ABSTRACT
Any holomorphic function in V can be viewed as a representative of a meromorphic function in V, and it is clear from the identity theorem for holomorphic functions on varieties that distinct holomorphic functions represent distinct meromorphic functions. Thus there is a natural inclusion OV???MV of OV as a subring of MV and consequently of course a natural inclusion C?? MV so that MV has the natural structure of a complex algebra. The zero element in the ring MV is the constant 0, and the unit element the constant 1. Before turning to a discussion of properties of meromorphic functions on arbitrary varieties, this chapter establishes the generally useful auxiliary result, which essentially means that as far as meromorphic functions are concerned, it is often possible to limit the consideration to irreducible holomorphic varieties. In some ways meromorphic functions are more flexible as well as being more plentiful than holomorphic functions.
