Envelopes of holomorphy and holomorphic convexity

Abstract

This paper is primarily a study of generalized notions of envelope of holomorphy and holomorphic convexity for special (algebraically restricted) subsets of C n {{\mathbf {C}}^n} and in part for arbitrary subsets of C n {{\mathbf {C}}^n} . For any special set S in C n {{\mathbf {C}}^n} , we show that every function holomorphic in a neighborhood of S not only can be holomorphically continued but also holomorphically extended to a neighborhood in C n {{\mathbf {C}}^n} of a maximal set S ~ \tilde {S} , the “envelope of holomorphy” of S, which is also a special set of the same type as S. Formulas are obtained for constructing S ~ \tilde {S} for any special set S. “Holomorphic convexity” is characterized for these special sets. With one exception, the only topological restriction on these special sets is connectivity. Examples are given which illustrate applications of the theorems and help to clarify the concepts of “envelope of holomorphy” and “holomorphic convexity."