Holomorphic extension and domains of holomorphy in several variables

Let $\mathrm{\Omega },\tilde{\mathrm{\Omega }}\subseteq {\mathbb{C}}^m$ be domains with $\mathrm{\Omega }\cap \tilde{\mathrm{\Omega }}\mathrm{\ne }\mathrm{\varnothing }$, and let $f:\mathrm{\Omega }\to \mathbb{C}$ be holomorphic (Holomorphic functions on an open subset of ${\mathbb{C}}^m$).

A holomorphic function $\tilde{f}:\tilde{\mathrm{\Omega }}\to \mathbb{C}$ is a holomorphic extension of $f$ to $\tilde{\mathrm{\Omega }}$ when there is a nonempty open set $W\subseteq \mathrm{\Omega }\cap \tilde{\mathrm{\Omega }}$ such that $\tilde{f}=f$ on $W$.

A domain $\mathrm{\Omega }$ is a domain of holomorphy when there do not exist domains $U_1,U_2\subseteq {\mathbb{C}}^m$ with

$\mathrm{\varnothing }\mathrm{\ne }{U}_1\subseteq U_2\cap \mathrm{\Omega },U_2\subseteq ̸\mathrm{\Omega },$

such that every holomorphic $f\in \mathcal{O}(\mathrm{\Omega })$ admits a holomorphic extension $F_f\in \mathcal{O}(U_2)$ satisfying $F_f=f$ on $U_1$.

This page uses the simultaneous-extension convention. To show that a domain is not a domain of holomorphy it is enough to find one fixed overlap $U_1\subseteq U_2\cap \mathrm{\Omega }$ from which every holomorphic function on $\mathrm{\Omega }$ extends to $U_2$. The continuation is part of the datum for each function, but the witnessing pair $U_1,U_2$ is common.

Agreement propagates only on a common connected domain. If two holomorphic functions are both defined on one connected domain and agree on a nonempty open subset, the several-variable identity theorem forces agreement there. In the definition above, however, $\mathrm{\Omega }\cap \tilde{\mathrm{\Omega }}$ can be disconnected: agreement on one component need not imply agreement on another. The witnessing overlap $W$ is therefore part of the extension datum and cannot in general be changed arbitrarily.