Chapter 1 Selected Problems in One Complex Variable The study of holomorphic functions of several complex variables involves the use of powerful tools from many areas of modern mathematics, areas such as commutative algebra, functional analysis, homological algebra, sheaf the- ory, and algebraic topology. For this reason, a course in several complex variables is a great opportunity to teach students how the seemingly sep- arate fields of pure mathematics can be used in concert to solve difficult problems and produce striking results. However, this fact also makes the study of holomorphic functions in several variables much more difficult and sophisticated than the study of other classes of functions - continuous func- tions, differentiate functions, holomorphic functions of a single variable - that students encounter in their early graduate work. When they begin to realize this, students tend to ask questions such as: Why are we developing all this machinery? Where is this headed? What is this good for? It is diffi- cult to answer these questions until much of the language and machinery of several complex variables has been developed. However, in this introductory section we will attempt to give some indication of where we are headed and why we are headed there, by discussing several problems from the theory of a single complex variable that illustrate some of the issues that will be central in the several variable theory. While the main purpose of this chapter is to illustrate and motivate what is to come, that is not its only purpose. Some of the results developed in this chapter will be needed later. This is true, for example, of the results on partitions of unity in section 1.3 and those on the inhomogeneous Cauchy- Riemann equation in section 1.4. 1
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