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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Locally conformally K\"ahler manifolds \\[15mm]
\small lecture 6: Orbifolds and cones}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf HSE and IUM, Moscow
\\[2mm]  March 17, 2014
}
\vfil

{\Large\bf \red NO LECTURE 24.03.2014! \\
NEXT LECTURE March 31.
}
\vfil
\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 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 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 $C(S)$. 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

\newpage

{\bf \blue Orbispaces }

\definition
{\bf \blue Groupoid} is a category with all morphisms invertible.

\definition
An action of a group on a manifold is {\bf \blue rigid} if
the set of points with trivial stabilizer is dense.

\definition
{\bf \blue An orbispace} is a topological space $M$, 
equipped with a structure of a groupoid (the points of $M$
are objects of the groupoid category), a covering $\{U_i\}$,
and continuous maps $\phi_i:\; V_i\arrow U_i$, where each
$V_i$ is equipped with a rigid action of a finite group $G_i$,
satisfying the following properties. \\
\hphantom{a}\ \ \ 1. $\phi_i:\; V_i \arrow V_i/G_i=U_i$ is the quotient map.\\
\hphantom{a}\ \ \ 2. For each $x\in M$ and $U_i\ni x$, 
the group $\Mor(x,x)$ is equal to the stabilizer of $x$ in $G_i$.

\remark
An orbispace is a topological space, locally obtained as
a quotient, {\bf \purple with the quotient structure remembered
via the groupoid structure.}

\newpage

{\bf \blue Orbifolds }

\definition
{\bf \blue An orbifold} is an orbispace
$(M, \{\phi_i:\; V_i \arrow V_i/G_i=U_i\})$,
where all $V_i$ are diffeomorphic to open balls in $\R^n$.

\example
Let $M=\C P^1/((x,y)\sim(x, -y))$. {\bf \purple This quotient
is homeomorphic to $\C P^1$.} {\bf \red However, it is a different orbifold}
if we consider the covering induced from $\C P^1/G$, $G=\{\pm 1\}$
and the groupoid structure where $\Mor(x,x)=\St_G(x)$.

\definition
{\bf \blue A smooth orbifold} is an orbifold $M$ equipped with
a sheaf of functions $C^\infty M$ in such a way that for each 
$U_i=V_i/G_i$, the corresponding ring of sections
$C^\infty U_i$ is identified with a 
ring of $G_i$-invariant smooth functions on $V_i$.

\definition
{\bf \blue A Riemannian metric} on a smooth orbifold is a
$G_i$-invariant metric on each $V_i$, compatible with the
gluing maps. 

\newpage

{\bf \blue Complex orbifolds }

\definition
{\bf \blue A complex orbifold} is an orbifold $M$ equipped with
a sheaf of functions $\calo_M$ in such a way that each $V_i$
is an open ball in $\C^n$, and for each 
$U_i=V_i/G_i$, the corresponding ring of sections
 $\calo_{U_i}$ is identified with a 
ring of $G_i$-invariant holomorphic functions on $V_i$.

\definition
{\bf \blue An underlying complex variety} of a 
complex orbifold is a complex variety with the topological
space $M$ and the structure sheaf $\calo_M$.


\example
Let $\C^*$ act on $\C^n$ as \[ h_t(x_1, ..., x_n)=
(t^{a_1}x_1, t^{a_2}x_2, ..., t^{a_n}x_n).\]
The quotient $(\C^n\backslash 0)/\C^*$ is called
{\bf \blue weighted projective space}, and denoted
$\C P^{n-1}(a_1, ..., a_n)$.

\exercise
{\bf \purple Prove that it is an orbifold.}


\newpage

{\bf \blue Projective orbifolds }

\definition
{\bf \blue A projective orbifold} is a complex
orbifold with the underlying complex variety projective.



\definition
{\bf \blue A holomorphic vector bundle} on a complex orbifold
is a $G_i$-equivariant vector bundle on each $V_i$, equipped
with the $G_i$-invariant gluing maps satisfying cocycle condition.

\theorem {\bf \blue (Baily)}\\
Let $M$ be a compact complex orbifold equipped with 
a holomorphic Hermitian vector bundle $L$. Assume that
the curvature of $L$ is positive definite on all $V_i$
(in this case $L$ is called {\bf \blue positive}). {\bf \red Then
$M$ is projective.}

{\small \it W. L. Baily, \emph{On the imbedding of V-manifolds
    in projective spaces}, Amer. J. Math. {\bf 79} (1957), 403-430.}

\newpage

{\bf \blue Quotients of torus action }

\theorem
Let $T^n$ be a compact torus acting on a manifold $M$
with all orbits of the same dimension. {\bf \red Then $M/T^n$ is an 
orbifold}.

{\bf \green The proof is futher in these slides.}

\lemma
{\bf \purple The set of compact subgroups of $T^n$ is countable.}
\endproof

\lemma
Let $M$ be a topological space with continuous action of $T^n$,
and $\St(x)$ the stabilizer of $x\in M$ in $T^n$.
{\bf \red Then the map $x\arrow \St(x)$ is semicontinuous:}
for any sequence $\{x_i\}\subset M$,
$\lim_i x_i =x$, one has $\St(x)\supset \lim_i \St(x_i)$,
where {\bf \blue $\lim_i \St(x_i)$ is the set of all limit points
of the sequences $\{t_i\}$, $t_i \in \St(x_i)$.}
\endproof

\proposition
{\bf \red For any sequence of compact subgroup of a torus $T_i \subset T^n$, the
limit $\lim_i T_i$ contains all $T_i$, except a finite number.}

\exercise {\bf \purple Prove this!}

\newpage

{\bf \blue Subgroups of a torus }

\proposition
{\bf \red For any sequence of compact subgroup of a torus $T_i \subset T^n$, the
limit $\lim_i T_i$ contains all $T_i$, except a finite number.}

{\bf \green Proof. Step 1:}  \lemma\\
Let $X, Y$ be subsets of a metric space, and
$\delta(X, Y):= \sup_{x\in X}d(x, Y)$. Fix a flat Riemannian
metric on a compact torus $T^n$. {\bf \purple Then for any compact
subgroup $G\subset T^n$ there exists a positive number $\epsilon(G)$
such that $\delta (G_1, G)> \epsilon(G)$ unless $G_1\subset G$.}

To see this, take $\epsilon(G) = \frac 2 3 R$, where
$R$ is a metric diameter of a smallest circle in the decomposition
$T^n/G= (S^1)^k$, where  $T^n/G$ is considered with a 
flat metric induced from $T^n$.
Then $\delta(G_1, G)\leq \delta(0, G_1/(G_1\cap G))\leq \epsilon(G)$.

{\bf \blue Step 2:} Each $T_i$ is a closure of a set
$\{\alpha_i, 2\alpha_i, 3\alpha_i, ... \}$,
where $\alpha_i$ is a sufficiently general point in $T_i$.
Then $T_\infty:= \lim_i T_i$ is the set of all limit points
of $\{ n\alpha_{m}\}$, $n, m\in \Z^{>0}$.
Therefore, for all $n, m$, except finitely many,
$d(n\alpha_{m}, T_\infty) < \epsilon(T_\infty)$,
giving $\delta(T_m, T_\infty)< \epsilon(T_\infty)$
and $T_m \subset T_\infty$. \endproof


\newpage

{\bf \blue Stratification associated with a torus action }


\corollary
Let $T^n$ be a compact torus acting on a topological space.
Consider a function $x \stackrel \psi \arrow \St(x)$.
Then there exist a stratification of $M$ by closed
strata $M_i$ such that {\bf \red $\Psi$ is constant on a complement
of $M_i$ by smaller strata, and $\Psi(M_i)\supset \Psi(M_j)$
whenever $M_j \supset M_i$.}

{\bf \green Proof:} Consider the set ${\goth A}$ of all compact
subgroups of $T^n$, and let 
$M_\alpha:= \{ x\in M \ \ |\ \ \Psi(x)\supset\alpha\}$,
where $\alpha \in {\goth A}$. By semicontinuity, $M_\alpha$ is closed
for each $\alpha$. Relation $\Psi(M_i)\supset \Psi(M_j)$
for smaller strata follows from the proposition above.
\endproof

\newpage

{\bf \blue Quotient orbifolds}

\theorem
Let Let $G= T^n$ be a torus acting on a complex manifold $G$
by biholomorphic maps, and $M_i$ the corresponding stratification.
Let $H_0:= \St(x)$, where $x$ is a general point of a maximal stratum.
{\bf \red Then the quotient $M/G$ is an orbifold,
and for each $x\in M/G$, the corresponding group $\Mor(x,x)$
is equal to $\St(x)/H_0$.}

{\bf \green Proof. Step 1:}  All orbits of $G$ are smooth. Indeed,
the Zariski tangent space to an orbit has constant dimension, because
it is a quotient of the Lie algebra of $G$ by $\Lie(H_0)$, and
{\bf \purple a variety with Zariski tangent space of constant dimension
is smooth.}

{\bf \green Step 2:} Define {\bf\blue a section} of an action 
of $G$ at $x\in M$ as a smooth submanifold $S\ni x$ defined locally
in some neighbourhood of $x$, transversal to the orbit
$G\cdot x$ and having complementary dimension.
{\bf \purple Clearly, a section exists at each $x\in M$.}


\newpage

{\bf \blue Quotient orbifolds (cont.)}


\theorem
Let Let $G= T^n$ be a torus acting on a complex manifold $G$
by biholomorphic maps, and $M_i$ the corresponding stratification.
Let $H_0:= \St(x)$, where $x$ is a general point of a maximal stratum.
{\bf \red Then the quotient $M/G$ is an orbifold,
and for each $x\in M/G$, the corresponding group $\Mor(x,x)$
is equal to $\St(x)/H_0$.}



{\bf \green Step 3:} {\bf \purple A section at $x$ can be always chosen
$\St(x)$-invariant. } To see that, chose a $G$-invariant
metric, let $W\subset T_x M$ be an orthogonal complement
of the tangent space to $G\cdot x$, and $S$ the union of 
all geodesics passing through $x$ and tangent to $W$,

{\bf \green Step 4:}
Let $H_x:= \St(x)$. 
Take a tubular neighbourgood $U$ of an orbit $G\cdot x$ given by
\[ U:=\bigcup\limits_{g\in G/H_x} gS.\] For $S$ sufficiently small,
this gives a decomposition $U=S\times (G/H_x)$. Therefore,
 $S\cap gS=\emptyset$ for all $g\notin H_x$. 
{\bf \purple This implies that the map $S\arrow M$ is a finite quotient
map, with $M$ locally isomorphic to $S/(H_x/H_0)$.}
\endproof


\newpage

{\bf \blue Sasakian and Vaisman manifolds and their projective orbifolds}

\corollary
Let $M$ be a quasiregilar Vaisman manifold,
$\Sigma$ its fundamental foliation, and $M/\Sigma$
the quotient space. {\bf \red Them $X:=M/\Sigma$ is a projective
orbifold.}

{\bf \green Proof:} $X$ is an orbifold as proven above.
Since it is a quotient of a complex space by a complex group
action, $X$ is a complex orbifold. By construction, the corresponding
conical K\"ahler manifold $\tilde M$ is a total space
of $\C^*$-bundle $L$ (in the orbifold sense).
The standard local argument implies that the
curvature of $L$ gives a K\"ahler orbifold metric on  
$X$. Baily's theorem implies that $X$ is projective.
\endproof

\corollary 
Let $S$ be a quasiregular Sasakian manifold, 
and $\Reeb$ its Reeb field. 
{\bf \red Then $X:=S/\Reeb$ is a projective orbifold, and
$S$ is a total space of $U(1)$-bundle over $X$ associated
with a positive  holomorphic line bundle.}

{\bf \green Proof:} $S\times S^1$ is Vaisman, and the
corresponding fundamental foliation is $TS^1\times \Reeb$.
\endproof


\newpage

{\bf \blue Conical K\"ahler structures and homotheties}

{\bf \green Proposition 1:}
Let $(M,  \omega)$ be a conical K\"ahler manifold,
and $X$ a vector field acting on $M$ by holomorphic, non-isometric 
homotheties, such that $IX$ also acts by homotheties,
and $e^{t X}$ is defined for any real $t$. Then \\
{\bf \red \phantom{a}\ (a) $dd^c \phi=  \omega$, where $\phi=|X|^2$.\\
\phantom{a}\ (b) Let $S_X:= \phi^{-1}(1)$. Then $S_X$ is Sasakian, \\and
$M$ is isometric to $C(S_X)$.\\
\phantom{a}\ (c) $S_X$ is quasiregular if and only if the action of $X$
integrates to a holomorphic $\C^*$-action.}

{\bf \green Proof. Step 1:} Since $X$, $IX$ act by homotheties, one has
a character $\chi:\; \langle X, IX\rangle\arrow \R$
such that $\Lie_Z\omega=\chi(Z)\omega$. {\bf \purple Replacing $X$ by
some linear combination of $X, IX$ if necessary, we 
may assume that $IX$ acts by isometries. }
Rescaling, we may assume that $\Lie_X g=2g$.

{\bf \green Step 2:} Define $X^\flat:= g(X, \cdot)$ {\bf \blue (``the dual 
1-form'').} Then $dX^\flat= \Lie_{IX}\omega=0$ and
$2 X^\flat = \Lie_X(X^\flat)=d\langle X, X^\flat\rangle=d|X|^2$.

{\bf \green Step 3:} $\Lie_X \omega= 2\omega$, which gives
$2\omega = d(\omega\cntrct X)= d(IX^\flat)= 2 dId|x|^2$
(last equation is proven in Step 2). {\bf \purple This proves
Proposition 1 (a).}

\newpage

{\bf \blue Conical K\"ahler structures and homotheties (cont.)}


{\bf \green Proposition 1:}
Let $(M,  \omega)$ be a conical K\"ahler manifold,
and $X$ a vector field acting on $M$ by holomorphic, non-isometric 
homotheties, such that $IX$ also acts by homotheties,
and $e^{t X}$ is defined for any real $t$. Then \\
{\bf \red \phantom{a}\ (a) $dd^c \phi=  \omega$, where $\phi=|X|^2$.\\
\phantom{a}\ (b) Let $S_X:= \phi^{-1}(1)$. Then $S_X$ is Sasakian, \\and
$M$ is isometric to $C(S_X)$.\\
\phantom{a}\ (c) $S_X$ is quasiregular if and only if the action of $X$
integrates to a holomorphic $\C^*$-action.}


{\bf \green Step 4:} Let $M\arrow S_X$ map $m$ to an intersection
of $e^{tX}m$ and $S_X$. {\bf \purple
This gives a decomposition $M=S_X \times \R^{>0}$,
compatible with the conical metric on $S_X \times \R^{>0}=C(S_X)$,}
as shown in the last lecture using the Vaisman manifolds local decomposition
into a product.

{\bf \green Step 5:} 
Let $C$ be the group generated by 
$e^{tX}, e^{tIX}$. Clearly, $C=\R^{>0}\times \{e^{tIX}\}$.
The Reeb orbits on $S_X$  
are orbits of $e^{tIX}$, {\bf \purple hence they are compact if and
only if $\{e^{tIX}\}$ is compact, equivalently,
iff $C=\C^*$.}
\endproof


\newpage

{\bf \blue Conical K\"ahler structures and $\C^*$-action}

\remark
For each holomorphic isometry $h$ of a Vaisman manifold,
$h$ lifts to a conformal automorphism of its K\"ahler
covering. However, {\bf \purple a conformal automorphism of a K\"ahler
manifold is a homothety,} because $d(f\omega) =df \wedge \omega$,
and this may vanish only when $df=0$.

{\bf \green Theorem 1:}
Let $C(S)$ be a conical K\"ahler manifold, $h_t$
the corresponding homothety action, and $X$ its vector field.
{\bf \red Then there exists a vector field $X_1$ arbitrarily close to $X$}
acting on $C(S)$ by holomorphic homotheties, with $IX_1$
also acting by homotheties, {\bf \red such that the action of $X_1$ integrates
to $\C^*$-action on $C(S)$. }

{\bf \green Proof:} Fix some $\lambda>1$, and let $M:= C(S)/h_\lambda$
be the corresponding Vaisman manifold, where $h_t$ acts isometrically.
Consider the Lie group $G\subset \Iso(M)$ 
obtained as the closure of $\{h_t\}$. {\bf \purple For each vector field
$X_1\in \Lie(G)$, $X_1$ acts on $M$ by holomorphic isometries,}
hence it acts on $C(S)$ by homotheties; non-isometrically when
$X_1$ is sufficiently close to $X$. 

Choosing $X'\in \Lie (G)$ rational and sufficiently close
to $X$, we obtain an isometry of $M$ which integrates to a $T^2$-action
on $M$ {\bf \purple 
and to non-isometric $\C^*$-action on its cone. } \endproof

\remark 
By Proposition 1,
{\bf \red his gives a new
cone structure on $C(S)$. }

\newpage

{\bf \blue Density of quasiregular Vaisman manifolds}

\corollary
Let $C(S)$ be a conical K\"ahler manifold,
with $S$ compact. {\bf \purple Then $C(S)$ is holomorphically 
isometric to a total space of non-zero sections 
of a positive line bundle over a 
projective orbifold.}
\endproof

\corollary {\bf \purple Any compact Vaisman manifold $(M,I)$ admits a deformation
$(M,I')$ which is quasi-regular.} Moreover, $I'$ can be chosen
arbitrarily close to $I$. 

{\bf \green Proof:} Take the conical K\"ahler manifold $C(S)$, and
replace the homothety vector field $X$ by a quasiregular one $X'$.
Then take a quotient $C(S)/\Z$ by $\Z$ acting as $e^{\lambda X'}$.
\endproof

\corollary {\bf \purple Any compact
Sasakian manifold $(M,I)$ admits a deformation
$(M,I')$ which is quasi-regular.} Moreover, $I'$ can be chosen
arbitrarily close to $I$. \endproof

\vfil

{\Large\bf \red NO LECTURE 24.03.2014! \\
NEXT LECTURE March 31.
}
\vfil


\end{document}

