ABSTRACT
The local parametrization theorem is of fundamental importance in the study of holomorphic subvarieties and varieties. This chapter discusses three different applications of that theorem. They are Hilbert's zero-theorem for holomorphic functions, the basic properties of finite holomorphic mappings, and some local topological properties of holomorphic varieties. The zero-theorem extends in a rather straightforward way to the corresponding assertion in the local ring of any germ of a holomorphic variety. Any germ of a finite branched holomorphic covering is a finite germ of a holomorphic mapping, but not conversely; finite germs of holomorphic mappings need not be surjective, for instance. The definition of a finite germ of a holomorphic mapping and the characterization of germs were of a rather geometric nature, so it may be of interest to see a complementary algebraic characterization of these germs. The local parametrization theorem can also be used to demonstrate several important general topological properties of holomorphic varieties.
