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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 $|D_w f_i|<C_\Delta$.}
However, $t^{-1}\lim_i |D_w f_i| =\lim_i |D_v f_i|=\infty$ --
contradiction. \endproof

\newpage

{\bf \blue $C^0$-topology on functions}


\definition
Let $C(M)$ be the space of functions on a topological space.
{\bf \blue Topology of uniform convergence on compacts} (also known
as {\bf \blue compact-open topology}; usually denoted as $C^0$) 
is a topology on $C(M)$ where a base of open sets
is given by 
\[ U(X, C):= \{f\in C(M)\ \ |\ \ \sup_K |f|< C\},
\] for all
compacts $K\subset M$ and $C>0$. 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}

