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\newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Observation \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcounter{exercise}[section] \setcounter{exercise}{0} \renewcommand{\theexercise}{\noindent{Exercise \thesection.\arabic{exercise}}} \newcommand{\exercise}{% \setcounter{exercise}{\value{Mycounter}} \refstepcounter{exercise} \stepcounter{Mycounter} {\bf \green EXERCISE:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \green QUESTION:\ }} \newcommand{\problem}{% {\bf \green PROBLEM:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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}