×

zbMATH — the first resource for mathematics

Interpolation by holomorphic automorphisms and embeddings in \({\mathbb{C}}^n\). (English) Zbl 0963.32006
Summary: Let \(n>1\) and let \(\mathbb{C}^n\) denote the complex \(n\)-dimensional Euclidean space. We prove several jet-interpolation results for nowhere degenerate entire mappings \(F:\mathbb{C}^n\to\mathbb{C}^n\) and for holomorphic automorphisms of \(\mathbb{C}^n\) on discrete subsets of \(\mathbb{C}^n\). We also prove an interpolation theorem for proper holomorphic embeddings of Stein manifolds into \(\mathbb{C}^n\). For each closed complex submanifold (or subvariety) \(M\subset\mathbb{C}^n\) of complex dimension \(m<n\) we construct a domain \(\Omega\subset \mathbb{C}^n\) containing \(M\) and a biholomorphic map \(F:\Omega \to \mathbb{C}^n\) onto \(\mathbb{C}^n\) with \(JF\equiv 1\) such that \(F(M)\) intersects the image of any nondegenerate entire map \(G:\mathbb{C}^{n-m} \to\mathbb{C}^n\) at infinitely many points. If \(m=n-1\), we construct \(F\) as above such that \(\mathbb{C}^n\setminus F(M)\) is hyperbolic. In particular, for each \(m\geq 1\) we construct proper holomorphic embeddings \(F:\mathbb{C}^m \to\mathbb{C}^{m+1}\) such that the complement \(\mathbb{C}^{m+1} \setminus F(\mathbb{C}^m)\) is hyperbolic.

MSC:
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32Q40 Embedding theorems for complex manifolds
32M05 Complex Lie groups, group actions on complex spaces
32Q28 Stein manifolds
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Andersén, E. Volume-preserving automorphisms of Cn,Complex Variables,14, 223–235, (1990). · Zbl 0705.58008
[2] Andersén, E. and Lempert, L. On the group of holomorphic automorphisms ofC n,Invent. Math.,110, 371–388, (1992). · Zbl 0770.32015 · doi:10.1007/BF01231337
[3] Buzzard, G. Kupka-Smale Theorem for Automorphisms ofC n,Duke Math. J.,93, 487–503, (1998). · Zbl 0946.32012 · doi:10.1215/S0012-7094-98-09317-6
[4] Buzzard, G. and Fornæss, J.E. An embedding ofC intoC 2 with hyperbolic complement,Math. Ann.,306, 539–546, (1996). · Zbl 0864.32013 · doi:10.1007/BF01445264
[5] Buzzard, G. and Forstneric, F. An interpolation theorem for holomorphic automorphisms ofC n. Preprint, 1996. · Zbl 0963.32014
[6] Chirka, E.Complex Analytic Sets, Kluwer, Dordrecht, 1989. · Zbl 0683.32002
[7] Eliashberg, Y. and Gromov, M. Embeddings of Stein manifolds of dimensionn into the affine space of dimension3n/2 + 1,Ann. Math.,136(2), 123–135, (1992). · Zbl 0758.32012 · doi:10.2307/2946547
[8] Forstneric, F. Approximation by automorphisms on smooth submanifolds ofC n,Math. Ann.,300, 719–738, (1994). · Zbl 0821.32028 · doi:10.1007/BF01450512
[9] Forstneric, F. Equivalence of real submanifolds under volume preserving holomorphic automorphisms ofC n,Duke Math. J.,77, 431–445, (1995). · Zbl 0831.32009 · doi:10.1215/S0012-7094-95-07713-8
[10] Forstneric, F. Holomorphic automorphism groups ofC n: A survey.The Proceedings Complex Analysis and Geometry, Ancona, V., Ballico, E., and Silva, A., Eds., 173–200,Lecture Notes in Pure and Applied Mathematics,173, Marcel Dekker, New York, 1996.
[11] Forstneric, F., Globevnik, J., and Rosay, J.-P. Non straightenable complex lines inC 2,Arkiv Math,34, 97–101, (1996). · Zbl 0853.58015 · doi:10.1007/BF02559509
[12] Forstneric, F., Globevnik, J., and Stensønes, B. Embedding holomorphic discs through discrete sets,Math. Ann.,304, 559–596, (1995). · Zbl 0859.32010
[13] Forstneric, F. and Rosay, J.-P. Approximation of biholomorphic mappings by automorphisms ofC n,Invent. Math.,112, 323–349, (1993). Correction,Invent. Math.,118, 573–574, (1994). · Zbl 0792.32011 · doi:10.1007/BF01232438
[14] Globevnik, J. and Stensønes, B. Holomorphic embeddings of planar domains intoC 2,Math. Ann.,303, 579–597, (1995). · Zbl 0847.32030 · doi:10.1007/BF01461006
[15] Hörmander, L.An Introduction to Complex Analysis in Several Variables, 3rd ed., North Holland, Amsterdam, 1990. · Zbl 0685.32001
[16] Jacobson, N.Basic Algebra I, Freeman, San Francisco, 1974. · Zbl 0284.16001
[17] Rosay, J.-P. and Rudin, W. Holomorphic maps fromC n toC n,Trans. Am. Math. Soc.,310, 47–86, (1988). · Zbl 0708.58003
[18] Rossi, H. The local maximum modulus principle,Ann. Math.,72(2), 1–11, (1960). · Zbl 0099.32703 · doi:10.2307/1970145
[19] Stolzenberg, G. Polynomially and rationally convex sets,Acta Math.,109, 259–289, (1963). · Zbl 0122.08404 · doi:10.1007/BF02391815
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.