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Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \setlength\paperheight {10in}% \setlength\paperwidth {13.5in} \setlength{\textwidth}{0.8\paperwidth} \setlength{\textheight}{0.8\paperheight} \setlength{\pdfpageheight}{\paperheight} \setlength{\pdfpagewidth}{\paperwidth} \addtolength{\topmargin}{-20mm} \addtolength{\leftmargin}{-25mm} \addtolength{\rightmargin}{-25mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Lemma, sublemma, corollary, proposition, theorem, % % definition,example defined there: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\proof}{{\bf \green Proof:\ }} \newcounter{section} \newcounter{Mycounter}[section] \newcounter{lemma}[section] \setcounter{lemma}{0} \renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}} \newcommand{\lemma}{% \setcounter{lemma}{\value{Mycounter}} \refstepcounter{lemma} \stepcounter{Mycounter} {\bf \green LEMMA:\ }} \newcounter{claim}[section] \setcounter{claim}{0} \renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}} \newcommand{\claim}{% \setcounter{claim}{\value{Mycounter}} \refstepcounter{claim} \stepcounter{Mycounter} {\bf \green CLAIM:\ }} \newcounter{corollary}[section] \setcounter{corollary}{0} \renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}} \newcommand{\corollary}{% \setcounter{corollary}{\value{Mycounter}} \refstepcounter{corollary} \stepcounter{Mycounter} {\bf \green COROLLARY:\ }} \newcounter{theorem}[section] \setcounter{theorem}{0} \renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}} \newcommand{\theorem}{% \setcounter{theorem}{\value{Mycounter}} \refstepcounter{theorem} \stepcounter{Mycounter} {\bf \green THEOREM:\ }} \newcounter{conjecture}[section] \setcounter{conjecture}{0} \renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}} \newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Observation \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcounter{exercise}[section] \setcounter{exercise}{0} \renewcommand{\theexercise}{\noindent{Exercise \thesection.\arabic{exercise}}} \newcommand{\exercise}{% \setcounter{exercise}{\value{Mycounter}} \refstepcounter{exercise} \stepcounter{Mycounter} {\bf \green EXERCISE:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \green QUESTION:\ }} \newcommand{\problem}{% {\bf \green PROBLEM:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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.} \end{document} \end{document}