ABSTRACT
Some holomorphic varieties V there are subvarieties W?V and uniformly bounded holomorphic functions on V – W that do not extend to holomorphic functions at points of W. This chapter discusses such functions, which are quite important in the investigation of holomorphic varieties. The extended form of Riemann's removable singularities theorem holds on complex manifolds. Since the extended form of Riemann's removable singularities theorem holds on normal varieties, so also do many of the consequences of that theorem, results derived and used so far only for regular varieties, as well as versions of these results that have separate implications about the singularities of normal varieties. A normal variety is irreducible—only that it is locally irreducible. A reducible normal variety must therefore be a disjoint union of its irreducible components, each of which is also normal.
