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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.

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See R. Narasimhan, Several Complex Variables, published as part of Chicago Lectures in Mathematics. It is quite a small book and discusses precisely the most important topics. –  Anweshi Jan 2 '10 at 21:43
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3 Answers 3

up vote 11 down vote accepted

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.

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I hadn't read Greg's answer when I wrote mine. It's quite encouraging that we get several common themes, but with different emphasis. –  Georges Elencwajg Dec 14 '09 at 10:30
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Along with TSS, look at G-R's companion book (self-contained) "Coherent Analytic Sheaves". One lesson it shows is the usefulness of commutative algebra (applied to the noetherian local rings) in the study of complex-analytic spaces, and the simplifications attained by leaving the world of manifolds and allowing the objects of study to have singularities (super-useful for reduction steps in various proofs, as well as to not get disoriented by arbitrary intersections, zero loci, etc.). –  BCnrd May 26 '10 at 6:38
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The question seems like something a red herring, because these different types of domains aren't really all that different. A Stein manifold is a holomorphically convex manifold, which also has enough holomorphic functions to separate points. A Stein domain is a technical generalization of a Stein manifold to domains with boundary.

To keep things simple, you can make it your goal to understand Stein manifolds. They are the manifolds that act like $\mathbb{C}^n$ itself in the theory of several complex variables. In one definition, a Stein manifold one which embeds as a closed analytic submanifold of $\mathbb{C}^n$ for some $n$.

Which functions should you study? Obviously you should study holomorphic functions. The real and imaginary parts of a holomorphic function are pluriharmonic functions; that is a strong reason to define that class of functions. Plurisubharmonic functions are a generalization of pluriharmonic functions that are used to establish geometric control. They are important, but my impression as an outsider is that you are not obliged to study them in their own right; you can wait for them until you need them.

For example, given a complex manifold, how would you conclude that it does embed in $\mathbb{C}^n$? The theorem is that the embedding definition of a Stein manifold is equivalent to the plurisubharmonic definition.

I think that the extension theorems are semi-separate topic from classes of domains. Each of these extension theorems looks cool to me, but I see no particular need to learn them any faster than one at a time. For instance, Hartogs' extension theorem is special to two or more dimensions, but the notion of a Stein manifold is the same in all dimensions.

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If you want an "extension theorem" that is not about pseudoconvexity, Bogolyubov's edge of the wedge theorem is a nice example. In all other respects, I agree with Greg.

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