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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 11: CR-geometry of Sasakian manifolds} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE and IUM, Moscow \\[2mm] April 28, 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 $(C(X),gt^2 + dt^2)$ is equipped with a complex structure, in such a way that the conical metric $gt^2 + dt^2$ 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)} \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 LCK manifolds with potential (reminder)} \definition Let $M$ be an LCK manifold, and $(\tilde M, \tilde \omega)$ its K\"ahler covering. It is called {\bf \blue LCK manifold with potential} if $\tilde M$ admits an automorphic K\"ahler potential $\phi:\; \tilde M \arrow \R^{>0}$, $dd^c \phi=\tilde \omega$, which is {\bf \blue proper} (preimage of a compact is again compact). \theorem {\bf \purple The property of being LCK with potential is stable under small deformations}. \theorem Let $M$ be an LCK manifold, $\Gamma\subset \R^{>0}$ the monodromy group, and $(\tilde M, \tilde \omega)$ its K\"ahler covering, with $\tilde M/\Gamma=M$. Assume that $\tilde \omega$ admits a $\Gamma$-automorphic K\"ahler potential $\phi$. {\bf \red The map $\phi$ is proper if and only if $\Gamma=\Z$.} \theorem Let $M$ be an LCK manifold with potential, and $\tilde M$ its K\"ahler $\Z$-covering. Then a metric completion $\tilde M_c$ {\bf \red admits a structure of a complex manifold,} compatible with the complex structure on $\tilde M \subset\tilde M_c$. Moreover, the monodromy action on $\tilde M$ is extended to a holomorphic automorphism of $\tilde M_c$. \theorem Let $M$ be an LCK manifold with potential, $\dim_\C M>2$. {\bf \red Then $M$ admits a holomorphic embedding to a linear Hopf manifold. } \newpage {\bf\blue CR-manifolds (reminder)} {\bf \green Definition:} Let $M$ be a smooth manifold, $B\subset TM$ a sub-bundle in a tangent bundle, and $I:\; B \arrow B$ an endomorphism satisfying $I^2=-1$. Consider its $\1$-eigenspace $B^{1,0}(M)\subset B\otimes \C \subset T_C M=TM\otimes \C$. Suppose that $[B^{1,0}, B^{1,0}]\subset B^{1,0}$. Then $(B,I)$ is called {\bf\blue a CR-structure on $M$}. {\bf \green Example:} A complex manifold is CR, with $B=TM$. Indeed, {\bf \purple $[T^{1,0}M, T^{1,0}M]\subset T^{1,0}M$ is equivalent to integrability of the complex structure} (Newlander-Nirenberg). {\bf \green Example:} Let $X$ be a complex manifold, and $M\subset X$ a hypersurface. Then $B:=\dim_\C TM \cap I(TM)=\dim_\C X-1$, hence $\rk B=n-1$. Since $[T^{1,0}X,T^{1,0}X]\subset T^{1,0}X$, {\bf \red $M$ is a CR-manifold.} {\bf \green Definition:} {\bf \blue A Frobenius form of a CR-manifold} is the tensor $B\otimes B \arrow TM/B$ mapping $X, Y$ to the $\Pi_{TM/B}([X,Y])$. It is an obstruction to integrability of the foliation given by $B$. \newpage {\bf \blue Contact CR-manifolds (reminder).} {\bf \green Definition:} Let $(M,B,I)$ be a CR-manifold, with $\codim B =1$. Then $M$ is called {\bf \blue a contact CR-manifold} if its Frobenius form is non-degenerate. {\bf \green Remark:} Since $[B^{1,0}, B^{1,0}]\subset B^{1,0}$ and $[B^{0,1}, B^{0,1}]\subset B^{0,1}$, the Frobenius form is a pairing between $B^{0,1}$ and $B^{1,0}$. This means that it is Hermitian. \definition This Hermitian form is called {\bf \blue Levi form} of a CR-manifold. {\bf \green Definition:} Let $(M,B,I)$ be a CR-manifold, with $\codim B =1$. Then $M$ is called {\bf \blue a strictly pseudoconvex CR-manifold} if its Levi form is positive definite. \proposition Let $M$ be a complex manifold, $\phi\in C^\infty M$ a smooth function, and $s$ a regular value of $\phi$. Consider $S:=\phi^{-1}(s)$ as a CR-manifold, with $B=TS\cap I(TS)$ and let $\Phi$ be its Levi form, taking values in $TS/B$. Then $d^c \phi:\; TS/B\arrow C^\infty S$ trivializes $TS/B$. Consider tangent vectors $u, v \in B_x S$. {\bf \red Then $-d^c \phi(\Phi(u,v))=dd^c\phi(x,y))$.} \corollary Let $M$ be a complex manifold, $\phi\in C^\infty M$ a strictly plurisubharmonic function, and $s$ a regular value of $\phi$. {\bf \red Then $S:=\phi^{-1}(s)$ is strictly pseudoconvex.} \newpage {\bf \blue Algebraic cones (reminder).} \definition {\bf\blue An algebraic cone} is an affine variety $\cac$ admitting a $\C^*$-action $\rho$ with a unique fixed point $x_0$, called {\bf \blue the origin}, and satisfying the following: (i) $\cac$ is smooth outside of $x_0$, (ii) $\rho$ acts on the Zariski tangent space $T_{x_0}\cac$ with all eigenvalues $|\alpha_i|<1$. {\bf \blue An open algebraic cone} is a closed algebraic cone without the origin. \theorem Let $M=\tilde M/A$ be LCK manifold with potential, and $\tilde M$ its K\"ahler $\Z$-covering. {\bf \red Then $\tilde M$ is an open algebraic cone.} \newpage {\bf \blue Pseudoconvex shells and logarithm (reminder)} \definition Let $\tilde M$ be an open algebraic cone, $\tilde M_c$ the corresponding closed cone, and $\vec r\in T\cac$ a holomorphic vector field such that for all $t>0$ the diffeomorphism $e^{t\vec r}$ is a holomorphic contraction of $\tilde M_c$ to origin. A strictly pseudoconvex hypersurface $S\subset \tilde M$ is called {\bf \blue a pseudoconvex shell} if $S$ intersects each orbit of $e^{t\vec r}$, $t\in \R$ exactly once. {\bf\green Theorem 1:} Let $\tilde M$ be an algebraic cone, $e^{t\vec r}$ a contraction, and $S\subset \tilde M$ a pseudoconvex shell. Then for each $\lambda\in \R$ there exists a unique function $\phi_\lambda$ such that $\Lie_{\vec r}\phi_\lambda = \lambda \phi_\lambda$ and $\phi_\lambda\restrict S=1$. Moreover, {\bf \red such $\phi_\lambda$ is strictly plurisubharmonic when $\lambda \gg 0$.} \corollary {\bf \blue (Gauduchon-Ornea)} \\ {\bf \red All linear Hopf manifolds are LCK with potential.} \theorem Let $M$ be an LCK manifold with potential, $\tilde M$ its K\"ahler its $\Z$-covering, and $M=\tilde M/\langle\gamma\rangle$. {\bf \red Then there exists $C\in \Z^{>0}$ and a holomorphic vector field $\vec r$ on $\tilde M$ such that $\gamma^C=\vec r$.} {\bf \green Proof:} Lecture 10. \definition Such a vector field is called {\bf \blue a logarithm of monodromy}. \newpage %{\bf\green Proof. Step 1:} %On $M$ the group $e^{\R \vec r}$ acts as a circle, and %maps $\omega$ to a form which is conformally equivalent to $\omega$ %Replacing $\omega$ by its average over $e^{\R \vec r}$, %we obtain a new LCK form $\omega'$ in the same conformal class and %$e^{\R \vec r}$-invariant. Further on, we replace %$\omega$ by an $e^{\R \vec r}$-invariant LCK form $\omega'$ if %necessary and assume $\mega$ is invariant. % %{\bf\green Step 2:} %Let $\phi:= \frac{\tilde \omega}{\pi^*\omega}$, where %$\phi:\; \tilde M \arrow M$ is the covering map, %and $d\omega=\omega\wedge \theta$. Then $0=d(\phi \omega)= %(d\phi -\phi\theta)\wedge \omega$, hence $\phi\theta=d\phi$. %This gives $\theta = d\log \phi$. % %{\bf\green Step 3:} Since $\Lie_{\vec r} (\phi \omega)= %\Lie_{\vec r} (\phi) \omega= \lambda\phi \omega$, we %obtain $\lambda\phi = \Lie_{\vec r} (\phi)= d\langle %\phi\theta, \vec r\rangle$, hence $\langle %\theta, \vec r\rangle=\const$. % %{\bf\green Step 4:} The Lie algebra $\langle \vec r, I %\vec r\langle$ acts on $\ % % % % %Let $\theta:= \tilde \omega\cntrct I(\vec r)$. %Clearly, $\Lie_{\vec r}\tilde \omega = d(\tilde %\omega\cntrct \vec r)= d (I\theta)$. Since \newpage {\bf \blue Theorem of Kamishima-Ornea} \theorem {\bf \blue (Kamishima-Ornea)} \\ Let $(M,\omega, \theta)$ be an LCK manifold equipped with a holomorphic conformal $\C$-action, which lifts to non-isometric homotheties on its K\"ahler covering $\tilde M$. {\bf \red Then $(M,\omega, \theta)$ is conformally equivalent to a Vaisman manifold.} {\bf \green Proof. Step 1:} Let $\vec r$ be a vector field of this $\C$-action, and $\tilde \omega$ the K\"ahler form of $\tilde M$. Then $\Lie_{\vec r} \tilde\omega=a\tilde\omega$ and $\Lie_{I\vec r} \tilde\omega=b\tilde\omega$. Replacing $\vec r$ by a linear combination of $\vec r$ and $I(\vec r)$, we obtain a vector field preserving $\tilde\omega$. Replacing $\vec r$ by an appropriate linear combination of $\vec r$ and $I\vec r$, {\bf \purple we can assume that $\Lie_{I\vec r} \tilde\omega=0$ and $\Lie_{\vec r} \tilde\omega=\tilde\omega$.} {\bf \green Step 2:} $\Lie_{\vec r} \tilde \omega= d(\tilde \omega\cntrct \vec r)=\tilde \omega$, and $\Lie_{I\vec r} \tilde \omega=d\eta=0$, where $\eta= I(\omega\cntrct \vec r)= \omega\cntrct (I\vec r)$. {\bf \green Step 3:} $\Lie_{\vec r}\eta=\eta=d(\eta \cntrct \vec r)= d\langle \eta, \vec r\rangle$. %and %$\Lie_{I\vec r}\eta=0=d(\eta \cntrct I\vec r)= %d\langle \eta, I\vec r\rangle$. This gives $\tilde \omega= dd^c \phi$, where $\phi=\langle \eta, \vec r\rangle$. {\bf \green Step 4:} The action by $\vec r$ multiplies the K\"ahler potential $\phi$ by a constant; the action by $I(\vec r)$ preserves $\phi$. Therefore, {\bf \purple $\tilde M$ is locally isometric to a K\"ahler cone,} and $\omega:=\phi^{-1}\tilde \omega$ is Vaisman (Lecture 3), that is, satisfies $\nabla\theta=0$. \endproof \newpage {\bf \blue Vaisman manifolds and homothety action} \definition Let $\tilde M$ be an algebraic cone, $\rho:=e^{t\vec r}$ a contraction, and $S\subset \tilde M$ a pseudoconvex shell. Consider the (necessarily unique) function potential $\phi_\lambda$ which satisfies $\Lie_{\vec r}\phi_\lambda = \lambda \phi_\lambda$. Assume that it is a K\"ahler potential (by Theorem 1, it is a K\"ahler potential for $\lambda \gg 0$). Then $\phi_\lambda$ is called {\bf \blue $\rho$-automorphic K\"ahler potential}, and $dd^c\phi$ {\bf \blue $\rho$-automorphic K\"ahler form}. \theorem Let $(M,\omega)$ be an LCK-manifold with potential, and $\tilde M$ its algebraic cone, $\tilde M/\langle \gamma\rangle =M$, and $\phi$ its K\"ahler potential. Then {\bf \red $\omega$ is conformally equivalent to a Vaisman metric if and only if there exists a logarithm $\vec r$ of $\gamma$ such that $\Lie_{I\vec r}\phi =0$.} {\bf \green Proof:} If $M$ is Vaisman, $\tilde M=C(S)$, where $S$ is Sasakian, $\vec r := t\frac{d}{dt}$ its logarithm, and $I\vec r$ its Reeb field, acting on $C(S)$ by holomorphic isometries. Conversely, if $\tilde M$ admits a logarithm $\vec r$ with such properties, then the corresponding holomorphic flow acts on $\tilde M$ by homotheties, and $M$ is Vaisman by Kamishima-Ornea. \endproof \newpage {\bf \blue Stein manifolds (reminder).} \definition A complex variety $M$ is called {\bf \blue holomorphically convex} if for any infinite discrete subset $S\subset M$, there exists a holomorphic function $f\in \calo_M$ which is unbounded on $S$. \definition A complex variety is called {\bf \blue Stein} if it is holomorphically convex, and has no compact complex subvarieties. \remark Equivalently, {\bf\red a complex variety is Stein if it admits a closed holomorphic embedding into $\C^n$.} \theorem (K. Oka, 1942) {\bf \purple A complex manifold $M$ is Stein} if and only $M$ admits a K\"ahler metric with a {\bf \purple K\"ahler potential which is positive and proper} (proper = preimages of compact sets are compact). \theorem (H. Cartan, 1951) {\bf \purple A complex variety $M$ is Stein} if and only if for any coherent sheaf $F$ on $M$, {\bf \purple its cohomology $H^i(F)$ vanish for all $i>0$.} \newpage {\bf \blue CR-holomorphic functions and vector fields} \definition Let $(S,B,I)$ be a CR-manifold. A function $f$ on $S$ is called {\bf \blue CR-holomorphic} if for any vector field $v\in B^{0,1}$, we have $\Lie_v f=0$. A vector field $v\in TM$ is called {\bf \blue CR-holomorphic} if the corresponding diffeomorphism flow preserves $B$ and $I$. \theorem {\bf \blue (Rossi-Andreotti-Siu)}\\ Let $S$ be a compact strictly pseudoconvex CR-manifold, $\dim_\R S\geq 5$, and $H^0(\calo_S)_b$ the ring of bounded CR-holomorphic functions. {\bf \red Then $S$ is a boundary of a Stein manifold $M$ with isolated singularities,} such that $H^0(\calo_S)_b = H^0(\calo_M)_b$, where $H^0(\calo_M)_b$ denotes the ring of bounded holomorphic functions. Moreover, $M$ is defined uniquely, $M=\Spec(H^0(\calo_S)_b)$. \corollary The Lie group $G:=\Aut(S)$ of CR-automorphisms is identified with the group of complex automorphisms of the corresponding Stein space $M$. {\bf \red Its Lie algebra (the algebra of holomorphic vector fields) is the Lie algebra of holomorphic vector fields on $M$.} \newpage {\bf \blue Burns-Lee theorem} \theorem {\bf \blue (Dan Burns, John M. Lee)}\\ Let $S$ be a compact strictly pseudoconvex CR-manifold, and $\Aut_0(S)$ the connected component of its group of automorphisms. {\bf \red Then $\Aut_0(S)$ is compact} unless $S$ is equivalent to the standard sphere $S^{2n-1}\subset \C^n$ with its induced CR-structure. In the latter case $\Aut_0(S)=U(1, n)$. {\bf \green This theorem would not be used.} %Further in this lecture, I prove Burns-Lee theorem when %$S$ admits a Sasakian structure and $\dim_\R S>3$. \newpage {\bf \blue CR-manifolds and Sasakian manifolds} \definition Let $(S,B,I)$ be a CR-manifold. We say that $S$ {\bf \blue admits a Sasakian structure} if it can be realized as a CR-hypersurface $S\subset C(S)$, where $C(S)$ is a conical K\"ahler manifold. \definition Let $(S,B,I)$ be a CR-manifold, with $TS/B$ oriented (for strictly pseudoconvex CR-manifolds, the Levi form defines the orientation on $TS/B$). A vector field $v\in TS$ is called {\bf \blue positive} if it is transversal to $B$ everywhere, and its projection to $TS/B$ is positive. \example {\bf \purple The Reeb field of a Sasakian manifold is always positive} (or negative, depending on the choice of orientation). Indeed, $I\Reeb$ is always normal to $S$, hence $\Reeb\notin B=TS\cap I(TS)$. \theorem Let $S$ be a strictly pseudoconvex compact CR-manifold, $\dim_\R S \geq 5$. {\bf \red Then $S$ admits a Sasakian structure if and only if $S$ admits a positive holomorphic vector field.} This vector field becomes a Reeb field of this Sasakian manifold. \remark The implication ``admits a Sasakian structure'' $\Rightarrow$ ``admits a positive holomorphic vector field'' is clear, because the Reeb vector field is positive and CR-holomorphic. \newpage {\bf \blue CR-manifolds and Sasakian manifolds (2)} Assume that a strictly pseudoconvex compact CR-manifold $S$ admits a CR-holomorphic positive vector field $R$. {\bf \red We need to construct a Sasakian metric on $S$ such that $R$ is its Reeb field.} \remark The argument here is essentially the same as used to embed an LCK manifold with potential to a Hopf manifold. {\bf \green Step 1:} By Rossi-Andreotti-Siu, $S=\6 M$, where $M=\Spec(H^0(\calo_S)_b)$ is a Stein variety with isolated singularities, and $R$ acts on $M$ by holomorphic automorphisms. {\bf \green Step 2:} Since $R$ is positive, $IR$ is transversal to $\6 M$; replacing $R$ by $-R$, we can always assume that $IR$ points toward interior of $M$, and {\bf \purple $A_\epsilon:=e^{\epsilon I R}$ for small $\epsilon$ maps $M$ to a subset $A_\epsilon(M)\subset M$ with compact closure.} {\bf \green Step 3:} Consider the ring ${\cal H}= H^0(\calo_M)_b$ of bounded holomprhic functions on $M$, with $\sup$-metric. Then ${\cal H}$ is a Banach ring. Since $A_\epsilon(M)$ has compact closure, {\bf \purple $A_\epsilon^* {\cal H}$ is a normal family, and $A_\epsilon^*$ is a compact operator.} \newpage {\bf \blue CR-manifolds and Sasakian manifolds (3)} {\bf \green Step 4:} By maximum principle, for any non-constant $f\in {\cal H}$, one has \\ $\sup_{A_\epsilon(M)} |f|<\sup_M |f|$. {\bf \purple Since any limit point $f_{\lim}$ of a sequence $(A_\epsilon^i)^* f$ satisfies $\sup {A_\epsilon(M)} |f|=\sup_M |f|$, it is constant.} A limit function $f_{\lim}$ exists, because $A_\epsilon^* {\cal H}$ is precompact. {\bf \green Step 5:} This implies that for each $z\in M$, a limit point $z_{\lim}$ of a sequence $\{z, A_\epsilon z,A_\epsilon^2 z, ...\}$ is unique and independent of $z$. Indeed, $f_{\lim}(z)= f(z_{\lim})$, but $f_{\lim}=\const$. This implies that {\bf \purple $A_\epsilon$ is a holomorphic contraction contracting $M$ to the origin point $x_0\subset M$.} {\bf \green Step 6:} Since $R\restrict {S_\epsilon}=A_\epsilon(R)$ is nowhere vanishing for each $\epsilon$, the vector field $\vec r:= IR$ is transversal to $S_\epsilon:= A_\epsilon(S)$ pointing to the origin. Therefore, through each point of $S$ passes a unique solution $\rho(t)$ of an equation $\frac{d\rho(t)}{dt}=\vec r$. {\bf \green Step 7:} Let $\phi_\lambda$ be a $\rho$-automorphic K\"ahler potential associated with this $S$ and $\rho$ as above. {\bf \purple The Lie algebra $\langle R, I R\rangle$ acts on $(M, dd^c \phi_\lambda)$ by holomorphic homotheties, hence it is a conical K\"ahler manifold} (Kamishima-Ornea). Therefore, $S$ is Sasakian. \endproof %\newpage % %{\bf \blue Kobayashi pseudometric and Kobayashi hyperbolic manifolds} % % %\definition %{\bf \blue Pseudometric} on $M$ %is a function $d:\; M \times M \arrow \R^{\geq 0}$ %which is symmetric: $d(x,y)=d(y,x)$ and satisfies the %triangle inequality $d(x,y)+d(y,z) \geq d(x,z)$. % %\remark %Let ${\goth D}$ be a set of pseudometrics. {\bf \purple Then %$d_{\max}(x,y):= \sup_{d\in {\goth D}}d(x,y)$ is also a pseudometric.} % %\definition %The {\bf \blue Kobayashi pseudometric} on a complex manifold $M$ %is $d_\max$ for the set ${\goth D}$ of all pseudometrics %such that any holomorphic map from the %Poincar\'e disk to $M$ is distance-decreasing. % %In other words, distance between points $x, y$ in Kobayashi pseudometric %is infimum of the distance over all sets of Poincare disks %connecting $x$ to $y$. % %\definition %A variety is called {\bf \blue Kobayashi hyperbolic} if %the Kobayashi pseudometric is non-degenerate. % %\exercise %{\bf \purple Prove that any bounded subset of $\C^n$ is Kobayashi hyperbolic.} %Use the Schwartz lemma. % %\remark %From this exercise it follows that any precompact subset of a Stein %manifold is Kobayashi hyperbolic. % %\definition %Let $M$ be a Stein manifold equipped with a finite volume form %$\Vol(M)$, and $L^2(\calo_M)$ the space of all $L^2$-integrable %holomorphic functions. A locally $L^2$-integrable holomorphic %function is holomorphic, by multi-dimensional Cauchy lemma. %Therefore, $L^2(\calo_M)$ is a Hilbert space. Consider %an orthonormal basis $\xi_i$ in $L^2(\calo_M)$. {\bf \blue %Bergman metric} on $M$ is $dd^c\sum |\xi_i|^2$. % %\exercise Let $M$ be a Stein manifold %which admits a closed holomorphic embedding to a bounded %open subset of $\C^n$. %Prove that the {\bf \blue Bergman kernel} %$\sum |\xi_i|^2$ is summable, and the %Bergman metric is nowhere degenerate and %depends only on the volume form $\Vol(M)$. % %\claim %Let $M$ be a Stein manifold %which admits a closed holomorphic embedding to a bounded %open subset of $\C^n$. Using the Kobayashi metric, we choose %the Hausdorff measure, giving a volume form $\Vol(M)$ on $M$. %{\bf \red Then the Bergman metric of $M$ is $\Aut(M)$-invariant.} % % %\newpage % %{\bf \blue Burns-Lee theorem for Sasakian manifolds} % % %\theorem %{\bf \blue (Dan Burns, John M. Lee)}\\ %Let $S$ be a compact strictly pseudoconvex CR-manifold %admitting a Sasakian structure, %and $\Aut_0(S)$ the connected component of its group of automorphisms. %{\bf \red Then $\Aut_0(S)$ is compact} unless $S$ is equivalent to %the standard sphere $S^{2n-1}\subset \C^n$ with its induced %CR-structure. In the latter case $\Aut_0(S)=U(1, n)$. % %{\bf \green Proof. Step 1:} Let $R$ be a positive CR-holomorphic %vector field on $S$, and $C(S)$ the corresponding conical K\"ahler %manifold. Consider its metric completion $\overline {C(S)}$ %with the complex structure obtained from Andreotti-Rossi-Siu, %and let $\rho$ be the corresponding homothety action. % %{\bf \green Step 2:} Put the Bergman metric $h$ associated with the %Kobayashi volume form on $M$. Clearly, $h$ is $\Aut_0(S)$-invariant. %The curvature $R^i_{jkl}$ of $h$ is $\rho$-invariant, %and {\bf \purple its absolute value $|R^i_{jkl}|$ computed with respect %to the K\"ahler metric $\rho$-automorphic, growing quadratically %as we go towards the origin $x_0$.} % %{\bf \green Step 2:} Unless $C(S)$ is flat (which happens precisely %when $S$ is a sphere), any automorphism \newpage {\bf \blue Jordan-Chevalley decomposition} \definition {\bf \blue An algebraic group} is a group object in the category of affine schemes. {\bf \blue A pro-algebraic group} is an inverse limit of algebraic groups. Further on, all algebraic groups are considered over $\C$. \definition An element of an algebraic group $G$ is called {\bf \blue semisimple} if its image is semisimple for any algebraic representation of $G$, and {\bf \blue unipotent} if its image is unipotent (that is, exponent of nilpotent) for any algebraic representation of $G$ \theorem {\bf \blue (The Jordan-Chevalley decomposition)}\\ Let $G$ be an algebraic group, and $a\in G$. {\bf \red Then there exists a unique decomposition $A= S U$ of $A$ onto a product of commuting elements $S$ and $U$, where $U$ is unipotent and $S$ semisimple.} \exercise {\bf \purple Prove this theorem.} \remark Since this decomposition is unique, it is functorial. {\bf \red Therefore, it is also true for all pro-algebraic groups.} \newpage {\bf \blue Semisimple LCK manifolds with potential} Recall that a linear operator is called {\bf \blue semisimple} if it is diagonalizable over an algebraic closure of the basic field. \definition Let $M$ be an LCK manifold with potential, and $j:\; M \arrow H$ a holomorphic embedding to a Hopf manifold $H = \C^n \backslash 0/\langle A\rangle$. Then $M$ is called {\bf \blue semisimple} if $A$ is semisimple. \remark Let $M$ be an LCK manifold with potential, $\tilde M$ its K\"ahler $\Z$-covering, $M= \tilde M/\langle A\rangle$, $j:\; M \arrow H$ a holomorphic embedding to a Hopf manifold $H = \C^n \backslash 0/\langle A\rangle$, and $\tilde M_c$ its completion. Let $R$ be a $\goth m$-adic completion of $\calo_{\tilde M_c}$ in the maximal ideal of the origin $x_0 \in \tilde M_c$, $R= \lim_{\leftarrow} \calo_{\tilde M_c}/ {\goth m}^k$. Let $G:= \Aut(R)$; clearly, \[ G= \lim_{\leftarrow} \Aut(\calo_{\tilde M_c}/ {\goth m}^k), \] hence it is a pro-algebraic group. \newpage {\bf \blue Semisimple LCK manifolds with potential (2)} \proposition Let $j:\; M \arrow H$ be an embedding of an LCK manifold $M= \tilde M/\langle A\rangle$ to a Hopf manifold $H = V \backslash 0/\langle A\rangle$, such that $V$ is an $A$-invariant subspace in $\calo_{\tilde M_c}$. {\bf \red Then the action of $A$ on $V$ is semisimple if and only if $A$ is semisimple as an element of the proalgebraic group $G= \lim\limits_{\leftarrow} \Aut(\calo_{\tilde M_c}/ {\goth m}^k)$.} \proof If $A$ is semisimple as an element of $G$, its action on $V$, considered as an $A$-invariant subspace in $R\supset \calo_{\tilde M_c}$, is also semisimple. Conversely, if $A$ is semisimple on $V$, $\calo_{\tilde M_c}$ is a subring in $R$, which is a quotient ring of $\C[[V]]$; the latter is an adic completion of the polynomial ring $\C[V]$, where $A$ is clearly semisimple. \endproof \newpage {\bf \blue Vaisman manifolds are semisimple} \theorem Let $M$ be an LCK manifold with potential. Then {\bf \red $M$ is semisimple $\Leftrightarrow$ it admits a Vaisman structure.} {\bf \green Proof of Vaisman $\Rightarrow$ semisimple:}\\ Let $M$ be Vaisman, $M= \tilde M/\langle A\rangle$, and $\tilde M_c$ its completion, equipped with the K\"ahler potential $\phi:\; \tilde M_c \arrow \R^{\geq 0}$. Consider a compact subset $\tilde M_c^a:= \phi^{-1}([0,a])$. Consider an $L^2$-structure on the ring $H^0(\calo_{\tilde M^a_c})_b$ of bounded holomorphic functions, $|f|^2 = \int_{\tilde M^a_c}|f|^2 \tilde\omega^n$. Let $\vec r:= t \frac{d}{dt}$ be the homothety vector field on $\tilde M = C(S)$. {\bf \purple Then $I\vec r$ acts on $H^0(\calo_{\tilde M^a_c})_b$ by isometries,} hence its action on each finite-dimensional subspace of $H^0(\calo_{\tilde M^a_c})_b$ is semisimple. \newpage {\bf \blue Semisimple LCK manifolds are Vaisman} {\bf \green Proof of semisimple $\Rightarrow$ Vaisman. Step 1:}\\ Since all subvarieties of Vaisman manifolds are again Vaisman, {\bf \purple it would suffice only to show that all semisimple Hopf manifolds are Vaisman.} {\bf \green Step 2:} Let $H=V \backslash 0/\langle A\rangle$ be a semisimple Hopf manifold, $e_i$ an eigenvalue basis in $V$, and $A(e_i)=\alpha_i u_i e_i$, with $\alpha_i \in ]0, 1[$ and $u_i \in U(1)$. Consider a unit sphere $S\subset V = C^n$, and let $\rho(t)(e_i):= \alpha_i^t e_i$. Then there exists a $\rho$-automorphic K\"ahler potential $\phi_\lambda$ on $V\backslash 0$. Since $A\circ \rho(-1)$ preserves $S$ and $A$ commutes with $\rho(t)$, the function $\phi_\lambda$ is $A$-automorphic. {\bf \green Step 3:} By Kamishima-Ornea, {\bf \purple this metric is Vaisman whenever $S$} (and, therefore, the potential $\phi_\lambda$, and the corresponding automorphic K\"ahler form) {\bf \purple is invariant with respect to $e^{t I\vec r}$,} where $r= \frac {d\rho(t)}{dt}$. However, $e^{t I\vec r}(e_i)= e^{\1 t\log \alpha_i} e_i$, and this operator is unitary. \endproof \end{document}