Duke Mathematical Journal

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

Frank Kutzschebauch and Sam Lodin

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Abstract

We construct holomorphic families of proper holomorphic embeddings of Ck into Cn (0<k<n1), so that for any two different parameters in the family, no holomorphic automorphism of Cn 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 Cn, we derive the existence of families of holomorphic C-actions on Cn (n5) 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 Cn (with prescribed linear part at a fixed point).

Article information

Source
Duke Math. J., Volume 162, Number 1 (2013), 49-94.

Dates
First available in Project Euclid: 14 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1358172074

Digital Object Identifier
doi:10.1215/00127094-1958969

Mathematical Reviews number (MathSciNet)
MR3011872

Zentralblatt MATH identifier
1266.32029

Subjects
Primary: 32M05: Complex Lie groups, automorphism groups acting on complex spaces [See also 22E10] 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions
Secondary: 32Q28: Stein manifolds 32Q40: Embedding theorems 32Q45: Hyperbolic and Kobayashi hyperbolic manifolds

Citation

Kutzschebauch, Frank; Lodin, Sam. Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space. Duke Math. J. 162 (2013), no. 1, 49--94. doi:10.1215/00127094-1958969. https://projecteuclid.org/euclid.dmj/1358172074


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