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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Locally conformally K\"ahler manifolds \\[15mm]
\small lecture 1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf HSE and IUM, Moscow
\\[2mm]  February 10, 2014
}
\end{center}

\newpage

{\bf \blue Complex manifolds}


{\bf\green DEFINITION:} Let $M$ be a smooth manifold. 
An {\bf \blue almost complex structure} is an operator
$I:\; TM \arrow TM$ which satisfies $I^2 = - \Id_{TM}$.

{\bf \purple The eigenvalues of this operator are $\pm \1$.}
The corresponding eigenvalue 
decomposition is denoted $TM=T^{0,1}M\oplus T^{1,0}(M)$.


{\bf\green DEFINITION:}
An almost complex structure is {\bf \blue integrable}
if $\forall X,Y \in T^{1,0}M$, one has $[X,Y]\in T^{1,0}M$.
In this case $I$ is called {\bf \blue a complex structure operator}.
A manifold with an integrable almost complex structure
is called {\bf \blue a complex manifold}. 

{\bf\green THEOREM:} (Newlander-Nirenberg)\\
{\bf \red This definition is equivalent to the usual one.}

\remark The commutator defines a $\C^\infty M$-linear map\\
$N:=\Lambda^2(T^{1,0})\arrow T^{0,1}M$, called {\bf \blue 
the Nijenhuis tensor} of $I$. {\bf \purple One can represent $N$ as a section 
of $\Lambda^{2,0}(M) \otimes T^{0,1}M$.}

\newpage


{\bf \blue K\"ahler manifolds}

{\bf\green DEFINITION:} An Riemannian metric $g$ on
an almost complex manifiold $M$ is called 
{\bf \blue Hermitian} if $g(Ix, Iy)= g(x,y)$.
In this case, $g(x, Iy)= g(Ix, I^2y) = - g(y, Ix)$,
hence $\omega(x,y):= g(x, Iy)$ is skew-symmetric.

{\bf\green DEFINITION:} The differential 
form $\omega\in \Lambda^2(M)$ is called
{\bf \blue the Hermitian form} of $(M,I,g)$.

\remark It is $U(1)$-invariant, hence {\bf \purple of Hodge type (1,1)}.

{\bf\green DEFINITION:} A complex Hermitian manifold $(M,I,\omega)$
is called {\bf \blue K\"ahler} if $d\omega=0$. 
The cohomology class $[\omega]\in H^2(M)$ of a form $\omega$ 
is called {\bf \blue the K\"ahler class} of $M$, and
$\omega$ {\bf \blue the K\"ahler form}. 


{\bf \green Definition:} Let $M=\C P^n$ be a complex projective
space, and $g$ a $U(n+1)$-invariant Riemannian form. It is called
{\bf \blue Fubini-Study form on $\C P^n$}. The Fubini-Study
form is obtained by taking arbitrary Riemannian form
and averaging with $U(n+1)$ using the Haar measure on $U(n+1)$.

\exercise
Prove that {\bf \red the Fubini-Study form is unique} (up to a constant
multiplier).

\newpage

{\bf\blue Examples of K\"ahler manifolds.}

{\bf \green Remark:} For any $x\in \C P^n$, the stabilizer
$St(x)$ is isomorphic to $U(n)$. Fubini-Study form on
$T_x\C P^n= \C^n$ is $U(n)$-invariant, hence unique up to a constant.

{\bf \green Claim:} {\bf \red Fubini-Study form is K\"ahler.}
Indeed, $d\omega\restrict x$ is a $U(n)$-invariant 3-form
on $\C^n$, but such a form must vanish, because $-\Id\in U(n)$

\remark 
{\bf \purple The same argument works for all symmetric spaces.}

\definition
{\bf \blue An almost complex submanifold} $X \subset M$
of an almost complex manifold $(M,I)$ is a smooth submanifold
which satisfies $I(TX)\subset TX$.

\exercise
Let $X \subset M$ be an almost complex submanifold of $(M,I)$,
where $I$ is integrable. {\bf \purple Prove that $(X, I\restrict{TX})$
is a complex manifold.}

\definition
In this situation, $X$ is called {\bf \blue a complex submanifold} of $M$.

{\bf \green Corollary:} {\bf \red Every projective manifold 
(complex submanifold of $\C P^n$) is K\"ahler.}
Indeed, a restriction of a closed form is again closed.

\newpage

{\bf \blue Menagerie of complex geometry}

Usually, in algebraic geometry one deals with
projective manifolds. There are two wider classes one has 
to consider necessarily when studying projective ones.

1. {\bf \blue Moishezon manifolds} are those which are birational
to projective. {\bf \purple Transcendence degree of a field $k(M)$ of 
global meromorphic function on a compact complex manifold $M$
satisfies $k(M)\leq M$;} as shown by Moishezon, equality here means 
that $M$ is Moishezon. 

To study birational category, one has necessarily to include 
Moishezon manifolds. {\bf \purple Any K\"ahler Moishezon manifold is
projective (Moishezon).}

2. {\bf\purple Small deformations of K\"ahler manifolds}
often result in non-projective K\"ahler ones (even for a 
torus and a K3). 


\newpage

{\bf \blue Fujiki class C manifolds}


A class which includes Moishezon and K\"ahler is called
{\bf \red Fujiki class C}. A manifold is {\bf \blue Fujiki class C}
if it is bimeromorphic to a K\"ahler manifold. As shown by Fujiki,
Fujiki class C manifolds are closed under all natural operations which occur
in algebraic geometry (such as taking moduli spaces or images).

{\bf \blue A K\"ahler minimal model program} would imply
that any K\"ahler manifold admits a sequence of bimeromorphic
fibrations with fibers which are either projective, hyperk\"ahler
or tori, hence {\bf \purple the class of K\"ahler manifolds
is probably very restricted.} It is known that {\bf \purple a fundamental
group of K\"ahler manifold is very special.}

By contrast, {\bf\purple the class of complex manifolds 
is extremely huge}.

\newpage

{\bf \blue Menagerie of complex geometry II}

\theorem
(Taubes, Panov-Petrunin)\\
For any finitely generated, finitely presented group $\Gamma$,
{\bf \purple
there exists a compact, complex 3-dimensional
manifold $M$ with $\pi_1(M)=\Gamma$.}

\hfill

\conjecture
Let $(M,I)$ be an almost complex manifold, $\dim_\C M\geq 3$.
{\bf \red Then $I$ can be deformed to a complex structure.}

\hfill

\remark 
(Non-)existence of a complex structure {\bf \red is 
highly non-trivial even in the simplest cases,} such as
$S^6$ (which is clearly almost complex).


\remark
We know that non-K\"ahler complex manifolds are much more abundant
than K\"ahler, except in complex dimension 2, where non-K\"ahler
manifolds are a few and much better understood than projective
ones. However, {\bf \purple it's very hard to
come with examples of compact,  non-K\"ahler complex manifolds.}


\newpage


{\bf \blue Examples of non-K\"ahler manifolds}

These listed below (and iterated fibrations of these)
are pretty much all examples known.

\example
{\bf \blue (Linear) Hopf manifold} is $\C^n\backslash 0/\langle A\rangle$,
where $A$ is an invertible linear map with all eigenvalues $|\alpha_i|<1$.
It's diffeomorphic to $S^{2n-1}\times S^1$, hence non-K\"ahler
(K\"ahler manifolds have even $b_{2k-1}$ Betti numbers).
It is locally conformally K\"ahler (LCK). 

\example {\bf \purple All complex subvarieties of a Hopf manifolds
are LCK.} For this reason, {\bf \red they are non-K\"ahler} (Vaisman).

\example {\bf \purple Twistor space} is a certain $\C P^1$-fibration over
a Riemannian 4-manifold. All such manifolds are 
rationally connected (connected by rational curves),
{\bf \red but never K\"ahler} except $\C P^3$ and flag space (Hitchin).
Theorem of Taubes is proved by constructing a twistor
space with prescribed fundamental group.

\example Homogeneous and locally homogeneous manifolds
(such as nilmanifolds) are often non-K\"ahler.

{\bf \red This explains importance of LCK manifolds} 
(defined below).


\newpage


{\bf \blue Connections}

{\bf \green Notation:}
Let $M$ be a smooth manifold, $TM$ its tangent bundle,
$\Lambda^iM$ the bundle of differential $i$-forms,
$C^\infty M$ the smooth functions. {\bf \purple The space of sections 
of a bundle $B$ is denoted by $B$.}

\definition
A {\bf\blue connection} on a vector bundle $B$ is
a map $B \stackrel \nabla \arrow \Lambda^1 M \otimes B$ which
satisfies \[ \nabla(fb) = df \otimes b + f \nabla b\]
for all $b\in B$, $f\in C^\infty M$.

\remark A connection $\nabla$ on $B$ gives
a connection $B^* \stackrel {\nabla^*} \arrow \Lambda^1 M \otimes B^*$
on the dual bundle, by the formula
\[
d(\langle b, \beta\rangle) = \langle \nabla b, \beta\rangle+
\langle b, \nabla^*\beta\rangle
\]
These connections are usually denoted {\bf \red by the same letter $\nabla$.}


\remark
For any tensor bundle 
${\cal B}_1:=
B^*\otimes B^* \otimes ... \otimes B^* \otimes B\otimes B \otimes ... \otimes B$
{\bf \green a connection on $B$ defines a connection on ${\cal B}_1$}
using the Leibniz formula:
\[
\nabla(b_1 \otimes b_2) = \nabla(b_1) \otimes b_2 + b_1 \otimes \nabla(b_2).
\]


\newpage

{\bf \blue Curvature of a connection}

Let $M$ be a   manifold, $B$ a bundle, $\Lambda^i M$ the
differential forms, and
$\nabla:\; B \arrow B \otimes \Lambda^1M$ 
a connection. We extend $\nabla$ to 
$B \otimes \Lambda^iM\stackrel\nabla \arrow B \otimes \Lambda^{i+1}M$ 
in a natural way, using the formula
\[
\nabla(b \otimes \eta) = \nabla(b)\wedge \eta + b \otimes d\eta,
\]
and define {\bf \blue the curvature $\Theta_\nabla$} of $\nabla$ as 
$\nabla\circ \nabla:\; B \arrow B\otimes \Lambda^2M$.

\claim {\bf \red This operator is $C^\infty M$-linear.}

\remark We shall consider $\Theta_\nabla$ as an element of 
$\Lambda^2M \otimes \End B$, that is, an $\End B$-valued 2-form.


\remark
Given vector fields $X, Y\in TM$, the curvature can be written
in terms of a connection as follows
\[
\Theta_\nabla(b)= \nabla_X\nabla_Yb - \nabla_Y\nabla_X B - \nabla_{[X,Y]}b.
\]

\claim Suppose that the structure group of $B$ is reduced
to its subgroup $G$, and let $\nabla$ be a connection
which preserves this reduction. This is the same as to say
that the connection form takes values in $\Lambda^1 \otimes \goth g(B)$.
{\bf \purple Then $\Theta_\nabla$ lies in $\Lambda^2M \otimes \goth g(B)$.}

\newpage

{\bf\blue Local systems}

\definition
A {\bf \blue local system} is a locally constant sheaf 
of vector spaces.

\theorem
A local system with fiber $B$ at $x\in M$
gives a homomorphism $\pi_1(M,x)\arrow \Aut(B)$.
{\bf \red This correspondence gives an equivalence of categories.}

{\bf\green Proof:} 
The etale space of a local system is a covering
of $M$, and the monodromy map from $\pi_1(M,x)$ to permutatons
of $B$ is by construction linear. 

To obtain a converse
correspondence, let $\pi:\; \tilde M \arrow M$ be the
universal cover, and $X:=\tilde M\times B/\pi_1(M)$
be a quotient where $\pi_1(M)$ acts on $\tilde M\times B$
diagonally. Let $y\in M$ and $U\ni y$ be a neighbourhood
for which $\pi^{-1}(U)$ is a union of several copies $U$. Then
$X$ is a product $B\times U$. This gives a local trivialization
of $\phi:\; X \arrow M$. {\bf \purple The sheaf of locally trivial
sections of $\phi$ is locally trivial}, and the corresponding
monodromy map is $\pi_1(M,x)\arrow \Aut(B)$ the one we started from.
\endproof


\newpage

{\bf\blue Flat bundles}

\definition
A bundle $(B, \nabla)$ is called {\bf \blue flat} if its curvature vanishes.


\definition
A section $b$ of $(B,\nabla)$ is called {\bf \blue parallel}
if $\nabla(b)=0$.

\claim
Let $(B, \nabla)$ be a flat bundle on $M$, and ${\cal B}$ be the
sheaf of parallel sections. {\bf \red Then ${\cal B}$ is a 
locally constant sheaf.}

\proof 
Indeed, through each point passes of the total space of $B$
passes a unique parallel section, which always exists locally.
\endproof

\theorem
This correspondence {\bf \red gives an equivalence of categories}
of flat bundles and local systems.

\proof 
Let ${\cal B}$ be a local system,
and $B:={\cal B}\otimes_\R C^\infty M$ the corresponding vector bundle.
Any section of $B$ can be written as $\sum f_i b_i$, where
$b_i$ are sections of ${\cal B}$, and $f_i\in C^\infty M$.
Write $\nabla(\sum f_i b_i):= \sum df_i\otimes b_i$.
Clearly, this connection is flat, and the corresponding
sheaf of parallel sections os ${\cal B}$.
\endproof

\definition
{\small Define {\bf \blue the $B$-valued de Rham differential}
on $d_\nabla \Lambda^i(M)\otimes B \arrow\Lambda^{i+1}(M)\otimes B$
as $d_\nabla(\eta\otimes b):= 
d\eta\otimes b + (-1)^{\tilde\eta}\eta\wedge \nabla b$.
{\bf \purple It is easy to check that $d_\nabla^2=0$.}

 \exercise
Show that {\bf \purple the cohomology of the complex
$(\Lambda^*M\otimes B, d_\nabla)$ are equal to the
cohomology of the local system ${\cal B}:=\ker \nabla$.}
}

\newpage

{\bf\blue LCS manifolds}

\definition
Let $L$ be an oriented real line bundle (one-dimensional bundle) on $M$,
equipped with a flat connection, and $\omega\in \Lambda^2(M)\otimes L$ 
an $L$-valued differential form. We say that $(M,\omega, L)$
is {\bf \blue locally conformally symplectic} (LCS) if $d_\nabla\omega=0$.
In this situation $L$ is called {\bf \blue the weight bundle}
of $(M,\omega)$, or {\bf \blue conformal weight}.

\claim {\bf \purple An oriented real line bundle $L$ can be smoothly 
trivialized.}

\proof Choose a Riemannian metric on $L$. Then the set of
of positive unit vectors is a nowhere degenerate section of $L$.
\endproof

\claim 
Let $(M,\omega, L)$ be an LCS manifold, $l$ a trivialization
of $L$, and $\theta\in \Lambda^1 M$ the corresponding connection
form, $\nabla(l)=l\otimes\theta$. {\bf \purple Then $d\omega_l=-\omega_l\wedge\theta$,
where $\theta$ is a closed 1-form,}  
and $\omega_l\in \Lambda^2(M)$ is $\omega$ considered as a differential
form after the identification $L\cong C^\infty M$ provided by $l$.

\proof After identifying $L$ and a trivial
line bundle, we obtain $0=d_\nabla(\omega)= d(\omega) + \omega \wedge \theta$.
\endproof

\newpage

{\bf\blue LCK manifolds}

\remark
We obtained that {\bf \purple 
the following two definitions are equivalent.}

1. LCS manifold is a manifold equipped with a non-degenerate 
2-form $\omega$ satisfying $d\omega=\omega\wedge\theta$,
where $\theta$ is a closed 1-form.

2. LCS manifold is a manifold equipped with a non-degenerate, 
closed 2-form $\omega$ taking values in a flat, oriented line bundle.

\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}.

\remark
Usually one silently assumes that $\theta$ is not exact.
Indeed, if $\theta=d\phi$, then $d(e^{-\phi}\omega)=
e^{-\phi}d\omega - e^{-\phi}\omega\wedge\theta=0$,
and $e^{-\phi}\omega$ is K\"ahler. In this case
$(M,I, \omega)$ is called {\bf \blue globally conformally
K\"ahler}.

\remark
As shown above, {\bf \purple 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}.

\exercise
Prove that {\bf \red $d\theta=0$ follows from $d\omega=\omega\wedge\theta$
when $\dim_\C M>2$.}

\newpage

{\bf\blue LCK manifolds and their K\"ahler covers}

\claim 
Let $L$ be a local system on $M$, and $\pi:\; \tilde M \arrow M$
is a universal cover. {\bf \purple Then $\pi^* L$ is a trivial local system.}

\proof Indeed, $\pi_1(\tilde M)=0$, hence all local systems on $\tilde M$
are trivial. \endproof

We obtain that a universal cover of an LCK manifold admits
a K\"ahler form taking values in a trivial bundle; {\bf \red this means
that it is K\"ahler.}

\remark Let $(M,I,\omega,\theta)$ be an LCK manifold,
$\pi:\; \tilde M \arrow M$ its universal cover, and $\phi$ a function
satisfying $d\phi=\pi^*\theta$. Then $d(e^{-\phi}\pi^*\omega)=
e^{-\phi}d\pi^*\omega - e^{-\phi}\pi^*\omega\wedge\pi^*\theta=0$,
hence {\bf \red the form $\tilde \omega:=e^{-\phi}\pi^*\omega$
is K\"ahler}.

\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 $\Aut_M(\tilde M)=\pi_1(M)$.}

\remark A deck transform maps $\pi^* \theta$ to itself, hence
it maps $\phi$ to $\phi+C$. This implies that a deck transform maps
$\tilde \omega$ to $e^C \tilde \omega$, {\bf \red 
acting on $\tilde M$ by K\"ahler homotheties.}

\newpage

{\bf\blue K\"ahler homotheties and LCK manifolds}

\definition
Let $(M,\omega)$ be an LCK manifold, $\tilde M$ its
K\"ahler cover, and $\pi_1(M)\cong \Aut_M(\tilde M)$
the deck transform maps.
{\bf \blue Homothety character} is a homomorphism
$\chi:\; \pi_1(M)\arrow \R^{>0}$ mapping a deck transform
$\gamma \in \Aut_M(\tilde M)$ to the number
$\frac{\gamma^*(\tilde \omega)}{\tilde\omega}$.

\remark Let $M$ be 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. Consider a local system $L$ on $M$ associated with
the homothety character $\chi:\; \pi_1(M)\arrow \R^{>0}$,
and let $\psi$ be its trivialization. For each
$\gamma\in \Aut_M(\tilde M)$, 
one has $\frac{\gamma^*\psi}{\psi}=\frac{\gamma^*(\tilde
  \omega)}{\tilde\omega}$. Therefore,
{\bf \purple $\psi^{-1}\tilde\omega$ is an 
$\Aut_M(\tilde M)$-form on $\tilde M$}.
Denote by $\omega$ the corresponding form on $M$.
Then 
\[ d\omega= d(\psi\tilde \omega)= d\psi\wedge
\tilde\omega=d\log\psi\wedge\omega.
\]
{\bf \red We obtained that the form $\omega$ satisfies 
$d\omega=\omega\wedge\theta$, where $\theta=d\log\psi$.}
{\bf \purple This brings one more definition of LCK manifolds.}

\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.

\newpage

{\bf\blue Examples of LCK manifolds}



\example 
{\bf \blue A classical Hopf manifold}
is $H:=\C^n \backslash 0/ \Z$, where $\Z$ acts as
a multiplication by a complex number $\lambda$, $|\lambda|>1$.

\remark {\bf \purple Its covering has a usual K\"ahler form, and
the mapping group acts by homotheties.}

\observation $H$ is diffeomorphic to $S^1 \times S^{2n-1}$,
and fibered over $\C P^{n-1}$ with fiber $\C^*/\langle \lambda \rangle$.

\observation
For any complex submanifold $X\subset \C P^{n-1}$,
its preimage in $H$ is a complex manifold.

\remark
Obviously, any complex submanifold of an LCK manifold
is again LCK. {\bf \purple This implies that $\sigma^* X\subset H$
is an LCK manifold.}

\remark
Next lecture I will prove that {\bf \red none of these manifolds
admits a K\"ahler form, if $n>1$.}





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