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

摘要

We construct holomorphic families of proper holomorphic embeddings of mathbb {C}^{k} into mathbb {C}^{n} (0extless kextless 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}^{*}-actions on mathbb {C}^{n} (nge5) 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}^{*}-actions on mathbb {C}^{n} (with prescribed linear part at a fixed point).