Domains of Holomorphy

The purpose of this chapter is to solve affirmatively the first two of the Three Big Problems. We first define the notion of domains of holomorphy, and then introduce holomorphically convex domains. We prove the Cartan–Thullen Theorem claiming the equivalence of the two domains in \({\mathbf{C}}^{n}\) . We solve the Approximation Problem by Oka’s Joku-Iko Principle. We then formulate the Continuous Cousin Problem and prove that it is solvable on domains of holomorphy. We give the solutions of the Cousin Problem (I and II) with the Oka principle. Similarly, we deal with the \({\bar{\partial}}\) -equation, and the interpolation problem. Lastly, we introduce unramified domains over \({\mathbf{C}}^{n}\) , on which we discuss the obtained results.