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\listok{8}{LCK manifolds 8: Stein manifolds and normal families}

\exercise
Let $X,Y$ -- complex manifold, and
$M(X,Y)$ the space of all holomorphic maps from $X$ to $Y$,
with open-compact topology.
\enum
\item Prove that $M(\C P^1, \C P^1)$ is compact,
or find a counterexample.
\item Let $X$ be a complex manifold, and $X$ a complex
curve of genus $>1$. Prove that $M(X,Y)$ is compact.
\item Let $X=Y=T^2$ be an elliptic curve. Prove that
$M(X,Y)$ is compact or find a counterexample.
\ee
\ez

\definition Let $V$ be a topological vector space.
A subset $K\subset V$ is called {\bf bounded} if for
any open neighbourhood $U\ni 0$, there exists
$a\in \R$ such that $aU\supset K$.
A topological vector space $V$ is called {\bf Montel}
if any closed bounded subset $K\subset V$
is compact.
\ed

\exercise
Let $M$ be a complex manifold, $B$ a vector bundle,
and $V=H^0(B)$ the space of holomorphic sections, 
with topology of uniform convergence on compacts.
\enum
\item Prove that $V$ is a Montel space.
\item Prove that any metrizable Montel space is finite-dimensional.
\item Show that $H^0(B)$ is always finite-dimensional
if $M$ is compact.
\ee
\ez

\definition
A complex variety $M$ is called {\bf  holomorphically convex}
if for any infinite discrete subset $S\subset M$,
there exists a holomorphic function $f\in \calo_M$ which
is unbounded on $S$. A holomorphically convex variety
without complex subvarieties is called {\bf Stein}.
\ed

\exercise
Let $U, V\subset \C^n$ be open, holomorphically convex subsets.
Prove that $U\cap V$ is also holomorphically convex.
\ez

\exercise
Consider the {\bf Reinhardt triangle}:
\[ D:= \{ (z_1, z_2) \in \C^2 \ \ |\ \  0< |z_1|<|z_2|<1\}.
\]
Prove that $D$ is holomorphically convex.
\ez

\exercise
Prove that any Stein manifold admits a proper holomorphic
map to $\C^n$.
\ez






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