Compact complex non-Kähler manifolds associated with totally real reciprocal units

Abstract

Using the theory of totally real number fields we construct a new class of compact complex non-Kähler manifolds in every even complex dimension and study their analytic and geometric properties.

1 Introduction

In this paper we construct a new class of compact complex non-Kähler manifolds in every complex dimension $n=2d, d\geqslant 1$, and investigate their complex analytic and topological properties.

Using totally real number fields we construct first 4d-dimensional real solvable Lie groups G admitting irreducible cocompact discrete subgroups $\Gamma$. This method works in general to produce real solv-manifolds. Next we show that these Lie groups admit left invariant complex structures. The left quotient $X:=\Gamma \backslash G$ is then a compact complex manifold. In the case $d=1$, one recovers the Inoue surfaces noted $S_{N}^{ {(+)}}$ in the famous paper [4], whose extension to higher dimensions had remained open since 1974. In 2005, Inoue surfaces of type $S_M$ were generalized in [6].

In the following sections we prove that the identity component of the holomorphic automorphism group ${{\,\mathrm{Aut}\,}}^0(X)$ is isomorphic to $(\mathbb {C}^{*})^{d}$ and that the whole group ${{\,\mathrm{Aut}\,}}(X)$ has infinitely many components if $d\geqslant 2$.

The action of $(\mathbb {C}^{*})^{d}$ is free, induces a holomorphic foliation ${\mathcal {F}}$ which is transversely hyperbolic, and is preserved by the whole automorphism group. If $d=2$, the restriction of certain automorphisms of X to the tangent bundle $T{{\mathcal {F}}}$ has an Anosov property in the sense that this bundle splits transversely into a stable and an unstable subbundle.

Furthermore we determine some topological invariants, prove that X is non-Kähler and show that the algebraic dimension is zero.

Open questions are whether some of the here constructed manifolds are locally conformally Kähler, whether they admit proper complex subvarieties, as well as whether they admit Anosov diffeomorphisms relative to ${\mathcal {F}}$ also for $d\geqslant 3$.

2 The construction

In this section we explain in detail the construction of a new class of compact complex manifolds. In the first two subsections we collect for the reader’s convenience a number of well-known facts about simply-connected nilpotent Lie groups, their rational structures and cocompact discrete subgroups, and the free 2-step nilpotent Lie algebra. In Sect. 2.3 we shall see how particular totally real number fields K allow the construction of irreducible rational structures on the d-fold product N of the three-dimensional real Heisenberg group. Then we extend N by an Abelian group in such a way that the corresponding solvable group possesses a cocompact discrete subgroup associated with the group of algebraic units in K, see Sect. 2.4. Finally, we show that the so obtained solv-manifolds carry a complex structure, which completes our construction.

2.1 Cocompact discrete subgroups of nilpotent Lie groups

In this section we recall some facts related to cocompact discrete subgroups of simply-connected nilpotent Lie groups. For proofs and more details we refer the reader to [7, Chapter II].

Let N be a simply-connected nilpotent real Lie group with Lie algebra ${\mathfrak {n}}$. A rational structure on N consists of a rational subalgebra ${\mathfrak {n}}_{\mathbb {Q}}$ of ${\mathfrak {n}}$ such that ${\mathfrak {n}}_{\mathbb {Q}}\otimes _{\mathbb {Q}} \mathbb {R}\cong {\mathfrak {n}}$. Equivalently, a rational structure on N is given by a basis ${\mathcal {B}}=(\xi _1,\dotsc ,\xi _n)$ of ${\mathfrak {n}}$ such that for all $1\leqslant k<l\leqslant n$ the coordinates of $[\xi _k,\xi _l]$ with respect to ${\mathcal {B}}$, i.e., the structure constants of ${\mathfrak {n}}$ with respect to ${\mathcal {B}}$, are rational.

Two rational structures on N are called isomorphic if the corresponding rational Lie algebras are isomorphic. A rational structure on N is called irreducible if ${\mathfrak {n}}_{\mathbb {Q}}$ is not isomorphic to the direct sum of two non-trivial ideals.

Remark

There are simply-connected nilpotent Lie groups N that do not admit any rational structure. It is also possible that N possesses several non-isomorphic rational structures, see [7, Remarks 2.14 and 2.15].

In order to explain how a rational structure on N yields cocompact discrete subgroups of N, we restate [7, Theorem 2.12] for the reader’s convenience. Let $\Lambda \subset {\mathfrak {n}}$ be any lattice of maximal rank contained in ${\mathfrak {n}}_{\mathbb {Q}}$. Then the group $\Gamma$ generated by $\exp (\Lambda )$ in N is a cocompact discrete subgroup of N. Any two discrete subgroups associated with the same rational structure are commensurable. Conversely, if $\Gamma \subset N$ is a cocompact discrete subgroup, then the $\mathbb {Z}$-span of $\exp ^{-1}(\Gamma )$ in ${\mathfrak {n}}$ is a lattice of maximal rank in ${\mathfrak {n}}$ and any basis of ${\mathfrak {n}}$ contained in this lattice defines a rational structure on N. If $\widetilde{\Gamma }$ is commensurable with $\Gamma$, then the associated rational structures are isomorphic.

Remark

It follows from the preceding considerations that the rational structure on N is irreducible, if and only if the associated cocompact discrete subgroup of N is not commensurable to the direct product of two non-trivial normal subgroups.

2.2 The free 2-step nilpotent Lie algebra

Let V be a 2d-dimensional real vector space and let $\bigwedge ^2V$ denote its exterior algebra. On the vector space $V\oplus \bigwedge ^2V$ we define a Lie bracket by

$$\begin{aligned} \bigl [(v,\alpha ),(w,\beta )\bigr ]:=(0,v\wedge w). \end{aligned}$$

The resulting Lie algebra is the free 2-step nilpotent Lie algebra ${\mathfrak {f}}_{2d}$ of dimension $2d+\left( {\begin{matrix}2d\\ 2\end{matrix}} \right) =2d^2+d$. We have

$$\begin{aligned} W:=\bigwedge ^2V={\mathfrak {f}}_{2d}'={\mathcal {Z}}({\mathfrak {f}}_{2d}), \end{aligned}$$

where ${\mathcal {Z}}({\mathfrak {f}}_{2d})$ denotes the center of ${\mathfrak {f}}_{2d}$.

Let $F_{2d}$ be the simply-connected nilpotent Lie group with Lie algebra ${\mathfrak {f}}_{2d}$.

Example

The Lie algebra ${\mathfrak {f}}_{2}$ is isomorphic to the 3-dimensional Heisenberg algebra ${\mathfrak {h}}_3$. An explicit isomorphism is given by

$$\begin{aligned} {\mathfrak {h}}_3\rightarrow {\mathfrak {f}}_{2},\quad \begin{pmatrix} 0\quad &{}\quad x\quad &{}\quad z\\ 0\quad &{}\quad 0\quad &{}\quad y\\ 0\quad &{}\quad 0\quad &{}\quad 0 \end{pmatrix} \mapsto (xe_1+ye_2,ze_1\wedge e_2)\in \mathbb {R}^2\oplus \bigwedge ^2\mathbb {R}^2. \end{aligned}$$

On the group level, one can realize the 3-dimensional Heisenberg group as

$$\begin{aligned} H_3:=\left\{ \begin{pmatrix} 1\quad &{}\quad x\quad &{}\quad z\\ 0\quad &{}\quad 1\quad &{}\quad y\\ 0\quad &{}\quad 0\quad &{}\quad 1 \end{pmatrix};\ x,y,z\in \mathbb {R} \right\} . \end{aligned}$$

The map

$$\begin{aligned} \begin{pmatrix} 1\quad &{}\quad x\quad &{}\quad z\\ 0\quad &{}\quad 1\quad &{}\quad y\\ 0\quad &{}\quad 0\quad &{}\quad 1 \end{pmatrix} \mapsto \left( x,y,z-\frac{xy}{2}\right) \end{aligned}$$

yields an explicit isomorphism with the realization of the Heisenberg group as the free nilpotent group $F_2$, with group structure given by

$$\begin{aligned}(x,y,z)({{\tilde{x}}},{{\tilde{y}}},{{\tilde{z}}})=\left( x +{{\tilde{x}}},y+\tilde{y},z+{{\tilde{z}}}+\frac{1}{2}(x{{\tilde{y}}} - {{\tilde{x}}} y)\right) .\end{aligned}$$

We shall therefore use in the sequel $H_3$ as a model for $F_2$.

The choice of any basis of V leads to a rational structure on $F_{2d}$ as follows. Let $(e_1,\dotsc ,e_{2d})$ be a basis of V and put $f_{k,l}:=[e_k,e_l]$. Then

$$\begin{aligned} (e_1,\dotsc ,e_{2d},f_{1,2},f_{1,3},\dotsc ,f_{2d-1,2d}) \end{aligned}$$

is a basis of ${\mathfrak {f}}_{2d}$ with respect to which the structure constants of ${\mathfrak {f}}_{2d}$ are rational. As explained above, this procedure yields cocompact discrete subgroups of $F_{2d}$.

2.3 Rational structures associated with totally real number fields

Let $\mathbb {Q}\subset K$ be a field extension of degree 2d, $d\geqslant 1$, and let ${\mathcal {O}}_K \subset K$ be its ring of algebraic integers. Choose elements $\omega _1,\dotsc ,\omega _{2d}\in {\mathcal {O}}_K$ such that

$$\begin{aligned} {\mathcal {O}}_K\cong \mathbb {Z}\omega _1\oplus \cdots \oplus \mathbb {Z}\omega _{2d} \end{aligned}$$

as $\mathbb {Z}$-modules.

We suppose that K is totally real, i.e., that all 2d embeddings $\sigma _1,\dotsc ,\sigma _{2d}:K\rightarrow \mathbb {R}\subset \mathbb {C}$ are real and consider the map $\sigma :K\rightarrow V:=\mathbb {R}^{2d}$ given by $\sigma (x)=\bigl (\sigma _1(x),\dotsc ,\sigma _{2d}(x)\bigr )$. It follows that $\Lambda _K:=\sigma ({\mathcal {O}}_K)\subset V$ is a lattice of maximal rank generated by $e_k:=\sigma (\omega _k)$ for $1\leqslant k\leqslant 2d$.

Now, as mentioned above, the basis ${\mathcal {B}}=\bigl (e_{1},\dotsc ,e_{2d}\bigr )$ of V yields the basis

$$\begin{aligned} \widetilde{{\mathcal {B}}}=(e_1,\dotsc ,e_{2d},f_{1,2},f_{1,3},\dotsc ,f_{2d-1,2d}) \end{aligned}$$

of ${\mathfrak {f}}_{2d}$ and induces therefore a rational structure on $F_{2d}$. In the following, $({\mathfrak {f}}_{2d})_{\mathbb {Q}}$ always denotes the corresponding rational Lie algebra.

Let ${\mathcal {O}}^*_K$ be the (multiplicative) group of units in ${\mathcal {O}}_K$. We say that a unit $u\in {\mathcal {O}}^*_K$ is totally positive if $\sigma _j(u)>0$ for all $1\leqslant j\leqslant 2d$, and we write ${\mathcal {O}}^{*,+}_K$ for the group of totally positive units. Due to Dirichlet’s theorem, the group ${\mathcal {O}}^{*,+}_K$ is isomorphic to $\mathbb {Z}^{2d-1}$.

The group ${\mathcal {O}}^{*,+}_K$ acts on $\Lambda _K$ as a group of $\mathbb {Z}$-module automorphisms via

$$\begin{aligned} u\cdot \sigma (x):=\sigma (ux). \end{aligned}$$

Extending $\sigma (x)\mapsto \sigma (ux)$ to an $\mathbb {R}$-linear map $\rho (u) :V\rightarrow V$ we obtain a representation

$$\begin{aligned} \rho :{\mathcal {O}}^{*,+}_K\rightarrow \mathrm{{SL}}(V). \end{aligned}$$

Note that we have $\det \rho (u)=1$ for all $u\in {\mathcal {O}}^{*,+}_K$ since the eigenvalues of $\rho (u)$ are the conjugates $\lambda _k=\sigma _{k}(u)$, $k=1,\dotsc ,2d$, of u and thus all positive.

Remark

Every element of $\mathrm{{GL}}(V)$ extends uniquely to an automorphism of the Lie algebra ${\mathfrak {f}}_{2d}$. If the matrix of an element in $\mathrm{{GL}}(V)$ with respect to ${\mathcal {B}}$ has rational coefficients, we obtain an automorphism of $({\mathfrak {f}}_{2d})_{\mathbb {Q}}$.

Note that by construction the matrix of $\rho (u)$ with respect to ${\mathcal {B}}$ lies in $\mathrm{{SL}}(2d,\mathbb {Z})$ for every $u \in {\mathcal {O}}^{*,+}_K$. Hence, $\rho ({\mathcal {O}}^{*,+}_K)$ is a discrete subgroup of $\mathrm{{SL}}(V)$. Furthermore, $\rho (u)$ induces an automorphism of ${\mathfrak {f}}_{2d}$ that leaves $({\mathfrak {f}}_{2d})_{\mathbb {Q}}$ invariant. We will denote this automorphism of ${\mathfrak {f}}_{2d}$ as well as its restriction to $({\mathfrak {f}}_{2d})_{\mathbb {Q}}$ by $\widehat{\rho }(u)$, where $\widehat{\rho }:{\mathcal {O}}^{*,+}_K\rightarrow {{\,\mathrm{Aut}\,}}\bigl (({\mathfrak {f}}_{2d})_{\mathbb {Q}} )\bigr )$. Clearly, $\widehat{\rho }(u)$ respects the decomposition

$$\begin{aligned} ({\mathfrak {f}}_{2d})_{\mathbb {Q}}=V_{\mathbb {Q}}\oplus W_{\mathbb {Q}} \end{aligned}$$

and the eigenvalues of the restriction of $\widehat{\rho }(u)$ to $W_{\mathbb {Q}}$ are the products $\sigma _{k}(u)\sigma _{l}(u)=\lambda _{k} \lambda _{l}$, $1\leqslant k<l \leqslant 2d$.

We formulate the following lemma in our setting, but it remains also valid in a more general form. Its proof is a slight adaptation of the proof of [1, Proposition 2.1.4].

Lemma 2.1

The set of all endomorphisms $\rho (u)$ of $V_{\mathbb {Q}},$ $u\in {\mathcal {O}}_K^*$ is simultaneously diagonalizable over the Galois closure L of K.

Proof

Since the number field K is totally real, there is a primitive unit $u_0\in {\mathcal {O}}^{*,+}$, see e.g. [9, Theorem 1.4]. Note that $V_{\mathbb {Q}}$ and K are isomorphic as rational vector spaces and that $\rho (u_0)$ corresponds to multiplication by $u_0$. It follows that the characteristic polynomial of $\rho (u_0)$ coincides with the minimal polynomial of $u_0$. Hence, $\rho (u_0)$ is diagonalizable over L and each of its eigenvalues has multiplicity 1. Since the group ${\mathcal {O}}^{*,+}$ is Abelian, it stabilizes every eigenspace, which proves the claim. $\square$

In order to proceed with the construction, we need the following lemma which was communicated to us with proof by Professor A. Dubickas. Recall that a unit $u\in {\mathcal {O}}^{*,+}_K$ is called reciprocal if u and $u^{-1}$ are conjugate, i.e., have the same minimal polynomial, which then is palindromic. Moreover, u is primitive if $K=\mathbb {Q}(u)$.

Lemma 2.2

For every $d\geqslant 1$ there exist totally real number fields of degree 2d which admit primitive reciprocal units.

Proof

Let $\alpha$ be a totally real algebraic integer of degree d and denote its minimal polynomial by P(X). Let L be an integer so large that for each of the d roots $\alpha _j$, $j=1,\dotsc ,d$, of P(X) we have $L+\alpha _j>2$. Consider then the polynomial $Q(X)=P(X+\frac{1}{X}-L)X^d$. It is clear that Q is a monic palindromic polynomial of degree 2d with 2d real roots, since $(L+\alpha _j)^2>4$, $j=1,\dotsc ,d$. Furthermore, for a generic choice of L, the polynomial Q is irreducible and its roots are totally real reciprocal units of degree 2d. $\square$

From now on we suppose that there exists a primitive reciprocal unit $u_0\in {\mathcal {O}}^{*,+}_K$. In other words, we suppose that $u_0$ and $u_0^{-1}$ are conjugate, as well as $K=\mathbb {Q}(u_0)$.

Proposition 2.3

Let $u_0\in {\mathcal {O}}^{*,+}_K$ be a primitive reciprocal unit. Then there exists a $\widehat{\rho }(u_0)$-invariant rational decomposition

$$\begin{aligned} W_{\mathbb {Q}}=(W_1)_{\mathbb {Q}}\oplus (W_2)_{\mathbb {Q}} \end{aligned}$$

where $W_1$ is the d-dimensional subspace of $\widehat{\rho }(u_0)$-fixed points in W.

Proof

Since the characteristic polynomial of $\rho (u_0)$ is palindromic, we can arrange the eigenvalues $\lambda _1,\dotsc ,\lambda _{2d}$ of $\rho (u_0)$ such that $\lambda _{2k}=\lambda _{2k-1}^{-1}$ for all $1\leqslant k\leqslant d$. Since the restriction of $\widehat{\rho }(u_0)$ to W is diagonalizable with eigenvalues $\lambda _k\lambda _l$ for $1\leqslant k<l\leqslant 2d$, we see that the subspace $W_1$ of $\widehat{\rho }(u_0)$-fixed points is of dimension d. Moreover, it follows that the characteristic polynomial of $\widehat{\rho }(u_0)|_W$ is divisible by $(x-1)^d$ in $\mathbb {Z}[x]$. This gives the desired decomposition of $W_{\mathbb {Q}}$ into two $\widehat{\rho }(u_0)$-stable rational subspaces, see [5, Theorem XI.4.1]. $\square$

Let us consider the rational Lie algebra

$$\begin{aligned} {\mathfrak {n}}_{\mathbb {Q}}:=({\mathfrak {f}}_{2d})_{\mathbb {Q}}/(W_2)_{\mathbb {Q}}. \end{aligned}$$

In the following we will view ${\mathfrak {n}}_{\mathbb {Q}}$ as $V_{\mathbb {Q}}\oplus \bigl (W_{\mathbb {Q}}/(W_2)_{\mathbb {Q}}\bigr )$. Note that $W_{\mathbb {Q}}/(W_2)_{\mathbb {Q}}$ coincides with the center of ${\mathfrak {n}}_{\mathbb {Q}}$.

Proposition 2.4

We have ${\mathfrak {n}}:={\mathfrak {n}}_{\mathbb {Q}}\otimes \mathbb {R}\cong {\mathfrak {h}}_3^d,$ i.e.,  the construction yields a rational structure on $N:=H_3^d.$ Moreover,  this rational structure on N is irreducible.

Proof

Let $(v_1,\dotsc ,v_{2d})$ be a basis of V such that $\rho (u_0)v_{2j-1}= \lambda _j v_{2j-1}$ and $\rho (u_0)v_{2j}=\lambda _j^{-1}v_{2j}$ for all $1\leqslant j\leqslant d$. Set $w_{j}:= [v_{2j-1},v_{2j}], \ j=1,\dotsc ,d$. Then

$$\begin{aligned} \mathbb {R}v_{2j-1}\oplus \mathbb {R}v_{2j}\oplus \mathbb {R}w_{j},\ j=1,\dotsc ,d, \end{aligned}$$

is a subalgebra of ${\mathfrak {f}}_{2d}$ isomorphic to ${\mathfrak {h}}_3$ which intersects $W_2$ trivially. Using the decomposition established in Proposition 2.3, one sees that these subalgebras commute pairwise modulo $W_2$, which proves the first claim.

In order to show that the rational structure on N is irreducible, suppose that we have a decomposition ${\mathfrak {n}}_{\mathbb {Q}}={\mathfrak {a}}_{\mathbb {Q}} \oplus {\mathfrak {b}}_{\mathbb {Q}}$ where ${\mathfrak {a}}_{\mathbb {Q}}$ and ${\mathfrak {b}}_{\mathbb {Q}}$ are non-trivial ideals of ${\mathfrak {n}}_{\mathbb {Q}}$. Then we have ${\mathfrak {n}}={\mathfrak {h}}_3^d={\mathfrak {a}}\oplus {\mathfrak {b}}$ for ${\mathfrak {a}}={\mathfrak {a}}_{\mathbb {Q}}\otimes _{\mathbb {Q}}\mathbb {R}$ and ${\mathfrak {b}}= {\mathfrak {b}}_{\mathbb {Q}}\otimes _{\mathbb {Q}}\mathbb {R}$.

If an ideal of ${\mathfrak {n}}$ contains an element of the form

$$\begin{aligned} \xi =\sum _{j=1}^d\bigr (\kappa _{j}v_{2j-1}+\mu _j v_{2j}+\nu _j w_{j} \bigr ) \end{aligned}$$

with $(\kappa _{j_{0}},\mu _{j_0})\not =(0,0)$, then this ideal contains also $w_{j_{0}}$. Define

$$\begin{aligned} J_{{\mathfrak {a}}}:=\left\{ 1\leqslant j_{0}\leqslant d;\ \exists \ \xi =\sum _{j=1}^d\bigr (\kappa _{j}v_{2j-1}+\mu _j v_{2j}+\nu _j w_{j}\bigr )\in {\mathfrak {a}}\text { with }(\kappa _{j_{0}},\mu _{j_{0}})\not =(0,0) \right\} \end{aligned}$$

and similarly $J_{{\mathfrak {b}}}$. Since ${\mathfrak {n}}={\mathfrak {a}}\oplus {\mathfrak {b}}$, the preceding observation implies that $J_{{\mathfrak {a}}}\cup J_{{\mathfrak {b}}}=\{1,\dotsc ,d\}$ and that this union is disjoint. Therefore, we can suppose that $J_{{\mathfrak {a}}}=\{1,\dotsc , k\}$ and hence get

$$\begin{aligned} \pi ({\mathfrak {a}})=\bigoplus _{j=1}^k(\mathbb {R}v_{2j-1}\oplus \mathbb {R}v_{2j})\quad \text {and}\quad \pi ({\mathfrak {b}})=\bigoplus _{j=k+1}^d(\mathbb {R}v_{2j-1}\oplus \mathbb {R}v_{2j}), \end{aligned}$$

where $\pi :{\mathfrak {h}}_3^d\rightarrow V$ is the projection along the center of ${\mathfrak {h}}_3^d$.

Since $\pi ({\mathfrak {a}})$ and $\pi ({\mathfrak {b}})$ are invariant under $\rho (u_0)$ and since $\pi$ is defined over $\mathbb {Q}$, we obtain the corresponding rational decomposition $V_{\mathbb {Q}}=\pi ({\mathfrak {a}}_{\mathbb {Q}})\oplus \pi ({\mathfrak {b}}_{\mathbb {Q}})$ into two rational $\rho (u_0)$-invariant subspaces. Now the claim follows from the fact that the characteristic polynomial of $\rho (u_0)$ is irreducible over $\mathbb {Q}$. $\square$

As explained in Sect. 2.1 we can now define a cocompact discrete subgroup of N as follows.

Definition 2.5

Let $\widehat{\Lambda }_K\subset V\oplus W/W_{2}$ be the full lattice generated by $\sigma (\omega _k)\in V$ for $1\leqslant k\leqslant 2d$ and the images of $f_{kl}=\bigl [\sigma (\omega _k),\sigma (\omega _l)\bigr ], k<l$ in $W_{\mathbb {Q}}/(W_2)_{\mathbb {Q}}$ for $1\leqslant k<l\leqslant 2d$. We define $\Gamma _N$ to be the discrete cocompact subgroup generated by $\exp (\widehat{\Lambda }_K)\subset N$.

Remark

Due to Proposition 2.4, the group $\Gamma _N$ is not commensurable to the product of two proper normal subgroups.

2.4 Solv-manifolds associated with totally real number fields

In this subsection, we construct an extension of N by an Abelian group which admits a cocompact discrete subgroup containing $\Gamma _N$.

Recall that ${\mathcal {O}}_K^{*,+}$ can be considered as a discrete Abelian subgroup of $\mathrm{{SL}}(2d,\mathbb {R})$ which leaves the lattice $\Lambda _K\subset V_{\mathbb {Q}}$ invariant. Moreover, its action extends to $V\oplus W$ leaving $W_{\mathbb {Q}}$ invariant.

Now we have

Proposition 2.6

The group ${\mathcal {O}}_K^{*,+}$ respects the decomposition $W_{\mathbb {Q}}= (W_1)_{\mathbb {Q}}\oplus (W_2)_{\mathbb {Q}}$. Consequently,  every $u\in {\mathcal {O}}_K^{*,+}$ acts on ${\mathfrak {n}}_{\mathbb {Q}}$ by an automorphism $\rho _{{\mathfrak {n}}}(u)$.

Proof

Due to Lemma 2.1, the transformations $\rho (u)\in {{\,\mathrm{End}\,}}(V)$ with $u\in {\mathcal {O}}^{*,+}_K$ are simultaneously diagonalizable. Thus the same holds for the transformations $\widehat{\rho }(u)\in \mathrm{{GL}}(V\oplus W)$. The claim follows from the fact that $W_1$ and $W_2$ are direct sums of eigenspaces for $\widehat{\rho }(u_0)$. In fact, $W_1$ is the eigenspace corresponding to the eigenvalue 1 and $W_2$ is a direct sum of eigenspaces with eigenvalues not equal to 1. $\square$

Since ${\mathcal {O}}_K^{*,+}$ acts on ${\mathfrak {n}}_{\mathbb {Q}}$ by automorphisms, it respects the decomposition ${\mathfrak {n}}_{\mathbb {Q}}=V_{\mathbb {Q}}\oplus {\mathcal {Z}}({\mathfrak {n}}_{\mathbb {Q}})$ where ${\mathcal {Z}}({\mathfrak {n}}_{\mathbb {Q}})\cong (W_1)_{\mathbb {Q}}$ as rational vector spaces. Consider the homomorphism

$$\begin{aligned} \psi :{\mathcal {O}}_K^{*,+}\rightarrow \mathrm{{SL}}\bigl ({\mathcal {Z}}( {\mathfrak {n}}_{\mathbb {Q}})\bigr ),\quad \psi (u):=\rho _{{\mathfrak {n}}}(u)|_{{\mathcal {Z}}({\mathfrak {n}}_{\mathbb {Q}})}. \end{aligned}$$

Note that by construction we have $u_0\in \ker (\psi )$.

Proposition 2.7

The subgroup $\ker (\psi )\subset {\mathcal {O}}^{*,+}_K$ has rank d,  hence is isomorphic to $\mathbb {Z}^d,$ and consists of reciprocal units in ${\mathcal {O}}^{*,+}_K$.

Proof

Let $T_n$ denote the group of diagonal matrices in $\mathrm{{SL}}(n,\mathbb {R})$ having strictly positive entries.

Due to Lemma 2.1, there exists an element $g_0\in \mathrm{{SL}}(2d,\mathbb {R})$ such that $g_0{\mathcal {O}}^{*,+}_Kg_0^{-1}$ is a discrete subgroup of $T_{2d}$. Every element $g_0^{-1}{{\,\mathrm{diag}\,}}(t_1,\dotsc ,t_{2d})g_0$ of $\widetilde{A}:=g_0^{-1}T_{2d}g_0$ induces a linear transformation on $W_1$ that is again diagonalizable with eigenvalues $(t_1t_2,t_3t_4,\dotsc ,t_{2d-1}t_{2d})$. This yields a homomorphism $\widehat{\psi }:\widetilde{A}\rightarrow \mathrm{{SL}}(d,\mathbb {R})$ which extends $\psi$ and whose image is conjugate to a subgroup of $T_d$.

According to Dirichlet’s theorem, we can view $\psi$ as a homomorphism from $\mathbb {Z}^{2d-1}$ to $\mathrm{{SL}}(d,\mathbb {Z})$. In particular, the image of $\psi$ is conjugate to a discrete subgroup of $T_d$ and therefore has rank at most $d-1$. This implies that the rank of $\ker (\psi )$ is at least d.

The Lie algebra $\widetilde{{\mathfrak {a}}}$ of $\widetilde{A}$ is conjugate to the set of trace zero diagonal matrices, hence $\widetilde{{\mathfrak {a}}}\cong \mathbb {R}^{2d-1}$. The derivative of $\widehat{\psi }:\widetilde{A}\rightarrow \mathrm{{SL}}(d,\mathbb {R})$ can be identified with the map $\mathbb {R}^{2d-1}\rightarrow \mathbb {R}^{d-1}$ given by

$$\begin{aligned} (x_1,\dotsc , x_{2d})\mapsto (x_1+x_2, x_3+x_4, \dotsc ,x_{2d-1}+x_{2d}), \end{aligned}$$

where we suppose that $x_1+\cdots +x_{2d}=0$. Since this map is surjective, its kernel is isomorphic to $\mathbb {R}^d$, which implies that the discrete subgroup $\ker (\psi )\subset A:=\ker (\widehat{\psi })$ is of rank at most d. $\square$

Let us summarize our construction. We have seen that we can view ${\mathcal {O}}^{*,+}_K\cong \mathbb {Z}^{2d-1}$ as a discrete subgroup of $\mathrm{{SL}}(2d,\mathbb {R})$ that normalizes $\Gamma _N$. The identity component of its real Zariski closure is $\widetilde{A}\cong (\mathbb {R}^{>0})^{2d-1}$ in $\mathrm{{SL}}(2,\mathbb {R})$, the elements of which are simultaneously diagonalizable. Moreover, the identity component of the real Zariski closure of $\Gamma _A:=\ker (\psi )\cong \mathbb {Z}^d$ is

$$\begin{aligned} A\cong \bigl \{(a_1,b_1,\dotsc ,a_d,b_d)\in (\mathbb {R}^{>0})^{2d};\ a_1b_1=\cdots =a_d b_d=1\bigr \}\cong (\mathbb {R}^{>0})^d. \end{aligned}$$

Consequently, $\Gamma _A$ acts on $\Gamma _N$ and we obtain the solvable discrete subgroup $\Gamma :=\Gamma _A\ltimes \Gamma _N$ which is cocompact in $A\ltimes N\cong (\mathbb {R}^{>0})^d\ltimes N$ and Zariski dense in $(\mathbb {R}^*)^d\ltimes N$.

Since for $a,b \in \mathbb {R}^{>0}$ and $x,y,z \in \mathbb {R}$ we have

$$\begin{aligned} \begin{pmatrix}ab \quad &{}\quad 0\quad &{} \quad 0 \\ 0 \quad &{}\quad b\quad &{}\quad 0 \\ 0 \quad &{} \quad 0 \quad &{} \quad 1\end{pmatrix} \begin{pmatrix}1 \quad &{}\quad x\quad &{} \quad z \\ 0\quad &{} \quad 1 \quad &{} \quad y \\ 0 \quad &{}\quad 0 \quad &{} \quad 1\end{pmatrix} \begin{pmatrix}a^{-1}b^{-1}\quad &{}\quad 0 \quad &{} \quad 0 \\ 0 \quad &{} \quad b^{-1} \quad &{} \quad 0 \\ 0 \quad &{} \quad 0 \quad &{} \quad 1 \end{pmatrix}= \begin{pmatrix}1\quad &{} \quad ax \quad &{} \quad abz \\ 0 \quad &{} \quad 1 \quad &{} \quad by \\ 0 \quad &{} \quad 0 \quad &{} \quad 1\end{pmatrix}, \end{aligned}$$

one can realise the Lie group ${\widetilde{G}}:={\widetilde{A}}\ltimes N\cong (\mathbb {R}^{>0})^{2d-1} \ltimes N$ as a matrix group isomorphic to

$$\begin{aligned} \left\{ \left. \begin{pmatrix} M_{1 } \quad &{}\quad 0 \quad &{} \quad \cdots \quad &{} \quad 0 \\ 0\quad &{} \quad M_{2 } \quad &{} \quad \cdots \quad &{} \quad 0\\ \vdots \quad &{} \quad \vdots \quad &{} \quad \ddots \quad &{} \quad \vdots \\ 0\quad &{} \quad 0\quad &{} \quad \cdots \quad &{} \quad M_{d} \end{pmatrix} \right| \ M_{i}= \begin{pmatrix} a_{i} b_{i}\quad &{}\quad x_{i}\quad &{}\quad z_{i}\\ 0\quad &{}\quad b_{i}\quad &{}\quad y_{i}\\ 0\quad &{}\quad 0\quad &{}\quad 1 \end{pmatrix},\ a_{i},b_{i}\in \mathbb {R}^{>0},x_{i},y_{i},z_{i}\in \mathbb {R}, \ i=1,\dotsc ,d \right\} . \end{aligned}$$

Under this isomorphism the group ${\mathcal {O}}^{*,+}_K$ corresponds to

$$\begin{aligned} \left\{ \left. \begin{pmatrix} D_{1 }\quad &{} \quad 0 \quad &{} \quad \cdots \quad &{} \quad 0 \\ 0\quad &{} \quad D_{2 } \quad &{} \quad \cdots \quad &{} \quad 0\\ \vdots \quad &{} \quad \vdots \quad &{} \quad \ddots \quad &{} \quad \vdots \\ 0\quad &{} \quad 0\quad &{} \quad \cdots \quad &{} \quad D_{d} \end{pmatrix} \right| \ D_{i}= \begin{pmatrix} \sigma _{2i-1}(u) \sigma _{2i}(u)\quad &{}\quad 0\quad &{}\quad 0\\ 0\quad &{}\quad \sigma _{2i}(u)\quad &{}\quad 0\\ 0\quad &{}\quad 0\quad &{}\quad 1 \end{pmatrix}, \ i=1,\dotsc ,d, \ u \in {\mathcal {O}}^{*,+}_K \right\} . \end{aligned}$$

Furthermore $G:=A\ltimes N\cong (\mathbb {R}^{>0})^d\ltimes N$ is the subgroup of ${\widetilde{G}} :={\widetilde{A}}\ltimes N$ given by the equations $a_{i} b_{i}=1, \ i = 1,\dotsc ,d$ and the subgroup $\Gamma _{A} \subset {\mathcal {O}}^{*,+}_K$ corresponds to

$$\begin{aligned} \left\{ \left. \begin{pmatrix} D_{1 } \quad &{}\quad 0 \quad &{} \quad \cdots \quad &{}\quad 0 \\ 0\quad &{} \quad D_{2 } \quad &{} \quad \cdots \quad &{} \quad 0\\ \vdots \quad &{} \quad \vdots \quad &{} \quad \ddots \quad &{} \quad \vdots \\ 0\quad &{} \quad 0\quad &{} \quad \cdots \quad &{} \quad D_{d} \end{pmatrix} \right| \ D_{i}= \begin{pmatrix} 1\quad &{}\quad 0\quad &{}\quad 0\\ 0\quad &{}\quad \sigma _{2i}(u)\quad &{}\quad 0\\ 0\quad &{}\quad 0\quad &{}\quad 1 \end{pmatrix}, \ i=1,\dotsc ,d, \ u \in A \right\} . \end{aligned}$$

(2.1)

2.5 Left-invariant complex structure on G

The matrix group

$$\begin{aligned} S:=\mathbb {R}^{>0}\ltimes H_3=\left\{ \left. \begin{pmatrix}1\quad &{} \quad x \quad &{} \quad v \\ 0\quad &{} \quad b \quad &{} \quad y \\ 0 \quad &{} \quad 0 \quad &{} \quad 1\end{pmatrix}\right| \ b\in \mathbb {R}^{>0}, \ x,y,v \in \mathbb {R} \right\} \end{aligned}$$

acts linearly on $\mathbb {C}^3$. The affine hyperplane $\mathbb {C}^2\times \{1\}$ of $\mathbb {C}^3$ is invariant under S. A direct calculation shows that the orbit through the point $z=(0,i,1)$ is open, has trivial isotropy and coincides with $\mathbb {C}\times \mathbb {H}^+\times \{1\}$ where $\mathbb {H}^+\subset \mathbb {C}$ is the upper half plane.

This proves the following result.

Proposition 2.8

The solvable real Lie group S admits a left-invariant complex structure with respect to which it is biholomorphic to $\mathbb {C}\times \mathbb {H}^+$.

Consider now the natural action of ${\widetilde{G}}$ on $\mathbb {C}^{3d}$. The orbit of the subgroup G through the point $(0,i,1,0,i,1,\dotsc ,0,i,1)\in \mathbb {C}^{3d}$ has trivial isotropy group, is biholomorphic to $(\mathbb {C} \times \mathbb {H}^+)^{d}$ and hence gives a left-invariant complex structure on the real Lie group G.

Therefore, the left quotient $X:=\Gamma \backslash G$ is a compact complex manifold.

2.6 A density property of $\Gamma _N$

We continue to consider $N\cong H_3^d$ and $G\cong S^d$ as matrix groups consisting of block diagonal matrices as written down in the closing of Sect. 2.4.

Let

$$\begin{aligned} H:=\left\{ \left. \begin{pmatrix} M_{1 } \quad &{} \quad 0 \quad &{} \quad \cdots \quad &{} \quad 0 \\ 0\quad &{} \quad M_{2 } \quad &{} \quad \cdots \quad &{} \quad 0\\ \vdots \quad &{} \quad \vdots \quad &{} \quad \ddots \quad &{} \quad \vdots \\ 0\quad &{} \quad 0\quad &{} \quad \cdots \quad &{} \quad M_{d} \end{pmatrix} \right| \ M_{i}= \begin{pmatrix} 1\quad &{}\quad x_{i}\quad &{}\quad z_{i}\\ 0\quad &{}\quad 1\quad &{}\quad 0\\ 0\quad &{}\quad 0\quad &{}\quad 1 \end{pmatrix} ,x_{i},v_{i}\in \mathbb {R}, \ i=1,\dotsc ,d \right\} . \end{aligned}$$

Proposition 2.9

The subgroup $\Gamma _{N} H$ is topologically dense in the Lie group N.

Proof

Let $K:=(\overline{ \Gamma _{N} H}^{\, \mathrm top})^{0}$ be the identity component of the topological closure of $\Gamma _{N} H$ in N. Since $\Gamma _{N} H$ is invariant under conjugation by elements of $\Gamma _{A}$, we see that the same is true for the subgroup K which has also the property that $(K\cap \Gamma _{N})\backslash K$ is compact. Therefore the Lie algebra ${\mathfrak {k}}$ of K is compatible with the rational structure of ${\mathfrak {n}}$ and the projection $V'$ of ${\mathfrak {k}}$ along the center ${\mathfrak {z}}({\mathfrak {n}})$ onto the vector space V has the same property. Furthermore, $V' \subset V$ is a $u_{0}$-invariant non-trivial subspace compatible with the rational structure and this implies that $V' = V$, see also the proof of Proposition 2.4. The proposition is proven. $\square$

3 Properties of the quotient manifolds

Let $G=A\ltimes N=(\mathbb {R}^{>0}\ltimes H_3)^d=S^{d}$ be the solvable Lie group equipped with the left-invariant complex structure such that $G\cong (\mathbb {C}\times \mathbb {H})^d$ and let $\Gamma =\Gamma _A\ltimes \Gamma _N$ be the cocompact discrete subgroup constructed above. In this section we establish a number of topological and complex geometric properties of the compact complex manifold $X=\Gamma \backslash G$.

3.1 The CR-fibration with Levi-flat fibers, the transversally hyperbolic foliation ${\mathcal {F}}$ and the Kodaira dimension

Let Z denote the center of G, which is also the center of N. We first remark that X considered as a real solv-manifold admits the following commutative diagram of equivariant fibrations, see [7, Proposition 2.17, Theorem 3.3, and Corollary 3.5]:

The group $\mathbb {C}^d$ acts on $G\cong (\mathbb {C}\times \mathbb {H})^d$ by translation in the $\mathbb {C}$-factors. One shows directly that this action commutes with the left multiplication by G and hence induces a holomorphic action of $\mathbb {C}^d$ on X. The kernel of this action is $\Gamma \cap Z$ and therefore we obtain an inclusion $(\mathbb {C}^*)^d\hookrightarrow {{\,\mathrm{Aut}\,}}(X)$. The orbits of this $(\mathbb {C}^*)^d$ are exactly the images of the $\mathbb {C}^d$-factors in the universal covering of X. As a consequence, we see that the action of $(\mathbb {C}^*)^d$ on X induces a transversally hyperbolic holomorphic foliation ${\mathcal {F}}$ of X.

Since the lift of p to the universal covering G of X coincides with the quotient map $G\rightarrow G/N\cong {\mathbb {R}}^d$, we see that p is $(\mathbb {C}^*)^d$-invariant. Moreover, the construction of the left-invariant complex structure on G shows that the N-orbits are generic CR-submanifolds of real dimension 3d and CR-dimension d in G. Since the complex tangent space to the N-orbits contains the $N'$-orbit, they are Levi-flat. It follows that p is a CR-map having Levi-flat fibers.

We determine the topological closure of the orbits of this $(\mathbb {C}^*)^d$-action in the following

Proposition 3.1

Let $x=\Gamma g\in \Gamma \backslash G=X$. The topological closure of $(\mathbb {C}^*)^d\cdot x$ in X coincides with the fiber of the projection p passing through the point x and is therefore isomorphic to the CR-nilmanifold $\Gamma _{N}\backslash N$. In particular,  X does not contain any proper $(\mathbb {C}^*)^d$-invariant analytic subset.

Proof

For the proof, it suffices to remark that the $(\mathbb {C}^*)^d$-orbits are exactly the right orbits in $\Gamma \backslash G=X$ of the normal subgroup H and to apply Proposition 2.9. $\square$

Corollary 3.2

Every holomorphic function on $\Gamma _N\backslash G$ is constant.

Proof

This follows from Proposition 3.1, since $\Gamma _{N} \backslash N$ is a generic CR-submanifold of $\Gamma _{N} \backslash G$. $\square$

This corollary implies the following

Corollary 3.3

The Kodaira dimension of X is $-\infty$.

Proof

Since $\Gamma$ acts by affine-linear transformations on $(\mathbb {C}\times \mathbb {H})^d$, the tangent bundle of X and all its induced vector bundles are flat. In particular, the canonical bundle of X and all its powers are flat, i.e., given by representations of $\Gamma$ in $\mathbb {C}^{*}$. This implies that the canonical bundle of a finite covering of $\Gamma _N\backslash G$ is holomorphically trivial, since for the commutator group one has $\Gamma ' \subset \Gamma _{N}$. Since every holomorphic function on $\Gamma _N\backslash G$ is constant, we see that $H^0(X,K_X^n)=0$ for all $n\geqslant 0$. Hence, ${{\,\mathrm{kod}\,}}X=-\infty$. $\square$

3.2 The identity component of ${{\,\mathrm{Aut}\,}}(X)$ and the non-Kähler property

In order to determine explicitly the holomorphic vector fields on X, let us give the action $G=A\ltimes N$ on each factor of $(\mathbb {C}\times \mathbb {H}^+)^d$ explicitly. For $g=\left( {\begin{matrix}1\quad &{}\quad a\quad &{}\quad c \\ 0\quad &{}\quad t\quad &{}\quad b\\ 0\quad &{}\quad 0\quad &{}\quad 1\end{matrix}}\right) \in \mathbb {R}^{>0}\ltimes H_3$ and $(z,w)\in \mathbb {C}\times \mathbb {H}^+$ we have

$$\begin{aligned} \begin{pmatrix} 1\quad &{}\quad a\quad &{}\quad c\\ 0\quad &{}\quad t\quad &{}\quad b\\ 0\quad &{}\quad 0\quad &{}\quad 1 \end{pmatrix}\cdot (z,w)=(z+aw+c,tw+b). \end{aligned}$$

Let $\pi :G\rightarrow {{\,\mathrm{Aut}\,}}\bigl ((\mathbb {H}^+)^d\bigr )$ be the natural projection. It follows from Proposition 2.9 that $\pi (\Gamma _N)$ is a countable, topologically dense subgroup of the unipotent radical of the Borel subgroup of affine transformations in ${{\,\mathrm{Aut}\,}}\bigl ((\mathbb {H}^+)^d\bigr )$. This observation allows us to carry over the proof of [4, Proposition 3(ii)] in order to obtain the following.

Proposition 3.4

We have $H^0(X,\Theta )=\mathbb {C}^d \cong \langle \frac{\partial }{\partial z_{1}},\dotsc ,\frac{\partial }{\partial z_{d}}\rangle _{\mathbb {C}}$ and therefore ${{\,\mathrm{Aut}\,}}^0(X)\cong (\mathbb {C}^*)^d$.

Corollary 3.5

The manifold X is not Kähler.

Proof

If X was Kähler, then due to [3] the group ${{\,\mathrm{Aut}\,}}^0(X)$ would act meromorphically on X and consequently its orbits would be locally closed, which is not the case. $\square$

3.3 Infinitely many connected components for $d \geqslant 2$

In this subsection we show that the whole group ${{\,\mathrm{Aut}\,}}(X)$ has infinitely many components for $d\geqslant 2$. Note that the automorphism groups of Inoue surfaces $S_{N}^{ {(+)}}$, (this is the case $d=1$) have only finitely many components.

First we note that the group $\widetilde{A}\cong (\mathbb {R}^{>0})^{2d-1}$ acts as a group of holomorphic transformations on $(\mathbb {C}\times \mathbb {H}^+)^d$ by

$$\begin{aligned} (\lambda _1,\mu _1,\dotsc ,\lambda _d,\mu _d)\cdot (z_1,w_1,\dotsc ,z_d,w_d) := (\lambda _1\mu _1 z_1,\mu _1 w_1,\dotsc ,\lambda _d\mu _d z_d,\mu _d w_d), \end{aligned}$$

where we suppose $\lambda _1\mu _1\cdots \lambda _d\mu _d=1$.

This action extends the A-action on $(\mathbb {C}\times \mathbb {H}^+)^d$ where A is embedded in $\widetilde{A}$ by

$$\begin{aligned} (\mu _1,\dotsc ,\mu _d)\mapsto (\mu _1^{-1},\mu _1,\dotsc ,\mu _d^{-1},\mu _d). \end{aligned}$$

In the next step we show that the $\widetilde{A}$-action normalizes the simply transitive G-action on $(\mathbb {C}\times \mathbb {H}^+)^d$. Writing $G=A\ltimes N$ as the d-fold product of the matrix group

$$\begin{aligned} S=\left\{ \begin{pmatrix} 1\quad &{} \quad a \quad &{} \quad c\\ 0 \quad &{} \quad \alpha \quad &{} \quad b \\ 0 \quad &{} \quad 0 \quad &{} \quad 1 \end{pmatrix};\ \alpha \in \mathbb {R}^{>0}, a,b,c\in \mathbb {R}\right\} , \end{aligned}$$

and defining $\varphi _{\lambda ,\mu }(z,w):=(\lambda \mu z,\mu w)$ for $\lambda ,\mu >0$ and $(z,w)\in \mathbb {C}\times \mathbb {H}^+$, we obtain

$$\begin{aligned} \varphi _{\lambda ,\mu }\bigl (g\cdot \varphi _{\lambda ,\mu }^{-1}(z,w)\bigr )= (z+\lambda a w + \lambda \mu c,\alpha w+\mu b). \end{aligned}$$

Consequently, the induced action of $\widetilde{A}$ on G coincides with the conjugation of $\widetilde{A}$ on the normal subgroup $A\ltimes N$ in $\widetilde{A} \ltimes N$.

It therefore follows that the action of the subgroup ${\mathcal {O}}^{*,+}_K$ of $\widetilde{A}$ on $(\mathbb {C}\times \mathbb {H}^+)^d$ normalizes $\Gamma =\Gamma _A \ltimes \Gamma _N$. This implies that the action of ${\mathcal {O}}^{*,+}_K$ descends holomorphically to the compact quotient $X=\Gamma \backslash G$.

3.4 An Anosov property of the foliation ${\mathcal {F}}$ in the case $d=2$

As we have seen in the previous subsection, the group ${\mathcal {O}}^{*,+}_K/ \Gamma _A$ embeds into ${{\,\mathrm{Aut}\,}}(X)$. It is clear that this discrete group of automorphisms stabilizes the foliation ${\mathcal {F}}$ of X. In this subsection we shall see that for $d=2$ non-trivial elements $\varphi$ of ${\mathcal {O}}^{*,+}_K/ \Gamma _A$ have an Anosov property relative to ${\mathcal {F}}$, i.e. that the bundle map $\varphi _*$ is Anosov when restricted to the involutive subbundle $T{\mathcal {F}}\subset TX$.

Suppose first that d is arbitrary and consider the bundle map $\varphi _*:TX\rightarrow TX$ given by the push-forward of tangent vectors. We shall trivialize first TG via left-invariant vector fields which trivialize then TX as well. Concretely, let $g\in G$ and consider $\varphi _*:T_gG\rightarrow T_{\varphi (g)}G$. Since $T_gG=(\ell _g)_*{\mathfrak {g}}$, we are led to consider

$$\begin{aligned} \bigl (\ell ^{-1}_{\varphi (g)}\circ \varphi \circ \ell _g\bigr )_*:{\mathfrak {g}}\rightarrow {\mathfrak {g}}. \end{aligned}$$

The map $G\rightarrow \mathrm{{GL}}({\mathfrak {g}})$ given by $g\mapsto \bigl (\ell ^{-1}_{\varphi (g)}\circ \varphi \circ \ell _g\bigr )_*$ encodes the action of $\varphi _*$ on TG. Moreover, since $\varphi$ normalizes the action of $\Gamma$ by left multiplication on G, it follows that we obtain a well-defined map

$$\begin{aligned} \rho _\varphi :X=\Gamma \backslash G\rightarrow \mathrm{{GL}}({\mathfrak {g}}) \end{aligned}$$

that encodes the action of $\varphi _*$ on TX. In particular, for $\varphi =\ell _{\gamma }$ with $\gamma \in \Gamma$ we have $\rho _\varphi (x)={{\,\mathrm{Id}\,}}_{\mathfrak {g}}$ for all $x\in X$.

Since the above defined matrix group S is an open subset of an affine subspace of $\mathbb {R}^{3\times 3}$, we have global coordinates on $G=S^d$ with respect to which we can explicitly calculate the map $S^d\rightarrow \mathrm{{GL}}({\mathfrak {g}})$. For $\varphi :S^d\rightarrow S^d$ given by

$$\begin{aligned} \varphi \begin{pmatrix} 1\quad &{}\quad x_i\quad &{}\quad z_i\\ 0\quad &{}\quad a_i\quad &{}\quad y_i\\ 0\quad &{}\quad 0\quad &{}\quad 1 \end{pmatrix}= \begin{pmatrix} 1\quad &{}\quad \lambda _i\mu _i x_i\quad &{}\quad \lambda _i\mu _i z_i\\ 0\quad &{}\quad \mu _i a_i\quad &{}\quad \mu y_i\\ 0\quad &{}\quad 0\quad &{}\quad 1 \end{pmatrix}, \quad i=1,\dotsc ,d, \end{aligned}$$

and $\xi _i=\left( {\begin{matrix}0\quad &{}p_i\quad &{} r_i\\ 0\quad &{} t_i\quad &{}q_i\\ 0\quad &{} 0\quad &{} d 0\end{matrix}} \right) \in {\mathfrak {s}}$ we obtain

$$\begin{aligned} \bigl (\ell ^{-1}_{\varphi (g)}\circ \varphi \circ \ell _g\bigr )_*\xi _i= \begin{pmatrix} 0\quad &{}\quad \lambda _i\mu _i p_i\quad &{}\quad \lambda _i\mu _i r_i\\ 0\quad &{}\quad t_i\quad &{}\quad q_i\\ 0\quad &{}\quad 0\quad &{}\quad 0 \end{pmatrix}. \end{aligned}$$

This shows that for all $\varphi \in {\mathcal {O}}^{*,+}_K/ \Gamma _A$ the action of $\varphi$ on the bundle $T{\mathcal {F}}$ is given by multiplication with the $\lambda _i\mu _i$ in the coordinate $\frac{\partial }{\partial z_{i}}$ for $i=1,\dotsc ,d$.

If $d=2$, let $\varphi \in {\mathcal {O}}^{*,+}_K/\Gamma _A$ be a non-trivial element and let $\lambda _{1},\mu _{1},\lambda _{2},\mu _{2} >0$ be the factors corresponding to $\varphi$. We have $\prod _{i=1}^{2} \lambda _{i}\mu _{i} =1$. If one of the products $\lambda _{i}\mu _{i}$ was equal to 1, the other one would also be equal to 1. Then $\varphi$ would be an element of $\Gamma _{A}$ and, considered as an element of ${{\,\mathrm{Aut}\,}}(X)$, would be the identity, a contradiction. Therefore the bundle $T{\mathcal {F}}$ enjoys the mentioned Anosov property with respect to $\varphi$.

For $d\geqslant 3$ it seems to be an interesting number theoretic question if there always exists an automorphism having this Anosov property with respect to the foliation ${\mathcal {F}}$.

3.5 Topological structure of X

Since G is simply-connected solvable and since the adjoint operators of A are diagonalizable over $\mathbb {R}$, the real cohomology of X may be computed via the Lie algebra cohomology of ${\mathfrak {g}}$.

Since $G=A\ltimes N$ is the identity component of the real-algebraic Lie group $(\mathbb {R}^*)^d\ltimes N$ and since $\Gamma$ is Zariski-dense in $(\mathbb {R}^*)^d\ltimes N$, we may apply [7, Corollary 7.29] in order to determine the deRham cohomology of X.

Proposition 3.6

We have $H^k(X,\mathbb {R})\cong H^k({\mathfrak {g}})$ for all $k\geqslant 0$. Moreover,  we have

$$\begin{aligned} H^k({\mathfrak {g}})=\bigoplus _{k_1+\cdots +k_d=k}\bigl (H^{k_1}(\mathbb {R}\oplus {\mathfrak {h}}_3) \otimes \cdots \otimes H^{k_d}(\mathbb {R}\otimes {\mathfrak {h}}_3)\bigr ), \end{aligned}$$

where $H^0(\mathbb {R}\otimes {\mathfrak {h}}_3)\cong H^4(\mathbb {R}\otimes {\mathfrak {h}}_3) \cong \mathbb {R},$ $H^1(\mathbb {R}\otimes {\mathfrak {h}}_3)\cong H^3(\mathbb {R}\otimes {\mathfrak {h}}_3) \cong \mathbb {R},$ and $H^2(\mathbb {R}\otimes {\mathfrak {h}}_3)=\{0\}$.

Remark

The above result shows that the topological Euler characteristic of X is zero. This can also be directly deduced from the fact that X is diffeomorphic to a tower of torus bundles over $(S^1)^d$. More precisely, the projection $G=A\ltimes N\rightarrow A$ induces a real fiber bundle $X\rightarrow \Gamma _A \backslash A\cong (S^1)^d$ with typical fiber $\Gamma _N\backslash N$ which in turn has the structure of a smooth fiber bundle over $(S^1)^d$ with fiber $(S^1)^{2d}$, cf. the diagram in Sect. 3.1.

3.6 Closed holomorphic 1-forms on X

In this subsection we give a second proof of the fact that X is not Kähler.

Proposition 3.7

There is no non-zero closed holomorphic 1-form on X. In particular,  X is not Kähler.

Proof

Let $\omega$ be a closed holomorphic 1-form on X and let $\xi$ be a holomorphic vector field on X induced by the $(\mathbb {C}^*)^d$-action. Then we have

$$\begin{aligned} {\mathcal {L}}_\xi (\omega )=\iota _\xi d\omega +d\iota _\xi \omega =0. \end{aligned}$$

Hence, every closed holomorphic 1-form on X must be $(\mathbb {C}^*)^d$-invariant. Pulling it back we get a $\Gamma$-invariant closed holomorphic 1-form $\omega$ on $(\mathbb {C}\times \mathbb {H})^d$ which must be of the form

$$\begin{aligned} \omega =\sum _{j=1}^d\lambda _jdz_j+\sum _{j=1}^d f_j(w_1,\dotsc ,w_d)dw_j. \end{aligned}$$

Since an element of N acts on $dz_j$ by $dz_j\mapsto dz_j+adw_j$, we conclude that in fact

$$\begin{aligned} \omega =\sum _{j=1}^d f_j(w_1,\dotsc ,w_d)dw_j. \end{aligned}$$

Now the claim follows from the fact that $\pi (\Gamma )$ contains a dense subgroup of the unipotent radical of the Borel subgroup of affine transformations in ${{\,\mathrm{Aut}\,}}\bigl ((\mathbb {H}^+)^d\bigr )$, see Proposition 2.9. $\square$

3.7 The algebraic dimension of X

We conclude by determining the algebraic dimension of X.

Theorem 3.8

Every meromorphic function on X is constant,  i.e.,  X has algebraic dimension zero.

Proof

Due to [2, Example 2], there exists a projective complex space Y and a holomorphic map $\pi :X\rightarrow Y$ such that every holomorphic map from X to any projective complex space factorizes through $\pi$. Consequently, $(\mathbb {C}^*)^d$ acts holomorphically on Y such that $\pi$ is equivariant.

We claim that the induced action of $(\mathbb {C}^*)^d$ on ${{\,\mathrm{Alb}\,}}(Y)$ is trivial. If this was not the case, the composed map $X\rightarrow Y\rightarrow {{\,\mathrm{Alb}\,}}(Y)$ would not be constant and hence we would obtain a non-zero closed holomorphic 1-form on X, contradicting Proposition 3.7.

Since $(\mathbb {C}^*)^d$ acts trivially on ${{\,\mathrm{Alb}\,}}(Y)$, it has a fixed point in Y due to [8]; the $\pi$-fiber over this fixed point is a $(\mathbb {C}^*)^d$-invariant analytic subset of X, hence X itself due to Proposition 3.1. It follows that Y is a point, i.e., every holomorphic map from X to a projective complex space is constant.

Now let us consider the algebraic reduction $a:X\dashrightarrow Z$ which is a priori only a meromorphic map. Since there are no proper $(\mathbb {C}^*)^d$-invariant analytic subsets of X, we obtain a holomorphic map $X\rightarrow \mathbb {P}_N$ by adding sufficiently many meromorphic functions. As we have seen above, this map must be constant, which proves the claim. $\square$

3.8 Possible extensions of the construction

In the same spirit as in Inoue’s original paper, we can modify the multiplicative action of $\Gamma _A$ as described in equation (2.1) as follows.

Firstly, every generator of $\Gamma _A$ may be combined with a central element that acts by translation on $\mathbb {C}^d$, compare the definition of $g_0$ in equation (18) on page 276 in [4].

Furthermore, we can construct analogs of the surfaces $S_N^{(-)}$ by adding any number of minus signs in the coordinates $(z_1,\dotsc ,z_d)\in \mathbb {C}^d$, compare the definition of $g_0$ in equation (21) on page 279 in [4]. Of course, such a choice will diminish the dimension of the automorphism group of the resulting quotient manifold.