This version: 28/02/2014 Chapter 6

and real analytic manifolds In Corollary 4.2.11 we saw that there will generally be significant restrictions on the character of holomorphic functions on holomorphic manifolds. In Chapter 3 we saw that there are domains in Cn (these are holomorphic manifolds, of course) for which it is possible to find holomorphic functions satisfying finitely many algebraic conditions. This raises the question of how one can distinguish between manifolds with few holomorphic functions and those with many holomorphic functions. It is this question we study in this chapter, adapting the methods of Chapter 3. The theory we present in this chapter originated in the work of Stein [1951], building on work of many, including [Behnke and Thullen 1934, Cartan 1931, Cartan and Thullen 1932, Oka 1939], Indeed, it is difficult to ascribe the development of the theory of so-called “Stein manifolds ” to one person. All of this aside, a detailed presentation of the theory is given by Grauert and Remmert [2004]. As with the material in the strongly related Chapter 3, the treatment in this chapter is mostly holomorphic. In Section 6.4 we apply holomorphic techniques to prov