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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Locally conformally K\"ahler manifolds \\[15mm]
\small lecture 13: automorphisms of LCK manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf HSE and IUM, Moscow
\\[2mm]  May 19, 2014
}
\end{center}

\newpage

{\bf\blue Local systems (reminder)}

\definition
A {\bf \blue local system} is a locally constant sheaf 
of vector spaces.

\theorem
A local system with fiber $B$ at $x\in M$
gives a homomorphism $\pi_1(M,x)\arrow \Aut(B)$.
{\bf \red This correspondence gives an equivalence of categories.}

\definition
A bundle $(B, \nabla)$ is called {\bf \blue flat} if its curvature vanishes.

\definition
A section $b$ of $(B,\nabla)$ is called {\bf \blue parallel}
if $\nabla(b)=0$.

\claim
Let $(B, \nabla)$ be a flat bundle on $M$, and ${\cal B}$ be the
sheaf of parallel sections. {\bf \red Then ${\cal B}$ is a 
locally constant sheaf.}

\theorem
This correspondence {\bf \red gives an equivalence of categories}
of flat bundles and local systems.

\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 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 Morse-Novikov cohomology (reminder)}

\definition
 Define {\bf \blue the $B$-valued de Rham differential}
$d_\nabla:\;  \Lambda^i(M)\otimes B \arrow\Lambda^{i+1}(M)\otimes B$
as $d_\nabla(\eta\otimes b):= 
d\eta\otimes b + (-1)^{\tilde\eta}\eta\wedge \nabla b$.
{\bf \purple It is easy to check that $d_\nabla^2=0$
if and only if the curvature of $\nabla$ vanishes.}

\claim
{\bf \purple The cohomology of the complex
$(\Lambda^*M\otimes B, d_\nabla)$ are equal to the
cohomology of the local system ${\cal B}:=\ker \nabla$.}

\remark
Let $B$ be a line bundle equipped with a flat connection,
$\phi$ its trivialization, and $\theta$ its connection form,
$\nabla(f\phi) = df\otimes \phi + f \theta \otimes \psi$.
Then $d_\nabla(\eta\otimes \psi)= d\eta\otimes \psi + 
\theta \wedge \eta\otimes \psi$. This is written as $d_\nabla=d+\theta$.

\definition
Cohomology of the complex $(\Lambda^*M, d_\theta:=d-\theta)$
are called {\bf \blue Morse-Novikov cohomology}, or
{\bf \blue Lichnerowicz cohomology}; the corresponding
complex -- {\bf \blue Morse-Novikov complex}.
They compute the cohomology of the local system $L^{-1}$

\newpage

{\bf \blue Automorphic forms (reminder)}

\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}, or {\bf
\blue weight} for $\eta$.

\definition
Let $L$ be an oriented real line bundle equipped with a flat connection
(we call $L$ {\bf \blue weight bundle}), and 
$\chi:\; \pi_1(M)\arrow \R^{>0}$ its monodromy.
{\bf \blue Automorphic form of weight $\lambda$}
is an automorphic form which satisfies
$\gamma^*\tilde \eta=\lambda^{-1}\chi(\gamma)\eta$
for each $\gamma\in \pi_1(M)$.
We denote the space of such forms by $\Lambda^*(M)_\lambda$.

\proposition
{\bf \red The Morse-Novikov complex
with coefficients in $L^\lambda$ is identified with the de Rham 
complex of automorphic forms of weight $\lambda$ on $\tilde M$.}

\newpage

{\bf \blue Morse-Novikov Dolbeault complex (reminder)}

\definition
Let $M$ be a complex manifold,
and $(L,\nabla)$ a flat, oriented, real line bundle.
Identifying sections of $L$ with automorphic
forms of weight 1 on $\tilde M$ as above, we consider {\bf \blue the
Hodge decomposition $d_\theta=\6_\theta+\bar\6_\theta$,}
where $d_\theta$ is the de Rham differential on 
automorphic forms, and $\6_\theta$, $\bar\6_\theta$ its Hodge components.

\proposition
Let $(L,\nabla)$ be a weight bundle on a complex manifold,
$\phi$ its trivialization, and $\theta$ a connection form.
Denote by $\tilde M\stackrel \pi \arrow M$ the universal covering,
and let $\Phi$ be a non-zero parallel section of $\pi^* L$.
Consider the equivalence 
\[ (\Lambda^*(M)\otimes L, d_\theta)\stackrel \Xi \arrow
(\Lambda^*(M)_\lambda, d)
\]
between Morse-Novikov complex and the
de Rham complex of automorphic forms. 
Then {\bf \purple this identification is compatible with 
Dolbeault decomposition, and gives an equivalence
between $\6_\theta$, $\bar\6_\theta$
and Dolbeault differentials on $\Lambda^*(M)_\lambda$.}

\corollary
This gives the following commutation relations:
$\{\6_\theta, \bar\6_\theta\}= \{\6_\theta, \6_\theta\}
=\{\bar\6_\theta, \bar\6_\theta\}=0= \{d_\theta, d_\theta^c\}$, and 
$-2\1\6_\theta\bar\6_\theta= d_\theta d_\theta^c$, where
$d_\theta^c= I d_\theta I^{-1}=
-\1(\6_\theta-\bar\6_\theta)=d^c - I(\theta)$.
\endproof

\newpage

{\bf \blue Lee class of an LCK manifold (reminder)}

\definition
Let $(M,\omega, \theta)$ be an LCK manifold.
The cohomology class $[\theta] \in H^1(M,\R)$ 
of its Lee form $\theta$ is called {\bf \blue the Lee class} of $M$.


\remark {\bf \blue Monodromy group} of an LCK manifold
$(M,\omega, \theta)$ 
is defined as the Galois group of the smallest covering
$\pi:\; \tilde M \arrow M$ such that $\pi^*\theta$ 
is exact. {\bf\blue Rank} of an LCK manifold is rank of
its monodromy group.

\proposition
Let $(M,\omega, \theta)$ be an LCK manifold
and $[\theta]$ its Lee class. Consider a
smallest rational subspace $V\subset H^1(M,\Q)$
such that $V\otimes_\Q \R$ contains $[\theta]$.
{\bf \red Then $\dim V$ is equal to the rank of $M$.}

\proof
The group $\Gamma$ is identified with an image of $\pi_1(M)$
under the map $[\theta]:\; \pi_1(M)\arrow \R$,
because it is equal to the monodromy of the weight bundle,
and the monodromy along a loop $\gamma$ is equal to
$e^{\int_\gamma \theta}$.
\endproof



\newpage

{\bf \blue Morse-Novikov class of an LCK manifold (reminder)}

\definition
Let $(M,\omega, \theta)$ be an LCK manifold,
$d\omega=\omega\wedge\theta$. Then $d_\theta(\omega)=0$.
The cohomology class $[\omega]_{MN}$ of $\omega$ in the Morse-Novikov
cohomology is called {\bf \blue Morse-Novikov class} of $M$.

\claim 
{\bf \red $[\omega]_{MN}$ vanishes for LCK manifolds with
potential and, hence, for Vaisman manifolds.}

{\bf \green Proof:}
Indeed, the corresponding automorphic
form $\tilde \omega=\Xi(\omega)$ is a differential of an automorphic form,
and the Morse-Novikov cohomology is cohomology of the
complex of automorphic forms.
\endproof

\remark  {\bf \purple $[\omega]_{MN}$ 
is known to be non-zero for some other LCK manifolds.}
All known examples of compact LCK manifolds
with vanishing Morse-Novikov class admit an LCK
metric with potential.



\newpage

{\bf \blue Bott-Chern cohomology (reminder)}

\definition
Let $M$ be a complex manifold, and $H^{p,q}_{BC}(M)$
the space of closed $(p,q)$-forms modulo $dd^c(\Lambda^{p-1,q-1}(M))$.
Then $H^{p,q}_{BC}(M)$ is called {\bf\blue the Bott-Chern
  cohomology} of $M$. 

\theorem
Let $M$ be a compact complex manifold. {\bf \red Then
$H^{p,q}_{BC}(M)$ is finite-dimensional.}

\definition
Let $M$ be a complex manifold, and $L$ a flat vector
bundle. Consider the corresponding differential
$d_\nabla=d_\theta$, and let $\6_\theta$, $\bar\6_\theta$
be its Hodge components.
{\bf \blue The weighted Bott-Chern cohomology}
are defined as 
\[ H^{p,q}_{BC}(M,L):= \frac{\ker
   d_\theta\restrict{\Lambda^{p,q}(M)\otimes L}}{\im
  \6_\theta\bar\6_\theta}.
\]

\theorem
Let $M$ be a compact complex manifold, and $L$ a flat vector
bundle. {\bf \red 
Then the group $H^{p,q}_{BC}(M,L)$ is finite-dimensional.}

\newpage

{\bf \blue Bott-Chern class (reminder)}

\definition
$(M, \omega,\theta)$ be an LCK manifold, and $L$ its
weight bundle. The cohomology class of $\omega$ in 
$H^{1,1}_{BC}(M,L)$ is called {\bf \blue Bott-Chern
 class of $M$}.

\remark {\bf \purple It is the best analogue of the K\"ahler class,}
and the following theorem (together with the Hopf
embedding result) is an LCK analogue of Kodaira embedding
theorem.

\theorem
Let $(M, \omega,\theta)$ be an LCK manifold.
Suppose that its Lee class $[\theta]$ is proportional to a
rational class in $H^1(M)$ and $[\omega]_{BC}=0$.
{\bf \red Then $(M, \omega,\theta)$ is an LCK manifold
  with potential.}

{\bf \green Proof:} Existence of an automorphic potential
is precisely vanishing of $[\omega]_{BC}$. Its
properness is equivalent to $\Gamma\cong \Z$, where
$\Gamma$ is a monodromy group of $M$. Since
rank of $\Gamma$ is equal to the dimension of a
smallest rational subspace generated by $[\theta]$,
it is equal 1. \endproof


\newpage

{\bf \blue Open questions (reminder)}


A weighted version of $dd^c$-lemma is known to be
wrong, even for Vaisman manifolds (Goto).
However, the following (very weak) version 
of $d_\theta d^c_\theta$-lemma could be true.

\problem
Let $M$ be a compact LCK manifold
with its Morse-Novikov class $[\omega]_{MN}$ equal zero.
{\bf \red Would it follow that $M$ has monodromy $\Z$? Would
it follow that $M$ admits an LCK metric with
potential, when its monodromy is $\Z$?}

\problem
Find an example of locally (but not globally)
conformally symplectic manifold of dimension 
$\geq 3$ not admitting LCK structure.

\problem
Prove that a compact torus with non-K\"ahler
complex structure does not admit an LCK metric,
or find one.

\newpage

{\bf \blue LCK manifolds with $S^1$-action: main theorem}

\theorem
Let $M$ be a compact complex manifold, equipped
with a holomorphic $S^1$-action and an LCK metric
(not necessarily compatible). Suppose that
the weight bundle $L$, restricted to a general
orbit of this $S^1$-action, is non-trivial
as a 1-dimensional local system. {\bf \red Then $M$ admits
an LCK metric with an automorphic potential.}

{\bf \green The proof takes the rest of this lecture.}

\remark
{\bf \red The converse statement is also true.} Indeed,
let $M=\tilde M/\Z$ be an LCK manifold with potential,
$\tilde M$ be its K\"ahler covering. As we have already
shown, the $\Z$-action on $\tilde M$ admits a 
logarithm, given by a holomorphic vector field $A\in T\tilde M$.
{\bf \purple Then $e^{tA}$ is a holomorphic $S^1$-action with the 
required properties.}


\newpage

{\bf \blue LCK manifolds with $S^1$-action: Lee form}

\remark
{\bf \blue Conformally equivalent metrics}
are metrics $g, g' = e^f g$. {\bf \blue Conformal class}
of a metric is its class of conformal equivalence.

\lemma
Let $M$ be a compact complex manifold, equipped
with a holomorphic $S^1$-action and an LCK metric
(not necessarily compatible). {\bf \red Then there exists
an LCK metric in the same conformal class with
$S^1$-invariant Lee form.}

\proof
Let $G$ be a compact subgroup of
$\mathrm{Aut}(M)$. Averaging the Lee form $\theta$ on $G$, we obtain a
closed $1$-form $\theta'$ which is $S^1$-invariant and stays in the same
cohomology class as $\theta$: $\theta'=\theta+df$. Then
$\omega'=e^{-f}\omega$ is a LCK form with Lee form $\theta'$ and conformal
to $\omega$.
\endproof


\newpage

{\bf \blue LCK manifolds with $S^1$-action: $S^1$-invariance}

{\bf \green Proposition 1:}
Let $(M,\omega, \theta)$ be a compact complex manifold, equipped
with a holomorphic $S^1$-action and an LCK metric
(not necessarily compatible). {\bf \red Then $M$ admits
an $S^1$-invariant LCK metric.}

{\bf \green Proof. Step 1:} Using the previous lemma, we 
chose a metric in the same conformal class with $S^1$-invariant
Lee form. {\bf \purple 
Therefore, we may assume $\theta$ is $S^1$-invariant.}

{\bf \green Proof. Step 2:} For each $t \in S^1$, let
$\omega_t:=\rho(t)^*\omega$. Then 
$d(\omega_t) =\omega_t\wedge \theta$.
Averaging $\omega_t$ with respect to $t$,
{\bf \purple we obtain a positive, $S^1$-invariant form
$\omega_{av}$ satisfying $d(\omega_{av}) =\omega_{av}\wedge \theta$.}
\endproof

\claim
Let $(M,\omega, \theta)$ be a compact complex manifold, equipped
with a holomorphic $S^1$-action by isometries.
Then, on a K\"ahler covering $\tilde M \arrow M$,
{\bf \purple the corresponding action of the covering
$\tilde S^1 = \R$ is by holomorphic homotheties. }

{\bf \green Proof:}
Indeed, if two
K\"ahler forms are conformally equivalent, they are 
proportional. \endproof


\newpage

{\bf \blue Holomorphic homotheties on K\"ahler manifolds}


{\bf \green Proposition 2:}
Let $A$ be a vector field acting on a
K\"ahler manifold $(\tilde M, \tilde \omega)$ by
holomorphic homotheties: $\Lie_A\tilde \omega
=\lambda\tilde\omega$, with $\lambda\neq 0$.
{\bf \red Then
\[
dd^c|A|^2= \lambda^2\tilde \omega  +\Lie_{A^c}^2\tilde \omega,
\]
where $A^c = I(A)$.}

{\bf \green Proof. Step 1:}
Let $\eta:= \tilde\omega\cntrct A= I(A)^\flat$
be the dual form to $A^c$. 
Replacing  $A$ by $\lambda^{-1}A$, we may
assume that $\lambda=1$. {\bf \purple By Cartan's formula,
\[
\tilde\omega=\Lie_A\tilde \omega  =d(\tilde\omega\cntrct A)=d\eta.
\]}
{\bf \green Step 2:}
Since $A$ and $A^c$ are holomorphic,
$\Lie_{A^c}$ commutes with $I$. This gives
\[ \Lie_{A^c}\tilde\omega=
\Lie_{A^c} I\tilde\omega=  I \Lie_{A^c}\tilde\omega=IdI^{-1}(\tilde\omega\cntrct A)=
d^c \eta.
\]
{\bf \green Step 3:} Since $\Lie_A$ commutes with $I$,
{\bf \purple one has $\{d^c, i_{A^c}\}= I\Lie_A I^{-1}=\Lie_A$,}
where $i_v(\alpha)= \alpha\cntrct v$ is the contraction operator.


\newpage

{\bf \blue Holomorphic homotheties on K\"ahler manifolds (2)}

{\bf \green Proposition 2:}
Let $A$ be a vector field acting on a
K\"ahler manifold $(\tilde M, \tilde \omega)$ by
holomorphic homotheties: $\Lie_A\tilde \omega
=\lambda\tilde\omega$, with $\lambda\neq 0$.
{\bf \red Then
\[
dd^c|A|^2= \lambda^2\tilde \omega  +\Lie_{A^c}^2\tilde \omega,
\]
where $A^c = I(A)$.}

{\bf \green Proof. Step 1:}
Let $\eta:= \tilde\omega\cntrct A$, and $\lambda=1$. Then
$\tilde\omega=\Lie_A\tilde \omega  =d(\tilde\omega\cntrct A)=d\eta$.

{\bf \green Step 2:}
$\Lie_{A^c}\tilde\omega=
\Lie_{A^c} I\tilde\omega=  I \Lie_{A^c}\tilde\omega=IdI^{-1}(\tilde\omega\cntrct A)=
d^c \eta.$

{\bf \green Step 3:}  $\{d^c, i_{A^c}\}= I\Lie_A I^{-1}=\Lie_A$,
where $i_v(\alpha)= \alpha\cntrct v$ is the contraction operator.


{\bf \green Step 4:} 
\[ \Lie_{A^c}^2 \tilde\omega= \Lie_{A^c} d^c\eta= 
i_{A^c}dd^c \eta+ d i_{A^c} d^c \eta \ \ \ \  (*)
\]
(Step 2 and Cartan's formula).
The first summand vanishes because  $dd^c \eta= -d^c d\eta= d^c \tilde\omega$
(Step 1). The second summand gives
\[
d i_{A^c} d^c \eta= dd^c \langle I(A), I(A)^\flat\rangle  -
d \{d^c, i_{A^c}\} \eta \ \ \ \ (**)
\]
Finally, $d \{d^c, i_{A^c}\} \eta= d\Lie_A\eta=\Lie_Ad\eta= \tilde\omega$
(Step 3 and Step 1). Therefore, (*) and (**) give
$\Lie_{A^c}^2 \tilde\omega=dd^c|A|^2- \omega$, for $\lambda=1$.
\endproof

\newpage

{\bf \blue LCK manifolds with $S^1$-action: main theorem (proof)}

\theorem
Let $M$ be a compact complex manifold, equipped
with a holomorphic $S^1$-action $\rho$ and an LCK metric
 (not necessarily compatible). Suppose that
the weight bundle $L$, restricted to a general
orbit of this $S^1$-action, is non-trivial
as a 1-dimensional local system. {\bf \red Then $M$ admits
an LCK metric with an automorphic potential.}

{\bf\green Proof. Step 1:} 
Using Proposition 1, {\bf \purple we may assume that the metric $\omega$
on $M$ is $S^1$-invariant.} Denote the corresponding K\"ahler
metric on $\tilde M$ by $\tilde \omega$, and let
$\tilde \rho$ be the lift of $S^1$-action to $\tilde M$.
Since conformally equivalent K\"ahler metrics are
proportional, {\bf \purple $\tilde \rho$ acts by homotheties.}

{\bf \green Step 2:}
Restriction of the flat connection in the weight bundle
$L$ to a loop has trivial monodromy whenever this
loop lifts to a homeomorphic loop in $\tilde M$.
Since $L$ is non-trivial on orbits of $\rho$,
the lift $\tilde \rho$ is an $\R$-action, not reducible
to $S^1$-action. Denote the kernel of the natural map 
$\im \tilde\rho\arrow \im \rho$ by $\Gamma$.
Since $\tilde M$ is a minimal
K\"ahler covering, $\Gamma$ acts on $\tilde M$
non-isometrically, hence $\tilde\rho$ acts by non-trivial 
homotheties. Rescaling, {\bf \purple we may assume that the vector field $A$
tangent to $\tilde\rho$ satisfies $\Lie_A \tilde\omega=\tilde\omega$.}


\newpage

{\bf \blue LCK manifolds with $S^1$-action: main theorem (proof, part 2)}


{\bf \green Step 3:} Proposition 2 gives 
\[ 
\tilde \omega=  dd^c|A|^2 -\Lie_{A^c}^2\tilde \omega,\ \ \ \ (***)
\]
where $A$ is the homothety vector field tangent to $\tilde \rho$, and
$A^c=I(A)$. Let $\mu_t:= \rho^c(t)^*[\omega]_{BC}$
be the Bott-Chern class of $e^{tA^c}(\omega)$.
By (***), $\mu_t$ satisfies the differential
equation $\mu_t''= -\mu_t$, hence 
{\bf \purple $\mu_t = a\sin(t)+ b \cos(t)$,
for some $a, b \in H^{1,1}_{BC}(M,L)$.}

{\bf \green Step 4:}
From Step 3 it follows that $\int_0^{2\pi} e^{t A^c}[\tilde \omega]dt=0.$
Consider the K\"ahler form $\tilde \omega_W:=
\int_0^{2\pi} e^{t A^c}(\tilde \omega) dt=0$
on $\tilde M$. This form is an average
of automorphic forms of the same character of automorphicity,
because $e^{tA^c}$ commutes with $e^{t'A}$.
The Bott-Chern class of $\omega_W$ vanishes, because
$\int_0^{2\pi}\sin(t) dt= \int_0^{2\pi}\cos(t) dt=0$. {\bf \purple Therefore, 
$\tilde \omega_W$ admits an automorphic potential.}

{\bf \green Step 5:}
To finish the proof, {\bf \purple it remains to show that the monodromy of 
$M$ is $\Z$.} This is implied by the theorem proven in Lecture 5.

\theorem
Let $(M,\omega,\theta)$ be a compact LCK manifold, and
$X$ a vector field acting on $M$ by isometries and on 
$\tilde M$ by non-isometric homotheties. {\bf \red Then $\Mon(M)=\Z$.}
\endproof

\end{document}

