Domain of holomorphy

For a domain ${\displaystyle \Omega }$  the following conditions are equivalent:

1. ${\displaystyle \Omega }$  is a domain of holomorphy
2. ${\displaystyle \Omega }$  is holomorphically convex
3. ${\displaystyle \Omega }$  is pseudoconvex
4. ${\displaystyle \Omega }$  is Levi convex - for every sequence ${\displaystyle S_{n}\subseteq \Omega }$  of analytic compact surfaces such that ${\displaystyle S_{n}\rightarrow S,\partial S_{n}\rightarrow \Gamma }$  for some set ${\displaystyle \Gamma }$  we have ${\displaystyle S\subseteq \Omega }$  (${\displaystyle \partial \Omega }$  cannot be "touched from inside" by a sequence of analytic surfaces)
5. ${\displaystyle \Omega }$  has local Levi property - for every point ${\displaystyle x\in \partial \Omega }$  there exist a neighbourhood ${\displaystyle U}$  of ${\displaystyle x}$  and ${\displaystyle f}$  holomorphic on ${\displaystyle U\cap \Omega }$  such that ${\displaystyle f}$  cannot be extended to any neighbourhood of ${\displaystyle x}$ 

Implications ${\displaystyle 1\Leftrightarrow 2,3\Leftrightarrow 4,1\Rightarrow 4,3\Rightarrow 5}$  are standard results (for ${\displaystyle 1\Rightarrow 3}$ , see Oka's lemma). The main difficulty lies in proving ${\displaystyle 5\Rightarrow 1}$ , i.e. constructing a global holomorphic function which admits no extension from non-extendable functions defined only locally. This is called the Levi problem (after E. E. Levi) and was first solved by Kiyoshi Oka, and then by Lars Hörmander using methods from functional analysis and partial differential equations (a consequence of ${\displaystyle {\bar {\partial }}}$ -problem).

• If ${\displaystyle \Omega _{1},\dots ,\Omega _{n}}$  are domains of holomorphy, then their intersection ${\displaystyle \Omega =\bigcap _{j=1}^{n}\Omega _{j}}$  is also a domain of holomorphy.
• If ${\displaystyle \Omega _{1}\subseteq \Omega _{2}\subseteq \dots }$  is an ascending sequence of domains of holomorphy, then their union ${\displaystyle \Omega =\bigcup _{n=1}^{\infty }\Omega _{n}}$  is also a domain of holomorphy (see Behnke-Stein theorem).
• If ${\displaystyle \Omega _{1}}$  and ${\displaystyle \Omega _{2}}$  are domains of holomorphy, then ${\displaystyle \Omega _{1}\times \Omega _{2}}$  is a domain of holomorphy.
• The first Cousin problem is always solvable in a domain of holomorphy; this is also true, with additional topological assumptions, for the second Cousin problem.

• Behnke–Stein theorem
• Levi pseudoconvex
• solution of the Levi problem
• Stein manifold

• Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.
• Boris Vladimirovich Shabat, Introduction to Complex Analysis, AMS, 1992

This article incorporates material from Domain of holomorphy on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.