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\listok{12}{LCK manifolds 12: Morse-Novikov and Bott-Chern cohomology}

\exercise
Find an example of compact LCK (non-K\"ahler)
manifold admitting symplectic structure.
\ez

\exercise
Let $M$ be an LCK manifold with potential,
and $\phi:\; \tilde M \arrow \R$ its potential.
Prove that $\phi\neq 0$.
\ez

\exercise
Let $M$ be a compact complex manifold,
$H^2(M)=H^1(M)=0$. Prove that $\dim H^{1,1}_{BC}(M)= 2 \dim H^1(\calo_M)$,
where $H^{p,q}_{BC}(M)$ denotes the Bott-Chern cohomology.
\ez

\exercise
Assume that the standard map $\bigoplus_{p,q} H^{p,q}_{BC}(M)\arrow H^*(M)$
to the de Rham cohomology is injective. Prove that it is surjective.
\ez

\exercise
Assume that the standard map 
$\bigoplus_{p,q} H^{p,q}_{BC}(M)\arrow \bigoplus_{p,q}H^{p,q}_{\bar \6}(M)$
to the Dolbeault cohomology is injective. Prove that it is surjective.
\ez

\exercise
Find an example of  a compact 
locally conformally symplectic manifold
 $(M,\omega,\theta)$ such that the cohomology class
of $\theta$ is non-zero, and the Morse-Novikov 
class $[\omega]_{MN}$ vanishes.
\ez

\exercise
Let $M$ be a Riemannian 4-manifold, and 
$\Lambda^2(M)=\Lambda^+(M)\oplus\Lambda^-(M)$ the decomposition
on eigenspaces of the Hodge $*$-operator. Consider the map $d_+:\; \Lambda^1(M) \arrow \Lambda^+(M)$
obtained by projecting de Rham differential to $\Lambda^+(M)$.
Compute symbols and prove that $\ker d^+/\im d$ is finite-dimensional.
\ez

\exercise
Let $G$ be a compact Lie group, and
$\omega$ a left-invariant locally conformally symplectic
form, $d\omega=\omega\wedge \theta$. Prove that
either $\theta=0$ or $[\omega]_{MN}=0$.
\ez

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