References for generalised Runge theorems

Question

I am currently working on a problem involving uniformly approximating locally defined holomorphic maps by globally defined ones. I am well aware of Runge's theorem for holomorphic maps of one complex variable, but I was wondering if there are generalisations of this theorem to several complex variables and if there are any textbooks for which these generalisations can be found. So for example, I would like to find an affirmative result to "if $f: U \to \mathbb{C}$ is holomorphic on an open set in $\mathbb{C}^2$, can I find a holomorphic map $g: \mathbb{C}^2 \to \mathbb{C}$ that uniformly approximates $f$ on compact subsets of $U$?"

Personally, I'm not particularly familiar with several complex variable theory, so ideally I'd much appreciate references that are more introductory than advanced. However, please don't hesitate to recommend texts on the more advanced side, because I am of the opinion that you can't ever have too many maths textbooks under your belt! Thanks!

Answer 1

The first such result you might encounter is the Oka-Weil approximation theorem.

Let $\Omega \subset \mathbb{C}^n$ be psuedoconvex, let $K$ be compactly contained in $\Omega$ and let $K$ be plurisubharmonically closed with respect to $\Omega$. If $f \in \mathcal{O}(\Omega')$ with $K$ compactly contained in $\Omega'$ which is open and compactly contained in $\Omega$, then $f$ can be uniformly approximated on $K$ by functions in $\mathcal{O}(\Omega)$.

https://en.wikipedia.org/wiki/Oka%E2%80%93Weil_theorem

As you can see, even the statement of the theorem is pretty technical. You don't get anything so pretty as Runge's theorem. In Several Complex Variables the geometry of the domain plays a much more important role. In a single complex variable every open set is a "domain of holomorphy" but this is not true in SVC.

There are many proofs of this theorem, but all of them rely on some pretty advanced machinery. Several Complex Variables is much harder than a single complex variable. One often needs to rely on some serious commutative algebra and sheaf theory, on functional analysis (Hörmander's approach to solving $\bar{\partial}$ problems), and there is always some differential geometry lurking in the background.

You could learn this stuff from a "sheafy" perspective from Joseph Taylor's Several Complex Variables book, or from a more "analytic" perspective from Steven Krantz'z book.

There is a whole industry of finding generalizations of this kind of approximation theorem! My thesis was a very small step in this direction (an $\mathcal{L}^2$ approximation theorem for domains with a Stein neighborhood basis).