Several complex variables and analytic spaces

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• ConceptSeveral complex variables

  A function of several complex variables is a function $f(z_1, \dots, z_n)$ whose arguments are $n$ complex numbers, that is, a function on a subset of $\C^n$. The theory of several complex variables studies such functions, mainly the holomorphic ones, and the domains and spaces on which they live. Many facts of one complex variable carry over, such as power series expansions and the Cauchy integral formula on a polydisc. Other facts change deeply when $n \ge 2$. Zeros are never isolated, some domains force holomorphic functions to extend to larger domains, and the unit ball and the polydisc are not biholomorphic.

• ConceptHolomorphic function of several variables

  A holomorphic function of several variables is a function on an open subset of $\C^n$ that is complex differentiable at every point. It extends the notion of a holomorphic function of one variable. Such a function is holomorphic in each variable separately and infinitely differentiable. It is locally the sum of a convergent power series in several variables. These functions are the main objects of the theory of several complex variables.

• ConceptPolydisc

  A polydisc in $\C^n$ is a product of $n$ open discs of the complex plane, one in each coordinate. It is the basic domain of several complex variables, playing the role that the disc plays in one variable. The Cauchy integral formula on a polydisc follows from the one-variable formula applied in each coordinate, and it leads to power series expansions. Its distinguished boundary is a torus, much smaller than its topological boundary.

• ConceptDistinguished boundary of a polydisc

  The distinguished boundary of a polydisc $P(a, r)$ in $\C^n$ is the torus of points with $\abs{z_j - a_j} = r_j$ for every $j$. It is the set over which the Cauchy integral formula on a polydisc integrates. A function holomorphic on the polydisc and continuous on its closure attains its largest modulus on this torus. It is the Shilov boundary of the algebra of such functions, and for $n \ge 2$ it is a small part of the topological boundary.

• ObjectUnit ball in C^n

  Let $n \ge 1$. The unit ball of $\C^n$ is $$\mathbb{B}^n = \set{z \in \C^n : \abs{z_1}^2 + \cdots + \abs{z_n}^2 < 1}.$$ It is the open ball of radius $1$ about $0$ for the norm $\abs{z} = (\sum_j \abs{z_j}^2)^{1/2}$. Writing $z_j = x_j + i y_j$, the condition becomes $\sum_j (x_j^2 + y_j^2) < 1$, so $\mathbb{B}^n$ is the Euclidean unit ball of $\R^{2n}$.

• TheoremCauchy integral formula on a polydisc

  Let $a \in \C^n$ and $r = (r_1, \dots, r_n)$ with each $r_j > 0$. Let $P = P(a, r)$ be the polydisc with centre $a$ and polyradius $r$, and let $T = T(a, r)$ be its distinguished boundary. Let $f$ be a holomorphic function on an open set of $\C^n$ that contains the closed polydisc $\overline{P}$. Then for every $z \in P$, $$f(z) = \frac{1}{(2\pi i)^n} \int_{T} \frac{f(\zeta)}{(\zeta_1 - z_1) \cdots (\zeta_n - z_n)} \, d\zeta_1 \cdots d\zeta_n,$$ where the integral is the torus integral over $T$.