Complex Analytic and Differential Geometry 2024, 48: Domains of Holomorphy

A domain of holomorphy is a complex open subset so that there is no part of its boundary across which all holomorphic functions can be extended. Every connected open set U meeting this boundary and every component V from the intersection has a holomorphic function which has no holomorphic extension from V to U. For n ≥ 2, and a connected open set w in n-1 dimensional complex space, and ω' an open subset of ω, then

A complex analytic manifold X and closed submanifold S with codimension ≥ 2, all holomorphic functions in X\S extend holomorphically to X. For complex manifolds X, and a compact subset K of X, the holomorphic holl of K in X is

A complex manifold X is holomorphically convex if the holomorphic hull of every compact set is compact. An open subset of complex space, domains of holomorphy are holomorphically convex, there is an holomorphic function on it which is unbounded on any neighborhood of any point on its boundary, and all countable subsets within that has no accumulation points has an holomorphic interpolation function that maps onto an arbitrary sequence of complex numbers.

A function on a topological space X is an "exhaustion" if all sublevel sets are relatively compact. It tends to infinity relatively to the filter of complements X \ K of compact subsets of X. A complex n-dimensional manifold is "weakly pseudoconvex" if there is a smooth PSH exhaustion, and "strongly pseudoconvex" if there is a smooth strictly PSH exhaustion function. Holomorphically convex manifolds are weakly pseudoconvex. The class of holomorphically convex manifolds contains domains of holomorphy, and compact complex manifolds. A complex manifold is a "Stein manifold" if it's holomorphically convex and its space of holomorphic functions locally separates points of the space. They are strongly pseudoconvex.

Holomorphic (pseudo)convexity are preserved under space multiplication. Closed complex submanifolds of (pseudo)convex spaces experience the same type of convexity. A collection of uniformly (pseudo)convex submanifolds of a complex manifold X whose intersection is a submanifold of X retains their (pseudo)convexity.