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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Locally conformally K\"ahler manifolds \\[15mm]
\small lecture 12: Morse-Novikov cohomology}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf HSE and IUM, Moscow
\\[2mm]  May 12, 2014
}
\end{center}

\newpage

{\bf\blue Local systems (reminder)}

\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.}

\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.}

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

\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 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 Semisimple LCK manifolds with potential (reminder)}

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.

\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 
$G= \lim\limits_{\leftarrow} \Aut(\calo_{\tilde M_c}/ {\goth m}^k)$.}

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

\newpage

{\bf \blue Morse-Novikov cohomology}

\definition
 Define {\bf \blue the $B$-valued de Rham differential}
$d_\nabla:\;  \Lambda^i(M)\otimes B \arrow\Lambda^{i+1}(M)\otimes B$
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 $d_\nabla^2=0$
if and only if the curvature of $\nabla$ vanishes.}

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


{\bf \green Proof:} The complex of sheaves
\[
\Lambda^0(M)\otimes B \stackrel {d_\nabla} \arrow \Lambda^1(M)\otimes B
\stackrel {d_\nabla}  \arrow\Lambda^2(M)\otimes B
\]
is a fine resolvent for the sheaf ${\Bbb B}$ of parallel section of
$(B, \nabla)$, hence its cohomology is $H^i(M, {\Bbb B})$.
\endproof

\remark
Let $B$ be a line bundle equipped with a flat connection,
$\phi$ its trivialization, and $\theta$ its connection form,
$\nabla(f\phi) = df\otimes \phi + f \theta \otimes \psi$.
Then $d_\nabla(\eta\otimes \psi)= d\eta\otimes \psi + 
\theta \wedge \eta\otimes \psi$. This is written as $d_\nabla=d+\theta$.

\definition
Cohomology of the complex $(\Lambda^*M, d-\theta)$
are called {\bf \blue Morse-Novikov cohomology}, or
{\bf \blue Lichnerowicz cohomology}; the corresponding
complex -- {\bf \blue Morse-Novikov complex}.
They compute the cohomology of the local system $L^{-1}$


\newpage

{\bf \blue Automorphic forms}

\definition
Let $M$ be a manifold, $\tilde M$ its Galois covering.
A form $\eta$ on $\tilde M$ is called {\bf \blue
  automorphic} if for any $\gamma\in \pi_1(M)$
acting on $\tilde M$ as usual, the form $\gamma^*\eta$
is proportional to $\eta$. The character 
$\chi_\eta(\gamma):=\frac{\gamma^*\tilde \eta}{\tilde \eta}$
 is called {\bf \blue the character of automorphy}, or {\bf
\blue weight} for $\eta$.

\definition
Let $L$ be an oriented real line bundle equipped with a flat connection
(we call $L$ {\bf \blue weight bundle}), and 
$\chi:\; \pi_1(M)\arrow \R^{>0}$ its monodromy.
{\bf \blue Automorphic form of weight $\lambda$}
is an automorphic form which satisfies
$\gamma^*\tilde \eta=\lambda^{-1}\chi(\gamma)\eta$
for each $\gamma\in \pi_1(M)$.
We denote the space of such forms by $\Lambda^*(M)_\lambda$.

\definition
Let $(L,\nabla)$ be a flat, oriented line bundle,
and $\phi$ a nowhere degenerate section, trivializing $L$.
Then $\nabla(f\phi)= f\theta\otimes \phi + df \otimes \phi$.
The form $\theta$ is called {\bf \blue connection 1-form
of $\nabla$}. For any real number $\lambda$, we define
{\bf\blue the tensor power}
$L^\lambda$ as a flat bundle with a connection form
$\lambda \theta$. Since $\nabla(\phi^k)=k \theta\otimes \phi^k$,
this definition is compatible with the usual one.

\remark
Let $(L, \nabla)$ be a weight bundle, 
$\phi$ its trivialization, and $\theta$ the corresponding
connection form. {\bf \purple Then $\theta$ is closed.}
Indeed, flatness of $\nabla$ means that 
$0=d_\nabla^2(v)= d\theta \otimes v$
for any section $v\in L$.


\newpage

{\bf \blue Automorphic forms (2)}


\proposition
Let $(L,\nabla)$ be a weight bundle on a manifold,
$\phi$ its trivialization, and $\theta$ a connection form.
Denote by $\tilde M\stackrel \pi \arrow M$ the universal covering,
and let $\Phi$ be a non-zero parallel section of $\pi^* L$.
Then \\
\phantom{UUU} 
a. {\bf\red $\frac {\pi^*\phi}\Phi$ is an automorphic function of
weight $\lambda$. }\\
\phantom{UUU} b. Moreover, for each $L^\lambda$-valued
differential form $\eta$, the differential form
$\frac {\pi^*\eta}{\pi^*\phi}\in \Lambda^*(\tilde M)$ is automorphic of weight 
$-\lambda$, giving {\bf\red equivalence $\Xi$ between the space of 
sections of $\Lambda^*(M)\otimes L$ and
$\Lambda^*(M)_\lambda$.} \\
\phantom{UUU} c. Under this equivalence, {\bf\red de Rham differential on
$\Lambda^*(M)_\lambda$ corresponds to $d_\nabla$.}

{\bf \green Proof:} (a) is clear, because monodromy  
acts on $\pi^*\phi$ trivially and on $\Phi$ with weight $\lambda$.
(b) is clear by the same reason: any section of 
$\Lambda^*(M)\otimes L$ produces an automorphic form,
and any automorphic form $\rho$ gives a section
$\rho \Phi^{-1}$ of $\pi^* L^\lambda$ which is fixed
by mondromy, hence obtained as a pullback.

To prove (c), take any $\rho \in \Lambda^*(M)\otimes L$,
then 
\[ d(\Xi(\rho))= d(\pi^*\rho \phi^{-1})=
\pi^* d_\nabla \rho \phi^{-1}-  \rho\wedge\nabla \phi\cdot \phi^{-1}=
\Xi(d_\nabla \rho)- \Xi(\rho \wedge \theta)
\]
where $\theta$ is a connection form in $L$.
\endproof

\remark
We obtain that {\bf \red Morse-Novikov complex
is identified with the de Rham 
complex of automorphic forms on $\tilde M$.}



\newpage

{\bf \blue Morse-Novikov Dolbeault complex}

\definition
Let $M$ be a complex manifold,
and $(L,\nabla)$ a flat, oriented, real line bundle.
Identifying sections of $L$ with automorphic
forms of weight 1 on $\tilde M$ as above, we consider {\bf \blue the
Hodge decomposition $d_\theta=\6_\theta+\bar\6_\theta$,}
where $d_\theta$ is the de Rham differential on 
automorphic forms, and $\6_\theta$, $\bar\6_\theta$ its Hodge components.

\proposition
Let $(L,\nabla)$ be a weight bundle on a complex manifold,
$\phi$ its trivialization, and $\theta$ a connection form.
Denote by $\tilde M\stackrel \pi \arrow M$ the universal covering,
and let $\Phi$ be a non-zero parallel section of $\pi^* L$.
Consider the equivalence 
\[ (\Lambda^*(M)\otimes L, d_\theta)\stackrel \Xi \arrow
(\Lambda^*(M)_\lambda, d)
\]
constructed above. {\bf \red Then $\6_\theta = \6 - \theta^{1,0}$
and $\bar\6_\theta = \bar\6 - \theta^{0,1}$.}

{\bf \green Proof:} The map $\Xi$ is compatible
with the Hodge decomposition.
\endproof

\corollary
The following commutation relations are clear:
$\{\6_\theta, \bar\6_\theta\}= \{\6_\theta, \6_\theta\}
=\{\bar\6_\theta, \bar\6_\theta\}=0= \{d_\theta, d_\theta^c\}$, and 
$-2\1\6_\theta\bar\6_\theta= d_\theta d_\theta^c$, where
$d_\theta^c= I d_\theta I^{-1}=
-\1(\6_\theta-\bar\6_\theta)=d^c - I(\theta)$.
\endproof

\newpage

{\bf \blue Lee class of an LCK manifold}

\definition
Let $(M,\omega, \theta)$ be an LCK manifold.
The cohomology class $[\theta] \in H^1(M,\R)$ 
of its Lee form $\theta$ is called {\bf \blue the Lee class} of $M$.

\example
Let $M= C(S)/\Z$ be a Vaisman manifold, $[\theta]$ its Lee
class. {\bf \red Then $\lambda [\theta]$ is also a Lee class of an
LCK structure, for any $\lambda >0$.}

{\bf \green Proof:} Let $\phi=t^2$ be an automorphic K\"ahler
potential on $C(S)= S \times \R^{>0}$. Then $dd^c \log \phi$ is
semipositive, and its zero eigenspace is generated by
$\frac d {dt}$. Then 
\[ dd^c \phi^\alpha= dd^c e^{\alpha\log\phi}
= \alpha \phi^\alpha dd^c\log\phi + 
\alpha^2 \phi^{\alpha} d\log\phi \wedge d^c\log\phi
\]
and this (1,1)-form is strictly positive for any
$\alpha>0$. Its Lee form is $\alpha dt$.
\endproof

\example 
Consider an LCK manifold $(M,\omega, \theta)$ with potential 
$\phi \in C^\infty M$. 
{\bf \red Then $a [\theta]$ is also a Lee class of an
LCK structure, for any $a >1$.}

{\bf \blue Proof:}
$
dd^c \phi^{a} = \phi^{a-2} (a \cdot \phi dd^c \phi + a(a-1)
d\phi\wedge d^c\phi),
$
hence $\phi^a$ is also an automorphic potential,
for any $a>1$.  Its Lee form is $\phi^{-a}d\phi^a=a\theta$.
\endproof

\conjecture
Let $(M,\omega, \theta)$ be a compact LCK manifold
such that $\lambda [\theta]$ is also a Lee class of an
LCK structure, for any $\lambda >0$ (or $\lambda >1$).
Then it admits a structure of 
a Vaisman manifold (or LCK manifold with potential).

\newpage

{\bf \blue Monodromy group and the Lee class}


\remark {\bf \blue Monodromy group} of an LCK manifold
$(M,\omega, \theta)$ 
is defined as the Galois group of the smallest covering
$\pi:\; \tilde M \arrow M$ such that $\pi^*\theta$ 
is exact. {\bf\blue Rank} of an LCK manifold is rank of
its monodromy group.

\proposition
Let $(M,\omega, \theta)$ be an LCK manifold
and $[\theta]$ its Lee class. Consider a
smallest rational subspace $V\subset H^1(M,\Q)$
such that $V\otimes_\Q \R$ contains $[\theta]$.
{\bf \red Then $\dim V$ is equal to the rank of $M$.}

\proof
The group $\Gamma$ is identified with an image of $\pi_1(M)$
under the map $[\theta]:\; \pi_1(M)\arrow \R$,
because it is equal to the monodromy of the weight bundle,
and the monodromy along a loop $\gamma$ is equal to
$\int_\gamma \theta$.
\endproof




\newpage

{\bf \blue Morse-Novikov class of an LCK manifold}

\definition
Let $(M,\omega, \theta)$ be an LCK manifold,
$d\omega=\omega\wedge\theta$. Then $d_\theta(\omega)=0$.
The cohomology class $[\omega]_{MN}$ of $\omega$ in the Morse-Novikov
cohomology is called {\bf \blue Morse-Novikov class} of $M$.

\claim 
{\bf \red $[\omega]_{MN}$ vanishes for LCK manifold with
potential and, hence, for Vaisman manifolds.}

{\bf \green Proof:}
Indeed, the corresponding automorphic
form $\tilde \omega=\Xi(\omega)$ is a differential of an automorphic form,
and the Morse-Novikov cohomology is cohomology of the
complex of automorphic forms.
\endproof

\remark  {\bf \purple $[\omega]_{MN}$ 
is known to be non-zero for some other LCK manifolds.}
All known examples of compact LCK manifolds
with vanishing Morse-Novikov class admit an LCK
metric with potential.



\newpage

{\bf \blue Bott-Chern cohomology}

\definition
Let $M$ be a complex manifold, and $H^{p,q}_{BC}(M)$
the space of closed $(p,q)$-forms modulo $dd^c(\Lambda^{p-1,q-1}(M))$.
Then $H^{p,q}_{BC}(M)$ is called {\bf\blue the Bott-Chern
  cohomology} of $M$. 

\remark
There are natural (and functorial) maps from the
Bott-Chern cohomology to the Dolbeault cohomology
$H^*(\Lambda^{*,*}(M), \bar\6)$ and to the de Rham
cohomology, but no morphisms between de Rham and
Dolbeault cohomology.

\theorem
Let $M$ be a compact complex manifold. {\bf \red Then
$H^{p,q}_{BC}(M)$ is finite-dimensional.}

\proof See below.


\newpage

{\bf \blue Differential operators}

\definition 
{\bf \blue (Grothendieck)}\\
Let $R$ be a commutative ring over a field $k$,
and $A, B$ $R$-modules.
{\bf \blue Differential operator of order 0}
from $A$ to $B$ is an $R$-linear map $\phi\in \Hom_R(A,B)$.
Differential operator of order $i>0$ is defined inductively:
$\alpha \in \Diff^i(A,B)$ if for any $r\in \R$,
the commutator $\alpha L_r-L_r\alpha$ belongs to $\Diff^{i-1}(A,B)$,
where $L_r(x)=rx$.

\definition
Given a vector bundle on a smooth manifold $M$, we
may consider its space of sections as an $C^\infty M$-module.
{\bf \blue Differential operators} $\Diff^i(F,G)$
on vector bundles $F$, $G$ are defined as differential
operators on the corresponding spaces of sections
in the sense of the Grothendieck's definition.
{\bf \blue Differential operator on $M$}
is an element of 
$\Diff^i(M):= \Diff^i(C^\infty M,C^\infty M)$.

\remark 
{\bf \purple This definition is equivalent to the usual one:}
locally (in coordinates) any differential operator
is expressed as a composition of derivations and
multiplications by $f\in C^\infty M$.

\newpage

{\bf \blue Symbols}

\theorem
Consider the filtration $\Diff^0(M)\subset
\Diff^1(M)\subset \Diff^2(M)\subset ...$. Then
{\bf \red its associated graded ring is isomorphic to 
$\bigoplus_i \Sym^i(TM)$,} identified with the
ring if fiberwise polynomial functions on $T^*M$.

\corollary
Let $F, G$ be vector bundles, 
and $\Diff^0(F,G) \subset \Diff^1(F,G) \subset \Diff^2(F,G)$
the corresponding spaces of differential operators.
{\bf \red 
Then \[
\Diff^i(F,G)/\Diff^{i-1}(F,G)=\Sym^i(TM)\otimes\Hom(F,G),
\]}
where $\Sym^i$ denotes the symmetric power (symmetric part
of the tensor power).

\definition
Let $F, G$ be vector bundles, 
and $D\in \Diff^i(F,G)$ a differential operator.
Consider its class in $\Diff^i(F,G)/\Diff^{i-1}(F,G)$ 
as a $\Hom(F,G)$-valued function on $T^*(M)$ (polynomial
of order $i$ on each cotangent space).
This function is called {\bf \blue symbol}
of $D$.

\exercise
Let $D:\; B \arrow B \otimes \Lambda^1 M$ be a first
order differential operator. {\bf \purple Prove that $D$ is
a connection if and only if its symbol
is equal to the identity operator} $\Id\in \Hom(\Lambda^1 M
\otimes (\Hom(B, B \otimes \Lambda^1 M))$

\exercise
Prove that the symbol of the Laplacian
operator $\Delta:\; \Lambda^*M \arrow \Lambda^* M$
on a Riemannian manifold $M$ at $\xi\in T^*M$ {\bf \purple is equal
to $|\xi|^2 \Id_{\Lambda^* M}$.}

\newpage

{\bf \blue Elliptic operators}

\definition
Let $F$, $G$ be vector bundles of the same rank.
A differential
operator $D:\; F \arrow G$ is called {\bf \blue elliptic}
if its symbol $\sigma(D)\in \Hom(F,G)\otimes \Sym^i(TM)$
is invertible at each non-zero $\xi \in T^*M$.

\definition
Let $F$ be a vector bundle on a compact manifold.
The {\bf \blue $L^2_p$-topology} on the space of sections
of $F$ is a topology defined by a quadratic form
$|f|^2=\sum_{i=0}^p \int_M |\nabla^if|^2$,
for some connection and scalar product on $F$ and $\Lambda^1M$.

\exercise
{\bf \purple Prove that this topology is independent from the
choice of a connection and a metric.}

\definition
A continuous operator $\psi:\; A \arrow B$ on topological
vector spaces is called {\bf \blue Fredholm}
if its kernel is finite-dimensional, and
its image is closed, and has finite codimension.

\theorem
Let $D:\; F \arrow G$ be an elliptic operator of order
$d$. Clearly, $D$ defines a continuous map
$L^2_{p}(F) \arrow L^2_{p-d}(G)$. {\bf \red Then this map is
Fredholm.}

\remark
{\bf \purple This difficult theorem is a foundation of Hodge theory}
(and many other things besides).


\newpage

{\bf \blue Elliptic complexes}

\definition
Let $F$, $G$, $H$ be vector bundles,
and $F\stackrel D \arrow G\stackrel D \arrow H$ a complex
of differential operators (that is, $D^2=0$).
It is called {\bf\blue  elliptic complex} if its symbols 
$F\stackrel {\sigma(D)} \arrow G\stackrel  {\sigma(D)} \arrow H$
give an exact sequence at each non-zero $\xi \in T^*M$.

\definition
Let $A$, $B$, $C$ be topological vector spaces
and $A\stackrel D \arrow B\stackrel D \arrow C$ a complex
of continuous maps. It is called {\bf \blue
Fredholm complex} if $\im D$ is closed, 
and $\frac{\ker D}{\im D}$ is finite-dimensional.

\theorem
Let $F\stackrel {D_1} \arrow G\stackrel {D_2} \arrow H$ be an 
elliptic complex of differential operators, with
$D_1$ of order $d_1$ and $D_2$ of order $d_2$.
{\bf \red Then the complex
$L^2_{p}(F) \stackrel{D_1}\arrow 
L^2_{p-d_1}(G)\stackrel{D_2}\arrow L^2_{p-d_1-d_2}(H)$
is Fredholm.}

\corollary
{\bf \purple 
Cohomology of any elliptic complex are finite-dimensional.}

\newpage

{\bf \blue Bott-Chern cohomology are finite-dimensional}

Now we can prove 

\theorem
Let $M$ be a compact complex manifold. {\bf \red Then
$H^{p,q}_{BC}(M)$ is finite-dimensional.}

\proof
It would suffice to show that the complex
\[
\Lambda^{p-1,q-1}(M)\stackrel{dd^c} \arrow
\Lambda^{p,q}(M) \stackrel{\6+\bar\6} \arrow \Lambda^{p+1,q}(M)
\oplus \Lambda^{p,q+1}(M)
\]
is elliptic. {\bf \purple At $\xi\in T^*M=\Lambda^{1,0}(M)$, symbol of $dd^c$ is 
equal to multiplication of a form by $\xi\wedge \bar\xi$,}
{\bf \purple the symbol of $\6$ is multiplication by $\xi$ and
the symbol of $\bar \6$ is multiplication by $\bar\xi$.}
Therefore, $\ker \sigma(\6) = \im \sigma(\6)$,
$\ker \sigma(\bar\6) = \im \sigma(\bar\6)$ (this
proves finite-dimensionality of Dolbeault cohomology),
and $\ker \sigma(\bar\6)\cap \ker \sigma(\6)=\im\sigma(\6\bar\6)$.
\endproof

\newpage

{\bf \blue Weighted Bott-Chern cohomology}

\definition
Let $M$ be a complex manifold, and $L$ a flat vector
bundle. Consider the corresponding differential
$d_\nabla=d_\theta$, and let $\6_\theta$, $\bar\6_\theta$
be its Hodge components.
{\bf \blue The weighted Bott-Chern cohomology}
are defined as 
\[ H^{p,q}_{BC}(M,L):= \frac{\ker
   d_\theta\restrict{\Lambda^{p,q}(M)\otimes L}}{\im
  \6_\theta\bar\6_\theta}.
\]

\theorem
Let $M$ be a compact complex manifold, and $L$ a flat vector
bundle. {\bf \red 
Then the group $H^{p,q}_{BC}(M,L)$ is finite-dimensional.}

\proof 
The complex 
\[
\Lambda^{p-1,q-1}(M)\otimes L\stackrel{d_\theta d^c_\theta} \arrow
\Lambda^{p,q}(M)\otimes L
\stackrel{\6_\theta+\bar\6_\theta} \arrow
\Lambda^{p+1,q}(M)
\oplus \Lambda^{p,q+1}(M)\otimes L
\]
is equal to the usual Bott-Chern complex up to terms of
lower order, hence it has the same symbols.
\endproof


\newpage

{\bf \blue Bott-Chern class}


\definition
$(M, \omega,\theta)$ be an LCK manifold, and $L$ its
weight bundle. The cohomology class of $\omega$ in 
$H^{1,1}_{BC}(M,L)$ is called {\bf \blue Bott-Chern
 class of $M$}.

\remark It is the best analogue of the K\"ahler class,
and the following theorem (together with the Hopf
embedding resukt) an LCK analogue of Kodaira embedding
theorem.

\theorem
Let $(M, \omega,\theta)$ be an LCK manifold.
Suppose that its Lee class $[\theta]$ is proportional to a
rational class in $H^1(M)$ and $[\omega]_{BC}=0$.
{\bf \red Then $(M, \omega,\theta)$ is an LCK manifold
  with potential.}

{\bf \green Proof:} Existence of an automorphic potential
is precisely vanishing of $[\omega]_{BC}=0$. Its
properness is equivalent to $\Gamma\cong \Z$, where
$\Gamma$ is a monodromy group of $M$. Since
rank of $\Gamma$ is equal to the dimension of a
smallest rational subspace generated by $[\theta]$,
it is equal 1. \endproof


\newpage

{\bf \blue Open questions}


A weighted version of $dd^c$-lemma is known to be
wrong, even for Vaisman manifolds (Goto).
However, the following (very weak) version 
of $d_\theta d^c_\theta$-lemma could be true.

\problem
Let $M$ be a compact LCK manifold
with its Morse-Novikov class $[\omega]_{MN}$ equal zero.
{\bf \red Would it follow that $M$ has monodromy $\Z$? Would
it follow that $M$ admits an LCK metric with
potential, when its monodromy is $\Z$?}

\problem
Find an example of locally (but not globally)
conformally symplectic manifold of dimension 
$\geq 3$ not admitting LCK structure.

\problem
Prove that a compact torus with non-K\"ahler
complex structure does not admit an LCK metric,
or find one.

 


\end{document}

