ABSTRACT
The ring M V A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750056/fc9af0ab-d5fc-41da-85f5-3bf0a7c9020e/content/eq3746.tif"/> of germs of meromorphic functions at a point A on a holomorphic variety V was defined in section B as the total quotient ring of the local ring V O A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750056/fc9af0ab-d5fc-41da-85f5-3bf0a7c9020e/content/eq3747.tif"/> of the variety V at A. It is thus the ring of all formal quotients f′/f″ of elements of V O A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750056/fc9af0ab-d5fc-41da-85f5-3bf0a7c9020e/content/eq3748.tif"/> for which f″ is not a zero-divisor, with the usual notion of equivalence and the usual ring operations. As the terminology suggests, this ring can be interpreted as the ring of germs of some globally defined meromorphic functions. The aim of the present discussion is to introduce these functions and to examine some of their basic properties. This endeavor is considerably easier on regular varieties than on arbitrary varieties, where there are some interesting subtleties at the singular points, so the present discussion will be limited to the case of regular varieties—primarily indeed just to the space ℂ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750056/fc9af0ab-d5fc-41da-85f5-3bf0a7c9020e/content/eq3749.tif"/> itself. Even in that case some care must be taken, since for example a quotient f′/f″ of holomorphic functions is not really at all well defined as a complex-valued function at points at which f″ vanishes. That suggests introducing the following definition, keeping as close as possible to the classical feeling that a meromorphic function should really after all be a function.
