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\listok{5}{LCK manifolds 5: Groups of automorphisms}


\exercise
Let $M$ be a compact Riemannian manifold.
\enum
\item Show that the group of isometries of $M$ is compact.
\item Prove that it is a Lie group, and its Lie algebra is 
an algebra of Killing vector fields.
\ee
\ez

\exercise
Let $M$ be a compact complex manifold, and $G$ its group of
holomorphic automorphisms, equipped with a natural
(compact-open) topology. Prove that $G$ is a Lie group,
and its Lie algebra is an algebra of holomorphic vector fields.
\ez


\exercise
Let $(M, I, \omega)$ be a compact K\"ahler manifold, and
$\g\subset TM$ a subalgebra consisting of all vector fields
$X\in TM$ such that $\Lie_X I=\Lie_X\omega=0$ and
$\Lie_{IX} I=\Lie_{IX}\omega=0$.
\enum
\item Prove that it is a subalgebra.
\item Prove that for all $X\in \g$, one has $\nabla(X)=0$.
\item Prove that $\g$ is Abelian.
\ee
\ez

\newcommand{\Alb}{\operatorname{Alb}}
\definition
Let $M$ be a compact K\"ahler manifold. Define an equivalence 
relation $x\sim y$ on $M$ as follows: two points $x, y$ are
equivalent if for any path $\gamma$ connecting $x$ to $y$
and any harmonic form $\alpha$ representing an integer
cohomology class, the number $\int_\gamma \alpha$ is integer.
The quotient $M/\sim$ is called {\bf the Albanese variety}
of $M$, denoted by $\Alb(M)$.
\ed

\exercise
Prove that $\Alb(M)$ is a compact complex torus, and
the projection $M \arrow \Alb(M)$ is a holomorphic map.
\ez


\exercise
Let $M$ be a compact K\"ahler manifold,
$\Aut(M)$ the group of its holomorphic automorphisms,
$\g\subset TM$ the Lie algebra of vector fields satisfying
$\nabla(X)=0$, and $G$ is Lie group.
\enum
\item Prove that $G\subset \Aut(M)$.
\item Prove that $G$ is a normal subgroup in $\Aut(M)$.
\item Prove that all its orbits are compact complex tori in $M$.
\item Let $G_1\subset \Aut(M)$ be the group of automorphisms
acting trivially on $\Alb(M)$. Prove that $G_1 \cap G=0$.
\item Prove that $\Aut(M)$ is a semidirect product of
$G$ and $G_1$.
\ee
\ez



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