Holomorphic classification of four-dimensional surfaces in~$\mathbb C^3$
Izvestiya. Mathematics , Tome 72 (2008) no. 3, pp. 413-427
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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.
@article{IM2_2008_72_3_a0,
author = {V. K. Beloshapka and V. V. Ezhov and G. Schmalz},
title = {Holomorphic classification of four-dimensional surfaces in~$\mathbb C^3$},
journal = {Izvestiya. Mathematics },
pages = {413--427},
publisher = {mathdoc},
volume = {72},
number = {3},
year = {2008},
language = {en},
url = {http://geodesic.mathdoc.fr/item/IM2_2008_72_3_a0/}
}
TY - JOUR AU - V. K. Beloshapka AU - V. V. Ezhov AU - G. Schmalz TI - Holomorphic classification of four-dimensional surfaces in~$\mathbb C^3$ JO - Izvestiya. Mathematics PY - 2008 SP - 413 EP - 427 VL - 72 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/IM2_2008_72_3_a0/ LA - en ID - IM2_2008_72_3_a0 ER -
V. K. Beloshapka; V. V. Ezhov; G. Schmalz. Holomorphic classification of four-dimensional surfaces in~$\mathbb C^3$. Izvestiya. Mathematics , Tome 72 (2008) no. 3, pp. 413-427. http://geodesic.mathdoc.fr/item/IM2_2008_72_3_a0/