Manifolds with infinite dimensional group of holomorphic automorphisms and the Linearization Problem

Abstract: We overview a number of precise notions for a holomorphic automorphism group to be big together with their implications, in particular we give an exposition of the notions of flexibility and of density property. These studies have their origin in the famous result of Anders\'en and Lempert from 1992 proving that the overshears generate a dense subgroup in the holomorphic automorphism group of $\C^n, n\ge 2$. There are many applications to natural geometric questions in complex geometry, several of which we mention here. Also the Linearization Problem, well known since the 1950 s and considered by many authors, has had a strong influence on those studies. It asks whether a compact subgroup in the holomorphic automorphism group of $\C^n$ is necessarily conjugate to a group of linear automorphisms. Despite many positive results, the answer in this generality is negative as shown by Derksen and the author. We describe various developments around that problem.