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

Kutzschebauch, Frank; Lodin, Sam (2013). Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space. Duke Mathematical Journal, 162(1), pp. 49-94. Duke University Press 10.1215/00127094-1958969

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We construct holomorphic families of proper holomorphic embeddings of \mathbb {C}^{k} into \mathbb {C}^{n} (0\textless k\textless 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} (n\ge5) 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).

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