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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 5: Structure theorem for Vaisman manifolds} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE and IUM, Moscow \\[2mm] March 10, 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.} {\it \small S. Sasaki, "On differentiable manifolds with certain structures which are closely related to almost contact structure", Tohoku Math. J. 2 (1960), 459-476.} \newpage {\bf \blue Contact manifolds (reminder)} {\bf \purple All manifolds are assumed to be oriented here.} {\bf \green Definition 1:} Let $C(S)=(S\times \R^>0)$ be a cone, equipped with the standard action $h_\lambda(x,t)=(x, \lambda t)$. Assume that $C(S)$ is equipped with a symplectic form $\omega$ such that $h_\lambda^*\omega=\lambda^2\omega$. Then $S$ is called {\bf \blue contact manifold}. {\bf \green Definition 2:} Let $S$ be an odd-dimensional manifold, and $B\subset TS$ an oriented sub-bundle of codimension 1, with Frobenius form $\Lambda^2 B \stackrel \Phi \arrow TS/B$ non-degenerate. Then $S$ is called {\bf \blue contact manifold}, $B\subset TS$ {\bf\blue the contact bundle}. {\bf \green Definition 3:} Let $S$ be manifold of dimension $2k+1$, $B\subset TS$ an oriented sub-bundle of codimension 1. Assume that for any nowhere vanishing 1-form $\theta\in \Lambda^1 S$, the form $\theta \wedge (d\theta)^k$ is a non-degenerate volume form. Then $(S,B)$ is called {\bf \blue a contact manifold}, and $\theta$ {\bf \blue a contact form}. \theorem {\bf \red These three definitions are equivalent.} \newpage {\bf \blue Reeb field (reminder)} \newcommand{\Reeb}{\operatorname{\sf Reeb}} \definition {\bf \blue A Sasakian manifold} is a contact manifold $S$ with a Riemannian structure, such that the symplectic cone $C(S)$ with its Riemannian metric is K\"ahler. \definition Let $S$ be a Sasakian manifold, $\omega$ the K\"ahler form on $C(S)$, and $r=t\frac{d}{dt}$ the homothety vector field. Then $\Lie_{Ir}t= \langle dt, Ir\rangle=0$, hence $iR$ is tangent to $S\subset C(S)$. This vector field (denoted by $\Reeb$) is called {\bf \blue the Reeb field} of a Sasakian manifold. \remark {\bf \purple The Reeb field is dual to the contact form} $\theta=\omega\cntrct r$. \theorem {\bf \red The Reeb field acts on a Sasakian manifold by contact isometries.} \definition A Sasakian manifold is called {\bf \blue regular} if the Reeb field generates a free action of $S^1$, {\bf \blue quasiregular} if all orbits of $\Reeb$ are closed, and {\bf \blue irregular} otherwise. \newpage {\bf\blue Examples of Sasakian manifolds.} {\bf \green Example:} Let $X\subset \C P^n$ be a complex submanifold, and $CX \subset \C^{n+1}\backslash 0$ the corresponding cone. The cone $CX$ is obviously K\"ahler and homogeneous, hence {\bf \red the intersection $CX\cap S^{2n-1}$ is Sasakian.} This intersection is an $S^1$-bundle over $X$. This construction gives many interesting contact manifolds, including Milnor's exotic 7-spheres, which happen to be Sasakian. \remark In other words, {\bf \blue a link of a homogeneous singularity is always Sasakian.} \remark {\bf \red Every quasiregular Sasakian manifold is obtained this way,} for some K\"ahler metric on $\C^{n+1}$ (Ornea-V., arXiv:math/0609617 ). \remark All 3-dimensional Sasakian manifolds are quasiregular (H. Geiges, 1997, F. Belgun, 2000). \remark{\bf \purple Every Sasakian manifold is diffeomorphic to a quasiregular one} (Ornea-V., arXiv:math/0306077). \remark {\bf \purple Every regular (quasiregular) Sasakian manifold is a total space of an $S^1$-bundle over a K\"ahler manifold (orbifold).} \newpage {\bf \blue Vaisman manifolds} \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.} {\bf \green Proof:} From Lecture 3, we know that $\tilde \omega$ is locally a conical K\"ahler metric. Let $\theta^\sharp$ be the {\bf \blue Lee field}, dual to $\theta$. Then $\Lie_{\theta^\sharp}\psi=2\psi$, hence the space of orbits of $e^{t\theta^\sharp}$-action is identified with $S:=\psi^{-1}(c)$. This gives $\tilde M =C(S)$. \endproof {\bf \purple Now we shall prove a global version of this result.} \newpage {\bf\blue Structure theorem for Vaisman manifolds} \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. \remark This gives {\bf \purple a Riemannian submersion $M \arrow S^1$} with Sasakian fibers. {\bf \green Proof. Step 1:} Since $\theta^\sharp$ is parallel and Killing, $M=X \times \R$ locally. Fix $x_0\in M$. Then the projection $M=X \times \R$ to $\R$ is induced by $x \arrow \int_{\gamma_{x_0,x}}\theta$, for $\gamma_{x_0,x}$ some path connecting $x$ and $x_0$. {\bf \purple Therefore, $M=X \times \R$ whenever $\theta$ is exact.} \newcommand{\Mon}{\operatorname{Mon}} \definition A {\bf\blue monodromy group} $\Mon(M)$ of an LCK manifold $M$ is the smallest group $\Gamma$ such that $M=\tilde M /\Gamma$ and $\tilde M$ is K\"ahler. \remark {\bf\red This is equivalent to the pullback of $\theta$ being exact}. \remark Monodromy group is an image of $\pi_1(M)$ in $\R^{>0}$ under a map associating to any $\gamma\in \pi_1(M)\subset \Aut(\tilde M)$ the number $\frac{\gamma^*\tilde\omega}{\tilde\omega}$. \newpage {\bf\blue The proof of Structure theorem for Vaisman manifolds} {\bf \green Proof. Step 2:} Let $\gamma_1, ..., \gamma_k\in H_1(M, \Z)$ be generators of homology, and $\alpha_i \int_{\gamma_i}\theta$ the corresponding periods. One has a map $M \arrow \R/\langle \alpha_1, ..., \alpha_k\rangle$, with a commutative diagram \begin{equation*} \begin{CD} \tilde M@>>> M \\ @VVV @VVV \\ \R @>>> \R/\langle \alpha_1, ..., \alpha_k\rangle \end{CD} \end{equation*} with vertical lines $x \arrow \int_{\gamma_{x_0,x}}\theta$. {\bf \red The Riemannian submersion to $S^1$ will be obtained if $\R/\langle \alpha_1, ..., \alpha_k\rangle=S^1$.} {\bf \green Step 3:} Let $G\subset \pi_1(M)$ be the group generated by all $\gamma\in \pi_1(M)$ such that $\int_\gamma\theta=0$. {\bf \green Then $\Gamma= \pi_1(M)/G$ is the monodromy group of $M$.} {\bf \red Therefore, $\R/\langle \alpha_1, ..., \alpha_k\rangle=S^1$ $\iff$ $\Mon(M)=\Z$.} \newpage {\bf\blue Computation of the monodromy group of a Vaisman manifold} \definition {\bf \blue Lee field} on a Vaisman manifold is the vector field $\theta^\sharp$ dual to the Lee form. Since locally a Vaisman manifold is a cone over Sasakian (as shown in Lecture 3), {\bf \purple $\theta^\sharp$ acts on $M$ by holomorphic isometries, and on $\tilde M$ by non-isometric homotheties.} {\bf \purple The following theorem finishes the proof of Structure Theorem.} \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 $\Mon(M)=\Z$.} This theorem is proven later today. \remark Let $G$ be a group obtained as a closure of one-parametric group $e^{tX}$, $t\in \R$. {\bf \red Since $X$ acts by isometries, $G$ is a compact torus, $G=(S^1)^{k}$}. \newpage {\bf\blue Computation of the monodromy group, part 2} \claim 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. Let $G=(S^1)^{k}$ be the group obtained as a closure of one-parametric group $e^{tX}$, $t\in \R$. Consider the group $\tilde G$ of pairs $\tilde f\in \Aut(\tilde M)$, $f\in G$, making the following diagram commutative. \begin{equation*} \begin{CD} \tilde M@>{\tilde f}>> \tilde M \\ @V{\pi}VV @VV{\pi}V \\ M@>{f}>> M \end{CD} \end{equation*} {\bf \red Then $\tilde G\cong (S^1)^{k-1} \times \R$.} \remark {\bf \purple From this claim, the isomorphism $\Mon(M)=\Z$ follows immediately.} Indeed, $\Mon(M)\subset \ker p:\; \tilde G\arrow G$. \newpage {\bf\blue Computation of the monodromy group of a Vaisman manifold (part 2)} {\bf \green Proof of $\tilde G\cong (S^1)^{k-1} \times \R$. Step 1:} $\tilde G$ is a covering of $G$, and {\bf \purple the kernel of this projection is $\tilde G\cap \Mon(M)$. } {\bf \green Step 2:} Let $\tilde G_0\subset \tilde G$ be a subgroup acting on $\tilde M$ by isometries. Since $\tilde G$ acts on $\tilde M$ by homotheties, {\bf \red $\tilde G_0$ has codimension 1}. Moreover, $\tilde G_0$ cannot intersect $\Mon(M)$ and {\bf \purple it maps injectively to $\Aut(M)\cong (S^1)^{k}$. } {\bf \green Step 3:} {\bf \purple We obtain that $\tilde G_0\cong (S^1)^{k-1}$} (it's codimension 1). {\bf \green Step 4:} Since $\tilde G_0$ meets every component of $\tilde G$, it is connected. {\bf \red Therefore, $\tilde G\cong \tilde G_0\times \R\cong (S^1)^{k-1} \times \R$.} \endproof \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 $dd^c f=\omega$. {\bf \green Theorem 1:} 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 $C(S)$. Moreover, the form $dd^c \log t$ vanishes on $\langle r, I(r)\rangle$ and the rest of its eigenvalues are positive.} {\bf \green Proof. Step 1:} $2\omega=\Lie_{r}\omega=d(I\theta)=d(tIdt)=dd^c(t^2)$ Therefore, $t^2$ is a K\"ahler potential. {\bf \green Step 2:} $dd^c \log t^2= \frac{\tilde \omega}{t^2}-\frac {dt\wedge Idt}{t^2}$. \endproof \newpage {\bf\blue The fundamental foliation} \definition Let $M$ be a Vaisman manifold, $\theta^\sharp$ its Lee field, and $\Sigma$ a 2-dimensional real foliation generated by $\theta^\sharp, I\theta^\sharp$. It is called {\bf \blue the fundamental foliation} of $M$. Clearly, $\Sigma$ is tangent to orbits of the one-parametric group of automorphisms of the covering $\tilde M$ generated by homotheties. Therefore, {\bf \purple $\Sigma$ is a holomorphic foliaton.} \theorem Let $M$ be a compact Vaisman manifold, and $\Sigma\subset TM$ its fundamental foliation. Then\\ {\bf \red \hphantom{a}\ \ 1. $\Sigma$ is independent from the choice of the Vaisman metric.\\ \hphantom{a}\ \ 2. There exists a positive, exact (1,1)-form $\omega_0$ with $\Sigma=\ker \omega_0$.\\ \hphantom{a}\ \ 3. For any complex subvariety $Z\subset M$, $Z$ is tangent to $\Sigma$. \\ \hphantom{a}\ \ 4. For any compact complex subvariety $Z\subset M$, the set of smooth points of $Z$ is Vaisman.} {\bf \green Proof of (2):} Let $\tilde M = C(X)$ be the conical K\"ahler manifold which covers $M$, and $\psi:\; \tilde M \arrow \R$ the function satisfying $d\psi=\theta$. {\bf \purple Then $\omega_0:=dd^c\psi$ is a pseudo-Hermitian form which vanishes on $\Sigma$ and positive on $TM/\Sigma$ (Theorem 1).} Also, $\omega_0=d(I\theta)$, hence this form is well defined on $M$. \newpage {\bf\blue The fundamental foliation (proofs)} {\bf \red \hphantom{a}\ \ \ \:1. $\Sigma$ is independent from the choice of the Vaisman metric.\\ \hphantom{a}\ \ 2. There exists a positive, exact (1,1)-form $\omega_0$ with $\Sigma=\ker \omega_0$.\\ \hphantom{a}\ \ 3. For any complex subvariety $Z\subset M$, $Z$ is tangent to $\Sigma$. \\ \hphantom{a}\ \ 4. For any compact complex subvariety $Z\subset M$, the set of smooth points of $Z$ is Vaisman.} {\bf \green Proof of (1):} The zero foliation of $\omega_0$ is independent from the choice of the Vaisman metric. Indeed, if tere are two Vaisman structures with $\omega_0$ and $\omega_0'$ vanishing on differenc 1-dimensional complex foliations, the sum $\omega_0+\omega'_0$ would be positive definite. However, $\int_M \omega_0^{\dim_\C M}$ vanishes, because $\omega_0$ is exact. {\bf \purple Since $\Sigma=\ker\omega_0$, $\Sigma$ is independent from the Vaisman structure.} {\bf \green Proof of (3):} For any compact subvariety $X\subset M$, the integral $\int_Z\omega_0^{\dim_\C M}$ vanishes, because $\omega_0$ is exact. {\bf \purple Therefore, $\omega_0\restrict {TZ}$ has one zero eigenvalue at each point of $Z$.} This means precisely that $\Sigma\subset TZ$ at this point. {\bf \green Proof of (4):} Since the Lee field is tangent ot $Z$, the covering $\tilde Z\subset C(S)$ is preserved by the homotheties. Therefore it is also a conical K\"ahler manifold. {\bf \purple Then $Z=\tilde Z/\Z$ is Vaisman.} \endproof \newpage {\bf\blue Regular Vaisman manifolds} \definition A Vaisman manifold $M$ is called {\bf\blue regular}, if the leaves of the fundamental foliation are orbits of the group $(S^1)^2$ freely acting on $M$, {\bf \blue quasiregular} if these leaves are compact, and {\bf \blue irregular} otherwise. \theorem {\bf \red A Vaisman manifold is regular if and only if it is a smooth elliptic fibration over a projective manifold,} obtained as a quotient of a total space of non-zero vectors in a positive bundle by the action of $\Z$ mapping $v$ to $\lambda v$, with $|\lambda|>1$. {\bf \blue Proof. Step 1:} Let $\tilde M:=C(S)$ be the corresponding conical K\"ahler manifold. Clearly, the leaves of $\Sigma$ are obtained from orbits of the Reeb field on $S$ by the Lee field acting on $C(S)$ as a standard homothety. {\bf \purple Therefore, $S$ is regular.}\\ {\bf \blue Step 2:} By structure theorem, $M= C(S)/\Z$ acting as $\langle (x, t) \mapsto (\phi(x), q t)\rangle$, where $q>1$, and $\phi$ is a Sasakian automorphism of $X$. The leaves of $\Sigma$ intersect with $S$ by a union several copies of $S^1$ numbered by $\langle \phi\rangle$. {\bf \purple Regularity of $M$ implies that $\phi$ has finite order, and the corresponding group acts freely on $S$.}\\ {\bf \blue Step 3:} Now, $S_1:=S/\langle \phi\rangle$ is a regular Sasakian manifold, hence it is a space of unit vectors in a positive line bundle $L$ over $X:=S_1/\Reeb$.\\ {\bf \blue Step 4:} By construction, $X$ is the space of leaves of $\Sigma$, hence $\tilde M$ is a $\C^*$-fibration over $X$. \endproof \end{document}