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Kutzschebauch, Frank
Publications (10 of 15) Show all publications
Kutzschebauch, F. & Lodin, S. (2013). Holomorphic families of non-equivalent embeddings and of holomorphic group actions on affine space. Duke mathematical journal, 162(1), 49-94
Open this publication in new window or tab >>Holomorphic families of non-equivalent embeddings and of holomorphic group actions on affine space
2013 (English)In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 162, no 1, p. 49-94Article in journal (Refereed) Published
Abstract [en]

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 >= 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).

Keywords
2 Complex-variables; Intrinsic Measures; Density Property; Stein Manifolds; Oka Principle; Automorphisms; Interpolation; Dimension; C(n); C-2
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-9633 (URN)10.1215/00127094-1958969 (DOI)000314079400002 ()2-s2.0-84873305556 (Scopus ID)
Available from: 2009-09-15 Created: 2009-09-15 Last updated: 2017-12-13Bibliographically approved
Borell, S. & Kutzschebauch, F. (2008). Embeddings through Discrete Sets of Discs. Arkiv för matematik, 46(2), 251-269
Open this publication in new window or tab >>Embeddings through Discrete Sets of Discs
2008 (English)In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 46, no 2, p. 251-269Article in journal (Refereed) Published
Abstract [en]

We investigate whether a Stein manifold M which allows proper holomorphic embedding into ℂ n can be embedded in such a way that the image contains a given discrete set of points and in addition follow arbitrarily close to prescribed tangent directions in a neighbourhood of the discrete set. The infinitesimal version was proven by Forstnerič to be always possible. We give a general positive answer if the dimension of M is smaller than n/2 and construct counterexamples for all other dimensional relations. The obstruction we use in these examples is a certain hyperbolicity condition.

National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-15026 (URN)10.1007/s11512-008-0079-8 (DOI)
Available from: 2011-12-05 Created: 2011-12-05 Last updated: 2017-12-08Bibliographically approved
Borell, S., Kutzschebauch, F. & Wold, E. F. (2008). Proper Holomorphic Disks in the Complement of Varieties in C2. Mathematical Research Letters, 15(4), 821-826
Open this publication in new window or tab >>Proper Holomorphic Disks in the Complement of Varieties in C2
2008 (English)In: Mathematical Research Letters, ISSN 1073-2780, E-ISSN 1945-001X, Vol. 15, no 4, p. 821-826Article in journal (Refereed) Published
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-15027 (URN)
Available from: 2011-12-05 Created: 2011-12-05 Last updated: 2017-12-08Bibliographically approved
Forstneric, F., Ivarsson, B. & Kutzschebauch, F. (2007). An interpolation theorem for proper holomorphic embeddings. Mathematische Annalen, 338(3), 545-554
Open this publication in new window or tab >>An interpolation theorem for proper holomorphic embeddings
2007 (English)In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 338, no 3, p. 545-554Article in journal (Refereed) Published
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-9635 (URN)10.1007/s00208-007-0087-1 (DOI)
Available from: 2009-09-15 Created: 2009-09-15 Last updated: 2017-12-13Bibliographically approved
Kaliman, S. & Kutzschebauch, F. (2007). Criteria for the density property of complex manifolds. Inventiones Mathematicae, 172, 71-87
Open this publication in new window or tab >>Criteria for the density property of complex manifolds
2007 (English)In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 172, p. 71-87Article in journal (Refereed) Published
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-9637 (URN)10.1007/s00222-007-0094-6 (DOI)
Available from: 2009-09-15 Created: 2009-09-15 Last updated: 2017-12-13Bibliographically approved
Kutzschebauch, F. & Kaliman, S. (2006). Criteria for the density property of complex manifolds. Bonn: Max-Planck inst
Open this publication in new window or tab >>Criteria for the density property of complex manifolds
2006 (English)Report (Other academic)
Abstract [en]

In this paper we suggest new effective criteria for the density property. This enables us to give a trivial proof of the original Anders\'en-Lempert result and to establish (almost free of charge) the algebraic density property for all linear algebraic groups whose connected components are different from tori or $\C_+$. As another application of this approach we tackle the question (asked among others by F. Forstneri\v{c}) about the density of algebraic vector fields on Euclidean space vanishing on a codimension 2 subvariety.

Place, publisher, year, edition, pages
Bonn: Max-Planck inst, 2006. p. 11
Series
Preprints of the Max-Planck-Institut für Mathematik ; 2006-89
Keywords
density property
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-5895 (URN)4376 (Local ID)4376 (Archive number)4376 (OAI)
Note
arXiv:0710.3864v1 To appear in Invent MathAvailable from: 2008-09-30 Created: 2009-09-15Bibliographically approved
Borell, S. & Kutzschebauch, F. (2006). Non-equivalent embeddings into complex Euclidean spaces. International Journal of Mathematics, 17(9), 1033-1046
Open this publication in new window or tab >>Non-equivalent embeddings into complex Euclidean spaces
2006 (English)In: International Journal of Mathematics, ISSN 0129-167X, Vol. 17, no 9, p. 1033-1046Article in journal (Refereed) Published
Abstract [en]

We study the number of equivalence classes of proper holomorphic embeddings of a Stein space X into ℂn. In this paper we prove that if the automorphism group of X is a Lie group and there exists a proper holomorphic embedding of X into ℂn, 0 < dim X < n, then for any k ≥ 0 there exist uncountably many non-equivalent proper holomorphic embeddings Φ: X × ℂk ℂn × ℂk. For k = 0 all embeddings will be proved to satisfy the additional property of ℂn\Φ(X) being (n - dim X)-Eisenman hyperbolic. As a corollary we conclude that there are uncountably many non-equivalent proper holomorphic embeddings of ℂk into ℂn whenever 0 < k < n.

Keywords
proper holomorphic embeddings, hyperbolicity, Eisenman, complex analysis
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-3420 (URN)10.1142/S0129167X06003795 (DOI)000242603500002 ()2-s2.0-33750928991 (Scopus ID)3404 (Local ID)3404 (Archive number)3404 (OAI)
Available from: 2008-09-30 Created: 2008-09-30 Last updated: 2017-12-12Bibliographically approved
Kaliman, S. & Kutzschebauch, F. (2006). The density property for hypersurfaces uv=p(x). Mathematische Zeitschrift, 258(1), 115-131
Open this publication in new window or tab >>The density property for hypersurfaces uv=p(x)
2006 (English)In: Mathematische Zeitschrift, ISSN 0025-5874, Vol. 258, no 1, p. 115-131Article in journal (Refereed) Published
Abstract [en]

We study hypersurfaces of Cn+2 ¯x,u,v given by equations of form uv = p( ¯x) where the zero locus of a polynomial p is smooth reduced. The main result says that the Lie algebra generated by algebraic completely integrable vector fields on such a hypersurface coincides with the Lie algebra of all algebraic vector fields. Consequences of this result for some conjectures of affine algebraic geometry and for the Oka-Grauert-Gromov principle are discussed.

Series
Max-Planck-Institut Preprint Series 2006 (90)
Keywords
density property
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-5894 (URN)10.1007/s00209-007-0162-z (DOI)4375 (Local ID)4375 (Archive number)4375 (OAI)
Available from: 2008-09-30 Created: 2008-09-30 Last updated: 2011-01-10Bibliographically approved
Forstneric, F., Ivarsson, B., Kutzschebauch, F. & Prezelj, J. (2005). An interpolation theorem for proper holomorphic embeddings.
Open this publication in new window or tab >>An interpolation theorem for proper holomorphic embeddings
2005 (English)Report (Other academic)
Abstract [en]

Given a Stein manifold X of dimension n>1, a discrete sequence a_j in X, and a discrete sequence b_j in C^m where m > [3n/2], there exists a proper holomorphic embedding of X into C^m which sends a_j to b_j for every j=1,2,.... This is the interpolation version of the embedding theorem due to Eliashberg, Gromov and Schurmann. The dimension m cannot be lowered in general due to an example of Forster.

Series
Erwin Schroedinger Institut, preprint ; 1735
Keywords
Stein spaces
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-5824 (URN)3494 (Local ID)3494 (Archive number)3494 (OAI)
Note
arXiv:math/0511122v3Available from: 2008-09-30 Created: 2008-09-30 Last updated: 2009-09-15Bibliographically approved
Kutzschebauch, F. (2005). Andersén-Lempert-theory with parameters: a representation theoretic point of view. Journal of Algebra and its Applications, 4(3), 325-340
Open this publication in new window or tab >>Andersén-Lempert-theory with parameters: a representation theoretic point of view
2005 (English)In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 4, no 3, p. 325-340Article in journal (Refereed) Published
Abstract [en]

We calculate the invariant subspaces in the linear representation of the group of algebraic automorphisms of ℂn on the vector space of algebraic vector fields on ℂn and more generally we do this in a setting with parameter. As an application to the field of Several Complex Variables we get a new proof of the Andersén–Lempert observation and a parametric version of the Andersén–Lempert theorem. Further applications to the question of embeddings of ℂk into ℂn are announced.

Keywords
Andersen-Lempert theory
National Category
Mathematics
Identifiers
urn:nbn:se:miun:diva-3442 (URN)10.1142/S0219498805001216 (DOI)3478 (Local ID)3478 (Archive number)3478 (OAI)
Available from: 2008-09-30 Created: 2009-09-15 Last updated: 2017-12-12Bibliographically approved

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