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Izv. RAN. Ser. Mat., 2008, Volume 72, Issue 3, Pages 3–18 (Mi izv696)  

This article is cited in 5 scientific papers (total in 5 papers)

Holomorphic classification of four-dimensional surfaces in $\mathbb C^3$

V. K. Beloshapkaa, V. V. Ezhovb, G. Schmalzc

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b University of Adelaide
c University of New England

Abstract: We use the method of model surfaces to study real four-dimensional submanifolds of $\mathbb C^3$. We prove that the dimension of the holomorphic symmetry group of any germ of an analytic four-dimensional manifold does not exceed 5 if this dimension is finite. (There are only two exceptional cases of infinite dimension.) The envelope of holomorphy of the model surface is calculated. We construct a normal form for arbitrary germs and use it to give a holomorphic classification of completely non-degenerate germs. It is shown that the existence of a completely non-degenerate CR-structure imposes strong restrictions on the topological structure of the manifold. In particular, the four-sphere $S^4$ admits no completely non-degenerate embedding into a three-dimensional complex manifold.

DOI: https://doi.org/10.4213/im696

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English version:
Izvestiya: Mathematics, 2008, 72:3, 413–427

Bibliographic databases:

UDC: 517.55+514.76
MSC: 32V40
Received: 30.04.2004
Revised: 02.03.2007

Citation: V. K. Beloshapka, V. V. Ezhov, G. Schmalz, “Holomorphic classification of four-dimensional surfaces in $\mathbb C^3$”, Izv. RAN. Ser. Mat., 72:3 (2008), 3–18; Izv. Math., 72:3 (2008), 413–427

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. K. Beloshapka, “A Counterexample to the Dimension Conjecture”, Math. Notes, 81:1 (2007), 117–120  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. Beloshapka V.K., Kossovskiy I.G., “Homogeneous hypersurfaces in $\mathbb C^3$, associated with a model CR-cubic”, J. Geom. Anal., 20:3 (2010), 538–564  crossref  mathscinet  zmath  isi  elib  scopus
    3. Beloshapka V.K., Kossovskiy I.G., “Classification of homogeneous CR-manifolds in dimension 4”, J. Math. Anal. Appl., 374:2 (2011), 655–672  crossref  mathscinet  zmath  isi  elib  scopus
    4. Burgués J.M., Dwilewicz R.J., “Geometry of semi-tube domains in $\mathbb C^2$”, Adv. Geom., 12:4 (2012), 685–702  crossref  mathscinet  zmath  isi  scopus
    5. Kolar M., Meylan F., Zaitsev D., “Chern-Moser Operators and Polynomial Models in Cr Geometry”, Adv. Math., 263 (2014), 321–356  crossref  mathscinet  zmath  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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