I would like to ask the following questions.
Let $X$ be a compact Kahler manifold. Denote by Aut(X) the group of all the biholomorphisms of $X.$
1) What can we say about this group? E.g, Is it a Lie group?
2) Does there exist a manifold $X$ with $Aut(X)$ trivial?
3) Let $A$ be an analytic subset of $X$ and $x_0\in A$ some point. Does there exist an automorphism $\gamma\in Aut(X)$ with $\gamma(x_0) \not\in A$?
4) Finally, is there a way to produce automorphisms of $X$?
Any help is appreciated. Thanks in advance.