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\listok{10}{LCK manifolds 10: Levi form}

From now until the end, $n >1$, and all hypersurfaces
are equipped with orientation.

\exercise
Consider {\bf the Penrose hypersurface} $S:=\{ z\in \C^3 \ \ |\ \ |z_1|^2+ 
|z_2|^2 = 1+ |z_3|^2\}$. Prove that its Levi form is non-degenerate
and find its signature.
\ez

\definition
A real-valued function $\phi$ on a complex manifold
is called {\bf pluriharmonic}
if $dd^c\phi=0$
\ed

\exercise
Let $\phi$ be a pluriharmonic function, $c$ its regular value, and $S:=
\phi^{-1}(c)$. Prove that the Levi form $\Phi$ of $S$ is vanishing
(such $S$ is called {\bf Levi-flat}).
\ez


\exercise
Let $\phi$ be a non-zero real function on a complex manifold
such that for some $a, b \in \R$, one has 
$a\phi\cdot  dd^c \phi + b \cdot d\phi \wedge d^c\phi=0$.
Prove that $\phi^\lambda$ is pluriharmonic, for some
non-zero $\lambda \in \R$.
\ez

\exercise
Let $M\subset \C^n$ be a holomorphically convex subset
with smooth boundary $S$. Prove that the Levi form of $S$
is semi-positive.
\ez


\exercise
Let $S\subset \C^n$ be a compact smooth real hypersurface,
such that the Levi form $\Phi$ of $S$ is non-degenerate. Prove that 
$\Phi$ is sign-definite.
\ez

\exercise
Let $S\subset \C^n$ be a smooth Levi-flat hypersurface.
Prove that $S$ is non-compact.
\ez

\exercise
Let $M$ be a complex manifold, and $\phi:\; M \arrow \R$
a smooth function satisfying $d\phi \wedge d^c \phi \wedge  dd^c\phi=0$. 
Prove that for any regular value $c$ of $\phi$,
the preimage $\phi^{-1}(c)$ is Levi-flat.
\ez


%\exercise
%Let $M$ be a complex manifold, and $\phi:\; M \arrow \R$
%a smooth function such that for any regular value $c$ of $\phi$,
%the preimage $\phi^{-1}(c)$ is pseudoconvex. Prove that 
%$dd^c \phi$ 
%\ez



\exercise
Let $M = C(S)/\langle\gamma\rangle$ be a Vaisman manifold,
with $\gamma(s, t) = (\phi(s), \lambda t)$, where
$\phi$ is a Sasakian automorphism.
\enum
\item Prove that there exists a holomorphic vector field $\vec r$ such that
$e^{\vec r}=\gamma$ if and only if $\phi$ lies in the connected
component of the Lie group of Sasakian automorphisms.
\item Find a Vaisman manifold $M = C(S)/\langle\gamma\rangle$
such that $\gamma\neq e^{\vec r}$ for any holomorphic $\vec r$.
\ee
\ez



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