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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 7: Immersion theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE and IUM, Moscow \\[2mm] March 31, 2014 } \end{center} \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 Conical K\"ahler manifolds (reminder)} \definition Let $(X,g)$ be a Riemannian manifold, and $C(X):= X \times \R^{>0}$, with the metric $t^2 g+ dt^2$, where $t$ is a coordinate on $\R^{>0}$. Then $C(X)$ is called {\bf \blue Riemannian cone} of $X$. {\bf \purple Multiplicative group $\R^{>0}$ acts on $C(X)$ by homotheties, $(m, t) \arrow (m, \lambda t)$.} \definition Let $(X,g)$ be a Riemannian manifold, $C(X):= X \times \R^{>0}$ its Riemannian cone, and $h_\lambda$ the homothety action. Assume that $(X,g)$ is equipped with a complex structure, in such a way that $g$ is K\"ahler, and $h_\lambda$ acts holomorphically. Then $C(X)$ is called {\bf \blue a conical K\"ahler manifold}. In this situation, $X$ is called {\bf \blue Sasakian manifold}. \remark A {\bf \blue contact manifold} is defined as a manifold $X$ with symplectic structure on $C(X)$, and $h_\lambda$ acting by homotheties. In particular, {\bf \purple Sasakian manifolds are contact}. {\bf \green Sasakian geometry is an odd-dimensional counterpart to K\"ahler geometry} \example Let $L$ be a positive holomorphic line bundle on a projective manifold. {\bf \purple Then the total space of its unit $S^1$-fibration is Sasakian.} \newpage {\bf \blue Reeb field (reminder)} \newcommand{\Reeb}{\operatorname{\sf Reeb}} \definition {\bf \blue A Sasakian manifold} is a contact manifold $S$ with a Riemannian structure, such that the symplectic cone $C(S)$ with its Riemannian metric is K\"ahler. \definition Let $S$ be a Sasakian manifold, $\omega$ the K\"ahler form on $C(S)$, and $r=t\frac{d}{dt}$ the homothety vector field. Then $\Lie_{Ir}t= \langle dt, Ir\rangle=0$, hence $iR$ is tangent to $S\subset C(S)$. This vector field (denoted by $\Reeb$) is called {\bf \blue the Reeb field} of a Sasakian manifold. \remark {\bf \purple The Reeb field is dual to the contact form} $\theta=\omega\cntrct r$. \theorem {\bf \red The Reeb field acts on a Sasakian manifold by contact isometries.} \definition A Sasakian manifold is called {\bf \blue regular} if the Reeb field generates a free action of $S^1$, {\bf \blue quasiregular} if all orbits of $\Reeb$ are closed, and {\bf \blue irregular} otherwise. \newpage {\bf \blue Vaisman manifolds (reminder)} \example For any given $\lambda\in \R^{>1}$, {\bf \purple the quotient $C(X)/h_\lambda$ of a conical K\"ahler manifold is locally conformally K\"ahler.} \definition An LCK manifold $(M, g, \omega, \theta)$ is called {\bf \blue Vaisman} if $\nabla\theta=0$, where $\nabla$ is the Levi-Civita connection associated with $g$. \theorem Let $M$ be a Vaisman manifold, $\tilde M$ its covering; the pullback of the Lee form $\theta$ to $\tilde M$ is denoted by the same letter $\theta$. Assume that $d\psi=\theta$ on $\tilde M$ (such $\psi$ exists, for example, if $\tilde M$ is a universal cover of $M$). Consider the form $\tilde\omega:=e^{-\psi}\omega$. {\bf \red Then $(\tilde M, \tilde \omega)$ is a K\"ahler manifold, isometric to a cone.} \theorem {\bf \red Every Vaisman manifold is obtained as $C(X)/\Z$,} where $X$ is Sasakian, $\Z= \bigg\langle (x, t) \mapsto (\phi(x), q t)\bigg\rangle$, $q>1$, and $\phi$ is a Sasakian automorphism of $X$. Moreover, the triple $(X, \phi, q)$ is unique. \newpage {\bf\blue K\"ahler potentials and plurisubharmonic functions (reminder)} \definition A real-valued smooth function on a complex manifold is called {\bf\blue plurisubharmonic (psh)} if the (1,1)-form $dd^c f$ is positive, and {\bf\blue strictly plurisubharmonic} if $dd^c f$ is an Hermitian form. \remark Since $dd^c f$ is always closed, {\bf \purple it is also K\"ahler when it is strictly positive.} \definition Let $(M,I,\omega)$ be a K\"ahler manifold. {\bf \blue K\"ahler potential} is a function $f$ such that $dd^c f=\omega$. \theorem Let $S$ be a Sasakian manifold, $C(S)=S \times \R^{>0}$ its cone, $t$ the coordinate along the second variable, and $r=t\frac{d}{dt}$. {\bf \red Then $t^2$ is a K\"ahler potential on $C(S)$. Moreover, the form $dd^c \log t$ vanishes on $\langle r, I(r)\rangle$ and the rest of its eigenvalues are positive.} \newpage {\bf\blue The fundamental foliation (reminder)} \definition Let $M$ be a Vaisman manifold, $\theta^\sharp$ its Lee field, and $\Sigma$ a 2-dimensional real foliation generated by $\theta^\sharp, I\theta^\sharp$. It is called {\bf \blue the fundamental foliation} of $M$. Clearly, $\Sigma$ is tangent to orbits of the one-parametric group of automorphisms of the covering $\tilde M$ generated by homotheties. Therefore, {\bf \purple $\Sigma$ is a holomorphic foliaton.} \theorem Let $M$ be a compact Vaisman manifold, and $\Sigma\subset TM$ its fundamental foliation. Then\\ {\bf \red \hphantom{a}\ \ 1. $\Sigma$ is independent from the choice of the Vaisman metric.\\ \hphantom{a}\ \ 2. There exists a positive, exact (1,1)-form $\omega_0$ with $\Sigma=\ker \omega_0$.\\ \hphantom{a}\ \ 3. For any complex subvariety $Z\subset M$, $Z$ is tangent to $\Sigma$. \\ \hphantom{a}\ \ 4. For any compact complex subvariety $Z\subset M$, the set of smooth points of $Z$ is Vaisman.} \definition A Vaisman manifold $M$ is called {\bf\blue regular}, if the leaves of the fundamental foliation are orbits of the group $(S^1)^2$ freely acting on $M$, {\bf \blue quasiregular} if these leaves are compact, and {\bf \blue irregular} otherwise. \newpage {\bf \blue Orbispaces (reminder)} \definition {\bf \blue Groupoid} is a category with all morphisms invertible. \definition An action of a group on a manifold is {\bf \blue rigid} if the set of points with trivial stabilizer is dense. \definition {\bf \blue An orbispace} is a topological space $M$, equipped with a structure of a groupoid (the points of $M$ are objects of the groupoid category), a covering $\{U_i\}$, and continuous maps $\phi_i:\; V_i\arrow U_i$, where each $V_i$ is equipped with a rigid action of a finite group $G_i$, satisfying the following properties. \\ \hphantom{a}\ \ \ 1. $\phi_i:\; V_i \arrow V_i/G_i=U_i$ is the quotient map.\\ \hphantom{a}\ \ \ 2. For each $x\in M$ and $U_i\ni x$, the group $\Mor(x,x)$ is equal to the stabilizer of $x$ in $G_i$. \remark An orbispace is a topological space, locally obtained as a quotient, {\bf \purple with the quotient structure remembered via the groupoid structure.} \newpage {\bf \blue Orbifolds (reminder)} \definition {\bf \blue An orbifold} is an orbispace $(M, \{\phi_i:\; V_i \arrow V_i/G_i=U_i\})$, where all $V_i$ are diffeomorphic to open balls in $\R^n$. \example Let $M=\C P^1/((x,y)\sim(x, -y))$. {\bf \purple This quotient is homeomorphic to $\C P^1$.} {\bf \red However, it is a different orbifold} if we consider the covering induced from $\C P^1/G$, $G=\{\pm 1\}$ and the groupoid structure where $\Mor(x,x)=\St_G(x)$. \definition {\bf \blue A smooth orbifold} is an orbifold $M$ equipped with a sheaf of functions $C^\infty M$ in such a way that for each $U_i=V_i/G_i$, the corresponding ring of sections $C^\infty U_i$ is identified with a ring of $G_i$-invariant smooth functions on $V_i$. \definition {\bf \blue A complex orbifold} is an orbifold $M$ equipped with a sheaf of functions $\calo_M$ in such a way that each $V_i$ is an open ball in $\C^n$, and for each $U_i=V_i/G_i$, the corresponding ring of sections $\calo_{U_i}$ is identified with a ring of $G_i$-invariant holomorphic functions on $V_i$. \newpage {\bf \blue Projective orbifolds (reminder)} \definition {\bf \blue An underlying complex variety} of a complex orbifold is a complex variety with the topological space $M$ and the structure sheaf $\calo_M$. \definition {\bf \blue A projective orbifold} is a complex orbifold with the underlying complex variety projective. \definition {\bf \blue A holomorphic vector bundle} on a complex orbifold is a $G_i$-equivariant vector bundle on each $V_i$, equipped with the $G_i$-invariant gluing maps satisfying cocycle condition. \theorem {\bf \blue (Baily)}\\ Let $M$ be a compact complex orbifold equipped with a holomorphic Hermitian vector bundle $L$. Assume that the curvature of $L$ is positive definite on all $V_i$ (in this case $L$ is called {\bf \blue positive}). {\bf \red Then $M$ is projective.} {\small \it W. L. Baily, \emph{On the imbedding of V-manifolds in projective spaces}, Amer. J. Math. {\bf 79} (1957), 403-430.} \newpage {\bf \blue Sasakian and Vaisman manifolds and their projective orbifolds} \theorem Let $T^n$ be a compact torus acting on a manifold $M$ with all orbits of the same dimension. {\bf \red Then $M/T^n$ is an orbifold}. \endproof \corollary Let $M$ be a quasiregilar Vaisman manifold, $\Sigma$ its fundamental foliation, and $M/\Sigma$ the quotient space. {\bf \red Them $X:=M/\Sigma$ is a projective orbifold.} {\bf \green Proof:} Being a $T^2$-quotient, $X$ is an orbifold. Since it is a quotient of a complex space by a complex group action, $X$ is a complex orbifold. By construction, the corresponding conical K\"ahler manifold $\tilde M$ is a total space of $\C^*$-bundle $L$ (in the orbifold sense). The standard local argument implies that the curvature of $L$ gives a K\"ahler orbifold metric on $X$. Baily's theorem implies that $X$ is projective. \endproof \corollary Let $S$ be a quasiregular Sasakian manifold, and $\Reeb$ its Reeb field. {\bf \red Then $X:=S/\Reeb$ is a projective orbifold, and $S$ is a total space of $U(1)$-bundle over $X$ associated with a positive holomorphic line bundle.} {\bf \green Proof:} $S\times S^1$ is Vaisman, and the corresponding fundamental foliation is $TS^1\times \Reeb$. \endproof \newpage {\bf \blue Conical K\"ahler structures and homotheties} {\bf \green Proposition 1:} Let $(M, \omega)$ be a conical K\"ahler manifold, and $X$ a vector field acting on $M$ by holomorphic, non-isometric homotheties, such that $IX$ also acts by homotheties, and $e^{t X}$ is defined for any real $t$. Then \\ {\bf \red \phantom{a}\ (a) $dd^c \phi= \omega$, where $\phi=|X|^2$.\\ \phantom{a}\ (b) Let $S_X:= \phi^{-1}(1)$. Then $S_X$ is Sasakian, \\and $M$ is isometric to $C(S_X)$.\\ \phantom{a}\ (c) $S_X$ is quasiregular if and only if the action of $X$ integrates to a holomorphic $\C^*$-action.} {\bf \green Proof. Step 1:} Since $X$, $IX$ act by homotheties, one has a character $\chi:\; \langle X, IX\rangle\arrow \R$ such that $\Lie_Z\omega=\chi(Z)\omega$. {\bf \purple Replacing $X$ by some linear combination of $X, IX$ if necessary, we may assume that $IX$ acts by isometries. } Rescaling, we may assume that $\Lie_X g=2g$. {\bf \green Step 2:} Define $X^\flat:= g(X, \cdot)$ {\bf \blue (``the dual 1-form'').} Then $dX^\flat= \Lie_{IX}\omega=0$ and $2 X^\flat = \Lie_X(X^\flat)=d\langle X, X^\flat\rangle=d|X|^2$. {\bf \green Step 3:} $\Lie_X \omega= 2\omega$, which gives $2\omega = d(\omega\cntrct X)= d(IX^\flat)= 2 dId|x|^2$ (last equation is proven in Step 2). {\bf \purple This proves Proposition 1 (a).} \newpage {\bf \blue Conical K\"ahler structures and homotheties (cont.)} {\bf \green Proposition 1:} Let $(M, \omega)$ be a conical K\"ahler manifold, and $X$ a vector field acting on $M$ by holomorphic, non-isometric homotheties, such that $IX$ also acts by homotheties, and $e^{t X}$ is defined for any real $t$. Then \\ {\bf \red \phantom{a}\ (a) $dd^c \phi= \omega$, where $\phi=|X|^2$.\\ \phantom{a}\ (b) Let $S_X:= \phi^{-1}(1)$. Then $S_X$ is Sasakian, \\and $M$ is isometric to $C(S_X)$.\\ \phantom{a}\ (c) $S_X$ is quasiregular if and only if the action of $X$ integrates to a holomorphic $\C^*$-action.} {\bf \green Step 4:} Let $M\arrow S_X$ map $m$ to an intersection of $e^{tX}m$ and $S_X$. {\bf \purple This gives a decomposition $M=S_X \times \R^{>0}$, compatible with the conical metric on $S_X \times \R^{>0}=C(S_X)$,} as shown in the last lecture using the local decomposition of Vaisman manifolds into a product of a Sasakian manifold and $\R$. {\bf \green Step 5:} Let $C$ be the group generated by $e^{tX}, e^{tIX}$. Clearly, $C=\R^{>0}\times \{e^{tIX}\}$. The Reeb orbits on $S_X$ are orbits of $e^{tIX}$, {\bf \purple hence they are compact if and only if $\{e^{tIX}\}$ is compact, equivalently, iff $C=\C^*$.} \endproof \newpage {\bf \blue Conical K\"ahler structures and $\C^*$-action} \remark For each holomorphic isometry $h$ of a Vaisman manifold, $h$ lifts to a conformal automorphism of its K\"ahler covering. However, {\bf \purple a conformal automorphism of a K\"ahler manifold is a homothety,} because $d(f\omega) =df \wedge \omega$, and this may vanish only when $df=0$. {\bf \green Theorem 1:} Let $C(S)$ be a conical K\"ahler manifold, $h_t$ the corresponding homothety action, and $X$ its vector field. {\bf \red Then there exists a vector field $X_1$ arbitrarily close to $X$} acting on $C(S)$ by holomorphic homotheties, with $I(X_1)$ also acting by homotheties, {\bf \red such that the action of $X_1$ integrates to $\C^*$-action on $C(S)$. } {\bf \green Proof:} Fix some $\lambda>1$, and let $M:= C(S)/h_\lambda$ be the corresponding Vaisman manifold, where $h_t$ acts isometrically. Consider the Lie group $G\subset \Iso(M)$ obtained as the closure of $\{h_t\}$. {\bf \purple For each vector field $X_1\in \Lie(G)$, $X_1$ acts on $M$ by holomorphic isometries,} hence it acts on $C(S)$ by homotheties; non-isometrically when $X_1$ is sufficiently close to $X$. Choosing $X'\in \Lie (G)$ rational and sufficiently close to $X$, we obtain an isometry of $M$ which integrates to a $T^2$-action on $M$ {\bf \purple and to non-isometric $\C^*$-action on its cone. } \endproof \remark By Proposition 1, {\bf \red his gives a new cone structure on $C(S)$. } \newpage {\bf \blue Density of quasiregular Vaisman manifolds} \corollary Let $C(S)$ be a conical K\"ahler manifold, with $S$ compact. {\bf \purple Then $C(S)$ is holomorphically isometric to a total space of non-zero sections of a positive line bundle over a projective orbifold.} \endproof \corollary {\bf \purple Any compact Vaisman manifold $(M,I)$ admits a deformation $(M,I')$ which is quasi-regular.} Moreover, $I'$ can be chosen arbitrarily close to $I$. {\bf \green Proof:} Take the conical K\"ahler manifold $C(S)$, and replace the homothety vector field $X$ by a quasiregular one $X'$. Then take a quotient $C(S)/\Z$ by $\Z$ acting as $e^{\lambda X'}$. \endproof \corollary {\bf \purple Any compact Sasakian manifold $(M,I)$ admits a deformation $(M,I')$ which is quasi-regular.} Moreover, $I'$ can be chosen arbitrarily close to $I$. \endproof \newpage {\bf \blue Immersion of conical K\"ahler manifolds} \corollary Let $C(S)$ be a conical K\"ahler manifold. {\bf \red Then there exists a holomorphic immersion $C(S)\arrow C(S^{2n-1})$ equivariant under homothety,} with $C(S^{2n-1})=\C^n \backslash 0$ the standard (flat) cone. \proof The manifold $C(S)$ is a space of non-zero vectors in a total space of a positive line bundle $L$ over a projective orbifold $X$. By Baily's theorem, $L^N$ is very ample, and there exists an embedding $X \stackrel j \hookrightarrow \C P^{n-1}$ such that $L^N=l^*(\calo(1))$. Consider a holomorphic map $\psi_0:\; C(S) \arrow \Tot(L^N)$ mapping $v$ to $v^N$. It is an $N$-sheeted covering. Now, define $\Psi:\; C(S)\arrow C(S^{2n-1})$ as $\Psi(v):= j(\psi_0(v))$. {\bf \purple Since $\psi_0$ is etale and $j$ an embedding, $\Psi$ is an immersion.} \endproof \newpage {\bf \blue Immersion of Vaisman manifolds} \definition {\bf\blue A linear Hopf manifold} is a quotient of $\C^n \backslash 0$ by a linear automorphism with all eigenvalues $|\alpha_i|<1$. \corollary Let $M$ be a quasiregular Vaisman manifold. {\bf \red Then $M$ admits an immersion into a linear Hopf manifold.} \proof Let $C(S)$ be a conical K\"ahler covering of $M$. Consider an immersion $\Psi:\; C(S)\arrow C(S^{2n-1})$, and let $\gamma:\; \Z \arrow \Aut(C(S))$ be the homothety action. Since $L$ is $\gamma$-equivariant, $\gamma$ actually induces a linear automorphism $\Gamma$ of a vector space $\C^n=H^0(L^N)$. Since $\gamma$ uniformly decreases the norm, the eigenvalues of $\Gamma$ are all $|\alpha_i|<1$. This gives a commutative square \begin{equation*} \begin{CD} C(S) @>{\Psi}>> C(S^{2n-1}) \\ @VV{/\gamma}V @VV{/\Gamma}V \\ M@>>> (\C^{n}\backslash 0)/\langle\Gamma\rangle \end{CD} \end{equation*} with the bottom arrow holomorphic immersion. \endproof \remark In fact, for each Vaisman manifold {\bf \purple there exists an embedding into a linear Hopf manifold.} \newpage {\bf \blue Kodaira stability theorem} \theorem {\bf \blue (Kodaira)} Let ${\cal X} \stackrel\pi\arrow B$ be a smooth, proper, holomorphic map, and $z\in B$ a point. Assume that the fiber $X_z:= \pi^{-1}(z)$ is K\"ahler (that is, admits K\"ahler structure). {\bf \red Then there exists a neighbourhood $W\ni z$ such that for each $y\in W$, the fiber $X_y:= \pi^{-1}(y)$ is K\"ahler.} {\bf \green Proof. Step 1:} Consider the relative Fr\"olicher spectral sequence \[ R^i\pi_*(\Omega^j_B{\cal X})\Rightarrow R^{i+j}\pi_*(\C_{\cal X})\ \ \ (*) \] Here $R^{i+j}\pi_*(\C_{\cal X})$ is the derived pushforward of a constant sheaf (that is, a graded local system over $B$ with the fibers of grading $k$ in $y\in B$ identified with $k$-th cohomology of $X_y$), and the $E_2$ term $R^i\pi_*(\Omega^j_B{\cal X})$ is a coherent sheaf obtained as a derived direct image of the fiberwise de Rham algebra $\Omega^j_B{\cal X}=\Omega^j({\cal X}/B)$. It is a relative (over $B$) version of the usual Fr\"olicher spectral sequence $H^i(\Omega^j M)\Rightarrow H^{i+j}(M,\C)$. This spectral sequence gives an inequality \[ \sum_{i+j=k}\dim H^i(\Omega^jX_y)\geq \sum_{i+j=k}\dim H^i(\Omega^jX_z) \ \ \ (**) \] \newpage {\bf \blue Kodaira stability theorem (part 2)} {\bf \green Proof. Step 1:} Consider the relative Fr\"olicher spectral sequence \[ R^i\pi_*(\Omega^j_B{\cal X})\Rightarrow R^{i+j}\pi_*(\C_{\cal X})\ \ \ (*) \] This spectral sequence gives an inequality \[ \sum_{i+j=k}\dim H^i(\Omega^jX_y)\geq \sum_{i+j=k}\dim H^i(\Omega^jX_z) \ \ \ (**) \] Since $X_z$ is K\"ahler, the Fr\"olicher spectral sequence for $X_z$ degenerates in $E_2$, giving $\sum_{i+j=k}\dim H^i(\Omega^jX_z)= h^k(X_z)$. By semicontinuity, \[ \sum_{i+j=k}\dim H^i(\Omega^jX_y)\leq \sum_{i+j=k}\dim H^i(\Omega^jX_z)\] in a sufficiently small neighbourhood $U$ of $z$. Comparing this with (**), we find that {\bf \purple rank of $H^i(\Omega^jX_y)$ is constant in $U$,} hence the inequality (**) is equality in $U$, and {\bf \purple the spectral sequence (*) degenerates.} \newpage {\bf \blue Kodaira stability theorem (part 3)} {\bf \green Step 2:} Consider the sheaf ${\cal H}:=R^1\pi_*(\Omega^1_B{\cal X})$. It is a coherent sheaf with fiber $H^{1,1}(X_y)$ at each $y\in B$. From Step 1, we obtain that ${\cal H}$ is locally free in $U$, and generated by fiberwise closed (1,1)-forms. Let $\Lambda^{1,1}_{cl}({\cal X}/B)$ be the sheaf of fiberwise closed vertical forms on ${\cal X}$, and $\pi_*\Lambda^{1,1}_{cl}({\cal X}/B)\stackrel \Xi\arrow {\cal H}$ the natural projection. Choose a Hermitian metric on ${\cal X}$, smoothly extending the K\"ahler metric $\omega_z$ on $X_z$, and let ${\cal H} \stackrel {\Xi^*}\arrow \pi_*\Lambda^{1,1}_{cl}({\cal X}/B)$ be the Hermitian conjugate map. By construction, $\Xi^*$ is an orthogonal projection of cohomology to closed (1,1)-forms along exact 2-forms. Therefore, {\bf \purple $\Xi^*$ maps the K\"ahler class $[\omega_z]$ to its harmonic representative $\omega_z$.} {\bf \green Step 3:} Let $\tilde \omega$ be a smooth section of ${\cal H}$ satisfying $\tilde\omega\restrict z=[\omega_z]$. {\bf \purple Then $\Xi^*(\tilde \omega)$ is a family of closed forms $\omega_y\in \Lambda^{1,1}_{cl}(X_y)$, depending smoothly on $y\in B$.} Since all eigenvalues of $\omega_z$ are positive, the same is true for $\omega_y$ for $y$ sufficiently close to $z$. However, a closed, positive $(1,1)$-form is K\"ahler. \endproof \remark Neither Vaisman manifolds nor LCK manifolds are stable under small deformations. However, a small deformation of Vaisman manifolds in LCK. Next lecture I will define a new class of LCK manifolds, called {\bf \blue LCK manifolds with potential} which is stable under small deformations and contains Vaisman manfilds. \end{document}