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\listok{11}{LCK manifolds 11: CR vs. Sasakian}


\newcommand{\Alb}{\operatorname{Alb}}
\newcommand{\Ham}{\operatorname{Ham}}
\definition
Let $M$ be a compact K\"ahler manifold. Define an equivalence 
relation $x\sim y$ on $M$ as follows: two points $x, y$ are
equivalent if for any path $\gamma$ connecting $x$ to $y$
and any harmonic form $\alpha$ representing an integer
cohomology class, the number $\int_\gamma \alpha$ is integer.
The quotient $M/\sim$ is called {\bf the Albanese variety}
of $M$, denoted by $\Alb(M)$.
\ed

\exercise
Let $(M, \omega)$ be a compact K\"ahler manifold, and $v$ a field
of holomorphic symplectomorphisms. Prove that $v$ is Hamiltonian
if and only if $v$ acts trivially on $\Alb(M)$.
\ez

\exercise
Let $S$ be a compact
regular Sasakian manifold, $S/\Reeb=X$ the corresponding
projective manifold. \enum
\item Prove that a holomorphic vector field  on $X$
can be lifted to a Sasakian isometry of $S$ if and only if it
is Hamiltonian.
\item Let $G_0$ be a connected component of the group of
Sasakian isometries of $S$, and $\Ham(X)$ the group of 
holomorphic Hamiltonian diffeomorphisms of $X$.
Consitruct an exact sequence $0 \arrow S^1 \arrow G_0 \arrow \Ham(X)\arrow 0$.
\ee
\ez

\hint Let $L$ be a Hermitian line bundle such that $S$ is its
space of unit vectors, and $\phi(\xi):= |\xi|^2$ its K\"ahler potential.
Prove that 
\[
\omega\cntrct v= 
(dd^c \log \phi)\cntrct v = \Lie_v d^c \log \phi- d\langle d^c\log \phi, v\rangle.
\]
\eh


\exercise
Let $S$ be a 3-dimensional Sasakian manifold not diffeomorphic
to a sphere or its quotient by a finite group. Prove that $S$
is quasiregular.
\ez


\exercise
Let $A$ be a local ring of a singular point on a complex variety.
Find an example of a singular point such that the Lie algebra 
of automorphisms of $A$ has dimension at least
2 and contains two non-proportional contractions.
\ez

\exercise
Find an example of a compact CR-manifold, not isomorphic
to a sphere, and admitting 2 non-proportional Sasakian metrics.
\ez

\hint Use the previous exercise.
\eh



\exercise
Let $A$ be a local ring of a singular point on a complex variety.
Find an example of a singular point such that the Lie algebra 
of automorphisms of $A$ contains a contraction which does not
lie in its center.
\ez


\exercise
Find an example of a compact Sasakian manifold 
admitting a CR-automorphism which is not an isometry.
\ez


\hint Use the previous exercise.
\eh



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