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\listok{4}{LCK manifolds 4: Sasakian manifolds}



\newcommand{\Reeb}{\operatorname{\sf Reeb}}
\exercise
Let $S$ be a regular Sasakian manifold, $X=S/\Reeb$,
$\pi_1(X)=0$, $b_2(X)=1$. Prove that $\pi_1(S)$
is a finite cyclic group.
\ez

\exercise 
Let $G$ be a finite cyclic group acting on a projective
manifold $X$. Find a Sasakian manifold $S$ such that
$S/\Reeb=X/G$.
\ez

\exercise
Let $S$ be a regular Sasakian manifold, $S/\Reeb=T^2$
(compact complex torus of dimension 1). Prove that
$\pi_1(S)$ is non-abelian.
\ez

\exercise Let $S$ be a regular Sasakian manifold,
and $X:=S/\Reeb$. Assume that $\pi_1(X)=\Z^2$, $\pi_2(X)=0$.
Prove that $\pi_1(S)$ is non-abelian.
\ez

\exercise
Let $S$ be a Sasakian manifold, $H$ a group of
Sasakian automorphisms, and $H_\C$ the corresponding
complex Lie group acting on $C(S)$ by holomorphic
automorphisms. Prove that $\dim_\R H=\dim_\C H_\C$.
\ez




\definition
Let $X$ be a projective manifold, $L$ a positive line bundle,
$\lambda>0$, and $\Tot(L^*)$ the space of all non-zero vectors in $L$.
The quotient $\Tot(L^*)/x\sim \lambda x$ is called 
{\bf a regular Vaisman manifold}; it is equipped
with a natural LCK structure (see Lecture 4).
\ed

\exercise
Let $(M,\omega, \theta)$ be a compact LCK manifold.
Since $d_\theta(\omega)=0$, $\omega$ represents a 
cohomology class $[\omega]\in H^2_\theta(M)$, called
{\bf the Morse-Novikov class of $M$}.
Prove that $[\omega]=0$ when $(M,\omega, \theta)$
is a regular Vaisman manifold.
\ez



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