Domain of holomorphy

Summary

A domain of holomorphy is an open set in $\C^n$ that is the natural domain of its holomorphic functions. No part of its boundary can be crossed by all of these functions at once. In one variable every open set has this property, but for $n \ge 2$ many open sets do not, by the Hartogs phenomenon. Domains of holomorphy are characterised by holomorphic convexity and, by the solution of the Levi problem, by pseudoconvexity. They are the domains in $\C^n$ on which the global theory of holomorphic functions works best.