Fundamental motivation for several complex variables [closed]

Question

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?

Answer 1

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!

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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).

Answer 6

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).

Answer 7

Siegel developed the theory of holomorphic functions of several complex variables in order to study hermitian symmetric spaces (especially the Siegel modular space).