+1},\ldots,\sigma_{l+2q}$; we can assume that $\sigma_{i}=\overline{\sigma_{q+i}}$ for $i\geqslant l+1$. The map

[TABLE]

is an embedding (known as geometric representation of the field $K$, see [3], Ch. II, §3). The map ${\mathscr{L}}:K^{*}\to{\mathbb{R}}^{l+q}$ defined as follows:

[TABLE]

[TABLE]

is called the logarithmic representation map.

The ring ${\mathbb{Z}}[t]/P$ is an order in $K$, denote it by ${\mathcal{O}}$. Dirichlet’s unit theorem (see [3], Ch. II, §4.3) guarantees the existence of $l+q-1$ elements $u_{1},\ldots,u_{l+q-1}$ in ${\mathcal{O}}$, which are invertible in ${\mathcal{O}}$ and generate a free abelian group in ${\mathcal{O}}^{*}$. Moreover, these units can be chosen so as their ${\mathscr{L}}$-images form a basis in the hyperplane

[TABLE]

see [3], Chapter II, §4.3. Denote by $p$ the projection of ${\mathbb{R}}^{l+q}$ onto the space ${\mathbb{R}}^{l}$ corresponding to the first $l$ coordinates. Since $q\geqslant 1$ we can assume (re-numbering the units $u_{i}$ if necessary), that the vectors $p({\mathscr{L}}(u_{1})),...,p({\mathscr{L}}(u_{l}))$ form a basis in ${\mathbb{R}}^{l}$.

Pick any polynomials $P_{i}\in{\mathbb{Z}}[t]$ (here $1\leqslant i\leqslant l$)  such that   $P_{i}(t)=u_{i}$ in ${\mathcal{O}}$. Every real embedding of $K$ sends $t$ to one of the real roots of $P$, so renumbering the roots $\gamma_{i}$ if necessary we can assume that $\sigma_{i}(t)=\gamma_{i}$. Therefore

[TABLE]

Replacing the units $u_{i}$ by their squares $u_{i}^{2}$ if necessary we can assume that $P_{i}(\gamma_{j})>0$. The property 2) of the statement of the Proposition follow. As for 3) it suffices to observe that the vector $\lg P_{i}(\overrightarrow{\gamma})$ equals $p({\mathscr{L}}(u_{i}))$. The point 4) is obvious. $\hfill\square$

Corollary 4.2**.**

Let $M\in SL(N,{\mathbb{Z}})$; assume that the characteristic polynomial $P$ of $M$ satisfies the assumptions of Proposition 4.1. Then there exists a Dirichlet family of polynomials for $M$.

*Proof. *The family $(P_{1},\ldots,P_{l})$ of polynomials from Proposition 4.1 satisfy the properties $\Delta 2)$ and $\Delta 3)$ (they follow from the properties 2) and 3) of Proposition 4.1). Further, observe that every matrix $P_{i}(M)$ is invertible. Indeed there exists a polynomial $T_{i}$  such that   $P_{i}T_{i}\equiv 1\ \mod(P)$. Therefore $P_{i}(M)T_{i}(M)=Id$ and $P_{i}(M)$ is invertible.

It remains only to show that $\det P_{i}(M)>0$. This property follows from the next Lemma, the proof of which is an easy consequence of the Jordan decomposition theorem and will be omitted.

Lemma 4.3**.**

Let $M$ be a square matrix with entries in ${\mathbb{R}}$, and $P\in{\mathbb{R}}[t]$. Assume that for every real eigenvalue $\alpha$ of $M$ we have $P(\alpha)\geqslant 0$. Then $\det P(M)\geqslant 0$. $\hfill\square$

$\hfill\square$

4.2. Dirichlet families: a particular case

In this subsection we begin the construction of Dirichlet families; we consider here a particular class of matrices. We will use the following terminology for polynomials with integer coefficients.

Definition 4.4**.**

Let $A,B\in{\mathbb{Z}}[t]$. We say that $A$ and $B$ are coprime if their greatest common divisor is $1$. We say that $A$ and $B$ are strongly coprime if the sum of the principal ideals $(A)$ and $(B)$ equals ${\mathbb{Z}}[t]$.

Definition 4.5**.**

An integer polynomial $C$ is said to be of type ${\mathcal{J}}_{1}$ if $C=B_{0}\cdot B_{1}$ with $B_{0},B_{1}\in{\mathbb{Z}}[t]$, and

$B_{0}$ has no real roots. 2. 2)

$B_{1}$ is irreducible, it has at least one real root and at least one imaginary root. 3. 3)

$B_{0}$ and $B_{1}$ are strongly coprime.

We say that a matrix $M\in SL(N,{\mathbb{Z}})$ is of type ${\mathcal{J}}_{1}$ if its characteristic polynomial is of type ${\mathcal{J}}_{1}$. $\hfill\triangle$

Remark 4.6**.**

This condition allows $B_{0}$ to have multiple complex roots.

Proposition 4.7**.**

If $M$ is of type ${\mathcal{J}}_{1}$, then there exists a Dirichlet family of polynomials for $M$.

*Proof. *Denote by $C$ the characteristic polynomial of $M$. The ring ${\mathbb{Z}}[t]ex:twidiag-tor/C$ has natural projections $\phi_{0}:{\mathbb{Z}}[t]/C\to{\mathbb{Z}}[t]/B_{0}$ and $\phi_{1}:{\mathbb{Z}}[t]/C\to{\mathbb{Z}}[t]/B_{1}$. The product of these projections

[TABLE]

is an isomorphism by the Chinese remainder theorem (see [5], Th 2.25, p. 131). Denote the real roots of $B_{1}$ by $\alpha_{1},\ldots,\alpha_{s}$. Apply Proposition 4.1 to $B_{1}$; we obtain polynomials $P_{1},\ldots,P_{s}\in{\mathbb{Z}}[t]$. For $1\leqslant i\leqslant s$ let $Q_{i}\in{\mathbb{Z}}[t]$ be any polynomial invertible $\mod B_{0}$, and $D_{i}\in{\mathbb{Z}}[t]$ be a polynomial  such that

[TABLE]

We claim that the family ${\mathcal{D}}=(D_{1},\ldots,D_{s})$ is a ${\mathcal{D}}$-family for $M$.

Since $D_{i}(\alpha_{j})=P_{i}(\alpha_{j})$, the formula (5) allows us to deduce the property $\Delta 2)$ from property 2) of Proposition 4.1. As for the property $\Delta 3)$ it follows immediately from property 3) of 4.1.

The polynomial $D_{i}$ is invertible in ${\mathbb{Z}}[t]/C$ since $\phi$ is an isomorphism and both $\phi_{0}(D_{i})$ and $\phi_{1}(D_{i})$ are invertible. Therefore $D_{i}(M)$ is invertible over ${\mathbb{Z}}$. Applying Lemma 4.3 we deduce that $\det D_{i}(M)=1$.

$\hfill\square$

Remark 4.8**.**

There is a large choice for polynomials $Q_{i}$. In particular the polynomials $Q_{i}(t)=t$ will do. Any Dirichlet family corresponding to this case will be called primary.

Here is one example of a polynomial of type ${\mathcal{J}}_{1}$.

Proposition 4.9**.**

Let $B_{1}(t)=t^{4}+3t^{3}+3t^{2}+3t+1,A(t)=t^{2}+1$. Let $q$ be any natural number $\geqslant 1$, and put $B_{0}(t)=(A(t))^{q}$. Then the polynomial $C(t)=B_{0}(t)B_{1}(t)$ is of type ${\mathcal{J}}_{1}$.

*Proof. *It is easy to check that $B_{1}$ is irreducible over ${\mathbb{Z}}$ and has two real roots (both negative), and two imaginary roots. The resultant $R(B_{1},A)$ equals $1$, therefore the ideal generated by $B_{0}$ and $A$ contains $1$, so that $B_{0}$ and $A$ are strongly coprime. We have also $R(B_{1},A^{q})=(R(B_{1},A))^{q}=1$, therefore $B_{0}(t)$ and $B_{1}(t)$ are coprime. $\hfill\square$

Using recent results from the theory of irreducible polynomials one can generalize the previous example to construct polynomials of type ${\mathcal{J}}_{1}$ of higher degrees. Here is one more example.

Proposition 4.10**.**

Let $B_{1}(t)=t^{12}+8t^{8}+20t^{7}+19t^{6}+20t^{5}+8t^{4}+1$, and $A(t)=t^{2}+1$ as before. Let $q$ be any natural number $\geqslant 1$, and put $B_{0}(t)=(A(t))^{q}$. Then the polynomial $C(t)=B_{0}(t)B_{1}(t)$ is of type ${\mathcal{J}}_{1}$.

*Proof. *The coefficients of $B_{1}$ are all positive, the degree of $B_{1}$ is less than 31, and the natural number $B_{1}(10)$ is prime. Therefore by a theorem of M. Filaseta and S. Gross ([7], Th. 1) the polynomial $B_{1}$ is irreducible. In particular, its roots are simple. We have $B_{1}(-1)=-3$, therefore $B_{1}$ has at least two real roots. The polynomial $f^{\prime}(x)$ is divisible by $x^{3}$, and can have at most 9 real roots, therefore $B_{1}$ has at least one imaginary root. The resultant $R(B_{1},B_{0})$ equals 1, similarly to the Proposition 4.9, and the proof is over. $\hfill\square$

4.3. Dirichlet families: general case

In this subsection we construct Dirichlet families in more general situation than in the previous subsection.

Definition 4.11**.**

We say that an integer polynomial $C$ is of type ${\mathcal{J}}$ if $C=B_{0}B_{1}\cdots B_{k}$ where $B_{i}$ have the following properties:

${\mathcal{J}}$1) $B_{0}$ has no real roots.

${\mathcal{J}}$2) For every $i\geqslant 1$ the polynomial $B_{i}$ is irreducible, it has at least one real root and at least one imaginary root.

${\mathcal{J}}$3) For every $i,j\geqslant 0$ the polynomials $B_{i},B_{j}$ are strongly coprime.

We say that a matrix $M\in SL(N,{\mathbb{Z}})$ is of type ${\mathcal{J}}$ if its characteristic polynomial is of type ${\mathcal{J}}$. $\hfill\triangle$

We have obviously ${\mathcal{J}}_{0}\Leftarrow{\mathcal{J}}\Leftarrow{\mathcal{J}}_{1}$. Here is one example of a polynomial of type ${\mathcal{J}}$ which is not of type ${\mathcal{J}}_{1}$.

Proposition 4.12**.**

Let $A(t)=t^{2}+1$ and

[TABLE]

Let $q$ be any natural number $\geqslant 1$, and put $B_{0}(t)=(A(t))^{q}$. Then the polynomial $C(t)=B_{0}(t)B_{1}(t)B_{2}(t)$ is of type ${\mathcal{J}}$.

*Proof. *It is easy to check that each of the polynomials $B_{1},B_{2}$ is irreducible over ${\mathbb{Z}}$ and has two real roots (both negative), and two imaginary roots. As for the resultants of these polynomials we have

[TABLE]

the proposition follows. $\hfill\square$

Theorem 4.13**.**

Let $M\in SL(N,{\mathbb{Z}})$ be a matrix of type ${\mathcal{J}}$. Then there exists a $D$-family of polynomials for $M$.

*Proof. *Denote by $s_{j}$ the number of real roots of $B_{j}$, let $s=\sum_{j=1}^{k}s_{j}$. The ring ${\mathbb{Z}}[t]/C$ has natural projections $\phi_{j}:{\mathbb{Z}}[t]/C\to{\mathbb{Z}}[t]/B_{j}$. Consider the product of these projections

[TABLE]

Since the polynomials $B_{j}$ are strongly pairwise coprime, $\phi$ is an isomorphisms by the Chinese remainder theorem (see [5], Th 2.25, p. 131).

For $1\leqslant j\leqslant k$ apply Proposition 4.1 to the polynomial $B_{j}$; we obtain polynomials $P_{j,i}$, where $1\leqslant i\leqslant s_{j}$. The polynomials of the Dirichlet family in construction will be indexed by couples $(j,i)$ with $1\leqslant j\leqslant k$ and $1\leqslant i\leqslant s_{j}$. Let $(j,i)$ be such couple.

Let $E_{j,i}\in{\mathbb{Z}}[t]$ be any polynomial invertible $\mod B_{0}$. Let $D_{j,i}$ be an integer polynomial  such that

[TABLE]

[TABLE]

[TABLE]

We can choose such polynomials since the map (6) is an isomorphism. We claim that the family ${\mathcal{D}}=\{D_{j,i}\}$ is a $D$-family for $M$.

Let us first prove the property $\Delta 2)$. Denote by $\alpha_{j,i}$ the real roots of $B_{j}$ (here $1\leqslant i\leqslant s_{j}$). We have $P_{j,i}(\alpha_{j,\lambda})>0$ for every $\lambda$; this implies $D_{j,i}(\alpha_{j,\lambda})>0$ since $D_{j,i}\equiv P_{j,i}\ {\rm mod}(B_{j})$ and $B_{j}(\alpha_{j,\lambda})=0$. Similarly, if $\mu\not=j$, then $D_{j,i}(\alpha_{\mu,\lambda})=1$ for every $\lambda$ and the property $\Delta 2)$ holds.

Let us proceed to the property $\Delta 1)$. For an integer polynomial $A$ we will denote by $\overline{A}$ its image in ${\mathbb{Z}}[t]/C$. Observe that for every $\mu\geqslant 0$ the element $\phi_{\mu}(\overline{D_{j,i}})$ is invertible, therefore $\overline{D_{j,i}}$ is also invertible, since $\phi$ is an isomorphism. This implies that the matrix $D_{j,i}(M)$ is invertible over ${\mathbb{Z}}$. Applying Lemma 4.3 we deduce that $\det D_{j,i}(M)=1$.

To prove $\Delta 3)$ observe that the $s\times s$ matrix $\log D_{j,i}(\alpha_{\mu,\lambda})$ is block-diagonal. It has $k$ blocks of sizes $s_{1},\ldots,s_{k}$ and the $j$-th block equals the $(s_{j}\times s_{j})$-matrix $\log D_{j,*}(\alpha_{j,*})$ with $1\leqslant*\leqslant s_{j}$. The determinants of these matrices are non zero by the property 3) of Proposition 4.1. The proof of Theorem 4.13 is now over. $\hfill\square$

Definition 4.14**.**

There is a large choice for polynomials $E_{j,i}$. In particular the polynomials $E_{j,i}(t)=t$ will do. Any Dirichlet family corresponding to this case will be called primary.

5. The manifold $T(M,{\mathcal{D}})$

Now we summarize the contents of the previous section to obtain the generalized OT-manifold. In this section $M$ denotes a matrix in $SL(N,{\mathbb{Z}})$ of type ${\mathcal{J}}$. This matrix has $s$ simple real eigenvalues with $s>0$ and at least one imaginary eigenvalue, and we have $N=s+2n$ with $n>0$. Let ${\mathcal{D}}=(D_{1},\ldots,D_{s})$ be any D-family for $M$. Such families exist by Theorem 4.13 of Subsection 4.3. The family ${\mathcal{D}}(M)=(D_{1}(M),\ldots,D_{s}(M))$ of integer matrices determines an action of ${\mathbb{Z}}^{s}$ on ${\mathbb{Z}}^{s+2n}$. Denote by $G_{\mathcal{D}}$ the group associated to this action, so that $G_{\mathcal{D}}\approx{\mathbb{Z}}^{s}\ltimes{\mathbb{Z}}^{s+2n}$. The next result is the main theorem of the present paper. It follows immediately from Proposition 3.9.

Theorem 5.1**.**

The group $G_{\mathcal{D}}$ acts by biholomorphisms on the space ${\mathbb{H}}^{s}\times{\mathbb{C}}^{n}$. This action is properly discontinuous and the quotient $T(M,{\mathcal{D}})$ is a compact complex $(n+s)$-dimensional manifold. We have a natural fibration $T(M,{\mathcal{D}})\to{\mathbb{T}}^{s}$ with fiber ${\mathbb{T}}^{s+2n}$, and the monodromy group isomorphic to ${\mathbb{Z}}^{s}$. The action of generators of the monodromy group on ${\mathbb{T}}^{s+2n}$ is given by the matrices $D_{1}(M^{\top}),\ldots,D_{s}(M^{\top})$. $\hfill\square$

5.1. Relation with generalized Inoue manifolds from [6]

In this subsection we explain the relation of manifolds $T(M,{\mathcal{D}})$ with the generalized Inoue manifolds, constructed in [6]. Recall from [6] that a matrix $M\in SL(2n+1,{\mathbb{Z}})$ is called matrix of type ${\mathcal{I}}$ if it has only one real eigenvalue which is simple, positive and irrational. In [6] we associated to each such matrix a complex non-Kähler manifold $T_{M}$ of complex dimension $n+1$ endowed with a fibration $T_{M}\to S^{1}$, whose fiber is a torus ${\mathbb{T}}^{2n+1}$.

A matrix of type ${\mathcal{I}}$ is clearly a matrix of type ${\mathcal{J}}_{0}$ with $s=1$. In this case the family consisting of one polynomial $D_{1}(t)=t$ is a Dirichlet family. The twisted-diagonal action of the semidirect product ${\mathbb{Z}}\ltimes{\mathbb{Z}}^{2n+1}$ corresponding to $D_{1}$ is the same as the action constructed in [6] and the manifold $T(M,{\mathcal{D}})$ is biholomorphic to the manifold $T_{M}$ from [6].

6. Mapping multi-tori

and their homological properties

From the topological point of view Inoue surfaces are just the mapping tori of the corresponding linear map of ${\mathbb{T}}^{3}$. In this section we discuss a generalization of the mapping torus construction, which leads to a simple description of the topological type of $T(M,{\mathcal{D}})$ and enables us to establish its basic topological properties.

Definition 6.1**.**

Let $X$ be a topological space and $\phi:{\mathbb{Z}}^{s}\to Homeo(X)$ a group homomorphism. Consider the diagonal action of ${\mathbb{Z}}^{s}$ on ${\mathbb{R}}^{s}\times X$ (${\mathbb{Z}}^{s}$ acting on ${\mathbb{R}}^{s}$ by translations). The mapping multi-torus of $\phi$ is the quotient of ${\mathbb{R}}^{s}\times X$ by this action. It is denoted by $MT(\phi)$.

Remark 6.2**.**

We have a natural fibration $p_{\phi}~{}:~{}MT(\phi)\to{\mathbb{T}}^{s}$ with fiber $X$. 2. 2)

If $s=1$ then the space $MT(\phi)$ is the mapping torus of the homeomorphism $\phi(1):X\to X$. 3. 3)

Choose a basis $t_{1},\ldots,t_{s}$ in ${\mathbb{Z}}^{s}$, let $G_{l}$ be the free abelian subgroup generated by $t_{1},\ldots,t_{l}$. Let $\phi_{k}=\phi~{}|~{}G_{k}$. The map $\phi(t_{k+1})$ determines a self-homeomorphism $\psi_{k+1}:MT(\phi_{k})\to MT(\phi_{k})$ and $MT(\phi_{k+1})$ is the mapping torus of $\psi_{k+1}$.

Definition 6.3**.**

A homomorphism $\ \phi:{\mathbb{Z}}^{s}\to Homeo(X)$ is called special if

The space $X$ is connected. 2. 2)

All the maps $\phi(g)$ for $g\in{\mathbb{Z}}^{s}$ have a common fixed point. 3. 3)

There is $g\in{\mathbb{Z}}^{s}$  such that   the homomorphism $(\phi(g))_{*}-\text{\rm Id}:H_{1}(X,{\mathbb{Q}})\to H_{1}(X,{\mathbb{Q}})$ is injective.

Observe that the property 2) from the above definition implies existence of a cross-section of the fibration $p_{\phi}$.

Proposition 6.4**.**

If $\phi$ is special, then $b_{1}(MT(\phi))=s$, and the fibration $p_{\phi}$ induces an isomorphism in $H_{1}(\ -\ ,{\mathbb{Q}})$.

*Proof. *Choose a basis $t_{1},\ldots,t_{s}\in{\mathbb{Z}}^{s}$ in such a way that $1$ is not an eigenvalue of the homomorphism  $(\phi(t_{1}))_{*}$ of $H_{1}(X,{\mathbb{Q}})$. Let $B_{l}=MT(\phi_{l})$. We will now prove two following assertions by induction in $l$ with $l\geqslant 1$.

1.   $b_{1}(B_{l},{\mathbb{Q}})=l$,   and

2. $(\phi(t_{l+1}))_{*}:H_{1}(B_{l},{\mathbb{Q}})\to H_{1}(B_{l},{\mathbb{Q}})$ equals Id.

Assuming that these assertions hold for a given value of $l\geqslant 1$ write the Milnor’s exact sequence (see [16])

[TABLE]

The space $B_{l}$ is connected, therefore $b_{1}(B_{l+1})\leqslant l+1$. The fibration $B_{l+1}\to{\mathbb{T}}^{l+1}$ has a cross-section, therefore $b_{1}(B_{l+1})\geqslant l+1$. The part 1) of the induction step follows. The part 2) follows from the fact that the cross-section induces isomorphism in $H_{1}$. The induction step is now complete. A similar argument proves the assertion for $l=1$, here we use the fact that 1 is not an eigenvalue of the homomorphism $(\phi_{1})_{*}$. $\hfill\square$

Proposition 6.5**.**

*Assume that the properties

1. and 2) of Definition 6.3 hold for a group homomorphism  $\phi:{\mathbb{Z}}^{s}\to Homeo(X)$. Then the group $\pi_{1}(MT(\phi))$ is the semidirect product ${\mathbb{Z}}^{s}\ltimes\pi_{1}(X)$ with respect to  the corresponding homomorphism   $\tilde{\phi}:{\mathbb{Z}}^{s}\to Aut(\pi_{1}(X))$. (Namely, $\tilde{\phi}$ sends any element $g\in{\mathbb{Z}}^{s}$ to the automorphism in $\pi_{1}$ induced by $\phi(g)$.)*

*Proof. *Follows immediately from the exact sequence of the fibration $p_{\phi}$. $\hfill\square$

Definition 6.6**.**

Let $\rho:{\mathbb{Z}}^{s}\to GL(k,{\mathbb{Z}})$ be a homomorphism. Denote by $G_{s,k}(\rho)$ the corresponding semidirect product ${\mathbb{Z}}^{s}\ltimes{\mathbb{Z}}^{k}$. We say that $\rho$ is special if there is $g\in{\mathbb{Z}}^{s}$  such that   $1$ is not an eigenvalue of $\rho(g)$. We say that $\rho$ is diagonalizable if for every $g\in{\mathbb{Z}}^{s}$ the matrix $\rho(g)$ is diagonalizable over ${\mathbb{C}}$.

Each homomorphism  $\rho:{\mathbb{Z}}^{s}\to GL(k,{\mathbb{Z}})$ determines a homomorphism  $\bar{\rho}:{\mathbb{Z}}^{s}\to Homeo({\mathbb{T}}^{k})$; if $\rho$ is special then $\bar{\rho}$ is also special.

Corollary 6.7**.**

If $\rho:{\mathbb{Z}}^{s}\to GL(k,{\mathbb{Z}})$ is special then $b_{1}(G_{s,k}(\rho))=s$. $\hfill\square$

Lemma 6.8**.**

Assume that a homomorphism  $\rho:{\mathbb{Z}}^{s}\to GL(k,{\mathbb{Z}})$ is special. Let $f_{1},f_{2}:G_{s,k}(\rho)\to{\mathbb{Z}}^{s}$ be epimorphisms. Then there is an isomorphism $\psi:{\mathbb{Z}}^{s}\to{\mathbb{Z}}^{s}$  such that   $f_{2}=f_{1}\circ\psi$.

*Proof. *Consider the Hurewicz homomorphism modulo torsion:

[TABLE]

The homomorphism  $f_{1}$ factors through $h$, so that $f_{1}=\psi_{1}\circ h$. The map $\psi_{1}$ is then an epimorphism of ${\mathbb{Z}}^{s}$ onto itself, therefore it is an isomorphism. Similarly $f_{2}=\psi_{2}\circ h$ and the proof is over. $\hfill\square$

Proposition 6.9**.**

Assume that $G_{s,k}(\rho)\approx G_{t,m}(\theta)$ where $\rho$ is special. Assume that $b_{1}(G_{t,m}(\theta))=s$. Then $s=t,\ k=m$ and there is an isomorphism $\psi:{\mathbb{Z}}^{s}\to{\mathbb{Z}}^{s}$  such that   the representations $\rho$ and $\theta\circ\psi$ are equivalent. In particular, if $\rho$ is diagonalizable, then $\theta$ is also diagonalizable.

*Proof. *Let $\Psi:G_{s,k}(\rho)\to G_{t,m}(\theta)$ be an isomorphism. Denote by $j:G_{t,m}\to{\mathbb{Z}}^{s}$ the Hurewicz homomorphism modulo torsion (similarly to (7)). Lemma 6.8 implies that there is an isomorphism $\psi$  such that   the following diagram is commutative.

[TABLE]

Then $\psi_{0}$ is necessarily an isomorphism, therefore $m=k$ and for each $g\in{\mathbb{Z}}^{s}$ the matrix $\rho(g)$ is conjugate to $\theta(\psi(g))$ via the isomorphism $\psi_{0}$. $\hfill\square$

Let us now return to generalized Inoue manifolds. Let $M\in SL(s+2n,{\mathbb{Z}})$ and ${\mathcal{D}}$ be a D-family for $M$. Every map $g_{i}=D_{i}(g_{0}):{\mathbb{R}}^{s}\times{\mathbb{C}}^{n}\to{\mathbb{R}}^{s}\times{\mathbb{C}}^{n}$ preserves the lattice ${\mathcal{L}}$ and determines a self-homeomorphism $\bar{g}_{i}$ of the torus $X={\mathbb{R}}^{s}\times{\mathbb{C}}^{n}/{\mathcal{L}}.$ Define a map $\phi:{\mathbb{Z}}^{s}\to Homeo(X)$ by $\phi(t_{i})=g_{i}$; then the manifold $T(M,{\mathcal{D}})$ is clearly homeomorphic to the mapping multi-torus $MT(\phi).$

Corollary 6.10**.**

The group $\pi_{1}(T(M,{\mathcal{D}}))$ is isomorphic to the semidirect product ${\mathbb{Z}}^{s}\ltimes{\mathbb{Z}}^{s+2n}$ where the action of ${\mathbb{Z}}^{s}$ on ${\mathbb{Z}}^{s+2n}\approx{\mathcal{L}}$ is given by the homomorphism $t_{i}\mapsto D_{i}(M^{\top})$.

Denote by $G({\mathcal{D}})$ the subgroup of $SL(s+2n,{\mathbb{Z}})$ generated by $D_{i}(M^{\top})$ (where $1\leqslant i\leqslant s$).

Definition 6.11**.**

The Dirichlet family ${\mathcal{D}}$ is called special, if in the subgroup $G({\mathcal{D}})$ there is at least one matrix without eigenvalue $1$.

For example in the case $s=1$ the Dirichlet family consisting of one polynomial $D_{1}(t)=t$ is special.

Proposition 6.12**.**

Let $M$ be a matrix of type ${\mathcal{J}}$ with characteristic polynomial $C=B_{0}\cdot B_{1}\cdot\ldots\cdot B_{k}$ (see Definition 4.11). Then any primary Dirichlet family is special. (In particular this is the case when $B_{0}=1$.)

*Proof. *The matrix

[TABLE]

has a non-zero determinant. Therefore some linear combination of its lines has only strictly positive coordinates. Thus for some $n_{1},\ldots,n_{s}\in{\mathbb{Z}}$ we have

[TABLE]

Let $N=\prod_{i=1}^{s}D^{n_{i}}_{i}(M^{\top})$. Then $N\in G({\mathcal{D}})$ and $1$ is not an eigenvalue of $N~{}|~{}V$. Further, $N~{}|~{}(W\oplus\bar{W})=M^{\top}~{}|~{}(W\oplus\bar{W})$ since the family ${\mathcal{D}}$ is primary. Therefore all eigenvalues of $N$ in $(W\oplus\bar{W})$ are imaginary. $\hfill\square$

Corollary 6.13**.**

Let ${\mathcal{D}}=(D_{1},\ldots,D_{s})$ be a special Dirichlet family for $M$. Then $b_{1}(T(M,{\mathcal{D}}))=s$. $\hfill\square$

7. LCK structures on $T(M,{\mathcal{D}})$

In this section we show that if $M$ is non-diagonalizable, and ${\mathcal{D}}$ is a primary Dirchlet family for $M$, then the manifold $T(M,{\mathcal{D}})$ does not admit a structure of a locally conformally Kähler manifold (in particular it is not Kähler). The argument follows basically the lines of the proof of the corresponding property of Oeljeklaus and Toma  manifolds ([17], Proposition 2.9), although we do not need the condition $s=1$ from [17].

Theorem 7.1**.**

Let $M$ be a matrix of type ${\mathcal{J}}$, and ${\mathcal{D}}$ be a primary Dirchlet family for $M$. Assume that at least one of the matrices $D_{i}(M)$ is non-diagonalizable. Then the manifold $T(M,{\mathcal{D}})$ does not admit a structure of a locally conformally Kähler manifold.

*Proof. *Denot $T(M,{\mathcal{D}})$ by $Y$ for brevity. We have a tower of coverings

[TABLE]

where $\pi$ is induced from the universal covering ${\mathbb{R}}^{s}\to{\mathbb{T}}^{s}$ via the fibration $Y\to{\mathbb{T}}^{s}$ from Theorem 5.1, and the covering $\chi$ is the universal covering for $\bar{Y}$. Assume that $Y$ admits an LCK-metric, let $\Omega$ be the corresponding $2$-form on $Y$ so that $d\Omega=\omega\wedge\Omega$, and $d\omega=0$. Then $d(\pi^{*}\Omega)=\pi^{*}\omega\wedge\pi^{*}\Omega$. The map $\pi$ induces the sero homomorphism  in $H_{1}(\cdot,{\mathbb{R}})$, therefore we have $\pi^{*}\omega=df$. The form $\theta=e^{-f}\pi^{*}\Omega$ on $\bar{Y}$ has the following properties: A) $\theta$ is closed.

B) For every $g\in{\mathbb{Z}}^{s}$ we have $g^{*}\theta=\rho(g)\cdot\theta$, where $\rho(g)=e^{-\langle[\omega],g\rangle}$.

C) The bilinear form $\tilde{\theta}$ associated to $\theta$ via the following formula:

[TABLE]

is positive definite.

The manifold $\bar{Y}$ is diffeomorphic to ${\mathbb{T}}^{s+2n}\times({\mathbb{R}}^{*}_{+})^{s}$, and so there is a natural action of the group ${\mathbb{T}}^{s+2n}$ on it. Denote by $\theta_{1}$ the 2-form obtained by averaging the form $\theta$ with respect to  this action and the usual Haar measure on ${\mathbb{T}}^{s+2n}$. It is clear that $\theta_{1}$ satisfies the conditions A) and C) above. The next lemma shows that the condition B) is also true for $\theta_{1}$.

Lemma 7.2**.**

For every $g\in{\mathbb{Z}}^{s}$ we have $g^{*}(\theta_{1})=\rho(g)\cdot\theta_{1}$.

*Proof. *Let $\xi\in{\mathbb{T}}^{s+2n}$. Observe that

[TABLE]

Therefore

[TABLE]

The diffeomorphism $\xi\mapsto g\xi\gamma^{-1}$ of ${\mathbb{T}}^{s+2n}$ onto itself preserves volumes, since all matrices $D_{i}(M)$ are in $SL(2n+s,{\mathbb{Z}})$. The changement of variables theorem implies that

[TABLE]

and we have $g^{*}\theta_{1}=\rho(g)\theta_{1}$. $\hfill\square$

The restriction of $\theta_{1}$ to any subspace ${\mathbb{T}}^{s+2n}\times v$ where $v\in({\mathbb{R}}^{*}_{+})^{s}$ has constant coefficients with respect to  the usual basis of $\Omega^{2}({\mathbb{T}}^{s+2n})$. Moreover let $\lambda=\chi^{*}(\theta_{1})$, then the coefficients of $\lambda$ with respect to  the usual basis of $\Omega^{2}({\mathbb{H}}^{s}\times{\mathbb{C}}^{n})$ do not depend on $\vec{z}$ (where $(\vec{w},\vec{z})$ denote the coordinates on ${\mathbb{H}}^{s}\times{\mathbb{C}}^{n}$). Thus the restriction of $\lambda$ to any subspace $w\times{\mathbb{C}}^{n}$ writes as

[TABLE]

Lemma 7.3**.**

The coefficients $\lambda_{ij}$ do not depend on $\vec{w}$.

*Proof. *Write

[TABLE]

[TABLE]

The coefficients of $d\lambda$ at $dw_{k}\wedge dz_{i}\wedge d\bar{z}_{j}$ and at $d\bar{w}_{k}\wedge dz_{i}\wedge d\bar{z}_{j}$

equal $\frac{\partial\lambda_{ij}(w)}{\partial w_{k}}$, and respectively $\frac{\partial\lambda_{ij}(w)}{\partial\bar{w}_{k}}$. Since $d\lambda=0$, the lemma follows. $\hfill\square$

Put $\lambda_{0}=\lambda~{}|~{}(1\times{\mathbb{C}}^{n})$. The symmetric bilinear form $\tilde{\lambda}_{0}$ associated to $\lambda_{0}$ via the formula (8) is a scalar product on ${\mathbb{C}}^{n}\approx{\mathbb{R}}^{2n}$. Since the family ${\mathcal{D}}$ is primary, the restriction of any matrix $D_{i}(M)$ to ${\mathbb{C}}^{n}$ equals $R^{\top}$ and is therefore non-diagonalizable. Put $U=D_{i}(M)$. Then we have $\mu(U\xi,U\eta)=C\mu(\xi,\eta)$ for $\xi,\eta\in{\mathbb{C}}^{n}$, so that the matrix $U_{0}=U/\sqrt{C}$ preserves the scalar product. This contradicts to the assumption that $U$ is non-diagonalizable. $\hfill\square$

8. Relations with the Oeljeklaus-Toma construction

This section is about relations of the manifolds $T(M,{\mathcal{D}})$ with the manifolds constructed by Oeljeklaus and Toma  in [17] (OT-manifolds for short). We begin by a brief recollection of Oeljeklaus-Toma construction, then we show that some of OT-manifolds are biholomorphic to manifolds $T(M,{\mathcal{D}})$. Then we show that in the non-diagonalizable case the manifold $T(M,{\mathcal{D}})$ is not homeomorphic to any of OT-manifolds.

8.1. Construction of OT-manifolds

Let $K$ be an algebraic number field. An embedding $K\hookrightarrow{\mathbb{C}}$ is called real if its image is in ${\mathbb{R}}$; an embedding which is not real is called complex. Denote by $s$ the number of real embeddings and by $n$ the number of complex embeddings. Then $(K:{\mathbb{Q}})=s+2n$. Let $\sigma_{1},\ldots,\sigma_{s}$ be the real embeddings and $\sigma_{s+1},\ldots,\sigma_{s+2n}$ be the complex embeddings; we can assume that $\sigma_{i}=\overline{\sigma_{n+i}}$ for $i\geqslant s+1$. The map

[TABLE]

is an embedding (known as geometric representation of the field $K$, see [3], Ch. II, §3). Let ${\mathscr{O}}$ be any order in $K$, then $\sigma({\mathscr{O}})$ is a full lattice in ${\mathbb{R}}^{s}\times{\mathbb{C}}^{n}.$ Denote by ${\mathscr{O}}^{*}$ the group of all units of ${\mathscr{O}}$. The Dirichlet Unit Theorem (see [3], Ch. II, §4 , Th. 5) says that the group ${\mathscr{O}}^{*}/Tors$ is a free abelian group of rank $s+n-1$. Assume that $n\geqslant 1$. Choose any elements $u_{1},\ldots,u_{s}$ of ${\mathscr{O}}^{*}$ generating in ${\mathscr{O}}^{*}/Tors$ a free abelian subgroup of rank $s$. A unit $\lambda\in{\mathscr{O}}$ will be called positive if $\sigma_{i}(\lambda)>0$ for every $i\leqslant s$. Replacing $u_{i}$ by $u_{i}^{2}$ if necessary we can assume that every $u_{i}$ is positive. The subgroup $U$ of ${\mathscr{O}}^{*}$ generated by $u_{1},\ldots,u_{s}$ acts on ${\mathscr{O}}$ and we can form the semidirect product ${\mathscr{P}}=U\ltimes{\mathscr{O}}$. The group ${\mathscr{P}}$ acts on ${\mathbb{C}}^{r}={\mathbb{C}}^{s}\times{\mathbb{C}}^{n}$ as follows:

• •

any element $\xi\in{\mathscr{O}}$ acts by translation by vector $\sigma(\xi)\in{\mathbb{R}}^{s}\times{\mathbb{C}}^{r}.$

• •

any element $\lambda\in U$ acts as follows:

[TABLE]

For $i\leqslant s$ the numbers $\sigma_{i}(\lambda)$ are real and positive, so the subset ${\mathbb{H}}^{s}\times{\mathbb{C}}^{n}$ is invariant under the action of ${\mathscr{P}}$. This action is properly discontinuous and the quotient is a complex analytic manifold of dimension $s+n$ which will be denoted by $X(K,{\mathscr{O}},U)$ . The notation $X(K,U)$ used in the article [17] pertains to the case when the order ${\mathscr{O}}$ is the maximal order of $K$.

8.2. OT-manifolds as manifolds of type $T(M,{\mathcal{D}})$

In this section we assume that there is a Dirichlet unit $\xi$ in $K$  such that   ${\mathbb{Q}}(\xi)=K$. This assumption holds for example when $s+n\geqslant 2$ and there are no proper subfields ${\mathbb{Q}}\subsetneq K^{\prime}\subsetneq K$; this is always the case if $s+n\geqslant 2$ and $2n+s$ is a prime number. Replacing $\xi$ by $\xi^{2}$ if necessary we can assume that $\xi$ is positive. Denote by $P$ the minimal polynomial of $\xi$, let $C_{P}$ be the companion matrix of $P$, and put $B_{P}=C_{P}^{\top}$. Denote by ${\mathscr{O}}$ the order ${\mathbb{Z}}[\xi]$, and let $U$ be any subgroup of the group of units of ${\mathbb{Z}}[\xi]$, isomorphic to ${\mathbb{Z}}^{s}$, and generated by positive units $u_{1},\ldots,u_{s}$.

Choose polynomials $D_{i}\in{\mathbb{Z}}[t]$,  such that   $u_{i}=D_{i}(\xi)$. The family ${\mathcal{D}}=(D_{1},\ldots,D_{s})$ is a Dirichlet family for the matrix $C_{P}$ (see Proposition 4.1 and Corollary 4.2).

Both $X(K,{\mathscr{O}},U)$ and $T(M,{\mathcal{D}})$ are obtained as quotients of actions of semidirect products of groups ${\mathbb{Z}}^{s}$ and ${\mathbb{Z}}^{s+2n}$ on ${\mathbb{H}}^{s}\times{\mathbb{C}}^{n}$. We are now going to describe these two actions in details and show that they are isomorphic. Let $\alpha_{1},\ldots,\alpha_{s},\beta_{1},\ldots,\beta_{n},\bar{\beta}_{1},\ldots,\bar{\beta}_{n}$ be the roots of $P$. Renumbering the roots of $P$ if necessary we can assume that $\sigma_{i}(\xi)=\alpha_{i}$ for $i\leqslant s$, and $\sigma_{i}(\xi)=\beta_{i},\ \sigma_{i+s}(\xi)=\overline{\beta_{i}}$ for $i>s$,

**1. **   The manifold  $T(B_{P},{\mathcal{D}}).$

The eigenvalues of the matrix $B_{P}$ are the same as those of the matrix $C_{P}$, that is, $\alpha_{1},\ldots,\alpha_{s},\beta_{1},\ldots,\beta_{n},\bar{\beta}_{1},\ldots,\bar{\beta}_{n},$. The corresponding eigenvectors of $B_{P}$ are:

[TABLE]

(where $1\leqslant r\leqslant s$ and $1\leqslant i\leqslant n$). We have therefore a decomposition of ${\mathbb{C}}^{s}\times{\mathbb{C}}^{2n}$ into a direct sum of invariant subspaces

[TABLE]

where $V$ is generated by the eigenvectors $a_{1},\ldots,a_{s}$ and $W$ is generated by the eigenvectors $b_{1},\ldots,b_{n}.$ The matrix $R$ is therefore diagonal with diagonal entries $\beta_{1},\ldots,\beta_{n}$. The vectors $v_{i}$ (see the formula (1)) are given by the formulas

[TABLE]

[TABLE]

The element $g_{0}$ (see the formula (2)) acts as follows:

[TABLE]

The elements $g_{i}$ (where $1\leqslant i\leqslant s$) act as follows:

[TABLE]

[TABLE]

**2. **   The manifold  $X(K,{\mathscr{O}},U).$ The $\sigma$-image of $\xi^{r}$ equals $(\alpha_{1}^{r},\ldots,\alpha^{r}_{s},\beta^{r}_{1},\ldots,\beta^{r}_{n})$; this vector equals $v_{r+1}$ (here $0\leqslant r\leqslant s+2n-1$). Thus the group of translations of ${\mathbb{H}}^{s}\times{\mathbb{C}}^{n}$ by elements of $\sigma({\mathbb{Z}}[\xi])$ is generated by the vectors $v_{i}$. The action of $\xi\in{\mathscr{O}}$ on ${\mathbb{H}}^{s}\times{\mathbb{C}}^{n}$ is given by

[TABLE]

[TABLE]

[TABLE]

Therefore the elements $D_{i}(\xi)$ generating the group $U$ act as follows:

[TABLE]

We deduce that the semidirect products of groups ${\mathbb{Z}}^{s}$ and ${\mathbb{Z}}^{2n+s}$ arising in the cases 1 and 2 are isomorphic and their actions on ${\mathbb{H}}^{s}\times{\mathbb{C}}^{n}$ are identical. Thus we arrive at the following conclusion.

Proposition 8.1**.**

We have a biholomorphism

[TABLE]

8.3. The non-diagonalizable case

Lemma 8.2**.**

Let $K$ be an algebraic number field admitting $s$ real and $n$ complex embeddings into ${\mathbb{C}}$ so that $(K:{\mathbb{Q}})=2n+s$. Let ${\mathscr{O}}$ be an order in $K$, denote by ${\mathscr{O}}^{*}$ the multiplicative group of units of ${\mathscr{O}}$. Let $U\approx{\mathbb{Z}}^{s}$ be a subgroup  such that   every $\gamma\in U$ is positive. Let $X=X(K,{\mathscr{O}},U)$ be the corresponding OT-manifold. Then the group $\pi_{1}(X)$ is a semidirect product ${\mathbb{Z}}^{s}\ltimes{\mathbb{Z}}^{2n+s}$ of diagonal type.

*Proof. *The group $\pi_{1}(X)$ is isomorphic to $G_{s,2n}(A)$ where $A:U\to GL(2n+s,{\mathbb{Z}})$ denotes the action of the group $U$ on the order ${\mathscr{O}}\approx{\mathbb{Z}}^{2n+s}$. Let $\gamma\in U$, and $P$ be the minimal polynomial of $\gamma$. The roots of $P$ are simple and different from 1. Since $P(A(\gamma))=0$, the minimal polynomial of the matrix $A(\gamma)$ has the same properties. Therefore $A(\gamma)$ is diagonalizable and Lemma is proved. $\hfill\square$

The following proposition is now a direct consequence of Proposition 6.9.

Proposition 8.3**.**

Let $M\in SL(2n+s,{\mathbb{Z}})$. Assume that there is a special Dirichlet family ${\mathcal{D}}=(D_{1},\ldots,D_{s})$ for $M$  such that   at least one of matrices $D_{i}(M)$ is non-diagonalizable over ${\mathbb{C}}$. Then the group $\pi_{1}(T(M,{\mathcal{D}}))$ is not isomorphic to the fundamental group of any of manifolds $X(K,{\mathscr{O}},U)$ constructed in [17]. Therefore $T(M,{\mathcal{D}})$ is not homeomorphic to any of manifolds $X(K,{\mathscr{O}},U)$.$\hfill\square$

The assumptions of the previous proposition hold if ${\mathcal{D}}$ is primary. Therefore we obtain the next Corollary.

Corollary 8.4**.**

Let $M\in SL(2n+s,{\mathbb{Z}})$ be a non-diagonalizable matrix of type ${\mathcal{J}}$, and ${\mathcal{D}}$ a primary Dirichlet family for $M$. Then $T(M,{\mathcal{D}})$ is not homeomorphic to any of manifolds $X(K,{\mathscr{O}},U)$.$\hfill\square$

9. Acknowledgements

The first author thanks the Nantes University and the DefiMaths program for the support and warm hospitality. The first author was partially supported by JSPS KAKENHI Grant Numbers 16K05142, 17H06128, 26287013, 19H01788. The second author thanks Professor F. Bogomolov for initiating him to the theory of Inoue surfaces in 2013, for several discussions on this subject and for support. The work on this article began in January 2018 when the second author was visiting the Tokyo Institute of Technology; many thanks for the warm hospitality and support. We are grateful to K. Conrad and to Th. Laffey, whose works [4], [13] helped a lot with the algebraic part of our project.