Holomorphic classification of four-dimensional surfaces in ℂ³

Abstract

We use the method of model surfaces to study real four-dimensional submanifolds of ℂ³. 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 S⁴ admits no completely non-degenerate embedding into a three-dimensional complex manifold.