# A generalization of Inoue surfaces $S^{+}$

David Petcu  University of Bucharest, Faculty of Mathematics and Computer
Science
14 Academiei Str., Bucharest, Romania
and
Institute of Mathematics “Simion Stoilow” of the Romanian Academy
21 Calea Grivitei Street, 010702, Bucharest, Romania
<EMAIL_ADDRESS>
###### Abstract.

Using Lie groups with left-invariant complex structure, we construct new
examples of compact complex manifolds with flat affine structure in arbitrarly
high dimensions. In the 2-dimensional case, we retrieve the Inoue surfaces
$S^{+}$.

###### Key words and phrases:

Solvmanifolds, non-Kähler manifolds

###### 2000 Mathematics Subject Classification:

53C55, 22E25, 32J18

The present work is partly supported by the PNRR-III-C9-2023-I8 grant CF
149/31.07.2023 Conformal Aspects of Geometry and Dynamics

##  1\. Introduction

The surfaces introduced by Inoue in 1974 are complex, compact, non-Kähler
manifolds. Alongside Hopf surfaces, Kodaira surfaces, and tori, they serve as
examples of surfaces that possess a natural flat affine structure. This means
that there exists a covering of the manifold with coordinate charts such that
all the transition maps between these charts are affine functions.

To construct manifolds with this property, one can begin with a real Lie group
that is endowed with a left-invariant complex structure [Wa]. By taking the
quotient of this Lie group by the left action of a discrete cocompact
subgroup, one obtains the desired manifold. In this paper, we utilize this
approach to introduce a higher-dimensional generalization of Inoue surfaces of
type $S^{+}$.

There have been several constructions proposed to date that generalize
different types of Inoue surfaces. For instance, Oeljeklaus-Toma ([OeTo]) and
Endo-Pajitnov ([EnPa]) manifolds generalize $S^{0}$ surfaces, while
Oeljeklaus-Miebach manifolds ([OeMi]) generalize $S^{+}$ surfaces. Despite the
differences in their specific constructions, all these manifolds share the
fundamental property of admitting an affine structure, as defined above.

By expanding on Inoue’s original work, our paper aims to further explore the
geometric and topological properties of these higher-dimensional analogues.
The main tool we use to achieve this is group cohomology. Specifically, we use
it to prove that the proposed manifolds do not contain hypersurfaces. We also
investigate the natural generalization of the left-invariant metric structure
that exists on the $S^{+}$ surfaces.

##  2\. Construction

Let $d=2m+1$ be a positive integer and let $G$ be the Lie group:

$G=\left\\{\begin{pmatrix}I_{d}&\vline&\begin{matrix}b_{d}\\\ \vdots\\\
b_{1}\end{matrix}&\vline&\begin{matrix}c_{1d}&\dots&c_{dd}\\\
\vdots&\ddots&\vdots\\\ c_{11}&\dots&c_{d1}\\\ \end{matrix}\\\ \hline\cr
0&\vline&\alpha&\vline&\begin{matrix}a_{1}&\dots&a_{d}\end{matrix}\\\
\hline\cr 0&\vline&0&\vline&I_{d}\end{pmatrix}|\
\alpha\in\mathbb{R}_{>};a_{i},b_{j},c_{ij}\in\mathbb{R}\right\\}$

Let $n=\frac{(d+1)^{2}}{2}-1$, there is a difeomorphism from $G$ to
$\mathbb{H}\times\mathbb{C}^{n}$ (as real manifolds):

$\begin{pmatrix}I_{d}&\vline&\begin{matrix}y_{d0}\\\ \vdots\\\
y_{10}\end{matrix}&\vline&\begin{matrix}x_{d0}&y_{d1}&x_{d1}&\dots&y_{dm}&x_{dm}\\\
\vdots&\vdots&\vdots&\ &\vdots&\vdots\\\
x_{10}&y_{11}&x_{11}&\dots&y_{1m}&x_{1m}\\\ \end{matrix}\\\ \hline\cr
0&\vline&w_{2}&\vline&\begin{matrix}w_{1}&y_{01}&x_{01}&\dots&y_{0m}&x_{0m}\end{matrix}\\\
\hline\cr
0&\vline&0&\vline&I_{d}\end{pmatrix}\to(w_{1}+iw_{2},x_{ij}+iy_{ij})\text{.}$

Thus, $G$ can be seen as a complex manifold. Moreover, the left actions by
elements of $G$ are holomorphic with respect to this structure.

The largest nilpotent subgroup of $G$ is

$\mathcal{N}=\left\\{\begin{pmatrix}I_{d}&\vline&(b_{j})&\vline&(c_{ij})\\\
\hline\cr 0&\vline&1&\vline&(a_{i})\\\ \hline\cr
0&\vline&0&\vline&I_{d}\end{pmatrix}|\
a_{i},b_{j},c_{ij}\in\mathbb{R}\right\\}$

This gives the short (split) exact sequence of groups:

(1) ${1}$${\mathcal{N}}$${G}$${{\mathbb{R}_{>}}}$${1}$

Denote by $\mathcal{Z}$ the first derived group of $\mathcal{N}$ (i.e.
$\mathcal{Z}=[\mathcal{N},\mathcal{N}]$):

$\mathcal{Z}=\left\\{\begin{pmatrix}I_{d}&\vline&0&\vline&(c_{ij})\\\
\hline\cr 0&\vline&1&\vline&0\\\ \hline\cr
0&\vline&0&\vline&I_{d}\end{pmatrix}|\ c_{ij}\in\mathbb{R}\right\\}.$

This subgroup is isomorphic to $\mathbb{R}^{d^{2}}$.

The following Lemma is immediate, but we insert its statement here as it plays
an important role in the sequel:

######  Lemma 2.1.

The subgroup $\mathcal{Z}$ is the center of $G$.

We get another short exact sequence of groups

(2) ${1}$${\mathcal{Z}}$${\mathcal{N}}$${\mathcal{V}}$${1}$$\scriptstyle{\pi}$

where the quotient Lie group $\mathcal{V}$ is isomorphic to $\mathbb{R}^{2d}.$

Next, we will search for a discrete cocompact subgroup $\Lambda$ of $G$. To do
this we will need the folllowing result, which is easy to prove:

######  Lemma 2.2.

Let

${1}$${{\mathcal{L}}}$${{\mathcal{M}}}$${{\mathcal{R}}}$${1}$$\scriptstyle{i}$$\scriptstyle{\pi}$

be a short exact sequence of Lie groups such that $i$ is a smooth embedding.
Let $\Lambda_{\mathcal{M}}$ be a subgroup of $\mathcal{M}$ and denote by
$\Lambda_{\mathcal{L}}$ the intersection
$\Lambda_{\mathcal{M}}\cap\mathcal{L}$ and by $\Lambda_{\mathcal{R}}$ the
image $\pi(\Lambda_{\mathcal{M}})\subset\mathcal{R}$. If
$\Lambda_{\mathcal{L}}$ is a discrete subgroup of $\mathcal{L}$ and
$\Lambda_{\mathcal{R}}$ is a discrete subgroup of $\mathcal{R}$, then
$\Lambda_{\mathcal{M}}$ is a discrete subgroup of $\mathcal{M}$.

Henceforth, in order to find the subgroup $\Lambda$, we look for a cocompact
discrete subgroup $\Lambda_{\mathcal{N}}\subset\mathcal{N}$ and a cocompact
discrete subgroup $\langle g_{0}\rangle$ of $\mathbb{R}_{>}$ (where
$\mathbb{R}_{>}$ is seen as a subgroup of $G$ via the natural splitting of 1)
such that $g_{0}\Lambda_{\mathcal{N}}g_{0}^{-1}=\Lambda_{\mathcal{N}}$. Then,
the group $\Lambda:=\langle\Lambda_{\mathcal{N}},g_{0}\rangle$ will be
cocompact and discrete in $G$, as desired. Indeed, we get discreteness by
applying lemma (2.2) while for cocompactness we observe that the action
$g_{0}$ descends to an action on the quotient $G/\Lambda_{\mathcal{N}}$, which
is difeomorphic to a fiber bundle over $\mathbb{R}_{>}$ with fibers
difeomorphic to $\mathcal{N}/\Lambda_{\mathcal{N}}$. Then,
$G/\Lambda_{\mathcal{N}}$ is difeomorphic to a fiber bundle with fiber
$\mathcal{N}/\Lambda_{\mathcal{N}}$ over the base $\mathbb{R}_{>}/\langle
g_{0}\rangle$.

To produce a cocompact lattice $\Lambda_{\mathcal{N}}\subset\mathcal{N}$ of
$\mathcal{N}$, first notice that
$D\coloneqq\Lambda_{\mathcal{N}}\cap\mathcal{Z}$ must be a cocompact subgroup
of $\mathcal{Z}$ (cf. e.g. [Ra], Prop 2.17). Assume $\Lambda_{\mathcal{N}}$ is
such a subgroup, let $h_{1}\dots h_{d^{2}}$ be the generators of the cocompact
lattice $D$ and denote $\pi(\Lambda_{\mathcal{N}})$ by
$\Lambda_{\mathcal{V}}$. Then, since $\Lambda_{\mathcal{N}}$ is cocompact in
$\mathcal{N}$ we get that
$\Lambda_{\mathcal{V}}\otimes\mathbb{R}=\mathcal{V}$, hence we only have to
check that $\Lambda_{\mathcal{V}}$ is discrete in $\mathcal{V}$. Before
continuing, we define the bilinear map
$[\cdot,\cdot]_{\mathcal{V}}:\mathcal{V}\times\mathcal{V}\to\mathcal{Z}$ as
$[\lambda,\lambda^{\prime}]_{\mathcal{V}}=[\lambda_{\mathcal{N}},\lambda^{\prime}_{\mathcal{N}}]$,
where $\lambda_{\mathcal{N}},\lambda^{\prime}_{\mathcal{N}}$ are elements in
$\mathcal{N}$ such that $\pi(\lambda_{\mathcal{N}})=\lambda$ and
$\pi(\lambda^{\prime}_{\mathcal{N}})=\lambda^{\prime}$. This map is indeed
well defined, as the group commutator on $\mathcal{N}$ depends only on the
images through $\pi$. Moreover, it is easy to see that for every nonzero
$\lambda\in\mathcal{V}$ there exists a $\lambda^{\prime}$ such that
$[\lambda,\lambda^{\prime}]_{\mathcal{V}}\neq 0$. Now, if
$\Lambda_{\mathcal{V}}$ is not discrete in $\mathcal{V}$, then
$[\Lambda_{\mathcal{V}},\Lambda_{\mathcal{V}}]_{\mathcal{V}}$ is not discrete
in $\mathcal{Z}$ and, as remarked before, this is a contradiction, therefore
$\Lambda_{\mathcal{V}}$ is a lattice. Choose $g_{1}\dots g_{2d}\in\mathcal{V}$
such that their images in $\mathcal{V}$ generate the lattice
$\Lambda_{\mathcal{V}}$. Thus, we have shown that $\Lambda_{\mathcal{V}}$ must
be of the form $\langle g_{1}\dots g_{2d},h_{1}\dots h_{d^{2}}\rangle$, where

$g_{k}=\begin{pmatrix}I_{d}&\vline&(b_{j}^{k})&\vline&(c_{ij}^{k})\\\
\hline\cr 0&\vline&1&\vline&(a_{i}^{k})\\\ \hline\cr
0&\vline&0&\vline&I_{d}\end{pmatrix}\text{, }k\in\\{1\dots 2d\\}\ \ \ \
h_{k}=\begin{pmatrix}I_{d}&\vline&0&\vline&(d_{ij}^{k})\\\ \hline\cr
0&\vline&1&\vline&0\\\ \hline\cr 0&\vline&0&\vline&I_{d}\end{pmatrix}\text{,
}k\in\\{1\dots d^{2}\\}$

Eventually, recall that the generator $g_{0}$ was chosen to be of the form

(3) $g_{0}=\begin{pmatrix}I_{d}&\vline&0&\vline&0\\\ \hline\cr
0&\vline&\alpha&\vline&0\\\ \hline\cr 0&\vline&0&\vline&I_{d}\end{pmatrix}$

Imposing that $g_{0}g_{k}g_{0}^{-1}\in\Lambda_{\mathcal{N}}$ for all $k$ gives

(4) $g_{0}g_{k}g_{0}^{-1}=g_{1}^{n_{k,1}}\dots
g_{2d}^{n_{k,2d}}h_{1}^{p_{k,1}}\dots h_{d^{2}}^{p_{k,d^{2}}}$

for some integers $n_{i,j}$ and $p_{i,j}$. This implies that

$\alpha a_{i}^{k}=\sum_{j=1}^{2d}n_{kj}a_{i}^{j}\text{ and
}\frac{1}{\alpha}b_{i}^{k}=\sum_{j=1}^{2d}n_{kj}b_{i}^{j}$

Thus, $\langle(a_{i}^{1}\dots a_{i}^{2d})|\ i\in\\{1\dots d\\}\rangle$ and
$\langle(b_{i}^{1}\dots b_{i}^{2d})|\ i\in\\{1\dots d\\}\rangle$ are the
eigenspaces of the matrix $N\coloneqq(n_{i,j})$ associated to the eigenvalues
$\alpha$ and $\frac{1}{\alpha}$ respectively. Moreover, the matrix $N$ is from
$SL_{2d}(\mathbb{Z})$ and has eigenvalues $\alpha$ and $\frac{1}{\alpha}$,
each with multiplicity $d$. The minimal polynomial of $N$ is
$P(X)=(X-\alpha)(X-\frac{1}{\alpha})$ so $\alpha$ is a real quadratic integer
and we may assume that $\alpha>1$.

######  Lemma 2.3.

For any choice of the matrices

$g_{k}=\begin{pmatrix}I_{d}&\vline&(b_{j}^{k})&\vline&(c_{ij}^{k})\\\
\hline\cr 0&\vline&1&\vline&(a_{i}^{k})\\\ \hline\cr
0&\vline&0&\vline&I_{d}\end{pmatrix}\text{, }k\in\\{1\dots 2d\\}\ $

satisfying the relations 4, the subgroup of $\mathcal{Z}$ generated by all
commmutators $g_{i}g_{j}g_{i}^{-1}g_{j}^{-1}$ is discrete.

###### Proof.

Let $\varphi$ be the nontrivial automorphism of the field
$\mathbb{Q}(\alpha)$, that is the automorphism sending $\sqrt{\delta}$ to
$-\sqrt{\delta}$, where $\delta$ is the discriminant of $\mathbb{Q}(\alpha)$.
We will first prove the lemma for a special choice of the matrices $g_{i}$:
assume the eigenvectors $v_{i}\coloneqq(a_{i}^{1}\dots a_{i}^{2d})$ are in
$\mathbb{Q}(\alpha)^{2d}$ and that $w_{i}\coloneqq(b_{i}^{1}\dots
b_{i}^{2d})=\varphi(a_{i}^{1}\dots a_{i}^{2d})$ for all $i$. If we set

$g_{s}g_{t}g_{s}^{-1}g_{t}^{-1}=\begin{pmatrix}I_{d}&\vline&0&\vline&(e_{ij}^{s,t})\\\
\hline\cr 0&\vline&1&\vline&0\\\ \hline\cr
0&\vline&0&\vline&I_{d}\end{pmatrix}\text{,}$

then

(5) $e_{ij}^{s,t}=b_{i}^{s}a_{j}^{t}-a_{j}^{s}b_{i}^{t}$

so $\varphi(e_{ij}^{s,t})=-e_{ji}^{s,t}$. Therefore, $(e_{ij})\in\langle
T_{ij}-T_{ji},\sqrt{\delta}T_{ij}+\sqrt{\delta}T_{ji}\rangle_{\mathbb{Q}}$,
where $T_{ij}$ is the matrix with $1$ in position $ij$ and $0$ everywhere
else. Thus $\mathcal{Z}$ is finitely generated subgroup of $\mathbb{Q}$-vector
space, and therefore a lattice. Now, we prove the lemma remains true for a
general choice of the matrices $g_{i}$, more precisely, we show that making a
different choice for the generators of the eigenspaces amounts to modifying
the above discrete subgroup of $\mathcal{Z}=\mathbb{R}^{d^{2}}$ by a linear
transformation: Consider $V_{i}\coloneqq(A_{i}^{1}\dots A_{i}^{2d})$ and
$W_{i}\coloneqq(B_{i}^{1}\dots B_{i}^{2d})$ be a different choice for the
bases of the eigenspaces of $\alpha$ and $\frac{1}{\alpha}$ respectively.
Then, if we set

$V_{i}=\sum_{j=1}^{d}l_{ij}v_{j}\text{, that is
}A_{i}^{s}=\sum_{j=1}^{d}l_{ij}a_{j}^{s}$

and

$W_{i}=\sum_{j=1}^{d}k_{ij}w_{j}\text{, that is
}B_{i}^{s}=\sum_{j=1}^{d}k_{ij}b_{j}^{s}$

we may define

$E_{ij}^{s,t}=B_{i}^{s}A_{j}^{t}-A_{j}^{s}B_{i}^{t}$

as before, then

$E_{ij}^{s,t}=\left(\sum_{p=1}^{d}k_{ip}b_{p}^{s}\right)\left(\sum_{r=1}^{d}l_{jr}a_{r}^{t}\right)-\left(\sum_{r=1}^{d}l_{jr}a_{r}^{s}\right)\left(\sum_{p=1}^{d}k_{ip}b_{p}^{t}\right)=$
$=\sum_{1\leq p,r\leq
d}k_{ip}l_{jr}(b_{p}^{s}a_{r}^{t}-a_{r}^{s}b_{p}^{t})=\sum_{1\leq p,r\leq
d}k_{ip}l_{jr}e_{pr}^{s,t}$

So the matrix $(E_{ij}^{s,t})$ can be expressed in terms of the matrix
$(e_{ij}^{s,t})$ as

$(E_{ij}^{s,t})=K(e_{ij}^{s,t})L^{\top}$

where $K=(k_{ij})$ and $L=(l_{ij})$. ∎

######  Corollary 2.1.

The intersection of $\widetilde{\Lambda_{\mathcal{N}}}\coloneqq\langle
g_{1}\dots g_{2d}\rangle$ with $\mathcal{Z}$ is a discrete cocompact subgroup
of $\mathcal{Z}$.

###### Proof.

In order to prove discreteness, by the previous lemma, it is enough to show
that every element in $\widetilde{\Lambda_{\mathcal{N}}}\cap\mathcal{Z}$ is
contained in the subgroup generated by the commutators
$\\{g_{i}g_{j}g_{i}^{-1}g_{j}^{-1}|i,j\in\\{1\dots 2d\\}\\}$. The group
$\Lambda_{\mathcal{V}}=\langle\widehat{g_{1}}\dots\widehat{g_{2d}}\rangle$
(where $\;\widehat{}\;$ denotes projection on $\mathcal{V}$) is the
abelianization of $\widetilde{\Lambda_{\mathcal{N}}}$, thus
$\widetilde{\Lambda_{\mathcal{N}}}\cap\mathcal{Z}=[\widetilde{\Lambda_{\mathcal{N}}},\widetilde{\Lambda_{\mathcal{N}}}]$.
We have to prove that any element in
$[\widetilde{\Lambda_{\mathcal{N}}},\widetilde{\Lambda_{\mathcal{N}}}]$ can be
written as a product of commutators $[g_{i},g_{j}]$: Let
$h,g,g^{\prime}\in\widetilde{\Lambda_{\mathcal{N}}}$ , then
$[h,gg^{\prime}]=h(gg^{\prime})h^{-1}(gg^{\prime})^{-1}=hgg^{\prime}h^{-1}g^{\prime-1}g^{-1}=hg(h^{-1}h)g^{\prime}h^{-1}g^{\prime-1}g^{-1}=hgh^{-1}(hg^{\prime}h^{-1}g^{\prime-1})g^{-1}=hgh^{-1}[h,g^{\prime}]g^{-1}=hgh^{-1}g^{-1}[h,g^{\prime}]=[h,g][h,g^{\prime}]$.
The second to last equality is due to the fact that all commutators are
central. Using the above relation we may decompose any commutator as desired.

We may identify $\mathcal{V}$ with $\mathbb{R}^{2d}$ by choosing a basis
$e_{1}\dots e_{d},f_{1}\dots f_{d}$ with respect to which
$\widehat{g_{s}}=(a_{1}^{s}\dots a_{d}^{s},b_{1}^{s}\dots b_{d}^{s})$, also
$\mathcal{Z}$ may be identified with the subspace $\langle e_{i}\wedge
f_{j}|i,j\in\\{1\dots d\\}\rangle$ of $\bigwedge^{2}\mathcal{V}$. Then
$[\widehat{g_{s}},\widehat{g_{t}}]_{\mathcal{V}}$ becomes the image of
$\widehat{g_{s}}\wedge\widehat{g_{t}}$ under the projection on $\mathcal{Z}$.
Since $\\{\widehat{g_{1}}\dots\widehat{g_{2d}}\\}$ is a basis in of
$\mathcal{V}$, then $\\{\widehat{g_{s}}\wedge\widehat{g_{t}}|1\leq s<t\leq
2d\\}$ is a basis of $\bigwedge^{2}\mathcal{V}$ so it generates a cocompact
lattice in $\bigwedge^{2}\mathcal{V}$. Therefore,
$[\widetilde{\Lambda}_{\mathcal{N}},\widetilde{\Lambda}_{\mathcal{N}}]=\widetilde{\Lambda}_{\mathcal{N}}\cap\mathcal{Z}$
is a cocompact subgroup of $\mathcal{Z}$. ∎

Finally, we choose the values for $(c_{ij}^{k})$ in $g_{k}$ such that

$g_{0}g_{k}g_{0}^{-1}=g_{1}^{n_{k,1}}\dots
g_{2d}^{n_{k,2d}}h_{1}^{p_{k,1}}\dots h_{d^{2}}^{p_{k,d^{2}}}$

is satisfied, that is for fixed $i$ and $j$, $(c_{ij}^{1}\dots c_{ij}^{2d})$
are the solutions to the following system of linear equations

(6) $\begin{pmatrix}c_{ij}^{1}\\\ \vdots\\\
c_{ij}^{2d}\end{pmatrix}=N\begin{pmatrix}c_{ij}^{1}\\\ \vdots\\\
c_{ij}^{2d}\end{pmatrix}+\begin{pmatrix}f_{ij}^{1}\\\ \vdots\\\
f_{ij}^{2d}\end{pmatrix}+P\begin{pmatrix}d_{ij}^{1}\\\ \vdots\\\
d_{ij}^{d^{2}}\end{pmatrix}$

$P$ being the matrix $(p_{i,j})$ and

$f_{ij}^{k}=\sum_{s=1}^{2d}\frac{n_{k,s}(n_{k,s}-1)}{2}b_{i}^{s}a_{j}^{s}+\sum_{1\leq
s<t\leq 2d}n_{k,s}n_{k,t}b_{i}^{s}a_{j}^{t}$

Since $1$ is not an eigenvalue of $N$, this has a unique solution.

To conclude, to construct a lattice $\Lambda$ of $G$, we proceed as follows:

  * •

pick a matrix $N\in SL_{2d}(\mathbb{Z})$ having eigenvalues $\alpha$ and
$1/\alpha$ (with $\alpha$ a quadratic positive algebraic integer) both of
multiplicities $d$;

  * •

pick basis $\langle(a_{i}^{1}\dots a_{i}^{2d})|\ i\in\\{1\dots d\\}\rangle$
and respectively $\langle(b_{i}^{1}\dots b_{i}^{2d})|\ i\in\\{1\dots
d\\}\rangle$ for the the eigenspaces (over $\mathbb{R}$) associated to the
eigenvalues $\alpha$ and $\frac{1}{\alpha}$ respectively.

  * •

Letting $e_{ij}^{s,t}$ be defined by (5) and let

$z^{s,t}:=\begin{pmatrix}I_{d}&\vline&0&\vline&(e_{ij}^{s,t})\\\ \hline\cr
0&\vline&1&\vline&0\\\ \hline\cr 0&\vline&0&\vline&I_{d}\end{pmatrix}.$

We see that
$\lambda_{\mathcal{Z}}\coloneqq\langle\\{z^{s,t}\\}_{s,t=1,2d}\rangle$ is
discrete lattice in $\mathcal{Z}$. Consider an arbitrary cocompact lattice
$D\coloneqq\\{h_{k}\\}_{k=1\dots d^{2}}$

$h_{k}=\begin{pmatrix}I_{d}&\vline&0&\vline&(d_{ij}^{k})\\\ \hline\cr
0&\vline&1&\vline&0\\\ \hline\cr 0&\vline&0&\vline&I_{d}\end{pmatrix}\text{,
}k\in\\{1\dots d^{2}\\}$

such that $D\supset\\{z^{s,t}\\}_{s,t\in\\{1\dots 2d\\}}$.

  * •

consider an arbitrary matrix $P\in{\rm Mat}_{2d,d^{2}}(\mathbb{Z})$ and let
$(c_{ij}^{1}\dots c_{ij}^{2d})$ be the solutions of the system (6). Define the
matrices

$g_{k}=\begin{pmatrix}I_{d}&\vline&(b_{j}^{k})&\vline&(c_{ij}^{k})\\\
\hline\cr 0&\vline&1&\vline&(a_{i}^{k})\\\ \hline\cr
0&\vline&0&\vline&I_{d}\end{pmatrix}\text{, }k\in\\{1\dots 2d\\}$

and let $g_{0}$ be defined as in (3). Then the subgroup

$\Lambda\coloneqq\langle
g_{0},\\{g_{k}\\}_{k=1,\dots,2d},\\{h_{k}\\}_{k=1,\dots d^{2}}\rangle$

is a cocompact lattice in $G$.

######  Definition 2.1.

Let $d$ be an odd integer, set $n=\frac{(d+1)^{2}}{2}$ and let $\Lambda$ be a
discrete cocompact subgroup of $G$ as above. We will denote by $X_{d,\Lambda}$
the compact complex manifold obtained as the quotient of
$G=\mathbb{H}\times\mathbb{C}^{n-1}$ by the left action of $\Lambda$.

######  Remark 2.1.

When $d=1$, the manifold $X_{d,\Lambda}$ is an Inoue surface of type
$S^{(+)}_{N,p,q,r;0}$. In this case, the cocompact lattice $D$ containing
$\lambda_{\mathcal{Z}}$ must be of the form
$D=\frac{1}{r}\lambda_{\mathcal{Z}}.$ The matrix $P$ is just
$P=\left(\begin{array}[]{ccc}p\\\ q\end{array}\right)$; see Inoue’s original
paper [In].

##  3\. Betti numbers

The manifold $X_{d,\Lambda}$ is difeomorphic to a fiber bundle over a circle
($X_{d,\Lambda}\to\mathbb{R}_{>}/\langle\alpha\rangle$) and the fiber is
$\mathcal{M}\coloneqq\mathcal{N}/\Lambda_{\mathcal{N}}$. In turn,
$\mathcal{M}$ is a fiber bundle over a $2d$-torus

$\mathbb{R}^{2d}/\langle(a_{1}^{k}\dots a_{d}^{k},b_{1}^{k}\dots
b_{d}^{k})|k\in\\{1\dots 2d\\}\rangle$

with fiber a $d^{2}$-torus ($\mathcal{Z}/D$). We will compute the first two
Betti numbers of $X_{d,\Lambda}$ using group cohomology and the spectral
sequences associated to these two fiber bundles. the key tool is the following
classical theorem (see [Mu]):

######  Theorem 3.1.

Let $X$ be a topological space and $G$ a discrete group, acting freely and
discontinously on $X$. Let $Y=X/G$ and $\pi:X\to Y$ be the projection, then
for any sheaf $\mathcal{F}$ on $Y$, there is a natural homomorphism

$\phi:H^{p}(G,\Gamma(X,\pi^{*}\mathcal{F}))\to H^{p}(Y,\mathcal{F})$

If $H^{i}(X,\pi^{*}\mathcal{F})=0$, $i\geq 1$, then the map $\phi$ is an
isomorphism. Furthermore, the maps $\phi$ are compatible with cup products.

We start by studying the cohomology of the fiber
$\mathcal{M}\simeq(\mathbb{R}^{2d}\times T^{d^{2}})/F$, where
$F(\simeq\mathbb{Z}^{2d})\subset\mathcal{V}$ is the group generated by
$\\{\widehat{g_{1}}\dots\widehat{g_{2d}}\\}$. Consider the spectral sequence
starting from the page zero given by $E_{0}^{p,q}\simeq
C^{p}(F,\Gamma(\mathcal{A}^{q}(\mathbb{R}^{2d}\times T^{d^{2}})))$ (where
${\mathcal{A}}^{q}$ is the sheaf of differential $q-$forms), $d$ denoting the
vertical differential and $\delta$ the horizontal one. Here, $E_{0}^{p,q}$ is
the set of all functions $F^{p}\to\Gamma(\mathcal{A}^{q}(\mathbb{R}^{2d}\times
T^{d^{2}}))$, $\delta$ is the differential of the bar resolution and $d$ is
the usual exterior differential of forms. By the previous theorem, taking
cohomology along the lines of the bicomplex will evaluate the cohomology of
the sheaves $\mathcal{A}^{q}$ on $\mathcal{M}$. Since these sheaves are fine,
we know that the spectral sequence will converge to the cohomology of
$\mathcal{M}$. The second page of the spectral sequence is $E_{2}^{p,q}\simeq
H^{p}(F,H^{q}(T^{d^{2}},\mathbb{C}))$ and $F$ acts trivially on the cohomology
of $T^{d^{2}}$, therefore $E_{2}^{p,q}\simeq H^{p}(F,\mathbb{C})\otimes
H^{q}(T^{d^{2}},\mathbb{C})$.

######  Remark 3.1.

If a cohomological spectral sequence with multiplicative structure has the
second page multiplicatively generated by elements of total degree one, then
$E_{3}^{p,q}\simeq E_{\infty}^{p,q}$. Moreover, in order to compute page $3$,
one only needs to know the map $d_{2}:E_{2}^{0,1}\to E_{2}^{2,0}$. This
applies to our spectral sequence and it is true in general for spectral
sequences coming from principal fiber bundles with a torus fiber over a torus
base.

The generators of $H^{0}(F,H^{1}(T^{d^{2}},\mathbb{C}))$ are represented by
$\\{\phi_{ij}:id_{F}\to dc_{ij}\\}$. Let $(k_{1}\dots
k_{2d})\in\mathbb{Z}^{2d}\simeq F$, then:

$\delta(\phi_{ij})(k_{1}\dots k_{2d})=(k_{1}\dots
k_{2d})\cdot\phi_{ij}-\phi_{ij}=((g_{1}^{k_{1}}\dots
g_{2d}^{k_{2d}})^{-1})^{*}dc_{ij}-dc_{ij}=\left(-\sum_{s=1}^{2d}k_{s}b_{j}^{s}\right)da_{i}$

Let $\psi_{ij}$ be such that $d(\psi_{ij})=\delta(\phi_{ij})$, for example:

$\psi_{ij}(k_{1}\dots
k_{2d})=\left(-\sum_{s=1}^{2d}k_{s}b_{j}^{s}\right)a_{i}$

Let $(l_{1}\dots l_{2d})\in\mathbb{Z}^{2d}\simeq F$, we compute
$\delta(\psi_{ij})$:

$\delta(\psi_{ij})((k_{1}\dots k_{2d}),(l_{1}\dots l_{2d}))=$
$=((g_{1}^{k_{1}}\dots g_{2d}^{k_{2d}})^{-1})^{*}\psi_{ij}(l_{1}\dots
l_{2d})-\psi_{ij}(k_{1}+l_{1}\dots k_{2d}+l_{2d})+\psi_{ij}(k_{1}\dots
k_{2d})=$
$=\left(-\sum_{s=1}^{2d}l_{s}b_{j}^{s}\right)\left(a_{i}-\sum_{t=1}^{2d}k_{t}a_{i}^{t}\right)-\left(-\sum_{s=1}^{2d}(k_{s}+l_{s})b_{j}^{s}\right)a_{i}+\left(-\sum_{s=1}^{2d}k_{s}b_{j}^{s}\right)a_{i}=$
$=\left(-\sum_{s=1}^{2d}l_{s}b_{j}^{s}\right)\left(-\sum_{t=1}^{2d}k_{t}a_{i}^{t}\right)$

For $\delta(\psi_{ij})$ determined above, we have
$d_{2}([\phi_{ij}]_{2})=[\delta\psi_{ij}]_{2}$. By identifying $E_{2}^{2,0}$
with $H^{2}_{DR}(T^{2d},\mathbb{C})$, $d_{2}([\phi_{ij}]_{2})$ becomes

$\sum_{1\leq s<t\leq
2d}(\delta(\psi_{ij})(\hat{g_{s}},\hat{g_{t}})-\delta(\psi_{ij})(\hat{g_{t}},\hat{g_{s}}))du_{s}\wedge
du_{t}=\sum_{1\leq s<t\leq
2d}(b_{j}^{s}a_{i}^{t}-b_{j}^{t}a_{i}^{s})du_{s}\wedge du_{t}$

where $u_{i}$ are the coordinate functions on $\mathbb{R}^{2d}$ with respect
to the basis $\\{\widehat{g_{i}}\\}$. Rewriting this in terms of $a_{i}$ and
$b_{i}$, we get

$\sum_{1\leq s<t\leq 2d}(b_{j}^{s}a_{i}^{t}-b_{j}^{t}a_{i}^{s})du_{s}\wedge
du_{t}=db_{j}\wedge da_{i}$

This proves that $E_{3}^{0,1}=0$ and $E_{3}^{2,0}=(\bigwedge^{2}\langle
da_{1}\dots da_{d}\rangle)\oplus(\bigwedge^{2}\langle db_{1}\dots
db_{d}\rangle)$, so $H^{1}(\mathcal{M},\mathbb{C})=\langle
da_{i},db_{j}\rangle$ and $H^{2}(\mathcal{M},\mathbb{C})=\bigwedge^{2}\langle
da_{i}\rangle\oplus\bigwedge^{2}\langle db_{j}\rangle\oplus\langle
da_{i}\otimes dc_{ij}\rangle\oplus\langle db_{j}\otimes dc_{ij}\rangle$.

We may now compute the first two Betti numbers of $X_{d,\Lambda}$. Consider
the following change in coordinates on $G$:
$b_{i}^{{}^{\prime}}=\frac{b_{i}}{\alpha}$, for $i\in\\{1\dots d\\}$ and all
the other coordinates remain unchanged. This presents the manifold
$G/\Lambda_{\mathcal{N}}$ as the product $\mathbb{R}_{>}\times\mathcal{M}$,
where $\mathcal{M}$ is as above. This time, we are interested in the spectral
sequence whose second page is
$E_{2}^{p,q}=H^{p}(\mathbb{Z},H^{q}(\mathcal{M},\mathbb{C}))$, where the
generator of $\mathbb{Z}$ acts as $g_{0}$. Since
$H^{1}(\mathcal{M},\mathbb{C})$ is finite-dimensional, the dimensions of
$H^{0}(\mathbb{Z},H^{1}(\mathcal{M},\mathbb{C}))$ and
$H^{1}(\mathbb{Z},H^{1}(\mathcal{M},\mathbb{C}))$ are equal - since they are
the kernel and cokernel of $(g_{0}^{-1})^{*}-id$ as an endomorphism of
$H^{1}(\mathcal{M},\mathbb{C})$. No elements of
$H^{1}(\mathcal{M},\mathbb{C})$ are invariant under the action of $g_{0}$
(since $(g_{0}^{-1})^{*}(da_{i})=\frac{1}{\alpha}da_{i}$ and
$(g_{0}^{-1})^{*}(db_{i}^{{}^{\prime}})=\alpha db_{i}^{{}^{\prime}}$), thus
$H^{0}(\mathbb{Z},H^{1}(\mathcal{M},\mathbb{C}))=H^{1}(\mathbb{Z},H^{1}(\mathcal{M},\mathbb{C}))=0$.
Similarly, $H^{0}(\mathbb{Z},H^{2}(\mathcal{M},\mathbb{C}))=0$ and
$H^{1}(\mathbb{Z},H^{0}(\mathcal{M},\mathbb{C}))=H^{1}(\mathbb{Z},\mathbb{C})=\mathbb{C}$.
We have proved the following:

######  Proposition 3.1.

The first two Betti numbers of $X_{d,\Lambda}$ are $b_{1}=1$ and $b_{2}=0$.

######  Corollary 3.1.

The manifolds $X_{d,\Lambda}$ are non-Kähler.

##  4\. Analytic properties

In this section we will study a certain class of manifolds of type
$X_{d,\Lambda}$. Let $\Lambda$ be a discrete subgroup cocompact of $G$ as in
the first section, consider the following properties of such a subgroup:

######  Definition 4.1.

1) We say that $\Lambda$ is of toroidal type if the quotient of
$\mathbb{C}^{n-m}$ by the subgroup $D$ is a toroidal group. Here,
$\mathbb{C}^{n-m}$ is the subspace of $G$ given by the variables $z_{ij}$ with
$i\neq 0$ and $m=\frac{d-1}{2}$, as before.
2) We say that $\Lambda$ is of algebraic type if all the entries in the
elements of $\Lambda$ are algebraic numbers.

######  Remark 4.1.

For $d=1$, the subgroup $\Lambda$ cannot be of toroidal type.

Examples:

  * i)

Let $d=3$, and let $\Lambda$ be chosen as in the special case of Lemma 2.3.
That is, if we represent the matrices in $D$ by their upper right $3\times 3$
block, then $D$ is of finite rank in $q\langle
T_{ij}-T_{ji},\sqrt{\delta}T_{ij}+\sqrt{\delta}T_{ji}|1\leq i,j\leq 3\rangle$,
where $q$ is a rational, nonzero number. To check if $\Lambda$ is of toroidal
type, we only need to check if the first columns of these matrices generate a
dense subgroup in $\mathbb{R}^{3}$ (see [AbKo]). The matrix formed with these
columns is:

$\begin{pmatrix}1&0&0&\sqrt{\delta}&0&0&0&0&0\\\
0&1&0&0&\sqrt{\delta}&0&0&0&0\\\ 0&0&0&0&0&0&\sqrt{\delta}&0&0\\\
\end{pmatrix}$

They do not generate a dense subgroup, so, in this case $\Lambda$ is not of
toroidal type.

  * ii)

Let $d$ and $N\in SL_{6}(\mathbb{Z})$ be the same as for the previous example,
but this time change the bases of the eigenspaces by the matrices $L$ and $K$
(again, see 2.3). This time, we shall look at the subgroup generated by the
first columns of the matrices
$K(T_{ij}-T_{ji})L^{\top},K(\sqrt{\delta}T_{ij}+\sqrt{\delta}T_{ji})L^{\top}$.
In fact, multiplying with $K$ from the right will not affect the property of
this subgroup of being dense, since it changes $\mathbb{R}^{3}$ by a linear
transformation, so we may consider $K=I_{3}$. As above, we now have:

$\begin{pmatrix}l_{11}&0&l_{12}&\sqrt{\delta}l_{11}&0&\sqrt{\delta}l_{12}&0&0&\sqrt{\delta}l_{13}\\\
0&l_{11}&-l_{13}&0&\sqrt{\delta}l_{11}&\sqrt{\delta}l_{13}&0&\sqrt{\delta}l_{12}&0\\\
-l_{13}&-l_{12}&0&\sqrt{\delta}l_{13}&\sqrt{\delta}l_{12}&0&\sqrt{\delta}l_{11}&0&0\\\
\end{pmatrix}$

which is plainly dense for generic choices of $L$ and $K.$ Notice that if $L$
and $K$ have algebraic entries then $\Lambda$ is of algebraic type.

If $\Gamma$ is a discrete subgroup of $\mathbb{C}^{n}$ such that
$\mathbb{C}^{n}/\Gamma$ is toroidal and all the entries of the period matrix
representing $\Gamma$ are algebraic numbers, then $\mathbb{C}^{n}/\Gamma$
satisfies Hodge decomposition. Moreover, this is also true for every domain
$U$ of $\mathbb{C}^{n}/\Gamma$ such that its inverse image $\tilde{U}$ in
$\mathbb{C}^{n}$ is convex (see [OtTo]). This result will be useful in proving
the following:

######  Proposition 4.1.

Let $X$ be a manifold of the form $G/\Lambda$, such that $\Lambda$ is of
toroidal and algebraic type, then the map $\zeta:H^{1}(X,\mathbb{C})\to
H^{1}(X,\mathcal{O}_{X})$, induced by the sheaf inclusion
$\mathbb{C}\to\mathcal{O}_{X}$, is an isomorphism.

###### Proof.

In order to study $H^{1}(X,\mathcal{O}_{X})$ we observe that it is, by
Mumford’s theorem 3.1, isomorphic to $H^{1}(\Lambda,A)$, where
$A=\Gamma(\mathcal{O}_{X})$ (a left $\Gamma$-module with action defined as
$ga=(g^{-1})^{*}a$, for $g\in\Lambda$ and $a\in A$). Using the short exact
sequence

${1}$${{\Lambda_{\mathcal{N}}}}$${\Lambda}$${{\langle g_{0}\rangle}}$${1}$

we may apply the Lyndon-Hochschild-Serre spectral sequence (LHS for short) and
get

${0}$${{H^{1}(\langle
g_{0}\rangle,A^{\Lambda_{\mathcal{N}}})}}$${{H^{1}(\Lambda,A)}}$${{H^{1}(\Lambda_{\mathcal{N}},A)^{\langle
g_{0}\rangle}}}$${{H^{2}(\langle g_{0}\rangle,A^{\Lambda_{\mathcal{N}}})}}$

First, we address the first and fourth term of this sequence. The elements of
$A^{\Lambda_{\mathcal{N}}}$ are holomorphic functions on the quotient
$G/\Lambda_{\mathcal{N}}$; let $f$ be such a function. For fixed $w_{2}$, $f$
is bounded since the (real) fibers of
$G/\Lambda_{\mathcal{N}}\to\mathcal{V}/\Lambda_{\mathcal{V}}$ are compact,
this implies that $f$ does not depend on the $z_{ij}$ coordinates. Moreover,
the subgroup generated by $\\{a_{1}^{i}|i\in\\{1\dots 2d\\}\\}$ in
$\mathbb{R}$ is dense so $f$, being continous, is constant with respect to
$w_{1}$, the real part of $w$, therefore $f$ is constant.

We have shown that $A^{\Lambda_{\mathcal{N}}}=\mathbb{C}$ (and henceforth the
action of $\langle g_{0}\rangle$ on it is trivial). Then

$H^{1}(\langle
g_{0}\rangle,A^{\Lambda_{\mathcal{N}}})=H^{1}(\mathbb{Z},\mathbb{C})=\mathbb{C}\
\ \text{ and }\ \ H^{2}(\langle
g_{0}\rangle,A^{\Lambda_{\mathcal{N}}})=H^{2}(\mathbb{Z},\mathbb{C})=0$

We see henceforth that $\zeta$ is injective, for example, by using the four
lemma on the following diagram

${0}$${{\mathbb{C}}}$${{H^{1}(\Lambda,\mathbb{C})}}$${0}$${0}$${{H^{1}(\langle
g_{0}\rangle,A^{\Lambda_{\mathcal{N}}})}}$${{H^{1}(\Lambda,A)}}$${{H^{1}(\Lambda_{\mathcal{N}},A)^{\langle
g_{0}\rangle}}}$$\scriptstyle{\zeta}$

The surjectivity of $\zeta$ is equivalent to
$H^{1}(\Lambda_{\mathcal{N}},A)^{\langle g_{0}\rangle}=0$. In the following
lemma we will show that the map $H^{1}(\Lambda_{\mathcal{N}},\mathbb{C})\to
H^{1}(\Lambda_{\mathcal{N}},A)$ is surjective. As
$H^{1}(\Lambda_{\mathcal{N}},\mathbb{C})$ is finite-dimensional it follows
that $H^{1}(\Lambda_{\mathcal{N}},A)$ is finite-dimensional too. Since the
action of $g_{0}$ on $H^{1}(\Lambda_{\mathcal{N}},\mathbb{C})$ is
diagonalisable and has not $1$ as an eigenvalue (see the arguments in the
previous section), it follows that the action of $g_{0}$ on
$H^{1}(\Lambda_{\mathcal{N}},A)$ also has no fixed points, henceforth
$H^{1}(\Lambda_{\mathcal{N}},A)^{\langle g_{0}\rangle}=0.$ ∎

Hence it remains to prove the

######  Lemma 4.1.

The following maps are surjective:

  * i)

$\beta:H^{1}(F,\mathbb{C})\to H^{1}(F,A^{D})$

  * ii)

$\gamma:H^{1}(D,\mathbb{C})\to(H^{1}(D,A))^{F}$

  * iii)

$\mu:H^{1}(\Lambda_{\mathcal{N}},\mathbb{C})\to
H^{1}(\Lambda_{\mathcal{N}},A)$

###### Proof.

i) To begin, recall that $F$ is generated by
$\widehat{g_{1}}\dots\widehat{g_{2d}}$ and that the action of these on
$\mathbb{H}\times\mathbb{C}^{m}$ (where $\mathbb{C}^{m}$ is the space
generated by $z_{01}\dots z_{0m}$) is done by translations. We can rewrite $F$
as the group generated by some $e_{1}\dots e_{d}$ and respectively
$e^{\prime}_{1}\dots e^{\prime}_{d}$ where the $e_{1}\dots e_{d}$ are chosen
in such a way that $\mathbb{H}\times\mathbb{C}^{m}/\langle e_{1}\dots
e_{d}\rangle$ is a (convex open subset of a) toroidal group.

The toroidality is equivalent to the condition that the action of $\langle
e_{1}\dots e_{d}\rangle$ on the real line $Re(w)$ has dense orbits, we see
that such a choice is possible because the group generated by $a_{1}^{1}\dots
a_{1}^{2d}$ is dense.

Moreover, since $\Lambda$ was chosen of algebraic type, we see that the
entries of the vectors $e_{i}$ are algebraic numbers (with respect to the
canonical basis of $\mathbb{C}\times\mathbb{C}^{m}$). Then the elements of

$H^{1}(\langle e_{1}\dots e_{d}\rangle,A^{D})$

are images of elements in $H^{1}(\langle e_{1}\dots e_{d}\rangle,\mathbb{C})$
(cf [OtTo], Thm. 3.1.).

Let $\omega\in H^{1}(F,A^{D}):$ then its image via the restriction map

$res:H^{1}(F,A^{D})\to H^{1}(\langle e_{1}\dots e_{d}\rangle,A^{D})$

takes constant values, henceforth $\omega$ takes constant values on elements
in $\langle e_{1},\dots,e_{d}\rangle$. Hence, we are left to show that
$\omega$ takes also constant values the remaining
$e^{\prime}_{1},\dots,e^{\prime}_{d}$. But $\omega$ is a cocycle, hence

$\omega(e_{i}+e^{\prime}_{j})=\omega(e_{i})+e_{i}\omega(e^{\prime}_{j})$

and also

$\omega(e_{i}+e^{\prime}_{j})=\omega(e^{\prime}_{j})+e^{\prime}_{j}\omega(e_{i})$

hence

$\omega(e_{i})+e_{i}\omega(e^{\prime}_{j})=\omega(e^{\prime}_{j})+e^{\prime}_{j}\omega(e_{i})$

As $\omega(e_{i})$ is constant, $e^{\prime}_{j}\omega(e_{i})=\omega(e_{i})$
hence we get

$e_{i}\omega(e^{\prime}_{j})=\omega(e^{\prime}_{j})$

hence $\omega(e^{\prime}_{j})$ is invariant w.r. to the $e_{1}\dots e_{d}$
hence from the toroidality assumption we get that $\omega(e^{\prime}_{j})$ is
also constant.

ii) $D$ acts by translations on $\mathbb{C}^{d(m+1)}$ where
$\mathbb{C}^{d(m+1)}$ is the space of the variables $z_{ij}$ with $i\not=0.$
Since, by our assumptions all the entries of the lattice $D$ are algebraic,
$\left(\mathbb{C}^{d(m+1)}/D\right)$ is a toroidal group of finite type, that
is all the cohomology groups

$H^{i}\left(\left(\mathbb{C}^{d(m+1)}/D\right),\mathcal{O}\right),i\geq 1$

are finite-dimensional. Next, notice that $\mathbb{H}\times\mathbb{C}^{n}/D$
can be written as
$(\mathbb{H}\times\mathbb{C}^{m})\times\left(\mathbb{C}^{d(m+1)}/D\right)$
that is, it is a product between a domain in $\mathbb{C}^{m+1}$ and a toroidal
group.

By Mumford’s theorem 3.1, $H^{1}(D,A)\simeq
H^{1}(\mathbb{H}\times\mathbb{C}^{n}/D,\mathcal{O}_{\mathbb{H}\times\mathbb{C}^{n}/D})$
and as $\mathbb{C}^{d(m+1)}/D$ is toroidal of finite type the dimension of
$H^{i}(\mathbb{C}^{d(m+1)}/D,\mathcal{O}_{\mathbb{C}^{d(m+1)}/D})$ is finite
for $i\geq 1$. As before, we know this because of the assumption that the
entries al algebraic. To determine $H^{1}(D,A)$, we will use the following
result by Kazama and Umeno ([KaUm]):

######  Theorem 4.1.

Let $T$ be a toroidal group of finite type and $U$ a polydisk in
$\mathbb{C}^{n}$ ($U=\left\\{(z_{1}\dots z_{n})||z_{i}|\leq
d_{i},i\in\left\\{1\dots n\right\\}\right\\}$, for some polyradius
$(d_{1}\dots d_{n})$), then

$H^{1}(T\times U,\mathcal{O}_{T\times U})\simeq
H^{1}(T,\mathcal{O}_{T})\otimes H^{0}(U,\mathcal{O}_{U})$

In fact, it is easy to see that the above theorem applies also when the some
of the radii $d_{i}$ are infinite. Applying the theorem to our case, we get

$H^{1}(\mathbb{H}\times\mathbb{C}^{n}/D,\mathcal{O}_{\mathbb{H}\times\mathbb{C}^{n}/D})\simeq
H^{1}(\mathbb{C}^{d(m+1)}/D,\mathcal{O}_{\mathbb{C}^{d(m+1)}/D})\otimes
H^{0}(\mathbb{H}\times\mathbb{C}^{m},\mathcal{O}_{\mathbb{H}\times\mathbb{C}^{m}}).$

By taking $F-$invariants, we get that
$H^{0}(\mathbb{H}\times\mathbb{C}^{m},\mathcal{O}_{\mathbb{H}\times\mathbb{C}^{m}})^{F}=\mathbb{C}$,
hence

$H^{1}(\mathbb{C}^{d(m+1)}/D,\mathcal{O}_{\mathbb{C}^{d(m+1)}/D})\simeq(H^{1}(D,A))^{F}$

and the conclusion follows since we know, by the assumption of toroidality of
finite type, that we have a surjection

$H^{1}(D,\mathbb{C})\to
H^{1}(\mathbb{C}^{d(m+1)}/D,\mathcal{O}_{\mathbb{C}^{d(m+1)}/D}).$

iii) Consider the restriction map $res:H^{1}(\Lambda_{\mathcal{N}},A)\to
H^{1}(D,A)^{F}$; by the surjectivity of $\gamma$, every element of
$H^{1}(D,A)^{F}$ is represented by a cocycle of constants. Thus, every element
of $H^{1}(\Lambda_{\mathcal{N}},A)$ is represented by a cocycle taking
constant values on $D$.

Let $\omega$ be such a cocycle and let $g\in\Lambda_{\mathcal{N}}$ and $h\in
D$, since $h$ is central, we have

$\omega(gh)=\omega(hg)\implies g\omega(h)+\omega(g)=h\omega(g)+\omega(h)$

$\omega(h)$ is constant and thus $\omega(g)=h\omega(g)$. So $\omega(g)\in
A^{D}$ (that is, a holomorphic function which depends only on the variables
$w,z_{01}\dots z_{0m}$). We have shown that the map
$H^{1}(\Lambda_{\mathcal{N}},A^{D})\to H^{1}(\Lambda_{\mathcal{N}},A)$ is
surjective.

We next study the space $H^{1}(\Lambda_{\mathcal{N}},A^{D})$. To do this,
consider the diagram:

(7)
${{H^{1}(F,\mathbb{C})}}$${{H^{1}(\Lambda_{\mathcal{N}},\mathbb{C})}}$${{H^{1}(D,\mathbb{C})}}$${{H^{1}(F,A^{D})}}$${{H^{1}(\Lambda_{\mathcal{N}},A^{D})}}$${{H^{1}(D,A^{D})^{F}}}$${{H^{1}(F,\Gamma(d\mathcal{O}_{\mathbb{H}\times\mathbb{C}^{m}}))}}$${{H^{1}(\Lambda_{\mathcal{N}},\Gamma(d\mathcal{O}_{\mathbb{H}\times\mathbb{C}^{m}}))}}$${{H^{1}(D,\Gamma(d\mathcal{O}_{\mathbb{H}\times\mathbb{C}^{m}}))^{F}}}$$\scriptstyle{\beta}$$\scriptstyle{\chi}$$\scriptstyle{inf}$$\scriptstyle{d}$$\scriptstyle{res}$$\scriptstyle{d}$$\scriptstyle{d}$$\scriptstyle{inf}$$\scriptstyle{res}$

First of all, some remarks about the diagram:

  * a)

The map $\beta$ is surjective, as proven in point (i) of the lemma.

  * b)

The map $\chi$ is an isomorphism. This is true because $A^{D}$ is a trivial
$D$-module and $D$ is contained in the center of $\Lambda_{\mathcal{N}}$, so,
by the definition of the action of $F$ on $H^{1}(D,A^{D})$, the term
$H^{1}(D,A^{D})^{F}$ contains precisely the cocycles taking constant values
(as they are the only $F$-invariant functions).

  * c)

The rows are exact, each being an inflation-restriction sequence (or low
degree terms in the LHS spectral sequence) coming from

${1}$${F}$${{\Lambda_{\mathcal{N}}}}$${D}$${1}$

  * d)

The columns are exact, for the first two this is clear from the long exact
sequence in cohomology induced by

${0}$${{\mathbb{C}}}$${{A^{D}}}$${{\Gamma(d\mathcal{O}_{\mathbb{H}\times\mathbb{C}^{m}})}}$${0}$

seen as a short exact sequence of $F$-modules, or
$\Lambda_{\mathcal{N}}$-modules respectively. For the last column, we start
with the same short exact sequence, this time treating them as trivial
$D$-modules and, after taking cohomology, we take $F$-invariants. This implies
that the last column is a complex and since $\chi$ is an isomorphism, we get
that it is in fact exact.

Let $\nu$ be a cocycle representing a class in $H^{1}(\Gamma_{H},A^{D})$,
then, since the bottom-right map labeled $d$ in (7) is zero (this is because
$\chi$ is surjective) we have:

$res(d(\nu))=d(res(\nu))=0$

Thus, $d(\nu)$ comes from the left (it is contained in the image of the
inflation map). By the forthcoming lemma (4.2), every element of
$H^{1}(F,\Gamma(d\mathcal{O}_{\mathbb{H}\times\mathbb{C}^{m}}))$ is
represented by a cocycle, taking values in $\langle dw,dz_{01}\dots
dz_{0m}\rangle$, so $d(\nu)$ is cohomologous to a cocycle taking values in
$\langle dw,dz_{01}\dots dz_{0m}\rangle$. From this, we conclude that, up to
cohomology, $\nu$ may be chosen to take values in
$\Gamma(\text{Aff}_{\mathbb{H}\times\mathbb{C}^{m}})$. We have shown that the
maps

$H^{1}(\Lambda_{\mathcal{N}},\Gamma(\text{Aff}_{\mathbb{H}\times\mathbb{C}^{m}}))\to
H^{1}(\Lambda_{\mathcal{N}},A^{D})\to H^{1}(\Lambda_{\mathcal{N}},A)$

are both surjective.

To finish the proof of point iii) in lemma 4.1, we will study the following
diagram:

(8)
${{H^{1}(\Lambda_{\mathcal{N}},\mathbb{C})}}$${{H^{1}(\Lambda_{\mathcal{N}},\Gamma(\text{Aff}_{\mathbb{H}\times\mathbb{C}^{m}}))}}$${{H^{1}(\Lambda_{\mathcal{N}},\Gamma(d\text{Aff}_{\mathbb{H}\times\mathbb{C}^{m}}))}}$${{H^{2}(\Lambda_{\mathcal{N}},\mathbb{C})}}$${{H^{1}(\Lambda_{\mathcal{N}},\mathbb{C})}}$${{H^{1}(\Lambda_{\mathcal{N}},A)}}$${{H^{1}(\Lambda_{\mathcal{N}},dA)}}$${{H^{2}(\Lambda_{\mathcal{N}},\mathbb{C})}}$$\scriptstyle{\eta}$$\scriptstyle{\epsilon}$

We want to show that the map $H^{1}(\Lambda_{\mathcal{N}},A)\to
H^{1}(\Lambda_{\mathcal{N}},dA)$ is zero. Since we know that the second
vertical map in (8) is surjective it is enough to prove that for any class
$[\omega]\in
H^{1}(\Lambda_{\mathcal{N}},\Gamma(d\text{Aff}_{\mathbb{H}\times\mathbb{C}^{m}}))$
such that $\eta([\omega])=0$, we have $\epsilon([\omega])=0$. Let the
representative $\omega$ be defined as follows:

(9) $\omega(g_{t})=S_{0}^{t}dw+S_{1}^{t}dz_{01}+\dots+S_{m}^{t}dz_{0m}\text{,
for all }t\in\\{1\dots 2d\\}$

This implies that

(10) $\omega(h_{s})=0\text{, for all }s\in\\{1\dots d^{2}\\}$

The cochain $\omega$ is closed and the action of $\Lambda_{\mathcal{N}}$ on
$\Gamma(d\text{Aff}_{\mathbb{H}\times\mathbb{C}})$ is trivial, thus
$\omega(g_{s}g_{t}g_{s}^{-1}g_{t}^{-1})=0$, for all $s,t\in\\{1\dots 2d\\}$.
As shown in the first section, for each element $h_{s}$ there exists a
positive integer $r$ such that $h_{s}^{r}$ is contained in the subgroup
generated by the commutators of the $g_{t}$’s. Thus, the above statement (10)
is true.

Also, denote by $\omega_{i}$ the $i$’th component of $\omega$ in the
decomposition given by (9), $\omega_{i}(g_{s})=S_{i}^{s}$ (notice that
$[\omega_{i}]$ can be viewed as a class in
$H^{1}(\Lambda_{\mathcal{N}},\mathbb{C})$). We may identify the classes
$da_{i}$, $db^{\prime}_{j}\in H^{1}_{DR}(G/\Lambda_{\mathcal{N}},\mathbb{C})$
with the group-cohomological classes represented by $\varphi_{i}$ and
$\psi_{j}$ respectively, where

$\varphi_{i}(g_{s})=-a_{i}^{s}\text{ and }\psi_{j}(g_{s})=-b_{j}^{s}.$

Then, the image of $[\omega]$ through the connecting map $\eta$ is represented
by the cocycle

$(\eta(\omega))(g_{s},g_{t})=-(S_{0}^{t}a_{1}^{s}+S_{1}^{t}(a_{2}^{s}+\sqrt{-1}a_{3}^{s})+\dots+S_{m}^{t}(a_{d-1}^{s}+\sqrt{-1}a_{d}^{s}))=$
$=\varphi_{1}\smile\omega_{0}+(\varphi_{2}+\sqrt{-1}\varphi_{3})\smile\omega_{1}+\dots+(\varphi_{d-1}+\sqrt{-1}\varphi_{d})\smile\omega_{m}$

where by $\smile$ we have denoted the cup product. Assuming that
$\eta(\omega)$ is exact, we get, by the computations in section 1, that

$\omega_{i}=B_{1}^{i}\psi_{1}+\dots
B_{d}^{i}\psi_{d}+A_{0}^{i}\varphi_{1}+A_{1}^{i}(\varphi_{2}+\sqrt{-1}\varphi_{3})+\dots+A_{m}^{i}(\varphi_{d-1}+\sqrt{-1}\varphi_{d})$

and $A_{j}^{i}=A_{i}^{j}$. We need to prove that that $\epsilon(\omega)$ is
exact. Let $\mu\in C^{0}(\Lambda_{\mathcal{N}},dA)$ be the $0$-cochain given
by the form:

$d\left(\sum_{\begin{subarray}{c}0\leq i\leq m\\\ 1\leq j\leq
d\end{subarray}}B_{i}^{j}z_{ji}\right)+d\left(\frac{\displaystyle\sum_{1\leq
i,j\leq m}A_{i}^{j}z_{0i}z_{0j}+\displaystyle\sum_{1\leq i\leq
m}(A_{i}^{0}+A_{0}^{i})wz_{0i}+A_{0}^{0}w^{2}}{2}\right)$

By a simple computation we can see that $\delta(\mu)=\epsilon(\omega)$, this
finishes the proof. ∎

######  Lemma 4.2.

Let $\Lambda$ be an additive subgroup of $\mathbb{C}^{n}$ (not necessarily
discrete) and $U\subseteq\mathbb{C}^{n}$ an open convex subset, invariant
under the action of $\Lambda$. If
$\Gamma(\mathcal{O}_{U})^{\Lambda}=\mathbb{C}$ and the map
$H^{1}(\Lambda,\mathbb{C})\to H^{1}(\Lambda,\Gamma(\mathcal{O}_{U}))$ is
surjective, then

$H^{1}(\Lambda,\Gamma(d{\rm Aff}_{U}))\to
H^{1}(\Lambda,\Gamma(d\mathcal{O}_{U}))$

is surjective. That is, every class in
$H^{1}(\Lambda,\Gamma(d\mathcal{O}_{U}))$ is represented by a cocycle taking
values in $\langle dz_{1}\dots dz_{n}\rangle$.

###### Proof.

We know that $\Gamma(\Omega^{1}_{U})$ and $(\Gamma(\mathcal{O}_{U}))^{n}$ are
isomorphic as vector spaces, and since $\Lambda$ acts by translations, they
are also isomorphic as $\Lambda$-modules. Consider the diagram:

${{H^{1}(\Lambda,\mathbb{C}^{n})}}$${{H^{1}(\Lambda,\Gamma(d\mathcal{O}_{U}))}}$${{H^{1}(\Lambda,(\Gamma(\mathcal{O}_{U}))^{n})}}$${0}$$\scriptstyle{\phi}$

The horizontal map is the one induced by the inclusion
$\Gamma(d\mathcal{O}_{U})\to\Gamma(\Omega^{1}_{U})\simeq(\Gamma(\mathcal{O}_{U}))^{n}$.
Then $\phi$ is induced by the map $\mathbb{C}^{n}\to\Gamma(d\mathcal{O}_{U})$,
sending $(a_{1}\dots a_{n})$ to $a_{1}dz_{1}+\dots+a_{n}dz_{n}$. We have to
show that $\phi$ is surjective.

The vertical map is surjective by hypotheis, hence if $\nu$ is a cocycle
representing a class in $H^{1}(\Lambda,(\Gamma(\mathcal{O}_{U}))^{n})$ then

$\nu(g)=(a_{1}^{g}\dots a_{n}^{g})+(g(f_{1}\dots f_{n})-(f_{1}\dots
f_{n}))\text{, for all }g\in\Lambda$

Here $g\to(a_{1}^{g}\dots a_{n}^{g})$ is cocycle in
$H^{1}(\Lambda,\mathbb{C}^{n})$ and $f_{1}\dots f_{n}$ are holomorphic
functions not depending on $g$. If $\nu$ comes from
$H^{1}(\Lambda,\Gamma(d\mathcal{O}_{U}))$ then

$\frac{\partial}{\partial z_{s}}(gf_{t}-f_{t})=\frac{\partial}{\partial
z_{t}}(gf_{s}-f_{s})\text{, for all }s,t\in\\{1\dots n\\}$

As $g$ is a translation, it commutes with the partial derivatives so we get

$g\left(\frac{\partial f_{t}}{\partial z_{s}}-\frac{\partial f_{s}}{\partial
z_{t}}\right)=\frac{\partial f_{t}}{\partial z_{s}}-\frac{\partial
f_{s}}{\partial z_{t}}$

Thus, $\frac{\partial f_{t}}{\partial z_{s}}-\frac{\partial f_{s}}{\partial
z_{t}}$ is a holomorphic function, invariant under the action of $\Lambda$ and
therefore constant.

The linear system formed by the equations $\frac{\partial f_{t}}{\partial
z_{s}}-\frac{\partial f_{s}}{\partial z_{t}}=c_{st}$ admits a solution
$(f_{1}^{\prime}\dots f_{n}^{\prime})$ with $f_{1}^{\prime}\dots
f_{n}^{\prime}$ linear. And the assciated system of homogenous equations
$\frac{\partial f_{t}}{\partial z_{s}}-\frac{\partial f_{s}}{\partial
z_{t}}=0$ has solutions of the form $(\frac{\partial F}{\partial
z_{1}}\dots\frac{\partial F}{\partial z_{n}})$, where $F$ is a holomorphic
function on $U$.

From the above computation, we get that

$\nu(g)=(a_{1}^{g}\dots a_{n}^{g})+(g(f_{1}^{\prime}\dots
f_{n}^{\prime})-(f_{1}^{\prime}\dots f_{n}^{\prime}))+(g\left(\frac{\partial
F}{\partial z_{1}}\dots\frac{\partial F}{\partial
z_{n}}\right)-\left(\frac{\partial F}{\partial z_{1}}\dots\frac{\partial
F}{\partial z_{n}}\right))$

The first two terms are cocycles taking constant values and the last term is
exact. ∎

Next, we will prove the following corollary of proposition (4.1)

######  Corollary 4.1.

  * 1)

Let $X$ be a manifold with the same properties as in proposition 4.1, then all
line bundles on $X$ are represented by a cocycle of constants.

  * 2)

The manifold $X$ does not contain any hypersurface.

###### Proof.

  * 1)

Consider the two exponential sequences on $X$

${0}$${{\mathbb{Z}}}$${{\mathbb{C}}}$${{\mathbb{C}^{*}}}$${0}$${0}$${{\mathbb{Z}}}$${{\mathcal{O}_{X}}}$${{\mathcal{O}_{X}^{*}}}$${0}$

by going to cohomology, we get:

${{H^{1}(X,\mathbb{Z})}}$${{H^{1}(X,\mathbb{C})}}$${{H^{1}(X,\mathbb{C}^{*})}}$${{H^{2}(X,\mathbb{Z})}}$${{H^{2}(X,\mathbb{C})}}$${{H^{1}(X,\mathbb{Z})}}$${{H^{1}(X,\mathcal{O}_{X})}}$${{H^{1}(X,\mathcal{O}_{X}^{*})}}$${{H^{2}(X,\mathbb{Z})}}$${{H^{2}(X,\mathcal{O}_{X})}}$

As seen in proposition 4.1, the second vertical arrow is an isomorphism, the
fifth arrow is injective since $b_{2}(X)=0$. By the five lemma, the map
$H^{1}(X,\mathbb{C}^{*})\to H^{1}(X,\mathcal{O}_{X}^{*})$ is an isomorphism.
This means that every line bundle is described by a cocycle of constants.

  * 2)

Let $L$ be a line bundle and let $\mu\in Hom(\Lambda,\mathbb{C}^{*})\simeq
H^{1}(X,\mathbb{C}^{*})$ be the character of $\Lambda$ representing $L$. A
holomorphic section of $L$ is a holomorphic function
$f\in\Gamma(\mathcal{O}_{G})$ satisfying $g^{*}f=\mu(g)f$, for all
$g\in\Lambda$. For any $h\in\Lambda_{H}$, the order of $h$ in
$\Lambda/[\Lambda,\Lambda]$ is finite, so $\mu(h)$ is of finite order in
$\mathbb{C}^{*}$. Let $n$ be such $\mu(h)^{n}=1$ for all $h\in\Lambda_{H}$,
then $f^{n}$ is invariant under the action of $\Lambda_{H}$, as shown in the
proof of proposition 4.1, this implies that $f^{n}$ is constant, so $f$ is
also constant. Then, if $L$ is not trivial, it does not admit any sections, so
there are no hypersurfaces contained in $X$.

∎

##  5\. Metric aspects

It is easy to see that the Lie algebra ${\mathfrak{g}}$ of $G$ has a natural
basis

$\\{T,A_{1}\dots A_{d},B_{1}\dots B_{d},C_{ij}|i,j\in\\{1\dots d\\}\\}$

with the following commutator relations:

$[A_{i},A_{j}]=0\ \ \ [B_{i},B_{j}]=0\ \ \ [B_{j},A_{i}]=C_{ij}\ \ \
[T,A_{i}]=A_{i}\ \ \ [T,B_{i}]=-B_{i}$

and the generators $C_{ij}$ are all central. If we rename the generators of
the Lie algebra ${\mathfrak{g}}$ such that $W_{i}=\frac{\partial}{\partial
w_{i}}$, $X_{ij}=\frac{\partial}{\partial x_{ij}}$ and
$Y_{ij}=\frac{\partial}{\partial y_{ij}}$ the left invariant complex structure
is defined by:

$\mathcal{J}(W_{1})=W_{2}\text{, }\mathcal{J}(W_{2})=-W_{1}\text{,
}\mathcal{J}(X_{ij})=Y_{ij}\text{, }\mathcal{J}(Y_{ij})=-X_{ij}$

######  Definition 5.1.

Let $(M,h)$ be a complex hermitian manifold, the metric is called locally
conformally Kähler (LCK) if the associated form $\omega$ satisfies

(11) $d\omega=\theta\wedge\omega$

for some closed $1$-form $\theta$, called the Lee form.

For more details see [OrVe].

######  Remark 5.1.

In the case $d=1$, there is an LCK metric on manifolds of type
$X_{d,\Lambda}$, that is the Tricerri LCK metric on surfaces of type
$S^{(+)}_{N,p,q,r;0}$ [Tr].

######  Proposition 5.1.

For $d>1$, there is no LCK metric on the manifolds $X_{\Lambda,d}$.

###### Proof.

Let $X$ denote the manifold $X_{\Lambda,d}$ with $d>1$. By the same arguments
as in [Bel], if $X$ has an LCK metric, then it induces a left invariant LCK
metric on $G$. Thus, it is enough to show that such structure cannot exist.
Assume $(\omega,\theta)$ is a left invariant LCK structure on $G$ and let $g$
be the riemannian metric associated to $\omega$. The following computations
are done at the level of the Lie algebra $\mathfrak{g}$. Since $\theta$ is
closed, it vanishes on commutators, henceforth
$\theta(A_{i})=\theta(B_{j})=\theta(C_{ij})=0$ for all $i,j.$ We prove that
$\theta(T)=0$ as well. To do this, we apply the LCK identity 11

$(d\omega)(A,B,C)=(\theta\wedge\omega)(A,B,C)$

for $A=W_{2}(=T),B=X_{11},C=Y_{11}.$ By direct computation, the LHS vanishes,
while the RHS equals $\theta(W_{2})\omega(X_{11},Y_{11})$. But as
$Y_{11}=\mathcal{J}(X_{11})$ we get that
$\omega(X_{11},Y_{11})=g(Y_{11},Y_{11})>0$, which implies $\theta(W_{2})=0.$

Hence, $\omega$ is a left invariant Kähler form on $G$. This implies that $X$
is Kähler, which is a contradiction ∎

######  Definition 5.2.

Let $(M,h)$ be a complex hermitian manifold of dimension $n$, the metric is
called locally conformally balanced (LCB) if the associated form $\omega$
satisfies

$d\omega^{n-1}=\theta\wedge\omega^{n-1}$

for some closed $1$-form $\theta$.

######  Proposition 5.2.

The manifolds $X_{\Lambda,d}$ admit a left invariant LCB metric.

###### Proof.

Consider the left invariant form

$\omega=W_{1}^{\vee}\wedge
W_{2}^{\vee}+\sum_{\begin{subarray}{c}\begin{subarray}{c}\begin{subarray}{c}0\end{subarray}\leq
i\leq 2m+1\\\ 0\leq j\leq m\\\
(i,j)\neq(0,0)\end{subarray}\end{subarray}}X_{ij}^{\vee}\wedge Y_{ij}^{\vee}$

This form is positive definite and of type $(1,1)$, so it comes from a
hermitian metric $h$. The metric $h$ defined this way is LCB. To prove this,
we will first determine $d\omega$ using the fact that
$d\omega(A,B,C)=-\omega([A,B],C)-\omega([C,A],B)-\omega([B,C],A)$, for any
left invariant vector fields $A,B,C$. For a general left invariant $2$-form,
we have:

$d\omega(A_{i},B_{j},B_{k})=-\omega([A_{i},B_{j}],B_{k})-\omega([B_{k},A_{i}],B_{j})=\omega(C_{ij},B_{k})-\omega(C_{ik},B_{j})\
\ ;$
$d\omega(A_{i},A_{j},B_{k})=-\omega([B_{k},A_{i}],A_{j})-\omega([A_{j},B_{k}],A_{i})=-\omega(C_{ik},A_{j})+\omega(C_{jk},A_{i})\
\ ;$
$d\omega(T,A_{i},A_{j})=-\omega([T,A_{i}],A_{j})-\omega([A_{j},T],A_{i})=2\omega(A_{j},A_{i})\
\ ;$
$d\omega(T,B_{i},B_{j})=-\omega([T,B_{i}],B_{j})-\omega([B_{j},T],B_{i})=-2\omega(B_{j},B_{i})\
\ ;$
$d\omega(T,A_{i},B_{j})=-\omega([T,A_{i}],B_{j})-\omega([B_{j},T],A_{i})-\omega([A_{i},B_{j}],T)=$
$=-\omega(A_{i},B_{j})-\omega(B_{j},A_{i})-\omega(-C_{ij},T)=\omega(C_{ij},T)\
\ ;$
$d\omega(A_{i},B_{j},C_{st})=-\omega([A_{i},B_{j}],C_{st})=\omega(C_{ij},C_{st})\
\ ;$ $d\omega(T,A_{i},C_{st})=-\omega([T,A_{i}],C_{st})=-\omega(A_{i},C_{st})\
\ ;$ $d\omega(T,B_{j},C_{st})=-\omega([T,B_{j}],C_{st})=\omega(B_{j},C_{st}).$

All the remaining possible combinations are zero directly by the commutator
relations. Applying these calculations to the form $\omega$ defined above, we
conclude that

$d\omega=\sum_{\begin{subarray}{c}1\leq i\leq 2m+1\\\ 1\leq j\leq
m\end{subarray}}X_{0j}^{\vee}\wedge Y_{i0}^{\vee}\wedge
Y_{ij}^{\vee}-\sum_{\begin{subarray}{c}1\leq i\leq 2m+1\\\ 1\leq j\leq
m\end{subarray}}Y_{0j}^{\vee}\wedge Y_{i0}^{\vee}\wedge
X_{ij}^{\vee}+\sum_{1\leq j\leq m}W_{2}^{\vee}\wedge X_{0j}^{\vee}\wedge
Y_{0j}^{\vee}$

Then,
$d\omega^{n-1}=(n-1)d\omega\wedge\omega^{n-2}=(n-1)\left(\displaystyle\sum_{1\leq
j\leq m}W_{2}^{\vee}\wedge X_{0j}^{\vee}\wedge
Y_{0j}^{\vee}\right)\wedge\omega^{n-2}$, since $X_{0j},Y_{i0}$ and $Y_{ij}$
come from three different $(X,Y)$ pairs (and so do $Y_{0j},Y_{i0}$ and
$X_{ij}$) and thus the first two terms of $d\omega$ vanish when multiplied
with $\omega^{n-2}$. Finally, we obtain

$(n-1)d\omega\wedge\omega^{n-2}=(n-1)(n-2)!(2m+1)W_{2}^{\vee}\wedge\bigwedge_{\begin{subarray}{c}\begin{subarray}{c}\begin{subarray}{c}0\end{subarray}\leq
i\leq 2m+1\\\ 0\leq j\leq m\\\
(i,j)\neq(0,0)\end{subarray}\end{subarray}}X_{ij}^{\vee}\wedge Y_{ij}^{\vee}=$
$=(2m+1)W_{2}^{\vee}\wedge\omega^{n-1}$

The form $\theta=(2m+1)W_{2}^{\vee}$ is closed because $W_{2}$ cannot be
obtained as the bracket of two other left invariant vector fields. ∎

## References

  * [AbKo] Y. Abe, K. Kopfermann,  Toroidal groups. Line bundles, cohomology and quasi-Abelian varieties. Lecture Notes in Mathematics. 1759. Berlin: Springer. viii, (2001).
  * [Bel] F.A. Belgun.  On the metric structure of non-Kähler complex surfaces. Math. Ann., 317(1), (2000) 1–40.
  * [EnPa] H. Endo, A. Pajitnov,  On generalized Inoue manifolds,  Proceedings of the International Geometry Center 13, 4 (2020), 24–39.
  * [In] M. Inoue,  On surfaces of class ${\rm VII}_{0}$,  Invent. Math. 24, (1974), 269–310.
  * [KaUm] H. Kazama,T. Umeno,  $\overline{\partial}$ Cohomology of Complex Lie Groups,  Publ. Res. Inst. Math. Sci. 26, No. 3, (1990), 473–484.
  * [Mu] D. Mumford,  Abelian varieties, Studies in Mathematics. Tata Institute of Fundamental Research 5. London: Oxford University Press. VIII, 242 p. (1970).
  * [OeMi] K. Oeljeklaus, C. Miebach,  Compact complex non-Kähler manifolds associated with totally real reciprocal units, Math. Zeitschrift 301, (2022), 2747–2760.
  * [OeTo] K. Oeljeklaus, M. Toma,  Non-Kähler compact complex manifolds associated to number fields,  Ann. Inst. Fourier 55 1 (2005), 161–171.
  * [OrVe] L. Ornea, M. Verbitsky,  Principles of Locally Conformally Kähler Geometry,  arXiv:2208.07188 (2022).
  * [OtTo] A. Otiman, M. Toma,  Hodge decomposition for Cousin groups and for Oeljeklaus-Toma manifolds,  Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 22, No. 2, (2021), 485–503.
  * [Ra] M.S. Raghunathan,  Discrete subgroups of Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 68. Berlin-Heidelberg-New York: Springer-Verlag.
  * [Tr] F. Tricerri,  Some examples of locally conformal Kähler manifolds,  Rend. Semin. Mat. Univ. Politecn. Torino 45 (1987), 117–123.
  * [Wa] C.T.C. Wall, Geometric structures on compact complex analytic manifolds, Topology 25 (1986), 119–153.


11institutetext: Jožef Stefan International Postgraduate School
22institutetext: Jožef Stefan Institute, Slovenia 33institutetext: Usher
Institute, Medical School, University of Edinburgh, Edinburgh
33email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# RaKUn: Rank-based Keyword extraction via Unsupervised learning and Meta
vertex aggregation

Blaž Škrlj 1122 Andraž Repar 1122 Senja Pollak 2233

###### Abstract

Keyword extraction is used for summarizing the content of a document and
supports efficient document retrieval, and is as such an indispensable part of
modern text-based systems. We explore how load centrality, a graph-theoretic
measure applied to graphs derived from a given text can be used to efficiently
identify and rank keywords. Introducing meta vertices (aggregates of existing
vertices) and systematic redundancy filters, the proposed method performs on
par with state-of-the-art for the keyword extraction task on 14 diverse
datasets. The proposed method is unsupervised, interpretable and can also be
used for document visualization.

###### Keywords:

keyword extraction graph applications vertex rankingload centrality
information retrieval

The final authenticated publication is available online at
10.1007/978-3-030-31372-2_26.

##  1 Introduction and related work

Keywords are terms (i.e. expressions) that best describe the subject of a
document [2]. A good keyword effectively summarizes the content of the
document and allows it to be efficiently retrieved when needed. Traditionally,
keyword assignment was a manual task, but with the emergence of large amounts
of textual data, automatic keyword extraction methods have become
indispensable. Despite a considerable effort from the research community,
state-of-the-art keyword extraction algorithms leave much to be desired and
their performance is still lower than on many other core NLP tasks [13]. The
first keyword extraction methods mostly followed a supervised approach [14,
24, 31]: they first extract keyword features and then train a classifier on a
gold standard dataset. For example, KEA [31], a state of the art supervised
keyword extraction algorithm is based on the Naive Bayes machine learning
algorithm. While these methods offer quite good performance, they rely on an
annotated gold standard dataset and require a (relatively) long training
process. In contrast, unsupervised approaches need no training and can be
applied directly without relying on a gold standard document collection. They
can be further divided into statistical and graph-based methods. The former,
such as YAKE [7, 6], KP-MINER [10] and RAKE [25], use statistical
characteristics of the texts to capture keywords, while the latter, such as
Topic Rank [3], TextRank [22], Topical PageRank [29] and Single Rank [30],
build graphs to rank words based on their position in the graph. Among
statistical approaches, the state-of-the-art keyword extraction algorithm is
YAKE [7, 6], which is also one of the best performing keyword extraction
algorithms overall; it defines a set of five features capturing keyword
characteristics which are heuristically combined to assign a single score to
every keyword. On the other hand, among graph-based approaches, Topic Rank [3]
can be considered state-of-the-art; candidate keywords are clustered into
topics and used as vertices in the final graph, used for keyword extraction.
Next, a graph-based ranking model is applied to assign a significance score to
each topic and keywords are generated by selecting a candidate from each of
the top-ranked topics. Network-based methodology has also been successfully
applied to the task of topic extraction [28].

The method that we propose in this paper, RaKUn, is a graph-based keyword
extraction method. We exploit some of the ideas from the area of graph
aggregation-based learning, where, for example, graph convolutional neural
networks and similar approaches were shown to yield high quality vertex
representations by aggregating their neighborhoods’ feature space [5]. This
work implements some of the similar ideas (albeit not in a neural network
setting), where redundant information is aggregated into meta vertices in a
similar manner. Similar efforts were shown as useful for hierarchical
subnetwork aggregation in sensor networks [8] and in biological use cases of
simulation of large proteins [9].

The main contributions of this paper are as follows. The notion of load
centrality was to our knowledge not yet sufficiently exploited for keyword
extraction. We show that this fast measure offers competitive performance to
other widely used centralities, such as for example the PageRank centrality
(used in [22]). To our knowledge, this work is the first to introduce the
notion of meta vertices with the aim of aggregating similar vertices,
following similar ideas to the statistical method YAKE [7], which is
considered a state-of-the-art for the keyword extraction. Next, as part of the
proposed RaKUn algorithm we extend the extraction from unigrams also to bigram
and threegram keywords based on load centrality scores computed for considered
tokens. Last but not least, we demonstrate how arbitrary textual corpora can
be transformed into weighted graphs whilst maintaining _global_ sequential
information, offering the opportunity to exploit potential context not
naturally present in statistical methods.

The paper is structured as follows. We first present the text to graph
transformation approach (Section 2), followed by the introduction of the RaKUn
keyword extractor (Section 3). We continue with qualitative evaluation
(Section 4) and quantitative evaluation (Section 5), before concluding the
paper in Section 6.

##  2 Transforming texts to graphs

We first discuss how the texts are transformed to graphs, on which RaKUn
operates. Next, we formally state the problem of keyword extraction and
discuss its relation to graph centrality metrics.

###  2.1 Representing text

In this work we consider _directed_ graphs. Let $G=(V,E)$ represent a graph
comprised of a set of vertices $V$ and a set of edges ($E\subseteq V\times
V$), which are ordered pairs. Further, each edge can have a real-valued weight
assigned. Let $\mathcal{D}$ represent a document comprised of tokens
$\\{t_{1},\dots,t_{n}\\}$. The order in which tokens in text appear is known,
thus $\mathcal{D}$ is a totally ordered set. A potential way of constructing a
graph from a document is by simply observing word co-occurrences. When two
words co-occur, they are used as an edge. However, such approaches do not take
into account the sequence nature of the words, meaning that the order is lost.
We attempt to take this aspect into account as follows. The given corpus is
traversed, and for each element $t_{i}$, its successor $t_{i+1}$, together
with a given element, forms a directed edge $(t_{i},t_{i+1})\in E$. Finally,
such edges are weighted according to the number of times they appear in a
given corpus. Thus the graph, constructed after traversing a given corpus,
consists of all local neighborhoods (order one), merged into a single joint
structure. Global contextual information is potentially kept intact (via
weights), even though it needs to be detected via network analysis as proposed
next.

###  2.2 Improving graph quality by meta vertex construction

A naïve approach to constructing a graph, as discussed in the previous
section, commonly yields noisy graphs, rendering learning tasks harder.
Therefore, we next discuss the selected approaches we employ in order to
reduce both the computational complexity and the spatial complexity of
constructing the graph, as well as increasing its quality (for the given down-
stream task).

First, we consider the following heuristics which reduce the complexity of the
graph that we construct for keyword extraction: Considered token length (while
traversing the document $\mathcal{D}$, only tokens of length
$\mu>\mu_{\textrm{min}}$ are considered), and next, lemmatization (tokens can
be lemmatized, offering spatial benefits and avoiding redundant vertices in
the final graph). The two modifications yield a potentially “simpler” graph,
which is more suitable and faster for mining.

Even if the optional lemmatization step is applied, one can still aim at
further reducing the graph complexity by merging similar vertices. This step
is called meta vertex construction. The motivation can be explained by the
fact, that even similar lemmas can be mapped to the same keyword (e.g.,
mechanic and mechanical; normal and abnormal). This step also captures
spelling errors (similar vertices that will not be handled by lemmatization),
spelling differences (e.g., British vs. American English), non-standard
writing (e.g., in Twitter data), mistakes in lemmatization or unavailable or
omitted lemmatization step.

Figure 1: Meta vertex construction. Sets of highlighted vertices are merged
into a single vertex. The resulting graph has less vertices, as well as edges.

The meta-vertex construction step works as follows. Let $V$ represent the set
of vertices, as defined above. A meta vertex $M$ is comprised of a set of
vertices that are elements of $V$, i.e. $M\subseteq V$. Let $M_{i}$ denote the
$i$-th meta vertex. We construct a given $M_{i}$ so that for each $u\in
M_{i}$, $u$’s initial edges (prior to merging it into a meta vertex) are
rewired to the newly added $M_{i}$. Note that such edges connect to vertices
which are not a part of $M_{i}$. Thus, both the number of vertices, as well as
edges get reduced substantially. This feature is implemented via the following
procedure:

  1. 1.

Meta vertex candidate identification. Edit distance and word lengths distance
are used to determine whether two words should be merged into a meta vertex
(only if length distance threshold is met, the more expensive edit distance is
computed).

  2. 2.

The meta vertex creation. As common identifiers, we use the stemmed version of
the original vertices and if there is more than one resulting stem, we select
the vertex from the identified candidates that has the highest centrality
value in the graph and its stemmed version is introduced as a novel vertex
(meta vertex).

  3. 3.

The edges of the words entailed in the meta vertex are next rewired to the
meta vertex.

  4. 4.

The two original words are removed from the graph.

  5. 5.

The procedure is repeated for all candidate pairs.

A schematic representation of meta vertex construction is shown in Figure 1.
The yellow and blue groups of vertices both form a meta vertex, the resulting
(right) graph is thus substantially reduced, both with respect to the number
of vertices, as well as the number of edges.

##  3 Keyword identification

Up to this point, we discussed how the graph used for keyword extraction is
constructed. In this work, we exploit the notion of load centrality, a fast
measure for estimating the importance of vertices in graphs. This metric can
be defined as follows.

####  3.0.1 Load centrality

The load centrality of a vertex falls under the family of centralities which
are defined based on the number of shortest paths that pass through a given
vertex $v$, i.e. $c(v)=\sum_{t\in V}\sum_{s\in
V}\frac{\sigma(s,t|v)}{\sigma(s,t)};t\neq s$, where $\sigma(s,t|v)$ represents
the number of shortest paths that pass from vertex $s$ to vertex $t$ via $v$
and $\sigma(s,t)$ the number of all shortest paths between $s$ and $t$ (see
[4, 11]). The considered load centrality measure is subtly different from the
better known betweenness centrality; specifically, it is assumed that each
vertex sends a package to each other vertex to which it is connected, with
routing based on a priority system: given an input of flow $x$ arriving at
vertex $v$ with destination $v$’, $v$ divides $x$ equally among all neighbors
of minimum shortest path to the target. The total flow passing through a given
$v$ via this process is defined as $v$’s load. Load centrality thus maps from
the set of vertices $V$ to real values. For detailed description and
computational complexity analysis, see [4]. Intuitively, vertices of the graph
with the highest load centrality represent key vertices in a given network. In
this work, we assume such vertices are good descriptors of the input document
(i.e. keywords). Thus, ranking the vertices yields a priority list of
(potential) keywords.

####  3.0.2 Formulating the RaKUn algorithm

We next discuss how the considered centrality is used as part of the whole
keyword extraction algorithm RaKUn, summarized in Algorithm 1.

Data: Document $D$, consisting of $n$ tokens $t_{1},\dots,t_{n}$

Parameters : General: number of keywords $k$, minimal token length $\mu$; Meta
vertex parameters: edit distance threshold $\alpha$, word length difference
threshold $l$, Multi-word keywords parameters: path length $p$, 2-gram
frequency threshold $f$

Result: A set of keywords $\mathcal{K}$

_corpusGraph_ $\leftarrow$ EmptyGraph;

$\triangleright$ Initialization.

1 for  _$t_{i}\in D$_ do

2 _edge_ $\leftarrow(t_{i},t_{i+1})$;

3 if  _edge not in corpusGraph and $\textrm{len}(t_{i})$ $\geq\mu$_ then

add _edge_ to _corpusGraph_ ;

$\triangleright$ Graph construction.

4

5 end if

updateEdgeWeight(_corpusGraph_ , _edge_) ;

$\triangleright$ Weight update.

6

7 end for

8 _corpusGraph_ $\leftarrow$ generateMetaVertices(_corpusGraph_ , $\alpha$,
$l$);

tokenRanks $\leftarrow$ loadCentrality(_corpusGraph_) ;

$\triangleright$ Initial token ranks.

scoredKeywords $\leftarrow$ generateKeywords($p$, $f$, tokenRanks) ;

$\triangleright$ Keyword search.

9 $\mathcal{K}$ = scoredKeywords[:$k$];

return _$\mathcal{K}$_

Algorithm 1 RaKUn algorithm.

The algorithm consists of three main steps described next. First, a graph is
constructed from a given ordered set of tokens (e.g., a document) (lines 1 to
8). The resulting graph is commonly very sparse, as most of the words rarely
co-occur. The result of this step is a smaller, denser graph, where both the
number of vertices, as well as edges is lower. Once constructed, load
centrality (line 10) is computed for each vertex. Note that at this point,
should the top $k$ vertices by centrality be considered, only single term
keywords emerge. As it can be seen from line 11, to extend the selection to 2-
and 3-grams, the following procedure is proposed:

2-gram keywords.

Keywords comprised of two terms are constructed as follows. First, pairs of
first order keywords (all tokens) are counted. If the support (= number of
occurrences) is higher than $f$ (line 11 in Algorithm 1), the token pair is
considered as potential 2-gram keyword. The load centralities of the two
tokens are averaged, i.e. $c_{v}=\frac{c_{1}+c_{2}}{2}$, and the obtained
keywords are considered for final selection along with the computed ranks.

3-gram keywords.

For construction of 3-gram keywords, we follow a similar idea to that of
bigrams. The obtained 2-gram keywords (previous step) are further explored as
follows. For each candidate 2-gram keyword, we consider two extension
scenarios: Extending the 2-gram from the left side. Here, the in-neighborhood
of the left token is considered as a potential extension to a given keyword.
Ranks of such candidates are computed by averaging the centrality scores in
the same manner as done for the 2-gram case. Extending the 2-gram from the
right side. The difference with the previous point is that all outgoing
connections of the rightmost vertex are considered as potential extensions.
The candidate keywords are ranked, as before, by averaging the load
centralities, i.e. $c_{v}=\frac{1}{3}\sum_{i=1}^{3}c_{i}$.

Having obtained a set of (keyword, score) pairs, we finally sort the set
according to the scores (descendingly), and take top $k$ keywords as the
result. We next discuss the evaluation the proposed algorithm.

##  4 Qualitative evaluation

Figure 2: Keyword visualization. Red dots represent keywords, other dots
represent the remainder of the corpus graph. Figure 3: Keyword visualization.
A close-up view shows some examples of keywords and their location in the
corpus graph. The keywords are mostly located in the central part of the
graph.

RaKUn can be used also for visualization of keywords in a given document or
document corpus. A visualization of extracted keywords is applied to an
example from wiki20 [21] (for dataset description see Section 5.1), where we
visualize both the global corpus graph, as well as a local (document) view
where keywords are emphasized, see Figures 2 and 3, respectively. It can be
observed that the global graph’s topology is far from uniform — even though we
did not perform any tests of scale-freeness, we believe the constructed graphs
are subject to distinct topologies, where keywords play prominent roles.

##  5 Quantitative evaluation

This section discusses the experimental setting used to validate the proposed
RaKUn approach against state-of-the-art baselines. We first describe the
datasets, and continue with the presentation of the experimental setting and
results.

###  5.1 Datasets

For RaKUn evaluation, we used 14 gold standard datasets from the list of [7,
6], from which we selected datasets in English. Detailed dataset descriptions
and statistics can be found in Table 1, while full statistics and files for
download can be found online111https://github.com/LIAAD/KeywordExtractor-
Datasets. Most datasets are from the domain of computer science or contain
multiple domains. They are very diverse in terms of the number of
documents—ranging from wiki20 with 20 documents to Inspec with 2,000
documents, in terms of the average number of gold standard keywords per
document—from 5.07 in kdd to 48.92 in 500N-KPCrowd-v1.1—and in terms of the
average length of the documents—from 75.97 in kdd to SemEval2017 with 8332.34.

Table 1: Selection of keyword extraction datasets in English language

Dataset |  Desc. | No. docs | Avg. keywords | Avg. doc length
---|---|---|---|---
500N-KPCrowd-v1.1 [18] |  Broadcast news transcriptions | 500 | 48.92 | 408.33
Inspec [15] |  Scientific journal papers from Computer Science collected between 1998 and 2002 | 2000 | 14.62 | 128.20
Nguyen2007 [23] |  Scientific conference papers | 209 | 11.33 | 5201.09
PubMed |  Full-text papers collected from PubMed Central | 500 | 15.24 | 3992.78
Schutz2008[26] |  Full-text papers collected from PubMed Central | 1231 | 44.69 | 3901.31
SemEval2010 [17] |  Scientific papers from the ACM Digital Library | 243 | 16.47 | 8332.34
SemEval2017 [1] |  500 paragraphs selected from 500 ScienceDirect journal articles, evenly distributed among the domains of Computer Science, Material Sciences and Physics | 500 | 18.19 | 178.22
citeulike180 [19] |  Full-text papers from the CiteULike.org | 180 | 18.42 | 4796.08
fao30 [20] |  Agricultural documents from two datasets based on Food and Agriculture Organization (FAO) of the UN | 30 | 33.23 | 4777.70
fao780 [20] |  Agricultural documents from two datasets based on Food and Agriculture Organization (FAO) of the UN | 779 | 8.97 | 4971.79
kdd [12] |  Abstracts from the ACM Conference on Knowledge Discovery and Data Mining (KDD) during 2004-2014 | 755 | 5.07 | 75.97
theses100 |  Full master and Ph.D. theses from the University of Waikato | 100 | 7.67 | 4728.86
wiki20 [21] |  Computer science technical research reports | 20 | 36.50 | 6177.65
www [12] |  Abstracts of WWW conference papers from 2004-2014 | 1330 | 5.80 | 84.08

###  5.2 Experimental setting

We adopted the same evaluation procedure as used for the series of results
recently introduced by YAKE authors [6]222We attempted to reproduce YAKE
evaluation procedure based on their experimental setup description and also
thank the authors for additional explanation regarding the evaluation. For
comparison of results we refer to their online repository
https://github.com/LIAAD/yake [7]. Five fold cross validation was used to
determine the overall performance, for which we measured Precision, Recall and
F1 score, with the latter being reported in Table 2.333The complete results
and the code are available at https://github.com/SkBlaz/rakun Keywords were
stemmed prior to evaluation.444This being a standard procedure, as suggested
by the authors of YAKE. As the number of keywords in the gold standard
document is not equal to the number of extracted keywords (in our experiments
$k$=10), in the recall we divide the correctly extracted keywords by the
number of gold standard keywords.

Selecting default configuration. First, we used a dedicated run for
determining the default parameters. The cross validation was performed as
follows. For each train-test dataset split, we kept the documents in the test
fold intact, whilst performing a grid search on the train part to find the
best parametrization. Finally, the selected configuration was used to extract
keywords on the unseen test set. For each train-test split, we thus obtained
the number of true and false positives, as well as true and false negatives,
which were summed up and, after all folds were considered, used to obtain
final F1 scores, which served for default parameter selection. The grid search
was conducted over the following parameter range Num keywords: $10$, Num
tokens (the number of tokens a keyword can consist of): Count threshold
(minimum support used to determine potential bigram candidates): Word length
difference threshold (maximum difference in word length used to determine
whether a given pair of words shall be aggregated): $[0,2,4]$, Edit length
difference (maximum edit distance allowed to consider a given pair of words
for aggregation): $[2,3]$, Lemmatization: [yes, no].

Even if one can use the described grid-search fine-tunning procedure to select
the best setting for individual datasets, we observed that in nearly all the
cases the best settings were the same. We therefore selected it as the
_default_ , which can be used also on new unlabeled data. The default
parameter setting was as follows. The number of tokens was set to 1, Count
threshold was thus not needed (only unigrams), for meta vertex construction
Word length difference threshold was set to 3 and Edit distance to 2. Words
were initially lemmatized. Next, we report the results using these selected
parameters (same across all datasets), by which we also test the general
usefulness of the approach. 555Note that at the time of writing of this paper,
repository of YAKE! did not include exact evaluation code, thus we attempted
to reproduce the results to the best of our knowledge. Recently published full
version of YAKE! implies the computation of recall differs, thus RaKUn’s
results are not directly comparable.

###  5.3 Results

The results are presented in Table 2, where we report on F1 with the default
parameter setting of RaKUn, together with the results from related work, as
reported in the github table of the YAKE
[7]666https://github.com/LIAAD/yake/blob/master/docs/YAKEvsBaselines.jpg
(accessed on: June 11, 2019).

Table 2: Performance comparison with state-of-the-art approaches.

Dataset |  RaKUn |  YAKE |  Single Rank |  KEA |  KP-MINER |  Text Rank |  Topic Rank |  Topical PageRank
---|---|---|---|---|---|---|---|---
500N-KPCrowd-v1.1 |  0.428 |  0.173 |  0.157 |  0.159 |  0.093 |  0.111 |  0.172 |  0.158
Inspec |  0.054 |  0.316 |  0.378 |  0.150 |  0.047 |  0.098 |  0.289 |  0.361
Nguyen2007 |  0.096 |  0.256 |  0.158 |  0.221 |  0.314 |  0.167 |  0.173 |  0.148
PubMed |  0.075 |  0.106 |  0.039 |  0.216 |  0.114 |  0.071 |  0.085 |  0.052
Schutz2008 |  0.418 |  0.196 |  0.086 |  0.182 |  0.230 |  0.118 |  0.258 |  0.123
SemEval2010 |  0.091 |  0.211 |  0.129 |  0.215 |  0.261 |  0.149 |  0.195 |  0.125
SemEval2017 |  0.112 |  0.329 |  0.449 |  0.201 |  0.071 |  0.125 |  0.332 |  0.443
citeulike180 |  0.250 |  0.256 |  0.066 |  0.317 |  0.240 |  0.112 |  0.156 |  0.072
fao30 |  0.233 |  0.184 |  0.066 |  0.139 |  0.183 |  0.077 |  0.154 |  0.107
fao780 |  0.094 |  0.187 |  0.085 |  0.114 |  0.174 |  0.083 |  0.137 |  0.108
kdd |  0.046 |  0.156 |  0.085 |  0.063 |  0.036 |  0.050 |  0.055 |  0.089
theses100 |  0.069 |  0.111 |  0.060 |  0.104 |  0.158 |  0.058 |  0.114 |  0.083
wiki20 |  0.190 |  0.162 |  0.038 |  0.134 |  0.156 |  0.074 |  0.106 |  0.059
www |  0.060 |  0.172 |  0.097 |  0.072 |  0.037 |  0.059 |  0.067 |  0.101

We first observe that on the selection of datasets, the proposed RaKUn wins
more than any other method. We also see that it performs notably better on
some of the datasets, whereas on the remainder it performs worse than state-
of-the-art approaches. Such results demonstrate that the proposed method finds
keywords differently, indicating load centrality, combined with meta vertices,
represents a promising research venue. The datasets, where the proposed method
outperforms the current state-of-the-art results are: 500N-KPCrowd-v1.1,
Schutz2008, fao30 and wiki20. In addition, RaKUn also achieves competitive
results on citeulike180. A look at the gold standard keywords in these
datasets reveals that they contain many single-word units which is why the
default configuration (which returns unigrams only) was able to perform so
well.

Four of these five datasets (500N-KPCrowd-v1.1, Schutz2008, fao30, wiki20) are
also the ones with the highest average number of keywords per document with at
least 33.23 keywords per document, while the fifth dataset (citeulike180) also
has a relatively large value (18.42). Similarly, four of the five well-
performing datasets (Schutz2008, fao30, citeulike180, wiki20) include long
documents (more than 3,900 words), with the exception being 500N-KPCrowd-v1.1.
For details, see Table 1. We observe that the proposed RaKUn outperforms the
majority of other competitive graph-based methods. For example, the most
similar variants Topical PageRank and TextRank do not perform as well on the
majority of the considered datasets. Furthermore, RaKUn also outperforms KEA,
a supervised keyword learner (e.g., very high difference in performance on
500N-KPCrowd-v1.1 and Schutz2008 datasets), indicating unsupervised learning
from the graph’s structure offers a more robust keyword extraction method than
learning a classifier directly.

##  6 Conclusions and further work

In this work we proposed RaKUn, a novel unsupervised keyword extraction
algorithm which exploits the efficient computation of load centrality,
combined with the introduction of meta vertices, which notably reduce corpus
graph sizes. The method is fast, and performs well compared to state-of-the-
art such as YAKE and graph-based keyword extractors. In further work, we will
test the method on other languages. We also believe additional semantic
background knowledge information could be used to prune the graph’s structure
even further, and potentially introduce keywords that are inherently not even
present in the text (cf.[27]). The proposed method does not attempt to exploit
meso-scale graph structure, such as convex skeletons or communities, which are
known to play prominent roles in real-world networks and could allow for
vertex aggregation based on additional graph properties. We believe the
proposed method could also be extended using the Ricci-Oliver [16] flows on
weighted graphs.

####  6.0.1 Acknowledgements

The work was supported by the Slovenian Research Agency through a young
researcher grant [BŠ], core research programme (P2-0103), and projects
Semantic Data Mining for Linked Open Data (N2-0078) and Terminology and
knowledge frames across languages (J6-9372). This work was supported also by
the EU Horizon 2020 research and innovation programme, Grant No. 825153,
EMBEDDIA (Cross-Lingual Embeddings for Less-Represented Languages in European
News Media). The results of this publication reflect only the authors’ views
and the EC is not responsible for any use that may be made of the information
it contains. We also thank the authors of YAKE for their clarifications.

## References

  * [1] Augenstein, I., Das, M., Riedel, S., Vikraman, L., McCallum, A.: Semeval 2017 task 10: Scienceie - extracting keyphrases and relations from scientific publications. CoRR abs/1704.02853 (2017)
  * [2] Beliga, S., Meštrović, A., Martinčić-Ipšić, S.: An overview of graph-based keyword extraction methods and approaches. Journal of information and organizational sciences 39(1), 1–20 (2015)
  * [3] Bougouin, A., Boudin, F., Daille, B.: Topicrank: Graph-based topic ranking for keyphrase extraction. In: International Joint Conference on Natural Language Processing (IJCNLP). pp. 543–551 (2013)
  * [4] Brandes, U.: On variants of shortest-path betweenness centrality and their generic computation. Social Networks 30(2), 136–145 (2008)
  * [5] Cai, H., Zheng, V.W., Chang, K.C.C.: A comprehensive survey of graph embedding: Problems, techniques, and applications. IEEE Transactions on Knowledge and Data Engineering 30(9), 1616–1637 (2018)
  * [6] Campos, R., Mangaravite, V., Pasquali, A., Jorge, A.M., Nunes, C., Jatowt, A.: A text feature based automatic keyword extraction method for single documents. In: Pasi, G., Piwowarski, B., Azzopardi, L., Hanbury, A. (eds.) Advances in Information Retrieval. pp. 684–691. Springer International Publishing, Cham (2018)
  * [7] Campos, R., Mangaravite, V., Pasquali, A., Jorge, A.M., Nunes, C., Jatowt, A.: Yake! collection-independent automatic keyword extractor. In: European Conference on Information Retrieval. pp. 806–810. Springer (2018)
  * [8] Chan, H., Perrig, A., Song, D.: Secure hierarchical in-network aggregation in sensor networks. In: Proceedings of the 13th ACM conference on Computer and communications security. pp. 278–287. ACM (2006)
  * [9] Doruker, P., Jernigan, R.L., Bahar, I.: Dynamics of large proteins through hierarchical levels of coarse-grained structures. Journal of computational chemistry 23(1), 119–127 (2002)
  * [10] El-Beltagy, S.R., Rafea, A.: Kp-miner: A keyphrase extraction system for english and arabic documents. Information Systems 34(1), 132–144 (2009)
  * [11] Goh, K.I., Kahng, B., Kim, D.: Universal behavior of load distribution in scale-free networks. Phys. Rev. Lett. 87, 278701 (Dec 2001)
  * [12] Gollapalli, S.D., Caragea, C.: Extracting keyphrases from research papers using citation networks. In: Twenty-Eighth AAAI Conference on Artificial Intelligence (2014)
  * [13] Hasan, K.S., Ng, V.: Automatic keyphrase extraction: A survey of the state of the art. In: Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). vol. 1, pp. 1262–1273 (2014)
  * [14] Hulth, A.: Improved automatic keyword extraction given more linguistic knowledge. In: Proceedings of the 2003 conference on Empirical methods in natural language processing. pp. 216–223. Association for Computational Linguistics (2003)
  * [15] Hulth, A.: Improved automatic keyword extraction given more linguistic knowledge. In: Proceedings of the 2003 Conference on Empirical Methods in Natural Language Processing. pp. 216–223. EMNLP ’03 (2003)
  * [16] Jin, M., Kim, J., Gu, X.D.: Discrete surface ricci flow: Theory and applications. In: IMA International Conference on Mathematics of Surfaces. pp. 209–232. Springer (2007)
  * [17] Kim, S.N., Medelyan, O., Kan, M.Y., Baldwin, T.: Semeval-2010 task 5: Automatic keyphrase extraction from scientific articles. In: Proceedings of the 5th International Workshop on Semantic Evaluation. pp. 21–26. SemEval ’10 (2010)
  * [18] Marujo, L., Viveiros, M., da Silva Neto, J.P.: Keyphrase cloud generation of broadcast news. CoRR abs/1306.4606 (2013)
  * [19] Medelyan, O., Frank, E., Witten, I.H.: Human-competitive tagging using automatic keyphrase extraction. In: Proceedings of the 2009 Conference on Empirical Methods in Natural Language Processing: Volume 3 - Volume 3. pp. 1318–1327. EMNLP ’09 (2009)
  * [20] Medelyan, O., Witten, I.H.: Domain-independent automatic keyphrase indexing with small training sets. Journal of the American Society for Information Science and Technology 59(7), 1026–1040
  * [21] Medelyan, O., Witten, I.H., Milne, D.: Topic indexing with wikipedia. In: Proceedings of the AAAI WikiAI workshop. vol. 1, pp. 19–24 (2008)
  * [22] Mihalcea, R., Tarau, P.: Textrank: Bringing order into text. In: Proceedings of the 2004 conference on empirical methods in natural language processing (2004)
  * [23] Nguyen, T.D., Kan, M.Y.: Keyphrase extraction in scientific publications. In: Goh, D.H.L., Cao, T.H., Sølvberg, I.T., Rasmussen, E. (eds.) Asian Digital Libraries. Looking Back 10 Years and Forging New Frontiers. pp. 317–326. Springer Berlin Heidelberg, Berlin, Heidelberg (2007)
  * [24] Nguyen, T.D., Luong, M.T.: Wingnus: Keyphrase extraction utilizing document logical structure. In: Proceedings of the 5th international workshop on semantic evaluation. pp. 166–169. Association for Computational Linguistics (2010)
  * [25] Rose, S., Engel, D., Cramer, N., Cowley, W.: Automatic keyword extraction from individual documents. Text mining: applications and theory pp. 1–20 (2010)
  * [26] Schutz, A.T., et al.: Keyphrase extraction from single documents in the open domain exploiting linguistic and statistical methods. Master’s thesis, National University of Ireland (2008)
  * [27] Škrlj, B., Kralj, J., Lavrač, N., Pollak, S.: Towards robust text classification with semantics-aware recurrent neural architecture. Machine Learning and Knowledge Extraction 1(2), 575–589 (2019)
  * [28] Spitz, A., Gertz, M.: Entity-centric topic extraction and exploration: A network-based approach. In: European Conference on Information Retrieval. pp. 3–15. Springer (2018)
  * [29] Sterckx, L., Demeester, T., Deleu, J., Develder, C.: Topical word importance for fast keyphrase extraction. In: Proceedings of the 24th International Conference on World Wide Web. pp. 121–122. ACM (2015)
  * [30] Wan, X., Xiao, J.: Single document keyphrase extraction using neighborhood knowledge. In: AAAI. vol. 8, pp. 855–860 (2008)
  * [31] Witten, I.H., Paynter, G.W., Frank, E., Gutwin, C., Nevill-Manning, C.G.: Kea: Practical automated keyphrase extraction. In: Design and Usability of Digital Libraries: Case Studies in the Asia Pacific, pp. 129–152. IGI Global (2005)