Multi-variable Complex Analysis and Domains of Holomorphy

Abstract

Unlike the 1-dimensional situation, the complex-analytic geometry of domains in complex n-space ℂⁿ is very complicated and exhibits many different phenomena, even among simply-connected domains of holomorphy. Therefore this chapter is devoted to a detailed description of the various geometric situations arising in the context of Toeplitz operators. A common theme is the emphasis on suitable Lie group actions on the domains, in particular the so-called polar decomposition, which will be crucial in later chapters.