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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Locally conformally K\"ahler manifolds \\[15mm]
\small lecture 9: holomorphic contractions and
Riesz-Schauder theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf HSE and IUM, Moscow
\\[2mm]  April 14, 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.}

\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}.

\remark For any complex submanifold $Z\subset M$
of an LCK manifold with potential, $Z$ is also 
an LCK manifold with potential.

\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 $\tilde M_c$. 


\newpage

{\bf \blue Normal families of functions (reminder)}

\definition
Let $C(M)$ be the space of functions on a topological space.
{\bf \blue Topology of uniform convergence on compacts} (also known
as {\bf \blue compact-open topology}; usually denoted as $C^0$) 
is a topology on $C(M)$ where a base of open sets
is given by 
\[ U(X, C):= \{f\in C(M)\ \ |\ \ \sup_K |f|< C\},
\] for all
compacts $K\subset M$ and $C>0$. A sequence $f_i$ of functions converges
to $f$ if it converges to $f$ uniformly on all compacts.


\definition
Let $M$ be a complex manifold, 
and ${\cal F}$ a family of holomorphic functions
$f_i \in H^0(\calo_M)$. We call ${\cal F}$ 
{\bf \blue a normal family} if for each compact
$K\subset M$ there exists $C_K>0$ such that
for each $f\in {\cal F}$, $\sup_K |f| \leq C_K$.

\theorem
Let $M$ be a complex manifold and ${\cal F}\subset H^0(\calo_M)$ 
a normal family of functions. Denote by $\bar{\cal F}$
its closure in $C^0$-topology. {\bf \red Then $\bar{\cal F}$
is compact and contained in $H^0(\calo_M)$.}


\newpage

{\bf \blue Banach space of holomorphic functions}

\definition 
A {\bf\blue Banach space} is a complete normed vector space.


\theorem
Let $M$ be a complex manifold, and $H^0_b(\calo_M)$
the space of all bounded holomorphic functions, equipped
with {\bf \blue the sup-norm} $|f|_\sup:=\sup_M |f|$.
{\bf \red Then it is a Banach space.}

{\bf \green Proof. Step 1:} 
Let $\{f_i\}\in H^0_b(\calo_M)$ be a Cauchy sequence in
$\sup$-norm. {\bf \purple Then $\{f_i\}$ converges to a continuous
function $f$} in $\sup$-topology.

{\bf \green Step 2:}
Since $\{f_i\}$ is a normal family, it
has a subsequence which converges in $C^0$-topology
to $\tilde f\in H^0(\calo_M)$. However, the {\bf \purple $C^0$-topology
is weaker than the $\sup$-topology, hence $\tilde f=f$.}
Therefore, $f$ is holomorphic. \endproof

\newpage

{\bf \blue Compact operators}


\definition
A {\bf \blue bounded subset} of a topological vector
space $V$ is a set $B\subset V$ such that for any
open neighbourhood $U \ni 0$, there exists $\lambda>0$
such that $\lambda B\subset U$

\remark 
Bounded subsets of normed spaces are subsets which
are contained in a ball of a sufficiently big radius.

\definition
A subset of a topological space is called {\bf \blue
precompact} if its closure is compact.

\definition
Let $V, W$ be topological vector spaces, and $\phi:\; V
\arrow W$ a continuous linear operator. It
is called {\bf \blue compact} if an image of
any bounded set is precompact.

\exercise
Let $V=H^0(\calo_M)$ be a space of holomorphic functions on a complex
manifold $M$ with $C^0$-topology. {\bf \purple Prove that any 
bounded subset of $V$ is precompact.} In this case, the
identity map is a compact operator.

\remark By Riesz theorem, {\bf \purple a closed ball in a normed
vector space $V$ is never compact,} unless $V$ is
finite-dimensional. This means that $(H^0(\calo_M), C^0)$
does not admit a norm. 

\newpage

{\bf \blue Holomorphic contractions}

\definition
{\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 $U$.

\theorem
Let $X$ be a complex variety, and 
$\gamma:\; X \arrow X$ a holomorphic contraction
such that $\gamma(X)$ is precompact. 
Consider the Banach space $V=H^0_b(\calo_X)$
with sup-metric. {\bf \red Then $\gamma^*:\; V \arrow V$
is compact,} and {\bf \red its operator norm
$\|\gamma^*\|:= \sup_{|v|\leq 1}|\gamma^*(v)|$
is strictly less than 1. }

{\bf \green Proof. Step 1:} Let $B_C:=\{ v\in V\ \ |
\ \  |v|_\sup \leq C\}$. Then 
\[|\gamma^* f|_\sup= \sup_{x\in \overline{\gamma(X)}}
|f(x)|.
\]
Therefore, {\bf \purple for any sequence $\{f_i\}$ converging in
$C^0$-topology, the sequence $\{\gamma^* f_i\}$ converges
in $\sup$-topology.} However, $B_C$ is precompact in
$C^0$-topology, because it is a normal family.
Then $\gamma^* B_C$ is precompact in
$\sup$-topology.


\newpage

{\bf \blue Holomorphic contractions (part 2)}


\exercise Prove {\bf \blue the maximum principle}:
{\bf \red a non-constant holomorphic function cannot have
any non-strict maxima.}



{\bf \green  Step 2:}
Since $\sup_X |\gamma^* f|= \sup_{\gamma(X)} |f| \leq \sup_X |f|$,
one has $\|\gamma^*\|\leq 1$. If this inequality is not
strict, for some sequence $f_i\in B_1$
one has $\lim_i \sup_{x\in \gamma(X)} |f_i(x)|=1$.
Since $B_1$ is a normal family, $f_i$ has a subsequence
converging in $C^0$-topology to $f$. Then $\gamma(f_i)$
converges to $\gamma(f)$ in $\sup$-topology, giving 
$\lim_i \sup_{x\in \gamma(X)} |f_i(x)|= \sup_{x\in  \gamma(X)} |f(x)|=1$.
{\bf \purple Since a holomorphic functions has no strict maxima, this means
that $|f(x)| >1$ somewhere on $X$.} Then $f$ cannot
be a $C^0$-limit of $f_i\in B_1$. \endproof


%\newpage
%
%{\bf \blue Logarithm in Banach spaces}
%
%\definition
%Let $A$ be a 
%Define its logarithm as a sum
%\[ \log(A) := \sum_{i=1}^\infty \frac{(1-A)^i}{i}.
%\]
%
%
%\theorem
%Let $\tilde M_c$ be a completion of a $\Z$-covering $\tilde M$ of
%an LCK manifold with potential, and $\gamma:\; \tilde M_c\arrow \tilde M_c$
%the holomorphic contraction, associated with the
%generator of the $\Z$-action. Consider the logarithm of $\gamma$
%defined as above. 
%{\bf \red 
%Then $\log\gamma^*$ converges in $C^0$-topology on $H^0(\tilde M_c)$.}
%Moreover, $r:=\log\gamma^*$ satisfies the Leibnitz identity, and
%$e^r=\gamma^*$.
%
%{\bf \green Proof:} 
%Convergency of a logarithm 


\newpage

{\bf \blue Hopf manifolds and finite vectors}



\definition
Let $A\in \End(\C^n)$ be an invertible linear endomorphism
with all eigenvalues $|\alpha_i| <1$. The quotient
$H:= (\C^n \backslash 0)/\langle A\rangle$ is called
{\bf \blue a linear Hopf manifold}.


{\bf \green Theorem 1:}
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. }

\definition
Let $\gamma$ be an endomorphism of a vector space $V$.
A vector $v\in V$ is called {\bf \blue $\gamma$-finite} if the
space $\langle v, \gamma(v), \gamma^2(v), ... \rangle$
is finite-dimensional.


{\bf \green Theorem 2:} Let $M$ be an LCK manifold with
potential, $\dim_\C M>2$, and $\tilde M$ its K\"ahler $\Z$-covering.
Consider a metric completion $\tilde M_c$ with its complex
structure and a contraction $\gamma:\; \tilde M_c \arrow \tilde M_c$ 
generating the
$\Z$-action. Let $H^0(\calo_{\tilde M_c})_f$ be the space of
functions which are $\gamma^*$-finite. {\bf \red Then
$H^0(\calo_{\tilde M_c})_f$ is dense in $\sup$-topology
on each compact subset of $\tilde M_c$.}

We deduce Theorem 1 from Theorem 2, and then prove Theorem
2 using Riesz-Schauder Theorem.


\newpage

{\bf \blue Embedding into Hopf manifolds}

{\bf \green Theorem 2 $\Rightarrow$ Theorem 1. Step 1:}\\
Let $W\subset H^0(\calo_{\tilde M_c})_f$ be an
$m$-dimensional $\gamma^*$-invariant subspace $W$ with basis
$w_1, ..., w_m$. Then the following diagram is
commutative:
\begin{equation*}
\begin{CD}
 \tilde M@>\Psi>> \C^m \\
@V{\gamma}VV  @VV{\gamma^*}V \\
M@>\Psi >>  \C^m
\end{CD},
\end{equation*}
where $\Psi(x)=(w_1(x), w_2(x), ..., w_m(x))$.

Suppose that the map $\Psi$ associated with a given 
$W\subset  H^0(\calo_{\tilde M_c})_f$ is injective. Then {\bf \purple the
quotient map gives an embedding 
$\Psi:\; \tilde M/\Z \arrow (\C^m\backslash 0)/\gamma^*$;}
all eigenvalues of $\gamma^*$ are $<1$ because its operator
norm is $<1$. 

{\bf \green Step 2:} To find an appropriate 
$W\subset  H^0(\calo_{\tilde M_c})_f$, choose a holomorphic embedding
$\Psi_1:\; \tilde M_c\hookrightarrow \C^n$, which exists because
$\tilde M_c$ is Stein. Let $\tilde w_1, ..., \tilde w_n$
be the coordinate functions of $\Psi_1$. 
Theorem 2 allows one to approximate $\tilde w_i$ by 
$w_i \in H^0(\calo_{\tilde M_c})_f$ in $C^0$-topology.
{\bf \purple Choosing $w_i$ sufficiently close to $\tilde w_i$
in a compact fundamental domain of $\Z$-action, we obtain
that $x\arrow (w_1(x), w_2(x), ..., w_n(x))$ is injective
in a compact fundamental domain of $\Z$.} To finish the argument,
take $W\subset H^0(\calo_{\tilde M_c})_f$ generated by 
the $\gamma^*$ from $w_1, ..., w_n$, and apply Step 1.
\endproof



\newpage

{\bf \blue Riesz-Schauder Theorem}

To find enough $\gamma^*$-finite vectors, we use the
Riesz-Schauder Theorem. It is a Banach analogue of 
spectral theorem which easily follows from Fredholm theory.

\theorem {\bf \blue (Riesz-Schauder)} \\
Let $F:\; V \arrow V$ be a compact operator on a Banach space.
Then for each non-zero $\mu \in \C$, there exists a sufficiently
big number $N\in \Z$ such that for each $n>N$ {\bf \red one has 
$V= \ker(F-\mu\Id)^n \oplus \overline{\im (F-\mu\Id)^n}$,
where $\overline{\im (F-\mu\Id)^n}$ is closure of the image.}
Moreover, $\ker(F-\mu\Id)^n$ is finite-dimensional and independent on $n$.

%\exercise Deduce from the Riesz-Schauder
% that {\bf \purple the same statement is true for any $F=P(K)$}
%where $K$ is compact, and $P$ a polynomial, .


\newpage

{\bf \blue Riesz-Schauder Theorem and adic filtration}

In our case the Riesz-Schauder theorem is especially effective.


\def\m{{{\goth m}}}

{\bf \green Proposition 1:}
Fix a precompact subset $\tilde M^a_c:= \phi^{-1}([0, a[)$,
where $\phi:\; \tilde M_c$ is the K\"ahler potential.
Let $A$ be the ring of bounded holomorphic functions
on $\tilde M^a_c$, and ${\goth m}$ the maximal ideal of the
origin point. Clearly, $\gamma^*$ preserves $\m$ and all 
its powers. Let $P_k(t)$ be the minimal polynomial
of $\gamma^*\restrict{A/\m^k}$. 
{\bf \red Then $\im(P_k(\gamma^*)\subset \m^k(A)$,
and $\ker P_k(\gamma^*)$ generates $A/\m^k$.}

{\bf \green Proof:} Since $P_k(t)$ is a minimal polynomial of
$\gamma^*$ on $A/\m^k$, the endomorphism $P_k(\gamma^*)$
acts trivially on $A/\m^k$, hence it maps $A$ to $\m^k$.

From Riesz-Schauder theorem applied to the Banach space
$A$ and $F=P_k(\gamma^*)$ 
it follows that $A=\im(P_k(\gamma^*)+  \overline{\im (F-\mu\Id)^n}$
hence $\ker P_k(\gamma^*)$ generates $A/\m^k$.
\endproof


\newpage

{\bf \blue $\m$-adic topology and $C^0$-topology}

\definition
Let $A$ be a ring and $\m$ an ideal.
The base of open sets in {\bf \blue $\m$-adic topology on $A$}
are $\m^k$ and their translates.

Proposition 1 implies the following result

\proposition
Let $H^0(\calo_{\tilde M_c})_f\subset H^0(\calo_{\tilde M_c})$
be the set of $\gamma^*$-finite functions and $\m$
the maximal ideal of the origin in $\tilde M_c$. {\bf \red Then
$H^0(\calo_{\tilde M_c})_f$ is dense in $\m$-adic topology.}

\proof A subspace $V\subset A$ is dense in $\m$-adic topology
in $A$ $\Leftrightarrow$ the quotient $V/v\cap \m^k$
surjects to $A/\m^k$. This is proven in Proposition 1
for the ring of bounded holomorphic functions
on $\tilde M^a_c$. However, any such function can be
extended to $\gamma^*$-finite function on $\tilde M_c$
using $\gamma^*$-action. \endproof

Now Theorem 2 is implied by the following claim.

\claim
Let $X$ be a connected complex variety,  
$A$ the ring of bounded holomorphic functions, $x\in X$ a point,
$\m\subset A$ its maximal ideal, and $R:\; A \arrow \hat A$
the natural map from $A$ to its $\m$-adic completion.
{\bf \red Then $R$ is continuous in $C^0$-topology and induces
homeomorphism of any bounded set  to its image.}


\newpage

{\bf \blue $\m$-adic topology and $C^0$-topology, part 2}



\claim
Let $X$ be a connected complex variety,  
$A$ the ring of bounded holomorphic functions, $x\in X$ a point,
$\m\subset A$ its maximal ideal, and $R:\; A \arrow \hat A$
the natural map from $A$ to its $\m$-adic completion.
{\bf \red Then $R$ is continuous in $C^0$-topology and induces
homeomorphism of any bounded set to its image.}

{\bf \green Proof:} Continuity is clear because $C^0$-topology
is equivalent to $C^1$-topology, $C^2$-topology and so on (Lecture 8). 
Therefore, taking successive derivatives in a point is continuous in
$C^0$-topology. However, $R$ takes a function and replaces
it by its Taylor series.

To see that $R$ is a homeomorphism, notice that any
bounded, closed subset of $A$ is compact, hence its image
under a continuous map is also closed. \endproof






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