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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 10: pseudoconvex shells} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE and IUM, Moscow \\[2mm] April 21, 2014 } \end{center} \newpage {\bf\blue LCK manifolds (reminder)} \definition Let $(M,I, \omega)$ be a Hermitian manifold, $\dim_\C M >1$. Then $M$ is called {\bf \blue locally conformally K\"ahler} (LCK) if $d\omega=\omega\wedge\theta$, where $\theta$ is a closed 1-form, called {\bf \blue the Lee form}. \definition {\bf \blue A manifold is locally conformally K\"ahler} iff it admits a K\"ahler form taking values in a positive, flat vector bundle $L$, called {\bf \blue the weight bundle}. \definition {\bf \blue Deck transform}, or {\bf \blue monodromy maps} of a covering $\tilde M \arrow M$ are elements of the group $\Aut_M(\tilde M)$. {\bf \purple When $\tilde M$ is a universal cover, one has $\Aut_M(\tilde M)=\pi_1(M)$.} \definition {\bf \blue An LCK manifold} is a complex manifold such that its universal cover $\tilde M$ is equipped with a K\"ahler form $\tilde \omega$, and the deck transform acts on $\tilde M$ by K\"ahler homotheties. \theorem {\bf \red These three definitions are equivalent}. \newpage {\bf \blue Conical K\"ahler manifolds (reminder)} \definition Let $(X,g)$ be a Riemannian manifold, and $C(X):= X \times \R^{>0}$, with the metric $t^2 g+ dt^2$, where $t$ is a coordinate on $\R^{>0}$. Then $C(X)$ is called {\bf \blue Riemannian cone} of $X$. {\bf \purple Multiplicative group $\R^{>0}$ acts on $C(X)$ by homotheties, $(m, t) \arrow (m, \lambda t)$.} \definition Let $(X,g)$ be a Riemannian manifold, $C(X):= X \times \R^{>0}$ its Riemannian cone, and $h_\lambda$ the homothety action. Assume that $(C(X),gt^2 + dt^2)$ is equipped with a complex structure, in such a way that the conical metric $gt^2 + dt^2$ is K\"ahler, and $h_\lambda$ acts holomorphically. Then $C(X)$ is called {\bf \blue a conical K\"ahler manifold}. In this situation, $X$ is called {\bf \blue Sasakian manifold}. \remark A {\bf \blue contact manifold} is defined as a manifold $X$ with symplectic structure on $C(X)$, and $h_\lambda$ acting by homotheties. In particular, {\bf \purple Sasakian manifolds are contact}. {\bf \green Sasakian geometry is an odd-dimensional counterpart to K\"ahler geometry} \example Let $L$ be a positive holomorphic line bundle on a projective manifold. {\bf \purple Then the total space of its unit $S^1$-fibration is Sasakian.} \newpage {\bf\blue 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 LCK manifolds with potential (reminder)} \definition Let $M$ be an LCK manifold, and $(\tilde M, \tilde \omega)$ its K\"ahler covering. It is called {\bf \blue LCK manifold with potential} if $\tilde M$ admits an automorphic K\"ahler potential $\phi:\; \tilde M \arrow \R^{>0}$, $dd^c \phi=\tilde \omega$, which is {\bf \blue proper} (preimage of a compact is again compact). \theorem {\bf \purple The property of being LCK with potential is stable under small deformations}. \theorem Let $M$ be an LCK manifold, $\Gamma\subset \R^{>0}$ the monodromy group, and $(\tilde M, \tilde \omega)$ its K\"ahler covering, with $\tilde M/\Gamma=M$. Assume that $\tilde \omega$ admits a $\Gamma$-automorphic K\"ahler potential $\phi$. {\bf \red The map $\phi$ is proper if and only if $\Gamma=\Z$.} \theorem Let $M$ be an LCK manifold with potential, and $\tilde M$ its K\"ahler $\Z$-covering. Then a metric completion $\tilde M_c$ {\bf \red admits a structure of a complex manifold,} compatible with the complex structure on $\tilde M \subset\tilde M_c$. Moreover, the monodromy action on $\tilde M$ is extended to a holomorphic automorphism of $\tilde M_c$. \theorem Let $M$ be an LCK manifold with potential, $\dim_\C M>2$. {\bf \red Then $M$ admits a holomorphic embedding to a linear Hopf manifold. } \newpage {\bf\blue CR-manifolds} {\bf \green Definition:} Let $M$ be a smooth manifold, $B\subset TM$ a sub-bundle in a tangent bundle, and $I:\; B \arrow B$ an endomorphism satisfying $I^2=-1$. Consider its $\1$-eigenspace $B^{1,0}(M)\subset B\otimes \C \subset T_C M=TM\otimes \C$. Suppose that $[B^{1,0}, B^{1,0}]\subset B^{1,0}$. Then $(B,I)$ is called {\bf\blue a CR-structure on $M$}. {\bf \green Example:} A complex manifold is CR, with $B=TM$. Indeed, {\bf \purple $[T^{1,0}M, T^{1,0}M]\subset T^{1,0}M$ is equivalent to integrability of the complex structure} (Newlander-Nirenberg). {\bf \green Example:} Let $X$ be a complex manifold, and $M\subset X$ a hypersurface. Then $B:=\dim_\C TM \cap I(TM)=\dim_\C X-1$, hence $\rk B=n-1$. Since $[T^{1,0}X,T^{1,0}X]\subset T^{1,0}X$, {\bf \red $M$ is a CR-manifold.} {\bf \green Definition:} {\bf \blue A Frobenius form of a CR-manifold} is the tensor $B\otimes B \arrow TM/B$ mapping $X, Y$ to the $\Pi_{TM/B}([X,Y])$. It is an obstruction to integrability of the foliation given by $B$. \newpage {\bf \blue Contact CR-manifolds.} Complex algebraic geometry is a rich source of contact structures. {\bf \green Definition:} Let $(M,B,I)$ be a CR-manifold, with $\codim B =1$. Then $M$ is called {\bf \blue a contact CR-manifold} if its Frobenius form is non-degenerate. {\bf \green Remark:} Since $[B^{1,0}, B^{1,0}]\subset B^{1,0}$ and $[B^{0,1}, B^{0,1}]\subset B^{0,1}$, the Frobenius form is a pairing between $B^{0,1}$ and $B^{1,0}$. This means that it is Hermitian. \definition This Hermitian form is called {\bf \blue Levi form} of a CR-manifold. {\bf \green Definition:} Let $(M,B,I)$ be a CR-manifold, with $\codim B =1$. Then $M$ is called {\bf \blue a strictly pseudoconvex CR-manifold} if its Levi form is positive definite. {\bf \green Example:} Let $h$ be a function on a complex manifold such that $\6\bar\6 h=\omega$ is a positive definite Hermitian form, and $X=h^{-1}(c)$ its level set. Then the Frobenius form of $X$ is equal to $\omega\restrict X$ (see the next slide). In particular, {\bf \purple $X$ is a strictly pseudoconvex CR-manifold}. \newpage {\bf \blue CR-manifolds and plurisubharmonic functions.} \proposition Let $M$ be a complex manifold, $\phi\in C^\infty M$ a smooth function, and $s$ a regular value of $\phi$. Consider $S:=\phi^{-1}(s)$ as a CR-manifold, with $B=TS\cap I(TS)$ and let $\Phi$ be its Levi form, taking values in \[ TS/B= \ker d\phi/\ker d\phi \cap I(\ker d\phi) \] Then $d^c \phi:\; TS/B\arrow C^\infty S$ trivializes $TS/B$. Consider tangent vectors $u, v \in B_x S$. {\bf \red Then $-d^c \phi(\Phi(u,v))=dd^c\phi(x,y))$.} {\bf \green Proof:} Extend $u, v$ to vector fields $u, v\in B= \ker d\phi\cap I(\ker d\phi)$. Then $-d^c\phi(\Phi(u,v))= -d^c\phi([u,v])=dd^c\phi(u, v)$. \endproof \corollary Let $M$ be a complex manifold, $\phi\in C^\infty M$ a strictly plurisubharmonic function, and $s$ a regular value of $\phi$. {\bf \red Then $S:=\phi^{-1}(s)$ is strictly pseudoconvex.} {\bf \green Proof:} By the above proposition, the Levi form of $S$ is expressed as $dd^c\phi(u, v)$, hence it is positive definite. \endproof \newpage {\bf \blue Algebraic cones.} \definition {\bf\blue An algebraic cone} is an affine variety $\cac$ admitting a $\C^*$-action $\rho$ with a unique fixed point $x_0$, called {\bf \blue the origin}, and satisfying the following: (i) $\cac$ is smooth outside of $x_0$, (ii) $\rho$ acts on the Zariski tangent space $T_{x_0}\cac$ with all eigenvalues $|\alpha_i|<1$. {\bf \blue An open algebraic cone} is a closed algebraic cone without the origin. \theorem Let $M=\tilde M/A$ be LCK manifold with potential, and $\tilde M$ its K\"ahler $\Z$-covering. {\bf \red Then $\tilde M$ is an open algebraic cone.} {\bf \green Proof. Step 1:} Let $\tilde M_c$ be a Stein completion of $\tilde M$ equipped with an $A$-equivariant embedding to $\C^n$, where $A$ acts as a linear operator with all eigenvalues $|\alpha_i|< 1$. Denote the ideal of $\tilde M_c$ in the local ring $\calo_{\C^n,0}$ as $I$. \newpage {\bf \blue Algebraic cones and LCK manifolds with potential} \theorem Let $M=\tilde M/A$ be LCK manifold with potential, and $\tilde M$ its K\"ahler $\Z$-covering. {\bf \red Then $\tilde M$ is an open algebraic cone.} {\bf \green Proof. Step 1:} [...] Denote the ideal of $\tilde M_c$ in the local ring $\calo_{\C^n,0}$ as $I$. {\bf \green Step 2:} Call an element $f\in \calo_{\C^n,0}$ {\bf \blue $A$-finite} if $\langle f, A^*f, {A^2}^* f, ...\rangle$ is finitely-dimensional. A polynomial function is clearly $A$-finite. The converse is also true, because a Taylor decomposition of an $A$-finite function $f$ can have only finitely many components, otherwise the eigenspace decomposition of $f$ is infinite. Therefore, {\bf \purple the ideal $I^A:=I\cap \calo_{\C^n,0}^A$ is finitely generated,} where $\calo_{\C^n,0}^A$ is a ring of $A$-finite functions (any ideal in the ring of polynomials is finitely generated, by Hilbert basis theorem). {\bf \green Step 3:} As we have shown in Lecture 9, the ring $\calo_{\tilde M_c,0}^A$ is dense in $\calo_{\tilde M_c,0}$ in $\goth m$-adic topology. in other words, it has the same associated graded ring with respect to the $\goth m^n$-filtration as $\calo_{\tilde M_c,0}$. Then the Nakayama's lemma implies that $I=I^A\otimes_{\calo_{\C^n,0}^A}\calo_{\C^n,0}$. {\bf \green Step 4:} Let $f_1, ..., f_n\subset \calo_{\C^n}$ be the polynomial generators of $I\subset \calo_{\C^n,0}$. Then $\tilde M_c$ is an affine variety defined by the ideal $\langle f_1, ..., f_n\rangle$. \endproof \newpage {\bf \blue Pseudoconvex shells} %\definition %Recall that {\bf\blue contraction} of a manifold $M$ to a point %$x\subset M$ is a morphism $\phi:\; M \arrow M$ such that for any %compact subset $K\subset M$ and any open $U\ni x$ %there exists $N>0$ such that for all %$n>N$, the map $\phi^n$ maps $K$ to $U$. \definition Let $\tilde M$ be an open algebraic cone, $\tilde M_c$ the corresponding closed cone, and $\vec r\in T\cac$ a holomorphic vector field such that for all $t>0$ the diffeomorphism $e^{t\vec r}$ is a holomorphic contraction of $\tilde M_c$ to origin. A strictly pseudoconvex hypersurface $S\subset \tilde M$ is called {\bf \blue a pseudoconvex shell} if $S$ intersects each orbit of $e^{t\vec r}$, $t\in \R$ exactly once. {\bf \green Theorem 1:} Let $\tilde M$ be an algebraic cone, $e^{t\vec r}$ a contraction, and $S\subset \tilde M$ a pseudoconvex shell. Then for each $\lambda\in \R$ there exists a unique function $\phi_\lambda$ such that $\Lie_{\vec r}\phi = \lambda \phi$ and $\phi_\lambda\restrict S=1$. Moreover, {\bf \red such $\phi_\lambda$ is strictly plurisubharmonic when $\lambda \gg 0$.} {\bf \green Theorem 2:} Any LCK manifold with potential admits a metric of this type. Theorem 1 (proven later in this lecture) implies the following corollary. \corollary {\bf \blue (Gauduchon-Ornea)} \\ {\bf \red All linear Hopf manifolds are LCK with potential.} \proof Let $M=(\C^n \backslash 0)/\langle A\rangle$, $\vec r=\log A$, and $S\subset \C^n$ be a unit sphere. Then $S$ is a pseudoconvex shell, and for $\lambda$ sufficiently big a plurisubharmonic function $\phi_\lambda$ gives an LCK-potential. \endproof \newpage {\bf \blue Pseudoconvex shells and plurisubharmonic functions} {\bf \green Theorem 1:} Let $\tilde M$ be an algebraic cone, $e^{t\vec r}$ a contraction, and $S\subset \tilde M$ a pseudoconvex shell. Then for each $\lambda\in \R$ there exists a unique function $\phi_\lambda$ such that $\Lie_{\vec r}\phi_\lambda = \lambda \phi_\lambda$ and $\phi_\lambda\restrict S=1$. Moreover, {\bf \red such $\phi_\lambda$ is strictly plurisubharmonic when $\lambda \gg 0$.} {\bf \green Proof. Step 1:} For each $\lambda$, {\bf \purple $\phi_\lambda$ is uniquely determined on each orbit of $e^{t\vec r}$,} $t\in \R$, because $\phi_\lambda$ restricted to this orbit is $e^{\lambda t}$. {\bf \green Step 2:} Let $B:= e^{\R \vec r}\cdot (TS \cap I(TS))\subset T\tilde M$ be a sub-bundle obtained from $TS \cap I(TS)$ by translations along $e^{t\vec r}$. {\bf \purple Then $dd^c \phi\restrict B$ is the Levi form of $S$,} hence it is positive definite. {\bf \green Step 3:} Replacing $\phi$ by $\phi^{2a}$ amounts to replacing $\lambda$ by $2a\lambda$. Then \[ dd^c \phi^{2a} = \phi^{2a-2} (2a \cdot \phi dd^c \phi + 2a(2a-1) d\phi\wedge d^c\phi). \] To prove Theorem 1 it would suffice to show that $dd^c \phi^{2a}\restrict S >0$ for $a$ sufficiently big. However, $S$ is compact, hence it is implied by the following lemma applied to $V=TM$, $W=B$, $h_1 = \phi dd^c \phi$, $h_2= d\phi\wedge d^c\phi$. \newpage {\bf \blue Positivity of Hermitian forms} \lemma Let $h_1, h_2$ be pseudo-Hermitian forms on a complex vector space $V$, and $W\subset V$ a subspace of codimension 1. Assume that $h_1 \restrict W$ is strictly positive, $h_2 \restrict W=0$, and $h_2 \restrict {V/W}$ is also strictly positive. {\bf \red Then there exists a number $T_0\in \R$ which depends continuously on $h_1, h_2$ such that $h_T:=h_1 + T h_2$ is positive definite for all $T>T_0$.} {\bf \green Proof:} We think of $h_1$, $h_2$ as of real valued bilinear symmetric forms. Let $y\in V$ be a vector which satisfies $h_2(y,y)=1$. Then any $x\in V$ can be written as $x=ay+z$, $z\in W$. This gives \[ h_T(x,x)= Ta^2+ a^2 h_1(y,y)+ h_1(z,z) +2 a h_1(z, y) \ \ \ (*) \] Consider (*) as a polynomial on $a$. Then (*) is positive definite for all $a$ if and only if \[ (h_1(z,y))^2 - (T+ h_1(y,y))\cdot h_1(z,z) <0. \ \ \ (**) \] Let $y'\in W$ be a vector which satisfies $h_1(z, y')=h_1(z,y)$ for all $z\in W$, and $T> h_1(y',y')- h_1(y,y)$. Then (**) becomes \[ (h_1(z,y'))^2 - h_1(y',y') h_1(z,z) <0 \] (Cauchy-Schwarz inequality). \endproof \newpage {\bf \blue Logarithm} \definition A {\bf \blue Banach ring} is a Banach space equipped with a commutative, continuous product. A Banach ring is {\bf \blue finitely generated} if it is a closure of a finitely-generated ring. \example A ring of bounded holomorphic function on a complex variety is a Banach ring. {\bf \green Proposition 1:} Let $R$ be a finitely generated, finitely presented Banach ring, and $R_1\subset R$ a finite-dimensional subspace containing unit, which generates $R$ multiplicatively. We write $R=\C[V]/I$, where $I$ is an ideal and $V=R_1$. Let $N$ a number such that $I\cap V^N$ generates $I$. Consider an automorphism $A$ of $R$ such that on $R_N:= R_1^N$ one has $\|A-\Id\|< 1$, where $\|\cdot\|$ is the operator norm. For each $x\in R_N$, define {\bf \blue the logarithm}: $\log(A)(x) := \sum_{i=1}^\infty \frac{(1-A)^i}{i}(x)$ (the series converges, because $\|A-\Id\|< 1$ on $R_N$). {\bf \red Then $\log A$ can be extended to a derivation on $R$ which satisfies $e^{\log A}=\Id$.} {\bf \green Proof:} For each $x,y, xy \in R_N$, one has $\log(A)(xy)=\log(A)(x) y + x \log(A)(y)$ by formal identities with logarithms. Since all relations are generated by elements of $V^N\cap I$, and $\log(A)=0$ on these by construction, the operator $\log(A)$ can be extended to $R$ using the Leibnitz identity. \endproof \newpage {\bf \blue Logarithm on LCK manifolds with potential} {\bf \green Lemma 1:} Let $\{a_1, ..., a_n\}$ be a finite set of complex numbers which satisfy $0<|a_i|<1$. {\bf \purple Then there exists an integer $C>0$ such that $|a_i^C-1|< 1$.} {\bf \green Proof:} Write $a_i = b_i u_i$, where $|u_i|=1$, $b_i \in \R$. For any given $\epsilon$ one can find $C$ such that $\arg(u_i^C)<\epsilon$ for all $i$. The statement of the lemma is obtained when $\epsilon = \frac \pi 3$. \endproof \theorem Let $M$ be an LCK manifold with potential, $\tilde M$ its K\"ahler its $\Z$-covering, and $M=\tilde M/\langle\gamma\rangle$. {\bf \red Then there exists $C\in \Z^{>0}$ and a holomorphic vector field $\vec r$ on $\tilde M$ such that $\gamma^C=\vec r$.} %\[ j:\; M \hookrightarrow %H=(\C^n\backslash 0)/\langle A \rangle %\] %a holomorphic embedding, {\bf \green Proof:} Let $\calo_{\tilde M_c}^\gamma$ be the ring of $\gamma$-finite functions (finitely generated and dense in $H^0(\calo_{\tilde M_c})$, as shown above), $R_1=V$ be a set of multiplicative generators of $\calo_{\tilde M_c}^\gamma$, containing unit, with $R=\C[V]/I$ and $N$ a number such that $I$ is generated by $V^N\cap I$. Define the Banach norm on $\calo_{\tilde M_c}^\gamma$ by taking $|f|=\sup_{x\in \phi^{-1}{[0,a]}}|f(x)|$, where $\phi$ is the LCK potential, and let $R$ be its Banach completion. Using Lemma 1, choose $C\in \Z^{>0}$ such that on $R_N:= R_1^N$ one has $|\gamma^C-\Id| <1$, and let $\log \gamma^C$ be the logarithm defined as in Proposition 1. Then $e^{\log \gamma^C}=\gamma^C$, hence we can take $\vec r :=\log \gamma^C$. \endproof \newpage {\bf \blue $\R$-automorphic LCK metrics} \remark Let $\phi$ be any LCK-potential on $\tilde M$, satisfying $(\gamma^k)^* \phi=e^\lambda \phi$, $\vec r$ a vector field constructed above, and $\rho(t) \arrow e^{-t\lambda}(e^{t\vec r})^*$ the corresponding endomorphism of $C^\infty M$. Since $\rho(k+t)(\phi)=\rho(t)\phi$, the orbit of $\phi$ is compact. {\bf \purple Averaging $\rho(t)\phi$ over $\R$, we obtain a $\rho(t)$-invariant K\"ahler potential $\phi_0$}. Then $\phi_0$ is obtained from a pseudoconvex shell $\phi_0^{-1}(1)$ and the vector field $\vec r$ as in Theorem 1. {\bf \red This proves Theorem 2.} \end{document}