\documentclass[12pt]{article}

% version 1.0, 10.02.2014
% version 1.1, 10.02.2014, H^*_\theta=0 -- error corrected
% version 1.2, 06.03. 2014, removed the problem about 
% regular Vaisman (put it to handout 4)

\newcommand{\version}{version 1.2,\ \   06.03.2014}
%\newcommand{\firstdate}{10.02.2013}


%\addtolength{\topmargin}{-5mm}
%\addtolength{\textheight}{10mm}
%\addtolength{\oddsidemargin}{-10mm}
%\addtolength{\textwidth}{20mm}

\input{defs-listki-en.tex}


\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\listok{2}{LCK manifolds 2: LCK manifolds and Hermitian structures}

\exercise
Let $\theta$ a closed 1-form on $M$
$d_\theta=d-\theta$, and $H^*_\theta(M)$ cohomology
of the complex $\Lambda^*(M), d_\theta$ (``Morse-Novikov cohomology'').
Prove that $H^i_\theta(M)=H^i(M)$ when $\theta$ is exact.
\ez

\exercise
Let $(M,\omega, \theta)$ be a compact LCK manifold, satisfying
$d^c\theta=0$. Prove that $\theta=0$.
\ez

\exercise
Let $(M,\omega, \theta)$ be a compact LCK manifold, satisfying
$dd^c\omega=0$, and $\dim_\C M> 2$. Prove that $\theta=0$.
\ez

%\exercise
%Prove that a line bundle of degree 0 on a complex curve.
%admits a Hermitian metric with flat Chern connection.
%\ez

\definition
Let $M$ be a complex manifold, $\dim_\C M=n$.
A Hermitian metric on $M$ is called {\bf balanced}
if $d\omega^{n-1}=0$. 
\ed

\exercise
Prove that a classical Hopf manifold $C^n/x\sim \lambda x$
does not admit a balanced metrics.
\ez

\exercise
Let $\omega$ be a non-degenerate 2-form on a
$2n$-dimensional smooth manifold, and $d(\omega^k)=0$ for
some $k$ satisfying $0<k<n-1$. Prove that $d\omega=0$.
\ez

\exercise
Let $(M,\omega, \theta)$ be a compact LCK manifold,
$\dim_\C M >2$, $dd^c\omega=0$. Prove that $\theta=0$.
\ez


\end{document}
