Pseudoconvexity, the Levi Problem and Vanishing Theorems

Abstract

Convexity was introduced in complex analysis by E.E. Levi around 1910 when he discovered that the (smooth) boundary of a domain of holomorphy in ℂn is not arbitrary but satisfies a certain condition of pseudoconvexity. The question whether conversely such a pseudoconvex domain is a domain of holomorphy became famous as the so-called “Levi problem” and influenced the development of complex analysis over several decades. The Levi problem was first solved in two variables by Oka (1942) and then in general by Oka, Norguet and Bremermann in the early fifties. A main tool for its solution was provided by the notion of plurisubharmonic (sometimes called pseudoconvex) functions. This generalizes the notion of subharmonic functions in one variable. Plurisubharmonic functions are today indispensable for complex analysts.KeywordsVector BundleLine BundleComplex SpacePseudoconvex DomainPlurisubharmonic FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.