several complex variables and Hartogs's phenomenon
/ HAR-tohgs /
What happens when you study holomorphic functions of more than one complex variable — functions f(z_1, z_2, ..., z_n) that are holomorphic in each variable separately? You might expect the theory to be 'one-variable complex analysis, repeated n times.' It is not. In two or more variables genuinely new and surprising things happen, and the first and most famous of them is Hartogs's phenomenon: in dimension two and above, holomorphic functions are forced to extend across small holes, so you cannot place an isolated singularity wherever you like the way you can on the plane.
Here is Hartogs's extension theorem in plain terms. Take a region in C^n (n at least 2), remove a small compact lump from its interior, and let f be holomorphic on what remains. Then f automatically continues holomorphically across the removed lump — the singularity fills itself in. In one variable this is wildly false: 1/z is holomorphic on the plane minus the origin and refuses to extend over that point. But in two variables a function holomorphic on a neighborhood minus a point must extend; there are no isolated singularities, and in fact zeros and singular sets are always 'large' (of complex codimension one). This forces a central new concept: a domain of holomorphy, a region that is the natural maximal home of some holomorphic function that genuinely cannot be extended past its boundary. In one variable every region is a domain of holomorphy; in several variables most regions are not, and characterizing the ones that are (they are exactly the pseudoconvex domains, by the solution of the Levi problem) is a core theorem of the subject.
This is the higher-dimensional frontier of complex analysis, and it is a living, geometric subject rather than a footnote. It grows into the theory of complex manifolds, sheaf cohomology, and complex algebraic and differential geometry — the language of much of modern mathematics and of string theory's Calabi-Yau spaces. The honest message is one of humility: the one-variable intuitions you have carefully built — isolated zeros and poles, Laurent expansions about a point, the Riemann mapping theorem (which fails completely, since the ball and the polydisk in C^2 are not biholomorphic) — do not transfer wholesale. Several complex variables is where complex analysis becomes geometry, and where many questions remain open.
In C^2 there is no function playing the role 1/z plays in C: you cannot make a holomorphic function on a ball that is singular only at the center, because Hartogs forces it to extend across that point. The natural singular sets in two variables are instead whole complex curves, like the set where z_1 = 0 (a one-complex-dimensional surface), never a single isolated point.
In C^2 and above, holomorphic functions fill in isolated holes — no isolated singularities.
Hartogs extension needs dimension at least 2 and a compact lump whose complement is connected near it; it is NOT a contradiction of the one-variable theory but a genuinely new dimensional effect, and it is precisely why the Riemann mapping theorem and isolated-singularity classification have no straightforward several-variable analogue.