For a new learner of several complex variables, the many domains (eg holomorphically convex, pseduconvex, Stein) and the many extension theorems (eg Riemann) and the many functions (plurisubharmonic) can be confusing.


Which domains, extension theorems and functions do you think are the most important for a learner to get up to reach in reading papers, and the most important relationships between these domains, extension theorems and functions?


As I am a new learner myself, if I question is too restrictive (eg other kinds of theorems are important too to understanding these domains) or too broad, please edit my question. Thank you in advance. 



Most important domains, extension theorems, and functions in several complex variables
Here are a few points to guide you into the beautiful subject you had the good taste to choose. 


1) Hartogs extension phenomenon :given two concentric balls in $ \mathbb C^n$, any holomorphic function $ B(0;M) \setminus B(0;m) \to \mathbb C$ extends to a holomorphic function $ B(0;M) \to \mathbb C$.This really launched the subject and showed that function theory in several variables is not just an extension of the theory in one variable.You should study a few such classes of examples. Key words: Hartogs figures, Reinhardt domains.


2) Domains for which such extensions do not exist are called holomorphy domains: balls are holomorphy domains but as we just saw "shells" $ B(0;M) \setminus B(0;m)$ are not.
If a region is not a domain of holomorphy, it has a holomorphic hull, but this is no longer included in $\mathbb C^n$ : you get étalé spaces ( Yes, you algebraic geometers out there, this is where they were introduced ! ). This is an important subject and you can test whether a domain is a holomorphy domain at its boundary. Key-words: Levi problem, plurisubharmonic functions. 


3)Holomorphic manifolds and Stein manifolds: these are abstractions of domains of $\mathbb C^n$ and holomorphic domains respectively . In retrospect they were inevitable because of the nature of holomorphy hulls (cf. 2). Stein manifolds are highly analogous to the affine varieties of algebraic geometers.


4) Sheaf theory, cohomology: these are all powerful techniques that you MUST master if you want to read anything at all in the subject. In particular you must understand coherent sheaves, which have a flavour of Noetherianness in them, but are a more subtle notion.
The most important result here is Cartan's theorem B : coherent sheaves have no cohomology in positive dimension.


To help you learn all this , I would recommend:


B.Kaup, L. Kaup: Holomorphic Functions of several Variables (de Gruyter). [Quite friendly]


H.Grauert R.Remmert: Theory of Stein Spaces (Springer). [The ultimate source by the Masters]


All my wishes for success in your study of complex geometry.


