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\listok{13}{LCK manifolds 13: Conformal symplectomorphisms}

\newcommand{\SConf}{\operatorname{\sf SConf}}
\newcommand{\Mon}{\operatorname{Mon}}
\newcommand{\sconf}{\operatorname{\goth {sconf}}}
\definition
A {\bf locally conformally symplectic} (LCS) manifold
is a manifold $M$, $\dim_\R M >2$ equipped with a non-degenerate 2-form
$\omega$ such that $d\omega= \omega\wedge\theta$, and
$\theta$ is closed. {\bf Conformal symplectomorphism}
of an LCS manifold is a diffeomorphism which maps
$\omega$ to $e^f \omega$. The group of conformal symplectomorphisms
is denoted $\SConf(M)$. We consider $\omega$ as a
symplectic form with coefficients in a flat line
bundle $L$, called {\bf the weight bundle} of $M$.
The Lie algebra $\sconf(M)\subset TM$ of all vector
fields $V$ such that $e^{tv}$ lies in $\SConf(M)$
is called {\bf the Lie algebra of conformally symplectic
vector fields}. {\bf Monodromy group} $\Mon(M)$ of an LCS manifold
is monodromy group of the flat connection on $L$.
\ed

\exercise
Let $(M,\omega)$ be a LCS manifold, and 
$\nu\in \SConf(M)$. Prove that $\nu$ acts as a symplectic
homothety on the symplectic cover $(\tilde M, \tilde\omega)$,
$\nu^*\tilde \omega = \chi(\nu)\tilde\omega$. 
Prove that $\chi:\; \SConf(M)\arrow \R^*$ is a 
group homomorphism.
\ez

\exercise
Prove that the weight bundle $L$ of an LCS manifold
is $\SConf(M)$-equivariant. Construct an $\SConf(M)$-equivariant
symplectic structure on the space of non-zero vectors in
$L\otimes \C$.
\ez

\exercise
Construct a character $\chi:\; \sconf(M)\arrow \R$ such that
$e^{\chi(v)}= \chi(e^v)$. Describe it explicitly in terms of
$\tilde M$.
\ez

\exercise
Suppose that an LCS manifold $M$ admits a vector field
$v\in \sconf(M)$, $\chi(v)\neq 0$, such that $e^tv$ induces
a circle action. Prove that the quotient $M/\langle e^{tv}\rangle$
is a contact orbifold.
\ez

\exercise
Let $(M,\omega, \theta)$ be an LCS-action, and $v\in \sconf(M)$
inducing a circle action $\rho(t):= e^{tv}$. Prove that
$\chi(v)=\int_{S}\theta$, where $S=S^1$
is an orbit of $\rho$.
\ez

\exercise
Let $M$ be an LCS-manifold, $v\in\sconf(M)$ a vector field,
$\chi(v)\neq 0$, and $\rho(t)=e^{tv}$ the corresponding
diffeomorphism flow. Suppose that the closure
$\overline{\im \rho}$ in the diffeomorphism group
(with $C^1$-topology) is compact. Prove that
$\Mon(M)=\Z$.
\ez

 






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