Domain of Holomorphy

Question

I found the following definition of domain of holomorphy in several places.

Def1: A connected open set $\Omega$ in the n-dimensional complex space ${\mathbb{C}}^n$ is called a domain of holomorphy if there do not exist non-empty open sets $U \subset \Omega$ and $V \subset {\mathbb{C}}^n$ where $V$ is connected, $V \not\subset \Omega$ and $U \subset \Omega \cap V$ such that for every holomorphic function $f$ on $\Omega$ there exists a holomorphic function $g$ on $V$ with $f = g$ on $U$.

From what I understand, intuitively speaking, $\Omega$ is a domain of holomorphy if we can find a function $g$ which is holomorphic on $\Omega$ such that it cannot be extended beyond the boundary of $\Omega$. Naively thinking, I would have written down the following definition for domain of holomorphy.

Def2: A connected open set $\Omega\subset \mathbb{C}^n$ is a domain of holomorphy if there is a $g$ which is homolorphic on $\Omega$ such for that any open $V\subset \mathbb{C}^n$ with $V\cap \partial\Omega\neq\phi$ there is no holomorphic function $F$ on $V$ with $F\vert_{V\cap\Omega}=g\vert_{V\cap\Omega}$.

Could someone please explain to me the need for the more complicated definition 1.

Answer 1

Further to Ben's answer, it might be useful to picture the situation in $\mathbb{C}$. (Of course in $\mathbb{C}$ every domain is a domain of holomorphy, but we can still exhibit the same phenomenon that causes us to need the more complicated definition.)

The principal branch of the logarithm $f := \operatorname{Log}$ is defined on the slit plane $\Omega :=\mathbb{C}\setminus (-\infty,0]$. The function $f$ does not extend holomorphically, or even continuously, to any point of $(-\infty,0]$.

However, let $x\in (\infty,0)$, set $\delta := -x$, and consider the disk $V := D(x,\delta)$ and the half-disk $$ U := \{z\in V: \operatorname{Im} z > 0\}.$$

Then the restriction of $f$ to $U$ extends analytically to a holomorphic function $g$ on $V$, where $g(z)=f(z)$ if $z\in U$, $g(z) = \log(-z)+i\pi$ if $z$ belongs to the diameter $(2x,0)$ of $V$, and $g(z)=\operatorname{Log}(z) + 2\pi i$ otherwise.

The point is that the slit plane is not a "natural" maximal domain of definition for the function $\operatorname{Log}$. (If we were looking for such a domain, it would be a spiralling Riemann surface spread over the punctured plane ...) In other words, suppose you have a domain of holomorphy for some analytic function, but you only know the germ of this function at some point. Then you can reconstruct the domain uniquely as the maximal domain to which the germ can be analytically continued.

Of course the interesting thing is that there are domains that are not domains of holomorphy, when $n\geq 2$.

Answer 2

Quoting Steven Krantz, Function Theory of Several Complex Variables, p. 6: the definition of domain of holomorphy is complicated because we must allow for the possibility (when dealing with an arbitrary open set $U$ rather than a smooth domain $\Omega$) that $\partial U$ may intersect itself.

Picture a cigar, and then bend it to make a necklace, making the two ends just touch. The interior of that set is an open set whose boundary, so to speak, touches itself.