Holomorphic Functions on Varieties

This chapter discusses a number of the standard properties of holomorphic functions on open subsets of Cⁿ that genuinely do not hold for holomorphic functions on arbitrary holomorphic varieties, one of the most basic being the extended form of Riemann's removable singularities theorem. Each irreducible component of a compact holomorphic variety is also compact, so any holomorphic function being continuous must attain its maximal value on each component. The chapter presents several results showing that certain families of submodules of families of free modules are locally finitely generated. The families considered were all essentially constructed in terms of a given finite collection of holomorphic functions, but in some circumstances the finiteness is not really required. Another category of semilocal results arises in the study of holomorphic functions themselves, rather than just germs of holomorphic functions, on varieties.