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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Locally conformally K\"ahler manifolds \\[15mm]
\small lecture 5: Structure theorem for Vaisman manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf HSE and IUM, Moscow
\\[2mm]  March 10, 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.}



{\it \small S. Sasaki, "On differentiable manifolds with certain
structures which are closely related to almost contact
structure", Tohoku Math. J. 2 (1960), 459-476.}


\newpage

{\bf \blue Contact manifolds (reminder)}

{\bf \purple All manifolds are assumed to be oriented here.}

{\bf \green Definition 1:} Let 
$C(S)=(S\times \R^>0)$ be a
cone, equipped with the standard action
$h_\lambda(x,t)=(x, \lambda t)$. Assume that
$C(S)$ is equipped with a symplectic form $\omega$
such that $h_\lambda^*\omega=\lambda^2\omega$.
Then $S$ is called {\bf \blue contact manifold}.

{\bf \green Definition 2:} Let $S$ be an odd-dimensional
manifold, and $B\subset TS$ an oriented sub-bundle of codimension 1,
with Frobenius form $\Lambda^2 B \stackrel \Phi \arrow TS/B$
non-degenerate. Then $S$ is called {\bf \blue contact 
manifold}, $B\subset TS$ {\bf\blue the contact bundle}.


{\bf \green Definition 3:} Let $S$ be manifold of
dimension $2k+1$,
$B\subset TS$ an oriented sub-bundle of codimension 1.
Assume that for any nowhere vanishing 1-form  $\theta\in
\Lambda^1 S$, the form $\theta \wedge (d\theta)^k$
is a non-degenerate volume form. Then $(S,B)$
is called {\bf \blue a contact manifold}, and
$\theta$ {\bf \blue a contact form}.

\theorem
{\bf \red These three definitions are equivalent.}

\newpage

{\bf \blue Reeb field (reminder)}
\newcommand{\Reeb}{\operatorname{\sf Reeb}}

\definition {\bf \blue A Sasakian manifold} is a contact manifold $S$ with
a Riemannian structure, such that the symplectic cone
$C(S)$ with its Riemannian metric is K\"ahler.

\definition
Let $S$ be a Sasakian manifold, $\omega$ the K\"ahler form
on $C(S)$, and $r=t\frac{d}{dt}$ the homothety vector field.
Then $\Lie_{Ir}t= \langle dt, Ir\rangle=0$, hence $iR$
is tangent to $S\subset C(S)$. This vector field 
(denoted by $\Reeb$) is called 
{\bf \blue the Reeb field} of a Sasakian manifold.

\remark {\bf \purple The Reeb field is dual to the contact 
form} $\theta=\omega\cntrct r$.

\theorem {\bf \red The Reeb field acts on a Sasakian manifold
by contact isometries.} 


\definition
A Sasakian manifold is called {\bf \blue regular} if the
Reeb field generates a free action of $S^1$, {\bf \blue
  quasiregular} if all orbits of $\Reeb$ are closed, and
{\bf \blue irregular} otherwise.



\newpage

{\bf\blue Examples of Sasakian manifolds.}

{\bf \green Example:} Let $X\subset \C P^n$ be a complex submanifold,
and $CX \subset \C^{n+1}\backslash 0$ the corresponding cone.
The cone $CX$ is obviously K\"ahler and homogeneous, hence {\bf \red the
intersection $CX\cap S^{2n-1}$ is Sasakian.} This
intersection  is an $S^1$-bundle over $X$. This construction
gives many interesting contact manifolds,
including Milnor's exotic 7-spheres, which happen to be Sasakian.

\remark In other words, 
{\bf \blue a link of a homogeneous singularity is
always Sasakian.}

\remark {\bf \red Every quasiregular Sasakian manifold is obtained
this way,} for some  K\"ahler metric on $\C^{n+1}$ (Ornea-V., 
arXiv:math/0609617 ).

\remark All 3-dimensional Sasakian manifolds are quasiregular
(H. Geiges, 1997, F. Belgun, 2000).

\remark{\bf \purple Every Sasakian manifold is diffeomorphic
to a quasiregular one} (Ornea-V., arXiv:math/0306077).

\remark
{\bf \purple Every regular (quasiregular)  Sasakian manifold
is a total space of an $S^1$-bundle over a K\"ahler manifold (orbifold).}

\newpage

{\bf \blue Vaisman manifolds}

\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.}

{\bf \green Proof:} From Lecture 3,
we know that $\tilde \omega$ is locally
a conical K\"ahler metric.
Let $\theta^\sharp$ be the {\bf \blue Lee field},
dual to $\theta$. Then $\Lie_{\theta^\sharp}\psi=2\psi$, hence
the space of orbits of $e^{t\theta^\sharp}$-action is identified
with $S:=\psi^{-1}(c)$. This gives $\tilde M =C(S)$. \endproof

{\bf \purple 
Now we shall prove a global version of this result.}


\newpage

{\bf\blue Structure theorem for Vaisman manifolds}

\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.

\remark This gives {\bf \purple a Riemannian
submersion $M \arrow S^1$} with Sasakian fibers.

{\bf \green Proof. Step 1:} Since $\theta^\sharp$ is parallel and Killing,
$M=X \times \R$ locally. Fix $x_0\in M$. Then 
the projection $M=X \times \R$ to  $\R$ is induced by 
$x \arrow \int_{\gamma_{x_0,x}}\theta$, for $\gamma_{x_0,x}$
some path connecting $x$ and $x_0$. {\bf \purple Therefore,
$M=X \times \R$ whenever $\theta$ is exact.}

\newcommand{\Mon}{\operatorname{Mon}}
\definition
A {\bf\blue monodromy group} $\Mon(M)$ of an LCK manifold $M$ is the smallest
group $\Gamma$ such that $M=\tilde M /\Gamma$ and $\tilde M$ is K\"ahler.

\remark {\bf\red This is equivalent to the 
pullback of $\theta$ being exact}.

\remark Monodromy group is an image of $\pi_1(M)$ in $\R^{>0}$
under a map associating to any $\gamma\in \pi_1(M)\subset \Aut(\tilde M)$
the number $\frac{\gamma^*\tilde\omega}{\tilde\omega}$.

\newpage


{\bf\blue The proof of Structure theorem for Vaisman manifolds}


{\bf \green Proof. Step 2:}
Let $\gamma_1, ..., \gamma_k\in H_1(M, \Z)$ be generators of homology, and
$\alpha_i \int_{\gamma_i}\theta$ the corresponding periods. One has a map
$M \arrow \R/\langle \alpha_1, ..., \alpha_k\rangle$, with 
a commutative diagram
\begin{equation*}
\begin{CD}
 \tilde M@>>>  M \\
@VVV  @VVV              \\
\R @>>>  \R/\langle \alpha_1, ..., \alpha_k\rangle
\end{CD}
\end{equation*}
with vertical lines  $x \arrow \int_{\gamma_{x_0,x}}\theta$.
{\bf \red The Riemannian submersion to $S^1$ will be obtained
if $\R/\langle \alpha_1, ..., \alpha_k\rangle=S^1$.}

{\bf \green Step 3:} 
Let $G\subset \pi_1(M)$ be the group generated by 
all $\gamma\in \pi_1(M)$ such that $\int_\gamma\theta=0$.
{\bf \green Then $\Gamma= \pi_1(M)/G$ is the monodromy group of $M$.}
{\bf \red Therefore, $\R/\langle \alpha_1, ..., \alpha_k\rangle=S^1$
$\iff$ $\Mon(M)=\Z$.}


\newpage

{\bf\blue Computation of the monodromy group of a Vaisman manifold}

\definition
{\bf \blue Lee field} on a Vaisman manifold is the vector field
$\theta^\sharp$ dual to the Lee form. Since locally a Vaisman manifold is a 
cone over Sasakian (as shown in Lecture 3), {\bf \purple
$\theta^\sharp$ acts on $M$ by holomorphic isometries, and on $\tilde M$
by non-isometric homotheties.}

{\bf \purple
The following theorem finishes the proof of Structure Theorem.}

\theorem
Let $(M,\omega,\theta)$ be a compact LCK manifold, and
$X$ a vector field acting on $M$ by isometries and on 
$\tilde M$ by non-isometric homotheties. {\bf \red Then $\Mon(M)=\Z$.}

This theorem is proven later today.

\remark Let $G$ be a group obtained as a closure of 
one-parametric group $e^{tX}$, $t\in \R$. 
{\bf \red Since $X$ acts by isometries, $G$ is a 
compact torus, $G=(S^1)^{k}$}.


\newpage

{\bf\blue Computation of the monodromy group, part 2}


\claim Let $(M,\omega,\theta)$ be a compact LCK manifold, and
$X$ a vector field acting on $M$ by isometries and on 
$\tilde M$ by non-isometric homotheties. Let
$G=(S^1)^{k}$ be the group obtained as a closure of 
one-parametric group $e^{tX}$, $t\in \R$.
Consider the group $\tilde G$ of pairs 
$\tilde f\in \Aut(\tilde M)$, $f\in G$, making the following diagram
commutative. 
\begin{equation*}
\begin{CD}
 \tilde M@>{\tilde f}>> \tilde M \\
@V{\pi}VV  @VV{\pi}V              \\
M@>{f}>>  M 
\end{CD}
\end{equation*}
{\bf \red Then $\tilde G\cong (S^1)^{k-1} \times \R$.}

\remark {\bf \purple 
From this claim, the isomorphism $\Mon(M)=\Z$ follows immediately.}
Indeed, $\Mon(M)\subset \ker p:\; \tilde G\arrow G$.


\newpage

{\bf\blue Computation of the monodromy group of a Vaisman manifold (part 2)}

{\bf \green Proof  of $\tilde G\cong (S^1)^{k-1} \times \R$. 

Step 1:} 
$\tilde G$ is a covering of $G$, and {\bf \purple the kernel of this projection
is $\tilde G\cap \Mon(M)$. }


{\bf \green Step 2:}
Let $\tilde G_0\subset \tilde G$ be a subgroup
acting on $\tilde M$ by isometries.
Since $\tilde G$ acts on $\tilde M$ by homotheties,
{\bf \red $\tilde G_0$ has codimension 1}. Moreover, $\tilde G_0$ cannot
intersect $\Mon(M)$ and {\bf \purple it maps injectively
to $\Aut(M)\cong (S^1)^{k}$. }

{\bf \green Step 3:} {\bf \purple We obtain that 
$\tilde G_0\cong (S^1)^{k-1}$} (it's codimension 1).


{\bf \green Step 4:}
Since $\tilde G_0$ meets every component of $\tilde G$,
it is connected. {\bf \red Therefore, 
$\tilde G\cong \tilde G_0\times \R\cong (S^1)^{k-1} \times \R$.}
\endproof

\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 $dd^c f=\omega$.

{\bf \green Theorem 1:}
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 $C(S)$. Moreover,
the form $dd^c \log t$
vanishes on $\langle r, I(r)\rangle$ and the rest of its
eigenvalues are positive.}

{\bf \green Proof. Step 1:}   
$2\omega=\Lie_{r}\omega=d(I\theta)=d(tIdt)=dd^c(t^2)$
Therefore, $t^2$ is a K\"ahler potential.

{\bf \green Step 2:}   
$dd^c \log t^2= \frac{\tilde \omega}{t^2}-\frac {dt\wedge  Idt}{t^2}$.
\endproof


\newpage

{\bf\blue The fundamental foliation}

\definition
Let $M$ be a Vaisman manifold, $\theta^\sharp$ its Lee field,
and $\Sigma$ a 2-dimensional real foliation generated by
$\theta^\sharp, I\theta^\sharp$. It is called {\bf \blue
  the fundamental foliation} of $M$. Clearly, $\Sigma$
is tangent to orbits of the one-parametric group of
automorphisms of the covering $\tilde M$ generated 
by homotheties. Therefore, {\bf \purple $\Sigma$ 
is a holomorphic foliaton.}

\theorem
Let $M$ be a compact Vaisman manifold, and 
$\Sigma\subset TM$ its fundamental foliation. Then\\
{\bf \red \hphantom{a}\ \ 1. $\Sigma$ is
independent from the choice of the Vaisman metric.\\
\hphantom{a}\ \ 2. There exists a positive, exact
(1,1)-form $\omega_0$ with $\Sigma=\ker \omega_0$.\\
\hphantom{a}\ \ 3. For any complex subvariety
$Z\subset M$, $Z$ is tangent to $\Sigma$. \\
\hphantom{a}\ \ 4. For any compact complex subvariety
$Z\subset M$, the set of smooth points of $Z$ is Vaisman.}


{\bf \green Proof of (2):} Let $\tilde M = C(X)$ be the
conical K\"ahler manifold which covers $M$, and $\psi:\;
\tilde M \arrow \R$ the function satisfying $d\psi=\theta$.
{\bf \purple Then $\omega_0:=dd^c\psi$ is a pseudo-Hermitian form which
vanishes on $\Sigma$ and positive on $TM/\Sigma$
(Theorem 1).} Also, $\omega_0=d(I\theta)$, hence this
form is well defined on $M$.

\newpage

{\bf\blue The fundamental foliation (proofs)}


{\bf \red \hphantom{a}\ \ \ \:1. $\Sigma$ is
independent from the choice of the Vaisman metric.\\
\hphantom{a}\ \ 2. There exists a positive, exact
(1,1)-form $\omega_0$ with $\Sigma=\ker \omega_0$.\\
\hphantom{a}\ \ 3. For any complex subvariety
$Z\subset M$, $Z$ is tangent to $\Sigma$. \\
\hphantom{a}\ \ 4. For any compact complex subvariety
$Z\subset M$, the set of smooth points of $Z$ is Vaisman.}

{\bf \green Proof of (1):} The zero foliation of
$\omega_0$ is independent from the choice of the Vaisman
metric. Indeed, if tere are two Vaisman structures
with $\omega_0$ and $\omega_0'$ vanishing on differenc
1-dimensional complex foliations, the sum 
$\omega_0+\omega'_0$ would be positive definite.
However, $\int_M \omega_0^{\dim_\C M}$
vanishes, because $\omega_0$ is exact.
{\bf \purple Since $\Sigma=\ker\omega_0$, $\Sigma$ 
is independent from the Vaisman structure.}


{\bf \green Proof of (3):} For any compact subvariety
$X\subset M$, the integral $\int_Z\omega_0^{\dim_\C M}$
vanishes, because $\omega_0$ is exact. {\bf \purple Therefore,
$\omega_0\restrict {TZ}$ has one zero eigenvalue at each
point of $Z$.} This means precisely that $\Sigma\subset TZ$
at this point.


{\bf \green Proof of (4):} Since the Lee field is tangent
ot $Z$, the covering $\tilde Z\subset C(S)$
is preserved by the homotheties. Therefore it is also a
conical K\"ahler manifold. {\bf \purple Then $Z=\tilde Z/\Z$
is Vaisman.} \endproof


\newpage

{\bf\blue Regular Vaisman manifolds}

\definition
A Vaisman manifold $M$ is called {\bf\blue regular}, if
the leaves of the fundamental foliation are orbits of
the group $(S^1)^2$ freely acting on $M$, {\bf \blue
  quasiregular} if these leaves are compact, and
{\bf \blue irregular} otherwise.

\theorem
{\bf \red A Vaisman manifold is regular if and only if
it is a smooth elliptic fibration over a projective manifold,}
obtained as a quotient of a total space of non-zero vectors in a
positive bundle by the action of $\Z$ mapping $v$ to
$\lambda v$, with $|\lambda|>1$.

{\bf \blue Proof. Step 1:} Let 
$\tilde M:=C(S)$ be the corresponding conical K\"ahler manifold.
Clearly, the leaves of $\Sigma$ are obtained from
orbits of the Reeb field on $S$ by the Lee field acting on 
$C(S)$ as a standard homothety. {\bf \purple Therefore, $S$ is regular.}\\
{\bf \blue Step 2:}
By structure theorem, $M= C(S)/\Z$ acting as
$\langle (x, t) \mapsto (\phi(x), q t)\rangle$, 
where $q>1$, and $\phi$ is a Sasakian automorphism of $X$.
The leaves of $\Sigma$ intersect with $S$ by a union
several copies of $S^1$ numbered by $\langle \phi\rangle$.
{\bf \purple Regularity of $M$ implies that $\phi$ has finite order,
and the corresponding group acts freely on $S$.}\\
{\bf \blue Step 3:} Now, $S_1:=S/\langle \phi\rangle$ is a 
regular Sasakian manifold, hence it is a space of unit 
vectors in a positive line bundle $L$ over $X:=S_1/\Reeb$.\\
{\bf \blue Step 4:} By construction, $X$ is the space of
leaves of $\Sigma$, hence $\tilde M$ is a $\C^*$-fibration
over $X$. 
\endproof

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