Geodesic
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. 
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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/

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