ABSTRACT
Holomorphic mappings that have been examined in any detail are finite holomorphic mappings, but by this point enough machinery has been developed to permit the fairly easy derivation of a number of useful properties of holomorphic mappings in general. The holomorphic mappings that are locally finite, in the sense that their germs at any points are the germs of finite holomorphic mappings, are precisely those holomorphic mappings for which the level sets are all zero-dimensional holomorphic subvarieties. When investigating holomorphic mappings between complex manifolds, the rank of the mapping, the rank of the Jacobian matrix of the mapping when described in terms of any local coordinate systems, is a very convenient tool. The simplest class of holomorphic mappings, the finite holomorphic mappings, can be characterized as those holomorphic mappings for which the level sets are all of dimension zero.
