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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Locally conformally K\"ahler manifolds \\[15mm]
\small lecture 11: CR-geometry of Sasakian manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf HSE and IUM, Moscow
\\[2mm]  April 28, 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 Reeb field (reminder)}


\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 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 (reminder)}


{\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 (reminder).}

{\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. 

\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 $TS/B$.
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 $-d^c \phi(\Phi(u,v))=dd^c\phi(x,y))$.}

\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.}

\newpage

{\bf \blue Algebraic cones (reminder).}


\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.}




\newpage

{\bf \blue Pseudoconvex shells and logarithm (reminder)}


\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 
$\Lie_{\vec r}\phi_\lambda = \lambda \phi_\lambda$ and $\phi_\lambda\restrict S=1$.
Moreover, {\bf \red such $\phi_\lambda$  is strictly 
plurisubharmonic when $\lambda \gg 0$.}

\corollary {\bf \blue (Gauduchon-Ornea)} \\
{\bf \red All linear Hopf manifolds are LCK with potential.}


\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
$\gamma^C=\vec r$.}

{\bf \green Proof:} Lecture 10.

\definition
Such a vector field is called {\bf \blue a logarithm of monodromy}.


\newpage


%{\bf\green Proof. Step 1:}
%On $M$ the group $e^{\R \vec r}$ acts as a circle, and
%maps $\omega$ to a form which is conformally equivalent to $\omega$
%Replacing $\omega$ by its average over $e^{\R \vec r}$,
%we obtain a new LCK form $\omega'$ in the same conformal class and 
%$e^{\R \vec r}$-invariant. Further on, we replace
%$\omega$ by an $e^{\R \vec r}$-invariant LCK form $\omega'$ if
%necessary and assume $\mega$ is invariant.
%
%{\bf\green Step 2:}
%Let $\phi:= \frac{\tilde \omega}{\pi^*\omega}$, where
%$\phi:\; \tilde M \arrow M$ is the covering map, 
%and $d\omega=\omega\wedge \theta$. Then $0=d(\phi \omega)=
%(d\phi -\phi\theta)\wedge \omega$, hence $\phi\theta=d\phi$.
%This gives $\theta = d\log \phi$.
%
%{\bf\green Step 3:} Since $\Lie_{\vec r} (\phi \omega)=
%\Lie_{\vec r} (\phi) \omega= \lambda\phi \omega$, we
%obtain $\lambda\phi = \Lie_{\vec r} (\phi)= d\langle
%\phi\theta, \vec r\rangle$, hence $\langle
%\theta, \vec r\rangle=\const$.
%
%{\bf\green Step 4:} The Lie algebra $\langle \vec r, I
%\vec r\langle$ acts on $\
%
%
%
%
%Let $\theta:= \tilde \omega\cntrct I(\vec r)$.
%Clearly, $\Lie_{\vec r}\tilde \omega = d(\tilde
%\omega\cntrct \vec r)= d (I\theta)$.  Since 

\newpage

{\bf \blue Theorem of Kamishima-Ornea}

\theorem
{\bf \blue (Kamishima-Ornea)} \\
Let $(M,\omega, \theta)$ be an LCK manifold equipped with a 
holomorphic conformal $\C$-action,
which lifts to non-isometric homotheties on 
its K\"ahler covering $\tilde M$. {\bf \red Then $(M,\omega, \theta)$
is conformally equivalent to a Vaisman manifold.}

{\bf \green Proof. Step 1:} 
Let $\vec r$ be a vector field of this $\C$-action,
and $\tilde \omega$ the K\"ahler form of $\tilde M$.
Then $\Lie_{\vec r} \tilde\omega=a\tilde\omega$ and
$\Lie_{I\vec r} \tilde\omega=b\tilde\omega$. Replacing
$\vec r$ by a linear combination of $\vec r$ and $I(\vec r)$,
we obtain a vector field preserving $\tilde\omega$.
Replacing $\vec r$ by an appropriate linear combination
of $\vec r$ and $I\vec r$, {\bf \purple we can assume that
$\Lie_{I\vec r} \tilde\omega=0$ and 
$\Lie_{\vec r} \tilde\omega=\tilde\omega$.}

{\bf \green Step 2:} $\Lie_{\vec r} \tilde \omega= 
d(\tilde \omega\cntrct \vec r)=\tilde \omega$, and
$\Lie_{I\vec r} \tilde \omega=d\eta=0$, where $\eta= I(\omega\cntrct \vec r)=
\omega\cntrct (I\vec r)$.

{\bf \green Step 3:} $\Lie_{\vec r}\eta=\eta=d(\eta \cntrct \vec r)=
d\langle \eta, \vec r\rangle$. 
%and 
%$\Lie_{I\vec r}\eta=0=d(\eta \cntrct I\vec r)=
%d\langle \eta, I\vec r\rangle$.
This gives $\tilde \omega= dd^c \phi$, where 
$\phi=\langle \eta, \vec r\rangle$. 

{\bf \green Step 4:} The action by $\vec r$ multiplies the
K\"ahler potential $\phi$ by a constant; the action by $I(\vec r)$ 
preserves $\phi$. Therefore, {\bf \purple $\tilde M$ is locally
isometric to a K\"ahler cone,} and $\omega:=\phi^{-1}\tilde \omega$
is Vaisman (Lecture 3), that is, satisfies
$\nabla\theta=0$.
\endproof


\newpage

{\bf \blue Vaisman manifolds and homothety action}





\definition
Let $\tilde M$
be an algebraic cone, $\rho:=e^{t\vec r}$ a contraction, and
$S\subset \tilde M$ a pseudoconvex shell. Consider the
(necessarily unique) function potential $\phi_\lambda$
which satisfies $\Lie_{\vec r}\phi_\lambda = \lambda
\phi_\lambda$. Assume that it is a K\"ahler potential
(by Theorem 1, it is a K\"ahler potential for 
$\lambda \gg 0$). Then $\phi_\lambda$ is called
{\bf \blue $\rho$-automorphic K\"ahler potential},
and $dd^c\phi$ {\bf \blue $\rho$-automorphic K\"ahler
  form}.

\theorem
Let $(M,\omega)$ be an LCK-manifold with potential,
and $\tilde M$ its algebraic cone, $\tilde M/\langle
\gamma\rangle =M$, and $\phi$ its K\"ahler potential.
Then {\bf \red $\omega$ is conformally equivalent to a Vaisman
metric if and only if there exists a logarithm 
$\vec r$ of $\gamma$ such that $\Lie_{I\vec r}\phi =0$.}


{\bf \green Proof:} If $M$ is Vaisman, $\tilde M=C(S)$,
where $S$ is Sasakian, $\vec r := t\frac{d}{dt}$ its
logarithm, and $I\vec r$ its Reeb field,
acting on $C(S)$ by holomorphic isometries.

Conversely, if $\tilde M$ admits a logarithm $\vec r$ 
with such properties, then the corresponding holomorphic
flow acts on $\tilde M$ by homotheties, and $M$ is
Vaisman by Kamishima-Ornea. \endproof


\newpage

{\bf \blue Stein manifolds (reminder).}

\definition 
A complex variety $M$ is called {\bf \blue holomorphically convex}
if for any infinite discrete subset $S\subset M$,
there exists a holomorphic function $f\in \calo_M$ which
is unbounded on $S$.

\definition
A complex variety is called {\bf \blue Stein}
if it is holomorphically convex, and 
has no compact complex subvarieties.

\remark 
Equivalently, {\bf\red a complex variety is Stein
if it admits a closed holomorphic embedding into $\C^n$.}

\theorem
(K. Oka, 1942) {\bf \purple A complex manifold $M$ is Stein}
if and only $M$ admits a K\"ahler metric with 
a {\bf \purple K\"ahler potential which is positive
and proper} (proper = preimages of compact sets are compact).

\theorem (H. Cartan, 1951)
{\bf \purple A complex variety $M$ is Stein}
if and only if for any coherent sheaf $F$ on $M$,
{\bf \purple its cohomology $H^i(F)$ vanish for all $i>0$.}


\newpage

{\bf \blue CR-holomorphic functions and vector fields}

\definition
Let $(S,B,I)$ be a CR-manifold. A function $f$ on $S$ is
called {\bf \blue CR-holomorphic} if for any vector field
$v\in B^{0,1}$, we have $\Lie_v f=0$. A vector field
$v\in TM$ is called {\bf \blue CR-holomorphic} if
the corresponding diffeomorphism flow preserves 
$B$ and $I$.

\theorem {\bf \blue (Rossi-Andreotti-Siu)}\\
Let $S$ be a compact strictly pseudoconvex CR-manifold,
$\dim_\R S\geq 5$, and $H^0(\calo_S)_b$ the ring of bounded
CR-holomorphic functions. {\bf \red Then $S$ is a boundary of a
Stein manifold $M$ with isolated singularities,} such that
$H^0(\calo_S)_b = H^0(\calo_M)_b$, where $H^0(\calo_M)_b$
denotes the ring of bounded holomorphic functions.
Moreover, $M$ is defined uniquely, $M=\Spec(H^0(\calo_S)_b)$.

\corollary 
The Lie group $G:=\Aut(S)$ of CR-automorphisms
is identified with the group of complex automorphisms
of the corresponding Stein space $M$. {\bf \red Its Lie algebra
(the algebra of holomorphic vector fields) is the Lie
algebra of holomorphic vector fields on $M$.}

\newpage

{\bf \blue Burns-Lee theorem}

\theorem
{\bf \blue (Dan Burns, John M. Lee)}\\
Let $S$ be a compact strictly pseudoconvex CR-manifold,
and $\Aut_0(S)$ the connected component of its group of automorphisms.
{\bf \red Then $\Aut_0(S)$ is compact} unless $S$ is equivalent to 
the standard sphere $S^{2n-1}\subset \C^n$ with its induced
CR-structure. In the latter case $\Aut_0(S)=U(1, n)$.

{\bf \green This theorem would not be used.}
%Further in this lecture, I prove Burns-Lee theorem when
%$S$ admits a Sasakian structure and $\dim_\R S>3$.



\newpage

{\bf \blue CR-manifolds and Sasakian manifolds}

\definition
Let $(S,B,I)$ be a CR-manifold.
We say that $S$ {\bf \blue admits a Sasakian structure}
if it can be realized as a CR-hypersurface $S\subset C(S)$, where 
$C(S)$ is a conical K\"ahler manifold.

\definition
Let $(S,B,I)$ be a CR-manifold, with $TS/B$ oriented
(for strictly pseudoconvex CR-manifolds, the Levi
form defines the orientation on $TS/B$).
A vector field $v\in TS$ is called {\bf \blue positive}
if it is transversal to $B$ everywhere, and
its projection to $TS/B$ is positive.

\example
{\bf \purple 
The Reeb field of a Sasakian manifold is always positive}
(or negative, depending on the choice of orientation).
Indeed, $I\Reeb$ is always normal to $S$, hence
$\Reeb\notin B=TS\cap I(TS)$.

\theorem
Let $S$ be a strictly pseudoconvex compact CR-manifold,
$\dim_\R S \geq 5$. {\bf \red Then $S$ admits a Sasakian structure
if and only if $S$ admits a positive holomorphic vector field.}
This vector field becomes a Reeb field of this Sasakian manifold.

\remark The
implication ``admits a Sasakian structure'' $\Rightarrow$
``admits a positive holomorphic vector field'' is clear, because 
the Reeb vector field is positive and CR-holomorphic.


\newpage

{\bf \blue CR-manifolds and Sasakian manifolds (2)}

Assume that a strictly pseudoconvex compact CR-manifold
$S$ admits a CR-holomorphic positive vector field $R$.
{\bf \red We need to construct a Sasakian metric on $S$ 
such that $R$ is its Reeb field.}

\remark
The argument here is essentially the same as used to 
embed an LCK manifold with potential to a Hopf manifold.

{\bf \green Step 1:} By Rossi-Andreotti-Siu,
$S=\6 M$, where $M=\Spec(H^0(\calo_S)_b)$
is a Stein variety with isolated singularities,
and $R$ acts on $M$ by holomorphic automorphisms.

{\bf \green Step 2:} Since $R$ is positive, $IR$ is transversal
to $\6 M$; replacing $R$ by $-R$, we can always assume that
$IR$ points toward interior of $M$, and 
{\bf \purple $A_\epsilon:=e^{\epsilon I R}$
for small $\epsilon$ maps $M$ to a subset $A_\epsilon(M)\subset M$ 
with compact closure.}


{\bf \green Step 3:}
Consider the ring ${\cal H}= H^0(\calo_M)_b$ of bounded 
holomprhic functions on $M$, with $\sup$-metric.
Then ${\cal H}$ is a Banach ring. Since $A_\epsilon(M)$
has compact closure, {\bf \purple $A_\epsilon^* {\cal H}$ is a normal
family, and $A_\epsilon^*$ is a compact operator.}


\newpage

{\bf \blue CR-manifolds and Sasakian manifolds (3)}

{\bf \green Step 4:} By maximum principle, for any
non-constant $f\in {\cal H}$, one has \\ $\sup_{A_\epsilon(M)} |f|<\sup_M |f|$.
{\bf \purple
Since any limit point $f_{\lim}$ of a sequence $(A_\epsilon^i)^* f$ satisfies
$\sup {A_\epsilon(M)} |f|=\sup_M |f|$, it is constant.} A limit 
function $f_{\lim}$ exists, because $A_\epsilon^* {\cal H}$ is precompact.

{\bf \green Step 5:} This implies that for each $z\in M$, a limit point $z_{\lim}$ 
of a sequence $\{z, A_\epsilon z,A_\epsilon^2 z, ...\}$ is unique and 
independent of $z$. Indeed, $f_{\lim}(z)= f(z_{\lim})$, but $f_{\lim}=\const$.
This implies that {\bf \purple $A_\epsilon$ is a holomorphic contraction
contracting $M$ to the origin point $x_0\subset M$.}

{\bf \green Step 6:} Since $R\restrict {S_\epsilon}=A_\epsilon(R)$
is nowhere vanishing for each $\epsilon$, the vector field 
$\vec r:= IR$ is transversal to $S_\epsilon:= A_\epsilon(S)$ 
pointing to the origin. Therefore, through each point of $S$
passes a unique solution $\rho(t)$ of an equation 
$\frac{d\rho(t)}{dt}=\vec r$.

{\bf \green Step 7:} Let $\phi_\lambda$ be a $\rho$-automorphic
K\"ahler potential associated with this $S$ and $\rho$ as above.
{\bf \purple 
The Lie algebra $\langle R, I R\rangle$ acts on $(M, dd^c \phi_\lambda)$
by holomorphic homotheties, hence it is a conical K\"ahler manifold}
(Kamishima-Ornea). Therefore, $S$ is Sasakian.
\endproof

%\newpage
%
%{\bf \blue Kobayashi pseudometric and Kobayashi hyperbolic manifolds}
%
%
%\definition
%{\bf \blue Pseudometric} on $M$ 
%is a function $d:\; M \times M \arrow \R^{\geq 0}$
%which is symmetric: $d(x,y)=d(y,x)$ and satisfies the
%triangle inequality $d(x,y)+d(y,z) \geq d(x,z)$.
%
%\remark 
%Let ${\goth D}$ be a set of pseudometrics. {\bf \purple Then
%$d_{\max}(x,y):= \sup_{d\in {\goth D}}d(x,y)$ is also a pseudometric.}
%
%\definition
%The {\bf \blue Kobayashi pseudometric} on a complex manifold $M$
%is $d_\max$ for the set ${\goth D}$ of all pseudometrics
%such that any holomorphic map from the 
%Poincar\'e disk to $M$ is distance-decreasing.
%
%In other words, distance between points $x, y$ in Kobayashi pseudometric
%is infimum of the distance over all sets of Poincare disks
%connecting $x$ to $y$.
%
%\definition
%A variety is called {\bf \blue Kobayashi hyperbolic} if
%the Kobayashi pseudometric is non-degenerate.
%
%\exercise
%{\bf \purple Prove that any bounded subset of $\C^n$ is Kobayashi hyperbolic.}
%Use the Schwartz lemma.
%
%\remark
%From this exercise it follows that any precompact subset of a Stein
%manifold is Kobayashi hyperbolic.
%
%\definition
%Let $M$ be a Stein manifold equipped with a finite volume form
%$\Vol(M)$, and $L^2(\calo_M)$ the space of all $L^2$-integrable
%holomorphic functions. A locally $L^2$-integrable holomorphic
%function is holomorphic, by multi-dimensional Cauchy lemma.
%Therefore, $L^2(\calo_M)$ is a Hilbert space. Consider
%an orthonormal basis $\xi_i$ in $L^2(\calo_M)$. {\bf \blue
%Bergman metric} on $M$ is $dd^c\sum |\xi_i|^2$. 
%
%\exercise Let $M$ be a Stein manifold
%which admits a closed holomorphic embedding to a bounded 
%open subset of $\C^n$.
%Prove that the {\bf \blue Bergman kernel} 
%$\sum |\xi_i|^2$ is summable, and the 
%Bergman metric is nowhere degenerate and 
%depends only on the volume form $\Vol(M)$.
%
%\claim
%Let $M$ be a Stein manifold
%which admits a closed holomorphic embedding to a bounded 
%open subset of $\C^n$. Using the Kobayashi metric, we choose
%the Hausdorff measure, giving a volume form $\Vol(M)$ on $M$.
%{\bf \red Then the Bergman metric of $M$ is $\Aut(M)$-invariant.}
%
%
%\newpage
%
%{\bf \blue Burns-Lee theorem for Sasakian manifolds}
%
%
%\theorem
%{\bf \blue (Dan Burns, John M. Lee)}\\
%Let $S$ be a compact strictly pseudoconvex CR-manifold
%admitting a Sasakian structure,
%and $\Aut_0(S)$ the connected component of its group of automorphisms.
%{\bf \red Then $\Aut_0(S)$ is compact} unless $S$ is equivalent to 
%the standard sphere $S^{2n-1}\subset \C^n$ with its induced
%CR-structure. In the latter case $\Aut_0(S)=U(1, n)$.
%
%{\bf \green Proof. Step 1:} Let $R$ be a positive CR-holomorphic
%vector field on $S$, and $C(S)$ the corresponding conical K\"ahler
%manifold. Consider its metric completion $\overline {C(S)}$
%with the complex structure obtained from Andreotti-Rossi-Siu,
%and let $\rho$ be the corresponding homothety action.
%
%{\bf \green Step 2:} Put the Bergman metric $h$ associated with the
%Kobayashi volume form on $M$. Clearly, $h$ is $\Aut_0(S)$-invariant.
%The curvature $R^i_{jkl}$ of $h$ is $\rho$-invariant,
%and {\bf \purple its absolute value $|R^i_{jkl}|$ computed with respect
%to the K\"ahler metric $\rho$-automorphic, growing quadratically
%as we go towards the origin $x_0$.}
%
%{\bf \green Step 2:}  Unless $C(S)$ is flat (which happens precisely
%when $S$ is a sphere), any automorphism

\newpage

{\bf \blue Jordan-Chevalley decomposition}

\definition
{\bf \blue An algebraic group} is a group object in the
category of affine schemes.
{\bf \blue A pro-algebraic group} is an inverse limit
of algebraic groups. Further on, all algebraic groups are
considered over $\C$.

\definition
An element of an algebraic group $G$ is called
{\bf \blue semisimple} if its image is semisimple
for any algebraic representation of $G$, and
{\bf \blue unipotent} if its image is unipotent
(that is, exponent of nilpotent) 
for any algebraic representation of $G$

\theorem {\bf \blue (The Jordan-Chevalley  decomposition)}\\
Let $G$ be an algebraic group, and $a\in G$.
{\bf \red Then there exists a unique decomposition $A= S U$
of $A$ onto a product of commuting elements $S$ and $U$,
where $U$ is unipotent and $S$ semisimple.}

\exercise {\bf \purple Prove this theorem.}

\remark
Since this decomposition is unique, it is functorial.
{\bf \red Therefore, it is also true for all pro-algebraic groups.}


\newpage

{\bf \blue Semisimple LCK manifolds with potential}

Recall that a linear operator is called {\bf \blue semisimple}
if it is diagonalizable over an algebraic closure of the basic field.

\definition
Let $M$ be an LCK manifold with potential, 
and $j:\; M \arrow H$ a holomorphic embedding to a Hopf
manifold $H = \C^n \backslash 0/\langle A\rangle$.
Then $M$ is called {\bf \blue semisimple} if $A$ is
semisimple.

\remark
Let $M$ be an LCK manifold with potential, $\tilde M$
its K\"ahler $\Z$-covering, $M= \tilde M/\langle
A\rangle$, $j:\; M \arrow H$ a holomorphic embedding to a Hopf
manifold $H = \C^n \backslash 0/\langle A\rangle$,
and $\tilde M_c$ its completion. Let $R$ be a $\goth
m$-adic completion of $\calo_{\tilde M_c}$ in
the maximal ideal of the origin $x_0 \in \tilde M_c$,
$R= \lim_{\leftarrow} \calo_{\tilde M_c}/ {\goth m}^k$.
Let $G:= \Aut(R)$; clearly, 
\[ 
G= \lim_{\leftarrow} \Aut(\calo_{\tilde M_c}/ {\goth m}^k),
\]
hence it is a pro-algebraic group.

\newpage

{\bf \blue Semisimple LCK manifolds with potential (2)}

\proposition
Let  $j:\; M \arrow H$ be an embedding of an LCK manifold
$M= \tilde M/\langle A\rangle$
to a Hopf manifold $H = V \backslash 0/\langle A\rangle$,
such that $V$ is an $A$-invariant subspace in $\calo_{\tilde M_c}$.
{\bf \red Then the action of $A$ on $V$ is semisimple if and only if
$A$ is semisimple as an element of the proalgebraic group 
$G= \lim\limits_{\leftarrow} \Aut(\calo_{\tilde M_c}/ {\goth m}^k)$.}

\proof If $A$ is semisimple as an element of $G$, its action
on $V$, considered as an $A$-invariant subspace in
 $R\supset \calo_{\tilde M_c}$, is also semisimple.

Conversely, if $A$ is semisimple on $V$, $\calo_{\tilde M_c}$
is a subring in $R$, which is a quotient ring of
$\C[[V]]$; the latter is an adic completion of the polynomial
ring $\C[V]$, where $A$ is  clearly semisimple.
\endproof

\newpage

{\bf \blue Vaisman manifolds are semisimple}

\theorem
Let $M$ be an LCK manifold with potential. Then {\bf \red $M$
is semisimple $\Leftrightarrow$ it admits a Vaisman
structure.}

{\bf \green Proof of Vaisman $\Rightarrow$ semisimple:}\\
Let $M$ be Vaisman, $M= \tilde M/\langle
A\rangle$, and $\tilde M_c$ its completion, equipped
with the K\"ahler potential $\phi:\; \tilde M_c \arrow
\R^{\geq 0}$. Consider a compact subset 
$\tilde M_c^a:= \phi^{-1}([0,a])$.
Consider an $L^2$-structure on  the ring 
$H^0(\calo_{\tilde M^a_c})_b$ of bounded holomorphic functions,
$|f|^2 = \int_{\tilde M^a_c}|f|^2 \tilde\omega^n$.
Let $\vec r:= t \frac{d}{dt}$
be the homothety vector field on $\tilde M = C(S)$.
{\bf \purple Then $I\vec r$ acts on $H^0(\calo_{\tilde M^a_c})_b$
by isometries,} hence its action on each finite-dimensional
subspace of $H^0(\calo_{\tilde M^a_c})_b$ is semisimple.


\newpage

{\bf \blue Semisimple LCK manifolds are Vaisman}


{\bf \green Proof of semisimple $\Rightarrow$ Vaisman. Step 1:}\\
Since all subvarieties of Vaisman manifolds are again Vaisman,
{\bf \purple it would suffice only to show that all semisimple Hopf manifolds
are Vaisman.}

{\bf \green Step 2:}
Let $H=V \backslash 0/\langle A\rangle$ be a semisimple Hopf manifold,
$e_i$ an eigenvalue basis in $V$, and $A(e_i)=\alpha_i u_i e_i$,
with $\alpha_i \in ]0, 1[$ and $u_i \in U(1)$.
Consider a unit sphere $S\subset V = C^n$, and 
let $\rho(t)(e_i):= \alpha_i^t e_i$. Then there exists
a $\rho$-automorphic K\"ahler potential $\phi_\lambda$
on $V\backslash 0$. Since $A\circ \rho(-1)$ preserves $S$
and $A$ commutes with $\rho(t)$,
the function $\phi_\lambda$ is $A$-automorphic.

{\bf \green Step 3:}
By Kamishima-Ornea, {\bf \purple this metric is Vaisman whenever
$S$} (and, therefore, the potential $\phi_\lambda$,
and the corresponding automorphic K\"ahler form) 
{\bf \purple is invariant with respect to $e^{t I\vec r}$,}
where $r= \frac {d\rho(t)}{dt}$. However,
$e^{t I\vec r}(e_i)= e^{\1 t\log \alpha_i} e_i$,
and this operator is unitary.
\endproof



\end{document}

