Holomorphic families of non-equivalent embeddings and of holomorphic group actions on affine space

Abstract

We construct holomorphic families of proper holomorphic embeddings of $\C^k$ into $\C^n$ ($0<k<n-1$), so that for any two different parameters in the family no holomorphic automorphism of $\C^n$ can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of $\C^n$ we derive the existence of families of holomorphic $\C^*$-actions on $\C^n$ ($n\ge 5$) so that different actions in the family are not conjugate. This result is surprising in view of the long standing Holomorphic Linearization Problem, which in particular asked whether there would be more than one conjugacy class of $\C^*$ actions on $\C^n$ (with prescribed linear part at a fixed point).