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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:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Locally conformally K\"ahler manifolds \\[15mm] \small lecture 1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE and IUM, Moscow \\[2mm] February 10, 2014 } \end{center} \newpage {\bf \blue Complex manifolds} {\bf\green DEFINITION:} Let $M$ be a smooth manifold. An {\bf \blue almost complex structure} is an operator $I:\; TM \arrow TM$ which satisfies $I^2 = - \Id_{TM}$. {\bf \purple The eigenvalues of this operator are $\pm \1$.} The corresponding eigenvalue decomposition is denoted $TM=T^{0,1}M\oplus T^{1,0}(M)$. {\bf\green DEFINITION:} An almost complex structure is {\bf \blue integrable} if $\forall X,Y \in T^{1,0}M$, one has $[X,Y]\in T^{1,0}M$. In this case $I$ is called {\bf \blue a complex structure operator}. A manifold with an integrable almost complex structure is called {\bf \blue a complex manifold}. {\bf\green THEOREM:} (Newlander-Nirenberg)\\ {\bf \red This definition is equivalent to the usual one.} \remark The commutator defines a $\C^\infty M$-linear map\\ $N:=\Lambda^2(T^{1,0})\arrow T^{0,1}M$, called {\bf \blue the Nijenhuis tensor} of $I$. {\bf \purple One can represent $N$ as a section of $\Lambda^{2,0}(M) \otimes T^{0,1}M$.} \newpage {\bf \blue K\"ahler manifolds} {\bf\green DEFINITION:} An Riemannian metric $g$ on an almost complex manifiold $M$ is called {\bf \blue Hermitian} if $g(Ix, Iy)= g(x,y)$. In this case, $g(x, Iy)= g(Ix, I^2y) = - g(y, Ix)$, hence $\omega(x,y):= g(x, Iy)$ is skew-symmetric. {\bf\green DEFINITION:} The differential form $\omega\in \Lambda^2(M)$ is called {\bf \blue the Hermitian form} of $(M,I,g)$. \remark It is $U(1)$-invariant, hence {\bf \purple of Hodge type (1,1)}. {\bf\green DEFINITION:} A complex Hermitian manifold $(M,I,\omega)$ is called {\bf \blue K\"ahler} if $d\omega=0$. The cohomology class $[\omega]\in H^2(M)$ of a form $\omega$ is called {\bf \blue the K\"ahler class} of $M$, and $\omega$ {\bf \blue the K\"ahler form}. {\bf \green Definition:} Let $M=\C P^n$ be a complex projective space, and $g$ a $U(n+1)$-invariant Riemannian form. It is called {\bf \blue Fubini-Study form on $\C P^n$}. The Fubini-Study form is obtained by taking arbitrary Riemannian form and averaging with $U(n+1)$ using the Haar measure on $U(n+1)$. \exercise Prove that {\bf \red the Fubini-Study form is unique} (up to a constant multiplier). \newpage {\bf\blue Examples of K\"ahler manifolds.} {\bf \green Remark:} For any $x\in \C P^n$, the stabilizer $St(x)$ is isomorphic to $U(n)$. Fubini-Study form on $T_x\C P^n= \C^n$ is $U(n)$-invariant, hence unique up to a constant. {\bf \green Claim:} {\bf \red Fubini-Study form is K\"ahler.} Indeed, $d\omega\restrict x$ is a $U(n)$-invariant 3-form on $\C^n$, but such a form must vanish, because $-\Id\in U(n)$ \remark {\bf \purple The same argument works for all symmetric spaces.} \definition {\bf \blue An almost complex submanifold} $X \subset M$ of an almost complex manifold $(M,I)$ is a smooth submanifold which satisfies $I(TX)\subset TX$. \exercise Let $X \subset M$ be an almost complex submanifold of $(M,I)$, where $I$ is integrable. {\bf \purple Prove that $(X, I\restrict{TX})$ is a complex manifold.} \definition In this situation, $X$ is called {\bf \blue a complex submanifold} of $M$. {\bf \green Corollary:} {\bf \red Every projective manifold (complex submanifold of $\C P^n$) is K\"ahler.} Indeed, a restriction of a closed form is again closed. \newpage {\bf \blue Menagerie of complex geometry} Usually, in algebraic geometry one deals with projective manifolds. There are two wider classes one has to consider necessarily when studying projective ones. 1. {\bf \blue Moishezon manifolds} are those which are birational to projective. {\bf \purple Transcendence degree of a field $k(M)$ of global meromorphic function on a compact complex manifold $M$ satisfies $k(M)\leq M$;} as shown by Moishezon, equality here means that $M$ is Moishezon. To study birational category, one has necessarily to include Moishezon manifolds. {\bf \purple Any K\"ahler Moishezon manifold is projective (Moishezon).} 2. {\bf\purple Small deformations of K\"ahler manifolds} often result in non-projective K\"ahler ones (even for a torus and a K3). \newpage {\bf \blue Fujiki class C manifolds} A class which includes Moishezon and K\"ahler is called {\bf \red Fujiki class C}. A manifold is {\bf \blue Fujiki class C} if it is bimeromorphic to a K\"ahler manifold. As shown by Fujiki, Fujiki class C manifolds are closed under all natural operations which occur in algebraic geometry (such as taking moduli spaces or images). {\bf \blue A K\"ahler minimal model program} would imply that any K\"ahler manifold admits a sequence of bimeromorphic fibrations with fibers which are either projective, hyperk\"ahler or tori, hence {\bf \purple the class of K\"ahler manifolds is probably very restricted.} It is known that {\bf \purple a fundamental group of K\"ahler manifold is very special.} By contrast, {\bf\purple the class of complex manifolds is extremely huge}. \newpage {\bf \blue Menagerie of complex geometry II} \theorem (Taubes, Panov-Petrunin)\\ For any finitely generated, finitely presented group $\Gamma$, {\bf \purple there exists a compact, complex 3-dimensional manifold $M$ with $\pi_1(M)=\Gamma$.} \hfill \conjecture Let $(M,I)$ be an almost complex manifold, $\dim_\C M\geq 3$. {\bf \red Then $I$ can be deformed to a complex structure.} \hfill \remark (Non-)existence of a complex structure {\bf \red is highly non-trivial even in the simplest cases,} such as $S^6$ (which is clearly almost complex). \remark We know that non-K\"ahler complex manifolds are much more abundant than K\"ahler, except in complex dimension 2, where non-K\"ahler manifolds are a few and much better understood than projective ones. However, {\bf \purple it's very hard to come with examples of compact, non-K\"ahler complex manifolds.} \newpage {\bf \blue Examples of non-K\"ahler manifolds} These listed below (and iterated fibrations of these) are pretty much all examples known. \example {\bf \blue (Linear) Hopf manifold} is $\C^n\backslash 0/\langle A\rangle$, where $A$ is an invertible linear map with all eigenvalues $|\alpha_i|<1$. It's diffeomorphic to $S^{2n-1}\times S^1$, hence non-K\"ahler (K\"ahler manifolds have even $b_{2k-1}$ Betti numbers). It is locally conformally K\"ahler (LCK). \example {\bf \purple All complex subvarieties of a Hopf manifolds are LCK.} For this reason, {\bf \red they are non-K\"ahler} (Vaisman). \example {\bf \purple Twistor space} is a certain $\C P^1$-fibration over a Riemannian 4-manifold. All such manifolds are rationally connected (connected by rational curves), {\bf \red but never K\"ahler} except $\C P^3$ and flag space (Hitchin). Theorem of Taubes is proved by constructing a twistor space with prescribed fundamental group. \example Homogeneous and locally homogeneous manifolds (such as nilmanifolds) are often non-K\"ahler. {\bf \red This explains importance of LCK manifolds} (defined below). \newpage {\bf \blue Connections} {\bf \green Notation:} Let $M$ be a smooth manifold, $TM$ its tangent bundle, $\Lambda^iM$ the bundle of differential $i$-forms, $C^\infty M$ the smooth functions. {\bf \purple The space of sections of a bundle $B$ is denoted by $B$.} \definition A {\bf\blue connection} on a vector bundle $B$ is a map $B \stackrel \nabla \arrow \Lambda^1 M \otimes B$ which satisfies \[ \nabla(fb) = df \otimes b + f \nabla b\] for all $b\in B$, $f\in C^\infty M$. \remark A connection $\nabla$ on $B$ gives a connection $B^* \stackrel {\nabla^*} \arrow \Lambda^1 M \otimes B^*$ on the dual bundle, by the formula \[ d(\langle b, \beta\rangle) = \langle \nabla b, \beta\rangle+ \langle b, \nabla^*\beta\rangle \] These connections are usually denoted {\bf \red by the same letter $\nabla$.} \remark For any tensor bundle ${\cal B}_1:= B^*\otimes B^* \otimes ... \otimes B^* \otimes B\otimes B \otimes ... \otimes B$ {\bf \green a connection on $B$ defines a connection on ${\cal B}_1$} using the Leibniz formula: \[ \nabla(b_1 \otimes b_2) = \nabla(b_1) \otimes b_2 + b_1 \otimes \nabla(b_2). \] \newpage {\bf \blue Curvature of a connection} Let $M$ be a manifold, $B$ a bundle, $\Lambda^i M$ the differential forms, and $\nabla:\; B \arrow B \otimes \Lambda^1M$ a connection. We extend $\nabla$ to $B \otimes \Lambda^iM\stackrel\nabla \arrow B \otimes \Lambda^{i+1}M$ in a natural way, using the formula \[ \nabla(b \otimes \eta) = \nabla(b)\wedge \eta + b \otimes d\eta, \] and define {\bf \blue the curvature $\Theta_\nabla$} of $\nabla$ as $\nabla\circ \nabla:\; B \arrow B\otimes \Lambda^2M$. \claim {\bf \red This operator is $C^\infty M$-linear.} \remark We shall consider $\Theta_\nabla$ as an element of $\Lambda^2M \otimes \End B$, that is, an $\End B$-valued 2-form. \remark Given vector fields $X, Y\in TM$, the curvature can be written in terms of a connection as follows \[ \Theta_\nabla(b)= \nabla_X\nabla_Yb - \nabla_Y\nabla_X B - \nabla_{[X,Y]}b. \] \claim Suppose that the structure group of $B$ is reduced to its subgroup $G$, and let $\nabla$ be a connection which preserves this reduction. This is the same as to say that the connection form takes values in $\Lambda^1 \otimes \goth g(B)$. {\bf \purple Then $\Theta_\nabla$ lies in $\Lambda^2M \otimes \goth g(B)$.} \newpage {\bf\blue Local systems} \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.} {\bf\green Proof:} The etale space of a local system is a covering of $M$, and the monodromy map from $\pi_1(M,x)$ to permutatons of $B$ is by construction linear. To obtain a converse correspondence, let $\pi:\; \tilde M \arrow M$ be the universal cover, and $X:=\tilde M\times B/\pi_1(M)$ be a quotient where $\pi_1(M)$ acts on $\tilde M\times B$ diagonally. Let $y\in M$ and $U\ni y$ be a neighbourhood for which $\pi^{-1}(U)$ is a union of several copies $U$. Then $X$ is a product $B\times U$. This gives a local trivialization of $\phi:\; X \arrow M$. {\bf \purple The sheaf of locally trivial sections of $\phi$ is locally trivial}, and the corresponding monodromy map is $\pi_1(M,x)\arrow \Aut(B)$ the one we started from. \endproof \newpage {\bf\blue Flat bundles} \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.} \proof Indeed, through each point passes of the total space of $B$ passes a unique parallel section, which always exists locally. \endproof \theorem This correspondence {\bf \red gives an equivalence of categories} of flat bundles and local systems. \proof Let ${\cal B}$ be a local system, and $B:={\cal B}\otimes_\R C^\infty M$ the corresponding vector bundle. Any section of $B$ can be written as $\sum f_i b_i$, where $b_i$ are sections of ${\cal B}$, and $f_i\in C^\infty M$. Write $\nabla(\sum f_i b_i):= \sum df_i\otimes b_i$. Clearly, this connection is flat, and the corresponding sheaf of parallel sections os ${\cal B}$. \endproof \definition {\small Define {\bf \blue the $B$-valued de Rham differential} on $d_\nabla \Lambda^i(M)\otimes B \arrow\Lambda^{i+1}(M)\otimes B$ as $d_\nabla(\eta\otimes b):= d\eta\otimes b + (-1)^{\tilde\eta}\eta\wedge \nabla b$. {\bf \purple It is easy to check that $d_\nabla^2=0$.} \exercise Show that {\bf \purple the cohomology of the complex $(\Lambda^*M\otimes B, d_\nabla)$ are equal to the cohomology of the local system ${\cal B}:=\ker \nabla$.} } \newpage {\bf\blue LCS manifolds} \definition Let $L$ be an oriented real line bundle (one-dimensional bundle) on $M$, equipped with a flat connection, and $\omega\in \Lambda^2(M)\otimes L$ an $L$-valued differential form. We say that $(M,\omega, L)$ is {\bf \blue locally conformally symplectic} (LCS) if $d_\nabla\omega=0$. In this situation $L$ is called {\bf \blue the weight bundle} of $(M,\omega)$, or {\bf \blue conformal weight}. \claim {\bf \purple An oriented real line bundle $L$ can be smoothly trivialized.} \proof Choose a Riemannian metric on $L$. Then the set of of positive unit vectors is a nowhere degenerate section of $L$. \endproof \claim Let $(M,\omega, L)$ be an LCS manifold, $l$ a trivialization of $L$, and $\theta\in \Lambda^1 M$ the corresponding connection form, $\nabla(l)=l\otimes\theta$. {\bf \purple Then $d\omega_l=-\omega_l\wedge\theta$, where $\theta$ is a closed 1-form,} and $\omega_l\in \Lambda^2(M)$ is $\omega$ considered as a differential form after the identification $L\cong C^\infty M$ provided by $l$. \proof After identifying $L$ and a trivial line bundle, we obtain $0=d_\nabla(\omega)= d(\omega) + \omega \wedge \theta$. \endproof \newpage {\bf\blue LCK manifolds} \remark We obtained that {\bf \purple the following two definitions are equivalent.} 1. LCS manifold is a manifold equipped with a non-degenerate 2-form $\omega$ satisfying $d\omega=\omega\wedge\theta$, where $\theta$ is a closed 1-form. 2. LCS manifold is a manifold equipped with a non-degenerate, closed 2-form $\omega$ taking values in a flat, oriented line bundle. \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}. \remark Usually one silently assumes that $\theta$ is not exact. Indeed, if $\theta=d\phi$, then $d(e^{-\phi}\omega)= e^{-\phi}d\omega - e^{-\phi}\omega\wedge\theta=0$, and $e^{-\phi}\omega$ is K\"ahler. In this case $(M,I, \omega)$ is called {\bf \blue globally conformally K\"ahler}. \remark As shown above, {\bf \purple 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}. \exercise Prove that {\bf \red $d\theta=0$ follows from $d\omega=\omega\wedge\theta$ when $\dim_\C M>2$.} \newpage {\bf\blue LCK manifolds and their K\"ahler covers} \claim Let $L$ be a local system on $M$, and $\pi:\; \tilde M \arrow M$ is a universal cover. {\bf \purple Then $\pi^* L$ is a trivial local system.} \proof Indeed, $\pi_1(\tilde M)=0$, hence all local systems on $\tilde M$ are trivial. \endproof We obtain that a universal cover of an LCK manifold admits a K\"ahler form taking values in a trivial bundle; {\bf \red this means that it is K\"ahler.} \remark Let $(M,I,\omega,\theta)$ be an LCK manifold, $\pi:\; \tilde M \arrow M$ its universal cover, and $\phi$ a function satisfying $d\phi=\pi^*\theta$. Then $d(e^{-\phi}\pi^*\omega)= e^{-\phi}d\pi^*\omega - e^{-\phi}\pi^*\omega\wedge\pi^*\theta=0$, hence {\bf \red the form $\tilde \omega:=e^{-\phi}\pi^*\omega$ is K\"ahler}. \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 $\Aut_M(\tilde M)=\pi_1(M)$.} \remark A deck transform maps $\pi^* \theta$ to itself, hence it maps $\phi$ to $\phi+C$. This implies that a deck transform maps $\tilde \omega$ to $e^C \tilde \omega$, {\bf \red acting on $\tilde M$ by K\"ahler homotheties.} \newpage {\bf\blue K\"ahler homotheties and LCK manifolds} \definition Let $(M,\omega)$ be an LCK manifold, $\tilde M$ its K\"ahler cover, and $\pi_1(M)\cong \Aut_M(\tilde M)$ the deck transform maps. {\bf \blue Homothety character} is a homomorphism $\chi:\; \pi_1(M)\arrow \R^{>0}$ mapping a deck transform $\gamma \in \Aut_M(\tilde M)$ to the number $\frac{\gamma^*(\tilde \omega)}{\tilde\omega}$. \remark Let $M$ be 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. Consider a local system $L$ on $M$ associated with the homothety character $\chi:\; \pi_1(M)\arrow \R^{>0}$, and let $\psi$ be its trivialization. For each $\gamma\in \Aut_M(\tilde M)$, one has $\frac{\gamma^*\psi}{\psi}=\frac{\gamma^*(\tilde \omega)}{\tilde\omega}$. Therefore, {\bf \purple $\psi^{-1}\tilde\omega$ is an $\Aut_M(\tilde M)$-form on $\tilde M$}. Denote by $\omega$ the corresponding form on $M$. Then \[ d\omega= d(\psi\tilde \omega)= d\psi\wedge \tilde\omega=d\log\psi\wedge\omega. \] {\bf \red We obtained that the form $\omega$ satisfies $d\omega=\omega\wedge\theta$, where $\theta=d\log\psi$.} {\bf \purple This brings one more definition of LCK manifolds.} \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. \newpage {\bf\blue Examples of LCK manifolds} \example {\bf \blue A classical Hopf manifold} is $H:=\C^n \backslash 0/ \Z$, where $\Z$ acts as a multiplication by a complex number $\lambda$, $|\lambda|>1$. \remark {\bf \purple Its covering has a usual K\"ahler form, and the mapping group acts by homotheties.} \observation $H$ is diffeomorphic to $S^1 \times S^{2n-1}$, and fibered over $\C P^{n-1}$ with fiber $\C^*/\langle \lambda \rangle$. \observation For any complex submanifold $X\subset \C P^{n-1}$, its preimage in $H$ is a complex manifold. \remark Obviously, any complex submanifold of an LCK manifold is again LCK. {\bf \purple This implies that $\sigma^* X\subset H$ is an LCK manifold.} \remark Next lecture I will prove that {\bf \red none of these manifolds admits a K\"ahler form, if $n>1$.} \end{document}