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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
