VI Domains of holomorphy and pseudoconvexity

Abstract

At the end of Chapter I and in Chapter III we met open sets in ℂⁿ on which any holomorphic function can be extended to a larger open set. The open sets which do not have this property are called domains of holomorphy: in this chapter we study such open sets. We start by giving a characterisation of domains of holomorphy in terms of holomorphic convexity (the Cartan–Thullen theorem). We then introduce the notion of pseudoconvexity in order to get a more analytic characterisation of domains of holomorphy. This requires us to define plurisubharmonic functions. We then prove that every domain of holomorphy is pseudoconvex: the converse, which is known as the Levi problem, is studied in Chapter VII.