About

The theory of several complex variables studies holomorphic (or analytic) functions defined over $\mathbb{C}^n$, where $n > 1$. Unlike the $n=1$ case, when $n > 1$ there is a strong lemma of Hartogs which states that all isolated singularities are removable and in particular there are domains that are not the domains of existence for holomorphic functions. In particular, there is a lot of interplay between the geometry of a domain $\Omega \subset \mathbb{C}^n$ and the function theory on $\Omega$.

Hartogs' lemma is just one of the many instances where analysis in several complex variables behaves very differently from complex analysis of a single variable. As an additional example, in one complex variable, Riemann's mapping theorem states that any simply connected domain (except the plane $\mathbb{C}$ itself) is biholomorphically equivalent to the unit disc. In several variables, there is nothing like Riemann's mapping theorem. The unit ball and the polydisc are for example not biholomorphically equivalent. In fact, an arbitrarily small perturbation of the unit ball is almost certainly not biholomorphic to the ball.

In real analysis, the theory in one and many dimensions generally behave similarly, except for when the algebraic structure of the real line as an ordered field comes into play, but as the examples above illustrate, the situation is very different in complex analysis. Therefore several complex variables is usually regarded as a distinct subject from complex analysis.

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