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\listok{1}{LCK manifolds 1: local systems and Morse-Novikov cohomology}

\exercise
Let $\lambda >0$ be a real number.
Define {\bf weight $\lambda$ homogeneous forms} on $\R^n\backslash 0$ 
as differential forms $\eta$ which satisfy $\rho_t^*\eta=\lambda^t\eta$,
where $\rho_t$ is a homothety map $z \arrow tz$, $t>0$.
Prove that a closed weight $\lambda$ form is exact for
any $\lambda\neq 1$.
\ez

\exercise
Let $M =\R^n\backslash 0/(x\sim 2x)$ be a Hopf manifold, 
$\theta$ a closed, non-exact 1-form, 
$d_\theta=d-\theta$, and $H^*_\theta(M)$ cohomology
of the complex $\Lambda^*(M), d_\theta$ (``Morse-Novikov cohomology'').
Prove that $H^i_\theta(M)=0$ for all $i$.
\ez

\hint
Use the previous exercise.
\eh

\exercise
Let $M=X\times S^1$, $\pi:\; M \arrow S^1$ the standard projection,
$\theta:= \pi^* dt$. Prove that  $H^i_\theta(M)=0$ for all $i$.
\ez

\exercise
Define {\bf projectively flat} connection on a vector bundle $B$
over $M$ as a connection with curvature $\Theta\in \Lambda^2 M \otimes \End(B)$
which satisfies $\Theta\in \Lambda^2M \otimes \Id_B$.
Prove that projectively flat connections on a bundle of rank $r$ 
are equivalent to representations $\pi_1(M) \arrow PGL(n)$.
\ez

\exercise
Let $M$ be a compact manifold, and $\theta$ a closed, non-exact, 
nowhere vanishing one-form. Prove that $H^*_{\lambda\theta}(M)=0$ for all
$\lambda \in \R$, except a finite number.
\ez

\exercise
(Moser stability for LCS)\\
Let $(M,\omega,\theta)$ be a compact 
locally conformally symplectic manifold, 
$\omega_t$ a continuous deformation of $\omega$
satisfying $d\omega_t=\omega_t\wedge\theta$
and $[\omega_t]=\const$, where
$[\omega_t]\in H^2_\theta(M)$ the Morse-Novikov class of $\omega$.
Find a flow of diffeomorphisms mapping $\omega$ to $\omega_t$.
\ez

\hint Use the usual Moser stability argument. Interpret
forms with coefficients in the weight bundle as equivariant
forms on the universal cover.
\eh

\end{document}
