Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space

Abstract

We construct holomorphic families of proper holomorphic embeddings of ${\mathbb{C}}^k$ into ${\mathbb{C}}^n$ ($0<k<n-1$), so that for any two different parameters in the family, no holomorphic automorphism of ${\mathbb{C}}^n$ 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 ${\mathbb{C}}^n$, we derive the existence of families of holomorphic ${\mathbb{C}}^{\ast }$-actions on ${\mathbb{C}}^n$ ($n\ge 5$) 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 ${\mathbb{C}}^{\ast }$-actions on ${\mathbb{C}}^n$ (with prescribed linear part at a fixed point).