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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} 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\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:\ }} \newcommand{\problem}{% {\bf \green PROBLEM:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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}