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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 8: LCK manifolds with potential} %normal families, CR-structures and Levy form} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE and IUM, Moscow \\[2mm] April 7, 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 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 $C(S)$.} \newpage {\bf\blue Deformational stability of LCK manifolds} \definition Let ${\goth A}$ be a property of compact complex manifolds, and ${\cal X}\stackrel \pi \arrow B$ a smooth, proper map, that is, a smooth family of compact manifolds. We say that ${\goth A}$ is {\bf \blue stable under small deformations} if the set of all $z\in B$ such that $X_z:=\pi^{-1}(z)$ has ${\goth A}$ is open in $B$. \example Property of admitting a K\"ahler metric is stable under small deformations (``Kodaira stability theorem''). One also says {\bf \blue ``K\"ahler manifolds are stable with respect to small deformations'',} or {\bf \blue a small deformation of K\"ahler manifold is again K\"ahler}. \example Call a complex manifold {\bf\blue Hermitian symplectic} if it admits a symplectic form $\omega$ such that its $(1,1)$-part is Hermitian. \exercise Prove that {\bf \purple a small deformation of a Hermitian symplectic manifold is again Hermitian symplectic} \observation {\bf \red LCK manifolds are not stable with respect to small deformations} (Belgun). Also, {\bf \red Vaisman manifolds are not stable with respect to small deformations.} \newpage {\bf\blue Deformations of Vaisman manifolds: automorphic forms and functions} {\bf \green Theorem 1:} 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 Vaisman (that is, admits a Vaisman metric). {\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 LCK.} {\bf \green For the proof, see the next slide}. \definition Let $M$ be an LCK manifold, $(\tilde M, \omega)$ is K\"ahler covering, equipped with an action of $\pi_1(M)$, and $\chi:\; \pi_1(M)\arrow \R^{>0}$ {\bf \blue the weight character}, or {\bf \blue the character of automorphy} which puts $\gamma\in \pi_1(M)$ to a number $\frac{\gamma^*\tilde \omega}{\tilde \omega}$. Its image $\Gamma\subset \R^{>0}$ is called {\bf \blue the monodromy group of an LCK manifold $M$}. We shall always chose $\tilde M$ in such a way that $\Gamma$ acts on $\tilde M$ and $M=\tilde M/\Gamma$. \definition Let $M$ be a manifold, $\tilde M$ its Galois covering. A form $\eta$ on $\tilde M$ is called {\bf \blue automorphic} if for any $\gamma\in \pi_1(M)$ acting on $\tilde M$ as usual, the form $\gamma^*\eta$ is proportional to $\eta$. The character $\chi_\eta(\gamma):=\frac{\gamma^*\tilde \eta}{\tilde \eta}$ is called {\bf \blue the character of automorphy} for $\eta$. \example Let $M$ be a Vaisman manifold, $\tilde M=C(S)$ its K\"ahler covering, and $\phi=t^2$ its K\"ahler potential. {\bf \purple Then $\phi$ is an automorphic function.} \newpage {\bf\blue Deformations of Vaisman manifolds} {\bf \green Theorem 1:} 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 Vaisman (that is, admits a Vaisman metric). {\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 LCK.} {\bf \green Proof. Step 1:} Let $\tilde X_z=C(S)$ be a conical K\"ahler covering of $X_z$. By Ehresmann fibration theorem, $\pi$ is a locally trivial fibration. Replacing $B$ by a sufficiently small open neighbourhood of $z$, {\bf \purple we may assume that $\pi$ is trivial as a smooth fibration: ${\cal X}=X_z\times B$.} Consider a covering $\tilde {\cal X}\arrow {\cal X}$ with $\tilde {\cal X}=\tilde X_z\times B$, and let $\tilde X_y$ denote the fibers of the projection $\tilde {\cal X}\stackrel {\tilde \pi} \arrow B$. {\bf \green Proof. Step 2:} Let $\phi$ be the automorphic K\"ahler potential of $\tilde X_z=C(S)$. Extend $\phi$ to $\tilde {\cal X} = \tilde X_z\times B$ using the projection $\tilde {\cal X}\arrow \tilde X_z$. Restricting $\phi$ to $\tilde X_y\subset \tilde {\cal X}$, {\bf \purple we obtain an automorphic function $\phi_y$ on any $\tilde X_y$.} {\bf \green Proof. Step 3:} The form $dd^c \phi_y$ is closed, automorphic and of type (1,1). Therefore, $\tilde X_y$ is LCK whenever the pseudo-Hermitian form $dd^c \phi_y$ is positive definite. However, the complex structure on $X_y$ smoothly depends on $y\in B$, hence the function $y \arrow dd^c \phi_y$ is continuous, and its eigenvalues continuously depend on $y\in B$. Therefore, {\bf \purple for $y$ sufficiently close to $z$, these eigenvalues remain positive,} and $\phi_y$ gives an automorphic K\"ahler potential. \endproof \newpage {\bf\blue LCK manifolds with potential} \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). \example Vaisman manifold is an example of an LCK manifold with potential. \remark {\bf \purple The property of being LCK with potential is stable under small deformations} (Theorem 1; same proof). \remark For any complex submanifold $Z\subset M$ of an LCK manifold with potential, $Z$ is also an LCK manifold with potential. \newpage {\bf\blue Monodromy group of LCK manifolds with potential} \proposition 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 $\Gamma=\Z$.} {\bf \green Proof. $\Gamma=\Z$ $\Rightarrow$ $\phi$ is proper:}\\ Let $\gamma$ be a generator of $\Z$, such that $\gamma^*\phi=\lambda\phi$, and $\pi:\; \tilde M \arrow \tilde M/\Gamma =M$ the quotient map. {\bf \purple Then $\phi^{-1}([1, \lambda[)$ is a fundamental domain of $\Gamma$-action.} Therefore $\pi:\;\phi^{-1}([1,\sqrt \lambda])\arrow M$ is bijective onto its image, which is compact, hence $\phi^{-1}([1,\sqrt \lambda])$ is also compact. This implies that {\bf \purple preimage of any closed interval is compact}. {\bf \green Proof. $\phi$ is proper $\Rightarrow$ $\Gamma=\Z$:}\\ Assume $\gamma\neq \Z$; then {\bf \purple $\Gamma$ is a dense subgroup of $\R^{>0}$.} Fix $x\in \tilde M$ and nonempty interval $]a,b[\subset \R^{>0}$, and let ${\goth H}\subset \Gamma$ be the set of all $\gamma\in \Gamma$ such that $\phi(\gamma(x))\in ]a, b[$ Since $\Gamma$ is dense, ${\goth H}$ is infinite. However, $\phi({\goth H}\cdot x)\subset [a, b]$, hence {\bf \purple an infinite discrete set ${\goth H}\cdot x$ is contained in a compact $\phi^{-1}([a,b])$.} Contradiction! \endproof \newpage {\bf\blue Linear Hopf manifold} \definition Let $A\in \End(\C^n)$ be an invertible linear endomorphism with all eigenvalues $|\alpha_i| <1$. The quotient $H:= (\C^n \backslash 0)/\langle A\rangle$ is called {\bf \blue a linear Hopf manifold}. When $A$ can be diagonalized, $H$ is called {\bf\blue a diagonal Hopf manifold}. \theorem {\bf \red A linear Hopf manifold is LCK with potential.} {\bf \green Proof in lecture 10.} \theorem Let $M$ be an LCK manifold with potential, $\dim_\C M>2$. Then $M$ admits a holomorphic embedding to a linear Hopf manifold. {\bf \green Proof in lecture 9.} \newpage {\bf \blue Stein manifolds.} \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 Rossi-Andreotti-Siu theorem.} \theorem (Rossi 1965, Andreotti-Siu 1970)\\ Let $M$ be a complex manifold with boundary, $\dim_\C M >2$, and $\phi$ a proper K\"ahler potential on $M$, taking values in $[a, \infty[$, and equal to $c$ in the boundary of $M$. {\bf \red Then there exists a Stein variety $M_a$} with isolated singularities, obtained by gluing a compact domain to $M$, and it is unique. Moreover, any holomorphic function on $M$ can be extended to $M_a$. {\bf \green Theorem 2:} 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 $\tilde M \subset\tilde M_c$. {\bf \green For the proof see the end of this lecture.} \definition In assumptions of Theorem 2, the manifold $\tilde M_c$ is called {\bf \blue the cone} of an LCK manifold with potential. \newpage {\bf \blue Cone of an LCK manifold with potential.} {\bf \green Theorem 2:} 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 $\tilde M \subset\tilde M_c$. {\bf \green Claim 1:} {\bf \red The complement $\tilde M_c\backslash \tilde M$ is just one point,} called {\bf \blue the origin}. \proof Indeed, let $z_i=\gamma^{n_i}(x_i)$ be a sequence of points in $\tilde M$, with each $x_i$ in the fundamental domain $\phi^{-1}([1,\lambda])$ of the $\Gamma=\Z$-action. Clearly, the distance between two fundamental domains $M_n:=\gamma^{n}\phi^{-1}([1,\lambda])=\phi^{-1}([\lambda^n,\lambda^{n+1}])$ and $M_{n+k+2}=\gamma^{n+k+2}\phi^{-1}([1,\lambda]) $ is written as \[ d(M_n,M_{n+k+2})= \sum_{i=0}^k \lambda^{n+i} v, \ \ \ \ (*) \] where $v$ is a distance between $M_0$ and $M_2$. Then, $z_i$ may converge only if $\lim_i n_i=-\infty$ or if all $n_i$, except finitely many, belong to a set $(p, p+1)$ for some $p$. The second case is irrelevant, because each $M_i$ is compact, and in the first case, $\{z_i\}$ is always a Cauchy sequence, as follows from (*). All such $\{z_i\}$ are therefore equivalent, hence {\bf \purple converge to the same point in the metric completion.} \endproof \newpage {\bf \blue Normal families of functions} \definition Let $M$ be a complex manifold, and ${\cal F}$ a family of holomorphic functions $f_i \in H^0(\calo_M)$. We call ${\cal F}$ {\bf \blue a normal family} if for each compact $K\subset M$ there exists $C_K>0$ such that for each $f\in {\cal F}$, $\sup_K |f| \leq C_K$. \lemma Let $M$ be a complex Hermitian manifold, ${\cal F}\subset H^0(\calo_M)$ a normal family, and $K\subset M$ a compact. {\bf \red Then there exists a number $A_K>0$ such that $\sup_K |f'| \leq A_K$.} \proof Assume otherwise. Then there exists $x\in K$, $v\in T_xM$, and a sequence $f_i \in {\cal F}$ such that $\lim_i |D_v f_i|=\infty$. Pick a disk $\Delta\stackrel j \hookrightarrow M$ with compact closure in $M$, tangent to $v$ in $x$, such that $j(0)=x$. Let $w=tv$ be the unit tangent vector. Then $\sup_\Delta |f_i| < C_\Delta$ by the normal family condition. {\bf \purple By Schwartz lemma, this implies $|D_w f_i|0$. A sequence $f_i$ of functions converges to $f$ if it converges to $f$ uniformly on all compacts. \remark When $M$ is locally compact, any sequence of continuous functions converging in $C^0$ {\bf \purple converges to a continuous function} {\bf \red (prove it!)} \remark In a similar way one defines {\bf \blue $C^0$-topology on the space of sections of a bundle.} \newpage {\bf \blue $C^1$-topology} \definition Let $B$ be a vector bundle on a smooth manifold $M$, and $\nabla:\; B \arrow B\otimes \Lambda^1 M$ a connection. Define {\bf \blue $C^1$-topology} on the space of sections of $B$ (denoted, as usual, by the same letter $B$) as one where a sub-base of open sets is given by $C^0$-open sets on $B$ and $\nabla^{-1}(W)$, where $W$ is an open set in $C^0$-topology in $B\otimes \Lambda^1 M$. \remark A sequence $f_i$ converges in $C^1$-topology if it converges uniformly on all compacts, and first derivatives $f_i'$ also converge uniformy on all compacts. {\bf \purple This can be seen as a definition of $C^1$-topology.} \exercise {\bf \purple Prove that $C^1$-topology is independent from the choice of a connection.} \exercise Prove that the topological vector space $C^1 M$ of 1-differentialble functions on a manifold {\bf \purple is complete in $C^1$-topology.} \newpage {\bf \blue Arzel\`a-Ascoli theorem for normal families} \theorem Let $M$ be a complex manifold and ${\cal F}\subset H^0(\calo_M)$ a normal family of functions. Denote by $\bar{\cal F}$ its closure in $C^0$-topology. {\bf \red Then $\bar{\cal F}$ is compact and contained in $H^0(\calo_M)$.} {\bf \green Proof. Step 1:} Let $\{f_i\}$ be a sequence of functions in ${\cal F}$. By Tychonoff theorem, for each compact $K$, there is a subsequence of $\{f_i\}$ which converges pointwise on a dense countable subset $Z\subset K$. Taking diagonal subsequence, we find a subsequence $\{f_{p_i}\}\subset \{f_i\}$ which converges pointwise on a dense countable subset $Z\subset M$. Since $|f'_i|$ is uniformly bounded on compacts, the limit $f:= \lim_i f_i$ is Lipschitz on all compact subsets of $M$. {\bf \purple Then it is continuous,} because a pointwise limit of Lipschitz functions is again Lipschitz. {\bf \green Step 2:} Since $|f'_i|$ is uniformly bounded on compacts, we can assume that $f'_i$ also converges pointwise in $Z$, and $f:= \lim_i f_i$ is differentiable. Since a limit of complex-linear operators is complex linear, $Df$ is complex linear, and $f$ is holomorphic. {\bf \purple This implies that $\bar{\cal F}\cap H^0(\calo_M)$ is compact. } \endproof \newpage {\bf \blue Metric completion of a $\Z$-covering} \theorem (Rossi 1965, Andreotti-Siu 1970)\\ Let $M$ be a complex manifold with boundary, $\dim_\C M >2$, and $\phi$ a proper K\"ahler potential on $M$, taking values in $[a, \infty[$, and equal to $a$ in the boundary of $M$. {\bf \red Then there exists a Stein variety $M_a$} with isolated singularities, obtained by gluing a compact domain to $M$, and it is unique. Moreover, any holomorphic function on $M$ can be extended to $M_a$. {\bf \green Theorem 2:} Let $M$ be an LCK manifold with potential $\phi:\; \tilde M \arrow \R^{>0}$, where $\tilde M$ is 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 $\tilde M \subset\tilde M_c$. {\bf \green Proof. Step 1:} Apply Rossi-Andreotti-Siu to $\phi^{-1}([a, \infty[)$, we obtain a Stein variety $\tilde M_a$ containing $\phi^{-1}([a, \infty[)$. Since $\tilde M_a$ contains $\phi^{-1}([a_1, \infty[)$ for any $a_1>a$, and the Rossi-Andreotti-Siu variety is unique, one has $\tilde M_a=\tilde M_{a_1}$. This implies that {\bf \purple $\tilde M_a=: \tilde M_c$ is independent from the choice of $a\in \R^{>0}$. } {\bf \green It remains to identify $\tilde M_c$ with a metric completion of $\tilde M$.} By Claim 1, {\bf \red this is equivalent to the complement $\tilde M_c\backslash \tilde M$ being a one-point set.} \newpage {\bf \blue Metric completion of a $\Z$-covering (part 2)} {\bf \green Theorem 2:} Let $M$ be an LCK manifold with potential $\phi:\; \tilde M \arrow \R^{>0}$, where $\tilde M$ is 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 $\tilde M \subset\tilde M_c$. {\bf \green It remains to identify $\tilde M_c$ with a metric completion of $\tilde M$.} By Claim 1, {\bf \red this is equivalent to the complement $\tilde M_c\backslash \tilde M$ being a one-point set.} {\bf \green Step 2:} {\bf \purple The monodromy group $\Gamma=\Z$ acts on $\tilde M_c$ by holomorphic automorphisms.} Indeed, any holomorphic function (hence, any holomorphic map) can be extended from $\tilde M$ to $\tilde M_c$ uniquely. {\bf \green Step 3:} Denote by $\gamma$ the generator of $\Gamma$ which decreases the metric by $\lambda<1$, and let $\tilde M^a_c$ be a Stein variety associated with $\phi^{-1}(]0, a])\subset\tilde M$ as above. Since $\gamma(\tilde M^a_c)=\tilde M^{\lambda a}_c$, for any holomorphic function $f$ on $\tilde M_c$, one has \[ \sup_{z\in\tilde M^a_c} |f(\gamma^n(z))|= \sup_{z\in\tilde M^{\lambda^n a}_c} |f(z)|\leq \sup_{z\in\tilde M^{a}_c} |f(z)| . \] {\bf \purple Therefore, $\{ f(\gamma^n(z))\}$ is a normal family.} \newpage {\bf \blue Metric completion of a $\Z$-covering (part 3)} {\bf \green Step 3:} Denote by $\gamma$ the generator of $\Gamma$ which decreases the metric by $\lambda<1$, and let $\tilde M^a_c$ be a Stein variety associated with $\phi^{-1}(]0, a])\subset\tilde M$ as above. Since $\gamma(\tilde M^a_c)=\tilde M^{\lambda a}_c$, for any holomorphic function $f$ on $\tilde M_c$, one has \[ \sup_{z\in\tilde M^a_c} |f(\gamma^n(z))|= \sup_{z\in\tilde M^{\lambda^n a}_c} |f(z)|\leq \sup_{z\in\tilde M^{a}_c} |f(z)| . \] {\bf \purple Therefore, $\{ f(\gamma^n(z))\}$ is a normal family.} {\bf \green Step 4:} Let $f_{\sf lim}$ be any limit point of the sequence $\{ f(\gamma^n(z))\}$. Since the sequence $t_i:=\sup_{z\in\tilde M^{\lambda^i a}_c} |f(z)|$ is non-increasing, it converges, and $\sup_{z\in\tilde M^a_c} f_{\sf lim} = \lim t_i$. Similarly, $\sup_{z\in\tilde M^{\lambda a}_c} f_{\sf lim} = \lim t_i$. By strong maximum principle, {\bf \purple a non-constant holomorphic function on a complex manifold with boundary cannot have local maxima (even non-strict) outside of the boundary.} Since $\tilde M^{\lambda a}_c$ does not intersect the boundary of $\tilde M^a_c$, the function $f_{\sf lim}$ must be constant. {\bf \green Step 5:} Consider now the complement $V:=\tilde M_c\backslash \tilde M$, and suppose it has two distinct points $x$ and $y$. Let $f$ be a holomorphic function which satisfy $f(x)\neq f(y)$. Replacing $f$ by an exponent of $\mu f$ if necessarily, we may assume that $|f(x)|< |f(y)$. Since $\gamma$ fixes $Z$, which is compact, {\bf \purple for any limit $f_{\sf lim}$ of the sequence $\{ f(\gamma^n(z))\}$, supremum $f_{\sf lim}$ on $Z$ is not equal to infimum of $f_{\sf lim}$ on $Z$.} This is impossible, hence $f=\const$ on $V$, and $V$ is one point. \endproof \end{document}