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I have 3 general abstract reasons to care about complex analysis in a single variable:

  1. The laplacian is, up to a constant multiple, the only isometry invariant PDO in the plane, and so it is abstractly very important. Holomorphic functions are intimately related to harmonic functions, so holomorphic functions are important.

  2. Holomorphic mappings are conformal if they have nonvanishing derivatives

  3. Many results which are real variable in nature are most easily understood in the light of complex analysis (factorization of real polynomials, radius of convergence of real power series, etc)

I currently have no such justification for several complex variables. The connection to harmonic functions mostly breaks down. Conformality breaks down. I have not seen applications to real variable phenomena.

Can anyone give me some insight into the big picture here? Where do the phenomena studied in several complex variables (domains of holomorphy and their ilk) come up in the rest of mathematics?

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Complex algebraic geometry? –  Qiaochu Yuan Nov 27 '12 at 4:15
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@fedja - I, and I suspect a lot of other mathematicians, like to have a general feeling for where and why the different fields of mathematics are useful. I may not have much use for cohomology in my day to day mathematical life, but I know that if I am confronted with a local to global problem, searching for a cohomological interpretation will be useful. Or if my problem is invariant under the action of some group, the representation theory of that group will probably be useful. I know SCV is an important field, I just don't know why yet. Maybe someone here can provide some insight. –  Steve Nov 27 '12 at 4:41
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"I have not yet seen a real application of SCV in complex algebraic geometry": you can look for example at almost any paper by J-P Demailly –  YangMills Nov 27 '12 at 5:25
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@Steve There are many ways to ask such questions and you chose the one to which my instinctive reaction was "And what motivation do I have to motivate you?". You see, "useful" is a dangerous word ("interesting" and "important" are other two). What interests me should not necessarily interest you and vice versa. There is no common "why". As to "where" I once used the Hormander L^2 bounds to reprove the Bourgain-Milman estimate for the volume product of a symmetric convex body, so I would say "Don't classify, grab whatever you can and try to get away with it". OK, I'll think of a better answer. –  fedja Nov 27 '12 at 5:37
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As far as I'm concerned, one-variable complex analysis doesn't need any "general abstract reasons" to justify caring about it. No matter which direction you come at it from, it's beautiful and somehow inevitable. Based on that alone, I trust that SCV is also beautiful and inevitable. I've never got around to studying SCV, but I wish I had. –  Tom Goodwillie Nov 27 '12 at 13:07
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On my opinion, there are two very general reasons why analytic functions are important.

  1. Solutions of many (almost all) important differential (and functional) equations are analytic. For example, all elementary and special functions arise in this way. Moreover, they are also usually analytic functions of parameters.

  2. Harmonic analysis. Fourier (Laplace) transforms. This includes generating functions as a special case. They are in many cases analytic.

I think almost every occurence of analytic functions in mathematics and in real life can be traced to one of these two general sources.

Both these fundamental motivations apply not only to analytic functions of one variable but to analytic functions of several variables.

Historically, it seems that the first analytic functions of several variables that were studied were Abelian functions, which occur in the inversion problem of Abelian integrals (that is they occur as solutions of certain systems of differential equations). These differential equations appear both in pure mathematics (inversion of Abelian integrals) and in physics (integrable systems). They were studied long before the general theory of several complex variables was developed.

Then comes the second source (already in XX century): Wiener-Paley theorem and its multi-dimensional generalizations, Fourier analysis, which itself was developed as a method of solving PDE.

You specifically mention applications in algebraic geometry? Just open Griffiths Harris book! And for applications in "real analysis", look at least at the table of contents of Hormander's Linear differential operators!

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Thank you for this answer. I will have to take some time to study and digest it. Maybe it would have been better to put a focus on domains of holomorphy in my question. Much of the research going on in SCV seems to deal with characterizing these domains. –  Steve Nov 28 '12 at 4:39
    
For an application of domains of holomorphy to physics, look under the title "Edge-of-the-wedge Theorem(s)". –  Alexandre Eremenko Dec 19 '12 at 22:16
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The reason I care about functions with several complex variables is the resolvent formalism. To solve a problem in linear algebra, you translate it into a problem in complex analysis (with several variables) and allow tools like Cauchy's Theorem and the Argument Principle (for functions of a single complex variable) to chew it up. The nicest proof I know that a matrix has a Jordan normal form proceeds in this way, and is nicely explained HERE. I presented this proof in the introductory Complex Analysis course I have just finished teaching as an illustration of the power of complex analysis, never mind the number of variables!

The nicest thing about resolvents is that they work just as well when the dimension is infinite (studying operators on Hilbert spaces instead of plain old linear transformations on finite dimensional vector spaces). They are thus a basis for Fredholm Theory.

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Thanks for showing this; it's a great proof that I hadn't seen before! However, isn't this just single variable complex analysis? –  Ralph Furmaniak Nov 27 '12 at 19:35
    
Indeed it seems that functional calculus (taking analytic functions of linear operators) does not fit to any of the "two sources of complex analysis" which I mentioned in my answer. Is this really a "third, independent source" of importance of complex analysis? –  Alexandre Eremenko Nov 27 '12 at 21:28
    
@Ralph Furmaniak: The resolvent has multiple complex variables (z is a vector), but, using a bit of analysis, the linked proof that a matrix has a Jordan normal form translates the problem into multiple problems each involving a single complex variable. Multiple single variable complex analysis ;-) –  Daniel Moskovich Nov 28 '12 at 1:51
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@Daniel - this "multiple single variable complex analysis" is mostly what I have seen applied - not the scv phenomena which are unique to that field. I think the proof you have offered is awesome though! –  Steve Nov 28 '12 at 4:37
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To say that $F$ is holomorphic is to say in a sense that it depends on only "half" of the variables $(\partial F/\partial\bar z_j=0)$. This is analogous to the wave functions of quantum mechanics depending only on $q$, i.e. only on "half" the phase space variables $(p,q)$. The analogy is made precise by the theory of polarizations in geometric quantization and the orbit method.

For instance, a reductive Lie group has some of its irreducible unitary representations attached to hyperbolic coadjoint orbits (consisting of matrices with real eigenvalues), others to elliptic orbits (consisting of matrices with imaginary eigenvalues). While the former admit real polarizations which lead to representations in $L^2$ sections over a space half the dimension (i.e., real analysis), the latter admit complex polarizations which lead to representations in holomorphic sections over the orbit (or Dolbeault cohomology thereof; i.e., complex analysis).

A classic example of the elliptic story is the Borel-Weil(-Bott) realization of all irreducible representations of compact Lie groups. This involves holomorphic functions on the complexified group in an essential way.

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To complement Alexandre's answer, there is an application of SCV in physics, in quantum field theory. One can prove that certain quantities ($n$-point functions $G(x_1,\ldots,x_n)$) appearing in (free) quantum field models are analytic in their $n$ $R^4$ arguments (while their Fourier transforms are essentially hyperfunctions). The analytic continuation into $n$ $\mathbb{C}^4$ arguments of the $n$-point functions and their Fourier transforms then allows one to prove some physically important results. For example, the proof of the spin-statistics theorem (fermions have half-integral spin, and bosons have integral spin) in the classic book of Streater & Wightman uses some results from SVC (Edge of the Wedge theorem and some analytic properties of Fourier transforms).

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So far, the motivations have been mostly of analytic flavor. Let me add an alien motivation.

Suppose you were an alien that only knows the complex number field $\mathbb{C}$ as a fundamental object (say, because it's Cauchy complete and algebraically closed) and considers $\mathbb{R}$ just as a minor object (much like we consider -say- the Eisenstein integers). And suppose you came up with the concept of increment and of derivative and then of differential: then, which Analysis would you study? And suppose you were able to imagine the notion of a manifold as a space with local coordinates in your favourite field: which geometry would you naturally study?

Of course, the answers are, respectively, "Complex Analysis" and "Holomorphic manifolds".

So, even if you're a Terrestrial, blinded by notions of linear order, isn't $\mathbb{C}$ a very fundamental object anyway? If so, then doing SCV and Complex Analytic Geometry is just doing the natural things with it.

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Even an alien lives in the same universe as us, in which time is noticeably $1$-real-dimensional. –  Qiaochu Yuan Nov 27 '12 at 19:06
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Yes, of course my answer was semi-tongue-in-cheek. The idea I wanted to convey is that, if you choose to start from $\mathbb{C}$ (which is doubtlessly a fundamental object) and try to play the same games you play over $\mathbb{R}$, then you are kind of forced to discover SCV and CG. –  Qfwfq Nov 27 '12 at 21:46
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Qiaochu: Is time really 1-real-dimensional? en.wikipedia.org/wiki/Imaginary_time Not to mention that time might be discrete... I'm not sure about bifurcations either (does time itself branch? Or does the world branch in time?). –  Daniel Moskovich Nov 28 '12 at 1:57
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Somebody told me the other day that he saw a popular science show on TV where a physicist was arguing that time did not exist. But that's not the same as saying that it's not real. –  Tom Goodwillie Nov 29 '12 at 15:07
    
Also, in a real world things are not always fair and equal, and we have to face $x < y$ :-) –  Suvrit Nov 29 '12 at 18:43
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Siegel developed the theory of holomorphic functions of several complex variables in order to study hermitian symmetric spaces (especially the Siegel modular space).

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If those are your reasons for caring about complex analysis, then I'm not sure to what extent I can convince you to care about SCV.

But here is why I like SCV: I like multivariable calculus, and I like complex numbers. I've always wondered what would happen if you put the two together. And apparently pretty wild things happen (compared with either the single-variable or real-variable case).

I'm also really interested in complex geometry. The local theory of complex manifolds involves SCV.

One final comment: Your reason (1) for caring about holomorphic functions is in fact my reason for caring about harmonic functions (for which I would otherwise desire motivation).

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Jesse, your second paragraph expresses my own feelings very well, but I have to argue with the last bit. There are lots of reasons for caring about harmonic functions, and not just in two variables or two dimensions. For one thing, they are local extrema for the energy (integral of $|grad f|^2$). –  Tom Goodwillie Nov 28 '12 at 14:54
    
That's fair. In truth, I'm still rather ignorant about harmonic functions and their properties. I don't mean to suggest that reason (1) is the only reason one would care about them, but simply that it's a primary reason that I do (again, given my ignorance). –  Jesse Madnick Nov 28 '12 at 23:04
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