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  \tiny M. Verbitsky }}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Locally conformally K\"ahler manifolds \\[15mm]
\small lecture 14: Oeljeklaus-Toma manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf HSE and IUM, Moscow
\\[2mm]  June 2, 2014
}
\end{center}

\newpage

{\bf\blue Local systems (reminder)}

\definition
A {\bf \blue local system} is a locally constant sheaf 
of vector spaces.

\theorem
A local system with fiber $B$ at $x\in M$
gives a homomorphism $\pi_1(M,x)\arrow \Aut(B)$.
{\bf \red This correspondence gives an equivalence of categories.}

\definition
A bundle $(B, \nabla)$ is called {\bf \blue flat} if its curvature vanishes.

\definition
A section $b$ of $(B,\nabla)$ is called {\bf \blue parallel}
if $\nabla(b)=0$.

\claim
Let $(B, \nabla)$ be a flat bundle on $M$, and ${\cal B}$ be the
sheaf of parallel sections. {\bf \red Then ${\cal B}$ is a 
locally constant sheaf.}

\theorem
This correspondence {\bf \red gives an equivalence of categories}
of flat bundles and local systems.

\newpage

{\bf\blue LCK manifolds (reminder)}


\definition
Let $(M,I, \omega)$ be a Hermitian manifold, $\dim_\C M >1$.
Then $M$ is called {\bf \blue locally conformally K\"ahler}
(LCK) if $d\omega=\omega\wedge\theta$, where $\theta$ is a closed
1-form, called {\bf \blue the Lee form}.

\definition
{\bf \blue A manifold is locally conformally K\"ahler}
iff it admits a K\"ahler form taking values in a positive,
flat vector bundle $L$, called {\bf \blue the weight bundle}.


\definition {\bf \blue Deck transform}, or {\bf \blue monodromy maps}
of a covering $\tilde M \arrow M$ are elements of the group $\Aut_M(\tilde M)$.
{\bf \purple
When $\tilde M$ is a universal cover, one has $\Aut_M(\tilde M)=\pi_1(M)$.}

\definition
{\bf \blue An LCK manifold} is a complex manifold such that
its universal cover $\tilde M$ is equipped with a K\"ahler
form $\tilde \omega$, and the deck transform acts on $\tilde M$ by
K\"ahler homotheties.

\theorem {\bf \red These three definitions are equivalent}.

\newpage

{\bf \blue Lee class of an LCK manifold (reminder)}

\definition
Let $(M,\omega, \theta)$ be an LCK manifold.
The cohomology class $[\theta] \in H^1(M,\R)$ 
of its Lee form $\theta$ is called {\bf \blue the Lee class} of $M$.


\remark {\bf \blue Monodromy group} of an LCK manifold
$(M,\omega, \theta)$ 
is defined as the Galois group of the smallest covering
$\pi:\; \tilde M \arrow M$ such that $\pi^*\theta$ 
is exact. {\bf\blue Rank} of an LCK manifold is rank of
its monodromy group.

\proposition
Let $(M,\omega, \theta)$ be an LCK manifold
and $[\theta]$ its Lee class. Consider a
smallest rational subspace $V\subset H^1(M,\Q)$
such that $V\otimes_\Q \R$ contains $[\theta]$.
{\bf \red Then $\dim V$ is equal to the rank of $M$.}

\proof
The group $\Gamma$ is identified with an image of $\pi_1(M)$
under the map $[\theta]:\; \pi_1(M)\arrow \R$,
because it is equal to the monodromy of the weight bundle,
and the monodromy along a loop $\gamma$ is equal to
$e^{\int_\gamma \theta}$.
\endproof

%\newpage
%
%{\bf \green Oeljeklaus-Toma manifolds: introduction}
%
%So far, for all examples of LCK manifolds the following
%was true: monodromy rank of $M$ was 1, and $H^1(M)$ was odd-dimensional.
%Vaisman conjectures that all LCK manifolds have $b_1$ odd.
%
%A counterexample is provided by Oeljeklaus-Toma manifold,
%which have sometimes even $b_1$ and arbitrary monodromy rank.
%
%{\bf \blue Plan.}
%
%{\bf \blue 1. Motivation:} 
%classification of 2-dimensional solvmanifolds,
%Inoue surfaces, Bogomolov's theorem.
%
%{\bf \blue 2. Number theory:} global and local fields,
%absolute value function, complex and real 
%embeddings, the Dirichlet's unit theorem.
%
%{\bf \blue 3. Inoue surfaces of class $S^0$.} 
%Curves on Inoue surfaces.
%
%{\bf \blue 4. Oeljeklaus-Toma manifolds}. Subvarieties
%in Oeljeklaus-Toma manifolds.
%
%{\bf \blue 5. The adele ring and the 
%strong approximation theorem}. 

\newpage

{\bf\blue Solvmanifolds}

\definition
Let $M$ be a smooth manifold equipped
with a transitive action of solvable Lie group $G$.
Then $M$ is called {\bf \blue a solvmanifold}.
If $G$ is nilpotent, $M$ is called {\bf \blue a nilmanifold}.

\remark
All solvmanifolds are obtained as quotient spaces,
$M=G/H$ (Mostow). All nilmanifolds are obtained as
quotient spaces  $M=G/\Gamma$, where $\Gamma$ is discrete
(Maltsev).

\definition
{\bf \blue An integrable complex structure} on a real Lie algebra
$\g$ is a subalgebra $\g^{1,0}\subset \g \otimes_\R \C$
such that $\g^{1,0}\oplus \overline{\g^{1,0}} = \g \otimes_\R \C$

\remark 
Right-invariant complex structures on a connected real Lie group
{\bf \purple are in 1 to 1 correspondence with integrable
complex structures} on its Lie algebra.

\definition
A {\bf \blue complex solvmanifold} is a solvmanifold
$M=G/H$ equipped with a complex structure, in such a way
that $G$ has a right-invariant complex structure,
and the projection $G \arrow M$ is holomorphic.

\remark 
{\bf \red Solvmanifolds are usually non-homogeneous}
(as complex manifolds).

%\newpage
%
%{\bf \blue Examples of 2-dimensional solvmanifolds}
%
%\remark
%{\bf \blue ``A surface''} here would always
%mean {\bf \red ``a compact complex manifold of complex dimension 2''.}
%
%\definition
%Let $T$, $T'$ be elliptic curves.
%{\bf \blue Kodaira surface} is a locally trivial
%holomorphic fibration over $T$ with fiber $T'$
%and non-trivial Chern class.
%
%{\bf \green A remark on terminology:}
%These are {\bf \blue ``primary''}  Kodaira surfaces.
%{\bf \blue ``Secondary''} ones are obtained by taking
%finite unramified quotients.
%
%\remark
%The Kodaira surface is diffeomorphic to a quotient
%$S^1 \times (G/G_\Z)$ where $G$ is a 3-dimensional 
%Heisenberg group. In particular, {\bf \purple 
%Kodaira surface is a nilmanifold}.
%
%
%\newpage
%
%{\bf \blue Kodaira surface is a complex nilmanifold}
%
%
%\proposition
%Let $M$ be a Kodaira surface. {\bf \red Then $M$ is Vaisman.}
%
%\proof
%Let $X=T^2$ be a base of the fibration $\pi:\; M \arrow X$
%Using the exponential exact sequence 
%$0\arrow \Z^2 \arrow \calo_{X}\arrow T^2 \arrow 0$, we get
%\[ 0 \arrow H^1(X, \Z^2)
%H^1(X, \calo_{X})\arrow H^1(X, T^2)\arrow H^2(X, \Z^2)\arrow 0,
%\] 
%This means that the space of Kodaira surfaces is parametrized
%by $\Pic_0(X)\times (\Z^2\backslash 0)$, 
%with the group $\Z^2\rtimes SL(2,\Z)$ of torus 
%automorphisms acting in a natural way. Regular Vaisman manifolds are
%similarly parametrized by $\Pic_0(X)\times (\Z\backslash 0)$. 
%{\bf \purple Therefore, any
%$SL(2,\Z)$-orbit contains a Vaisman manifold,} and all Kodaira surfaces
%are Vaisman.
%\endproof
%
%\proposition {\bf \red Kodaira surface $M$ is a complex nilmanifold.}
%
%\proof Since $M$ is Vaisman, its $\Z$-covering $\tilde M$ is a total
%space of principal $\calo^*_X$-bundle $L$ over $X=T^2$.
%The $X=T^2$-action on $X$ is lifted to a central extension
%of $T^2$ acting on $L$, which is obviously homogeneous.
%\endproof
%
%

%\newpage
%
%{\bf \blue Inoue surfaces}
%
%\definition
%("Bogomolov's theorem'') 
%{\bf\blue Inoue surface} is a complex surface without
%curves and with $b_2=0$.
%
%\remark Original definiton of Inoue was constructive,
%in terms of explicit action by matrices,
%and the above result is a theorem proven by Bogomolov
%in 1976.
%
%Bogomolov, F. A.
%{\blue \em Classification of surfaces of class VII${}_0$ with $b_{2}=0$}. 
%Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 2, 273-288, 469. 
%
%{\bf \small \green Math. Reviews:}
%{\em \small ...This unreadable paper contains several new ideas and
%claims two important results; unfortunately many of the
%arguments in the proof are subject to doubt. It would seem
%highly desirable to know whether proofs along the lines
%given here can be made to work. ... Without going into
%detailed criticism of the author's written style, the
%reviewer would like to comment that in the parts of the
%paper which he has been able to understand the author's
%disorganised shorthand and bestial notation put a burden
%on the reader which he considers unacceptable in a
%published paper.} -- Miles Reid
%
%Bogomolov's proof uses the action
%of the Galois group $[\C: \Q]$.
%
%
%\newpage
%
%{\bf \blue History of Inoue surfaces}
%
%In 1991, a new proof appeared, based on Yang-Mills theory:\\
%Li, J.; Yau, S.-T.; Zheng, F. {\em \blue A simple proof of
%Bogomolov's theorem on class ${\rm VII}_0$  surfaces
%with $b_2=0$.}  Illinois J. Math.  34  (1990),  no. 2,
%217-220.
%
%This proof was wrong.
%
%Finally, correct proofs were obtained.
%
%Teleman, Andrei Dumitru
%{\em \blue Projectively flat surfaces and Bogomolov's theorem on
%class ${\rm VII}_0$ surfaces.}
%Internat. J. Math. 5 (1994), no. 2, 253-264. 
%
%Li, Jun; Yau, Shing-Tung; Zheng, Fangyang
%{\em \blue On projectively flat Hermitian manifolds.}
%Comm. Anal. Geom. 2 (1994), no. 1, 103-109. 
%
%\newpage
%
%{\bf \blue Complex solvmanifolds of dimension 2}
%
%\theorem (Hasegawa)
%Let $M$ be a complex surface which is diffeomorphic to a solvmanifold.
% Then $M$ is (up to a finite unramified quotient) 
%{\bf \red isomorpic to one the following.}
%
%{\bf \green
%1. Compact complex torus
%
%2. Kodaira surface
%
%3. Inoue surface.
%}
%
%{\bf \purple This theorem directly follows from
%Bogomolov's theorem, Hasegawa's result on K\"ahler
%solvmanifolds, and Kodaira's classification}.
%
%
%To define the Inoue surfaces explicitly, we use
%{\bf \red the number theory}.

\newpage

{\bf \blue Normed fields}

\definition
{\bf \blue An absolute value} on a field $k$ is a function $|\cdot |:\; k
\arrow \R^{\geq 0}$, satisfying the following

{\bf \purple 1.  Zero:} $|x|=0$ $\Leftrightarrow$ $x=0$.

{\bf \purple 2.  Multiplicativity:} $|xy|= |x||y|$.

{\bf \purple 3.  There exists
$c>0$ such that $|\cdot |^c$ satisfies the triangle inequality. }

\example The usual absolute value on $\Q$, $\R$, $\C$.

\example
Let $p$ -- be a prime number, and $m,n\in \Z$  coprime with $p$.
Define {\bf \blue $p$-adic absolute value} on $\Q$ via
$|\frac m n p^k|:= p^{-k}$.

\remark
$p$-adic absolute value satisfies an additional 
``non-archimedean axiom'': $|x+y| \leq \max(|x|, |y|)$.
Such absolute values are called {\bf \blue non-archimedean}.

\remark
Any power of non-archimedean absolute value is again
non-archimedean, and satisfies the triangle inequality.

\newpage

{\bf \blue Normed fields and topology}


\definition
Let $|\cdot |$ be an absolute value on a field $F$.
Consider topology on $F$ with open sets generated by
\[ B_\epsilon (x) := \{y \in k \ \ |\ \ |x-y| < 
   \epsilon\}.
\]
Absolute values are called {\bf \blue equivalent}
if they induce the same topology.

\theorem
{\bf \red Absolute values $|\cdot |_1, |\cdot |_2$
are equivalent if and only if
$|\cdot |_1= |\cdot |_2^c$
for some $c>0$.}

\theorem
(Ostrowski) 
{\bf \red Every absolute value on $\Q$
is equivalent to the usual  ("archimedean") 
one or to $p$-adic one.}

\definition
{\bf\blue A completion} of a field $k$ under an absolute value
 $|\cdot |$ is a completion of $k$ in a metric $|\cdot |^c$,
where $c>0$ is a constant such that $|\cdot |^c$ 
satisfies the triangle inequality.

\remark
A completion of a field is again a field.

\example A completion of $\Q$ under the $p$-adic
absolute value is called {\bf \blue a field of $p$-adic
numbers}, denoted $\Q_p$.

\newpage

{\bf \blue Local fields}


\definition 
{\bf \blue A finite extension} $K\!\!:\!\!k$ of fields
is a field $K \supset k$ which is finite-dimensional
as a vector space over $k$. {\bf \blue A number field} 
is a finite extension of  $\Q$. {\bf \blue Functional field}
is a finite extension of ${\Bbb F}_p(t)$.
{\bf\blue Global field} is a number or functional field. 
{\bf \blue Local field} is a completion of a global field
under a non-trivial absolute value.

\theorem
Let  $\bar k$ be a field which is complete and locally
compact under some absolute value. {\bf \red Then $\bar k$
is a local field}. 

\definition
Let $K\!\!:\!\!k$ be a finite extension, and
$x\in K$. Consider the multiplication by $x$ as a
$k$-linear endomorphism of $K$. Define 
{\bf\blue the norm $N_{K/k}(x)$} as a determinant
of this operator.

\remark
Norm defines a homomorphism of multiplicative groups
$K^* \arrow k^*$.

\remark For Galois extensions, the norm  $N_{K/k}(x)$ {\bf
 \purple is a
product of all elements conjugate to $x$.}

\theorem
Let $\bar K\!:\! \bar k$ be a finite extension of 
local fields, degree $n$. {\bf \red Then an absolute
value on $\bar k$ is uniquely extended to $\bar K$.} Moreover,
{\bf \red this extension is expressed as
$|x|:= \left|N_{K/k}(x)\right|^{\frac 1 n}$.}

\newpage

{\bf \blue Absolute values and extensions of global fields}

\claim
Let $A,B$ be extensions of a field $k$, $\Char k=0$,
where $A\!\!:\!\!k$ is finite. Consider $A\otimes_k B$ 
as an $k$-algebra. {\bf \red Then $A\otimes_k B$ 
is a direct sum of fields, containing $A$ and $B$.}

\theorem
Let $k$ be a number field,
$|\cdot|$ an absolute value, $K\!\!:\!\!k$ a finite extension, and  
$\bar k$ -- its completion. Consider a decomposition $K
\otimes_k \bar k$ into a direct sum of fields 
$K \otimes_k \bar k:= \bigoplus_i \bar K_i$.
{\bf \purple Then each extension of an absolute value
$|\cdot|$ from $k$ to $K$ is induced from some  
$\bar K_i$, and all such extensions are non-equivalent.}

\remark
When $k=\Q$, and $|\cdot|$ is the usual
(archimedean) absolute value, we obtain that
all $K_i$ are extensions of $\R$, 
that is, isomorphic to $\R$ or $\C$. This gives

\corollary
{\bf \red For each number field $K$ of degree $n$
over $\Q$, there exists only a finite number
of different homomorphisms $K\hookrightarrow \C$},
all of them injective. 
Denote by $s$ the number of embeddings
whose image lies in $\R \subset \C$
(such an embedding is called {\bf \blue real}),
and $2t$ the number of embedding, whose image
does not lie in $\R$ (``{\bf \blue complex embeddings}). 
Then $s+2t =n$.


\newpage

{\bf \blue Dirichlet unit theorem}

\definition
Let $K\!:\!\Q$ be a number field of degree $n$.
{\bf \blue The ring of integers} $\calo_K \subset K$ 
is an integral closure of $\Z$ in $K$, that is, 
the set of all roots in $K$ of monic polynomials
$P(t)= t^n + a_{n-1} t^{n-1} +a_{n-2}t^{n-2} + ... + a_0$
with integer coefficients $a_i \in \Z$.

\claim
{\bf \purple An additive group $\calo_K^+$ 
is a finitely generated abelian group of rank $n$.} 

\definition
{\bf\blue A unit} of a ring $\calo_K$ is an element
$u\in \calo_K$, such that $u^{-1}$ also belongs to $\calo_K$.

\remark
$u \in \calo_K$ is a unit if and only if 
the norm $N_{K/\Q}(x)\in \Z$ is invertible, that is,
$N_{K/\Q}(x)=\pm 1$.

{\bf \green Dirichlet's unit theorem:}
Let $K$ be a number field which has $s$ real
embeddings and $2t$ complex ones. Then {\bf \red the group of
units  $\calo_K^*$ is isomorphic to $G \times \Z^{t+s-1}$,}
where $G$ is a finite group of roots of unity contained 
in $K$. Moreover, if $s>0$,  one has $G= \pm 1$.

\remark
For a quadratic field, the group of units is 
a group of solutions of Pell's equation.

\newpage

{\bf \blue Cubic fields and complex surfaces}


Let $K\!:\!\Q$ be a cubic extension of
$\Q$ which has 2 complex embeddings
$\tau$, $\bar\tau$ and one real one 
$\sigma$ (such an extension is obtained by adding
all roots of a cubic polynomial which has exactly
one real root).

\remark Due to Dirichlet theorem, 
$\calo_K^*$ is isomorphic to $\Z \times \{\pm 1\}$.
Let $\calo_K^{*,+}:= \sigma^{-1}(\R^{>0}) \cap \calo_K^*$.
Then {\bf \purple the group $\calo_K^{*,+}$ is isomorphic 
to $\Z$.}


Consider the action of $\calo_K^+\cong \Z^3$ on $\R^3 = \C \times \R$
\[
  \rho^+(x)(z, t):= (z+ \tau(x), t + \sigma(x)).
\]
Let $\Gamma$ be a semidirect product
$\calo^+_K\rtimes \calo_K^{*,+}$, defined from the 
natural action of $\calo_K^{*,+}$ on $\calo^+_K$.
{\bf \purple Define an action of $\Gamma$ on $\C \times {\H}$,
where ${\H}$ is an upper halfplane}, as follows. 

The subgroup $\calo^+_K\subset \Gamma$ acts
on $\C \times {\H}=\C \times \R\times \R^{>0}$ by
translations as above (trivially on the last argument), and 
Á $\calo_K^{*,+}$ acts multiplicatively as
\[ 
 \rho^*(\xi)(z, z'):= (\tau(\xi)z, \sigma(\xi) z').
\]
\newpage

{\bf \blue Inoue surfaces of type $S^0$}

\definition
The {\bf \blue Inoue surface of type $S^0$}
is a quotient $(\C \times {\H})/\Gamma$.

{\bf \green Its properties:}
1. It is a compact, complex solvmanifold


2. Inoue surface {\bf \purple admits a flat connection
preserving the complex structure} (by construction).

3. Its cohomology are the same as of $S^3\times S^1$

\theorem
The Inoue surface $M:= (\C \times {\H})/\Gamma$ 
{\bf \red is locally conformally K\"ahler.}

\proof Let $z, u$ be coordinates on $\C \times {\H}$,
and $\phi(z,u):= |z|^2 + \Im(u)^{-1}$. Since 
$dd^c\phi = \1 dz\wedge d\bar z+ 2\1 \Im(u)^{-3}du \wedge du \bar u$,
it is a K\"ahler form.
Clearly, $dd^c\phi$ is $\calo^+_K$-invariant.
Let $\epsilon \in \Z=\calo_K^{*,+}$ be a unit.
Since $N(\epsilon)$ is integer and invertible, $N(\epsilon)=1$.
This implies that $\tau(\epsilon)^2=\sigma(\epsilon)^-1$.
However, $\epsilon^*(\phi)= |\tau(\epsilon)z|^2 + \Im(\sigma(\epsilon)u)^{-1}$,
hence this $\epsilon^*(\phi)=\tau(\epsilon)^2\phi=\sigma(\epsilon)^{-1}\phi$.
We have found an K\"ahler form on $\C \times {\H}$.
\endproof

\newpage

{\bf \blue Curves on Inoue surface}


\theorem
The Inoue surface $M:= (\C \times {\H})/\Gamma$ 
{\bf \red has no complex curves}

{\bf\green Proof. Step 1:}
Consider on $\C \times {\H}$
a function $\phi(z, z'):= \log \Im(z')$.
Since $\Gamma$ multiplies $\Im(z')$
by a number, {\bf \purple the form $d\phi$ is $\Gamma$-invariant.}
Let $\theta$ be the corresponding 1-form on $M$.

{\bf\green Step 2:}  The 2-form $\omega_0:=d(I\theta)$ 
has Hodge type (1,1) 
and {\bf \red is positive definite on the leaves of
the foliation $\{z\}\times {\H}\subset \C \times {\H}$}. Indeed,
\[
\omega_0 = \1\6\bar\6\log \phi = 
\1 \frac{dz'\wedge d\bar z'} {|\im z'|^2},
\]
where $\omega_0$ is the Poincare metric on ${\H}$.

{\bf\green Step 3:} Let $\Sigma \subset TM$
be the null-space of the form $\omega_0$. {\bf \purple 
It is a holomorphic, involutive foliation,}
whose leaves are obtained from
$\C\times \{z'\}\subset \C \times {\H}$

{\bf\green Step 4:} For any complex curve
$C$ on $M$,  $\int_C \omega_0=0$, because
$\omega_0$ is exact. Therefore, $C$ is tangent
to a leaf of $\Sigma$.
{\bf \red It remains to show that $\Sigma$
has no compact leaves.}

\newpage

{\bf \blue Curves on Inoue surfaces}


{\bf\green Step 3:} Let $\Sigma \subset TM$
be the null-space of the form $\omega_0$. {\bf \purple 
It is a holomorphic, involutive foliation,}
whose leaves are obtained from
$\C\times \{z'\}\subset \C \times {\H}$

{\bf\green Step 4:} For any complex curve
$C$ on $M$,  $\int_C \omega_0=0$, because
$\omega_0$ is exact. Therefore, $C$ is tangent
to a leaf of $\Sigma$.
{\bf \red It remains to show that $\Sigma$
has no compact leaves.}

{\bf\green Step 5:}
Let $\Sigma_0$ be a leaf of $\Sigma$.
Its preimage in $\C \times {\H}$
contains the set
\[
 \tilde \Sigma_0:=\bigcup_{z\in \C, \zeta\in \calo_K^+}
 \bigg (z, (z'+\sigma(\zeta))\bigg)
\]
where $z'\in {\H}$ is a fixed point.
Since the image of  $\sigma$ is dense in $\R$,
{\bf \purple the closure $\tilde \Sigma_0$ contains
$\C \times \R \times \Im (z')$.}

{\bf\green Step 6:} Therefore,
 {\bf \purple the closure $\Sigma_0\subset M$
is at least 3-dimensional,} hence
{\bf \red $\Sigma$ has no compact leaves}.

\newpage

{\bf \blue Oeljeklaus-Toma manifolds}

Let $K$ be a number field which has $2t$
complex embedding denoted $\tau_i, \bar \tau_i$ and 
$s$ real ones denoted $\sigma_i$,  $s>0$, $t>0$.

Let $\calo_K^{*,+}:= \calo_K^*\cap \bigcap_i \sigma^{-1}_i(\R^{>0})$.
Choose in $\calo_K^{*,+}\cong\Z^{t+1-1}$
a free abelian subgroup $\calo_K^{*,U}\cong\Z^s$
 such that the quotient
$\R^s/\calo_K^{*,U}$ is compact, where
$\calo_K^{*,U}$ is mapped to $\R^t$ as
$\xi \arrow \big(\log(\sigma_1(\xi)), ..., \log(\sigma_t(\xi))\big).$
Let $\Gamma:= \calo^+_K\rtimes \calo_K^{*,U}$.

\definition
{\bf \blue An Oeljeklaus-Toma manifold} is a quotient
$\C^t \times {\H}^s/\Gamma$, where
$\calo^+_K$ acts on $\C^t \times {\H}^t$
as 
\[ \zeta(x_1,..., x_t, y_1, ..., y_s) = \bigg(x_1+
   \tau_1(\zeta), ..., x_t + \tau_t(\zeta), 
   y_1+\sigma_1(\zeta), ..., y_s+\sigma_s(\zeta)\bigg),
\]
and $\calo_K^{*,U}$ as 
\[
  \xi(x_1,..., x_t, y_1, ..., y_s)=
  \bigg (\tau_1(\xi)x_1,..., \tau_t(\xi)x_t,\sigma_1(\xi) y_1,  ..., \sigma_t(\xi) y_t\bigg)
\]

\newpage

{\bf \blue Oeljeklaus-Toma manifolds are LCK}

\theorem (Oeljeklaus-Toma) 
The OT-manifold  $M:=\C^t \times {\H}^s/\Gamma$
 {\bf \red is a compact complex solvmanifold}. When $t=1$, it is locally
conformally K\"ahler. When $s=1, t=1$, it is an Inoue surface
of class $S^0$.

\proof We write the automorphic K\"ahler metric on $\C \times {\H}^s$
as $dd^c\phi$, where 
$\phi(x, \zeta_1, ..., \zeta_s)= |x|^2 + \prod_{i=1}^s \Im(\zeta_i)^{-1}$.
The function $\phi$ is clearly 
plurisobharmonic (it is Poincare metric on each $\H$, and Euclidean on $\C$),
and $dd^c\phi$ is $\calo_K^{+}$-invariant. Any $\xi\in \calo_K^{*,+}$
multiplies $|x|^2$ by $A:=\tau(\xi)^2$ and $\prod_{i=1}^s \Im(\zeta_i)^{-1}$
by $B^{-1}$, where $B:=\prod_{i=1}^s \sigma_i(\xi)^{-1}$. However, $AB=N(\xi)=1$,
because the norm $N(\xi)$ is integer and invertible. \endproof



\newpage

{\bf \blue Complex geometry of Oeljeklaus-Toma manifolds}

\theorem 
Let $K$ be a number field which has $s$ real embeddings
and $2t$ complex ones, $t=1$, $s>0$.
{\bf \red Then the corresponding Oeljeklaus-Toma manifold
has no non-trivial complex subvarieties}.

{\bf \green Proof. Step 1:}
Consider on $\C \times {\H}^t$
a function $\phi(z, z_1, ..., z_s):= \prod_i \Im(z_i)$.
Since $\Gamma$ multiplies $\Im(z_i)$
by a number, {\bf \purple the form $d\log \phi$ is
$\Gamma$-invariant.}
Let $\theta$ denote the corresponding 1-form on $M=\C \times {\H}^s/\Gamma$.

{\bf \green  Step 2:}
The 2-form $\omega_0:=d(I\theta)= dd^c \log \phi$ has Hodge type (1,1) and
{\bf \red positive definite on the leaves of the foliation
$\{z\}\times {\H}^t\subset \C \times {\H}^t$}
\[
\omega_0 = \1\6\bar\6\log \phi = 
\1 \sum_i\frac{dz_i\wedge d\bar z_i} {|\im z_i|^2}.
\]
Also, $\omega_0 \geq 0$.

{\bf\green Step 3:} Let $\Sigma \subset TM$
be the null-foliation of $\omega_0$ (the foliation
generated by the null eigenspace). {\bf \purple 
It is a holomorphic, involutive, smooth 1-dimensional foliation,}
with the leaves which are obtained from
$\C\times \{(z_1, ..., z_s)\}\subset \C \times {\H}^s$.

\newpage

%{\bf \blue Complex geometry of Oeljeklaus-Toma manifolds (2)}


{\bf\green Step 4:} 
For any complex $k$-dimensional subvariety
$C\subset M$, the integral $\int_C \omega_0^k=0$, because
$\omega_0$ is exact. Therefore, $C$ is at each point
tangent to a leaf of $\Sigma$.
{\bf \purple Since $\Sigma$ is 1-dimensional, this
means that $C$ contains at least one leaf of $\Sigma$.}

{\bf\green Step 5:} 
{\bf \red It remains to show that any variety
which contains a leaf of $\Sigma$ coincides with $M$.}

{\bf\green Step 6:}
Let $\Sigma_0$ be a leaf of $\Sigma$.
Its preimage in $\C \times {\H}^s$
contains a set
\[
 \tilde \Sigma_0(z_1,..., z_s) :=\bigcup_{z\in \C, \zeta\in \calo_K^+}
 \bigg (z, (z_1+\sigma_1(\zeta), ..., z_s + \sigma_s(\zeta))\bigg)
\]
where $z_1, ..., z_s\in {\H}^s$ is some fixed point.

{\bf \green Step 7:} {\bf \red We reduced the theorem to the
following statement}

\claim
A closure of $\tilde\Sigma_0(z_1,..., z_s)$ contains a set
\[
Z_{\alpha_1, ..., \alpha_s}:=
 \{ (\zeta, \zeta_1, ..., \zeta_s)\ \ 
|\ \ \im \zeta_i=\alpha_i, i = 1, ..., s\}
\]
where $\alpha_i = \im z_i$.

Indeed, {\bf \purple the smallest complex subspace containing
$T_xZ_{\alpha_1, ..., \alpha_s}$ is $T_xM$.}

\newpage

{\bf \blue The adele ring}

The previous claim is immediately implied by the
following statement, applied to {\bf \blue the set  
$\rho_1, ..., \rho_m$ of all real embedings.}

{\bf \green Theorem 1}
Let $K\!\!:\!\!\Q$ be a number field with has
$2t$ complex embeddings $\tau_1, \bar\tau_1, ...$ and
$s$ real ones, $\sigma_1, ..., \sigma_t$,
Á $\rho_1, ..., \rho_m$ -- embeddings $K$ 
to $\C$ or $\R$, and each of $\tau_i$
and $\sigma_i$ appears  once, except one. Consider the map
$R:\; K \arrow \R^a \times \C^b$,
 $R(\xi):=\rho_1(\xi), ..., \rho_m(\xi)$.
{\bf \red Then the image of $\calo_K$ is dense in $\R^a \times \C^b$.}

The proof is based on the {\bf \blue strong approximation
  theorem} (which is a ``modern version'' of Chinese remainders
theorem).

\definition
{\bf \blue Adelic group} ${\cal A}_K$
is a subset of the product $\prod_\nu K_\nu$ 
of all completions of $K$ for all equivalence
classes $\nu$ of absolute value functions,
consisting of sequences
$(x_{\nu_1}, ..., x_{\nu_n}, ...)$
where $|x_{\nu_i}|\leq 1$ for all $i$
except the finite number.

\remark
Tikhonov's theorem {\bf \red implies that ${\cal A}_K$
is locally compact.}

\newpage

{\bf \blue The strong approximation theorem}

{\bf \green Strong approximation theorem:}
Consider the natural embedding
$K\subset {\cal A}_K$. {\bf \purple Then its
image is a discrete, cocompact subgroup.}
Moreover, the projection of
${\cal A}_K \stackrel{P_{\nu_0}}\arrow \prod_{\nu\neq \nu_0} K_\nu$
to the product of all completions except one
{\bf \red maps $K$ to a dense subset of
$R_{\nu_0}({\cal A}_K)$.}

\remark
Further on, $K$ is considered as a {\bf \purple subring of
${\cal A}_K$}.

{\bf \green Proof of Theorem 1. Step 1:}
Let $\calo_{A_K}$ be a ring of all {\bf \blue integer
  adeles}, that is, such $(x_{\nu_1}, ..., x_{\nu_n},
...)\in {\cal A}_K$, that
$|x_{\nu_i}|\leq 1$ for each non-archimedean absolute value.
{\bf \purple Then  $\calo_K= K \cap \calo_{A_K}$.}


{\bf \green Step 2:} Let now $P:\; {\cal A}_K \arrow {\cal A}_1$
be a projection of ${\cal A}_K$ to the product of all 
completions except one archimedean. {\bf \red 
Since $\calo_{A_K}$ is open in ${\cal A}_K$,
its projection to ${\cal A}_1$ is open in ${\cal A}_1$}
(the projection {\bf \purple is an open map}).

{\bf \green Step 3:} We obtain that the image $P(K)\cap P(\calo_{A_K})$ 
is dense in $P(\calo_{A_K})$. {\bf \purple From Step 1, 
we obtain that $P(K)\cap P(\calo_{A_K})$ coinsides with
$P(\calo_K)$.}

{\bf \green Step 4:} We proved that
$P(\calo_K)$ is dense in
${\cal A}_1\cap P(\calo_{A_K})$. {\bf \red 
Therefore, its projection to the product
of all archimedean completions except one is also dense.}






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