Functions of Several Complex Variables — Mathematics Knowledge System

Overview

The theory of functions of several complex variables (SCV) extends one-variable complex analysis to $\mathbb{C}^n$ for $n \geq 2$. Many results from one variable generalize, but significant new phenomena appear: the Hartogs extension theorem shows that isolated singularities cannot exist in $n \geq 2$, and domains of holomorphy replace the arbitrary domains of one-variable theory.

Intuition

In one complex variable, you can prescribe a region with an isolated singularity on the boundary. In $\mathbb{C}^n$ with $n \geq 2$, singularities cannot be isolated: if a function is holomorphic outside a small ball in $\mathbb{C}^n$, Hartogs's theorem forces it to extend holomorphically inside the ball. This rigidity means the geometry of domains of holomorphy in $\mathbb{C}^n$ is far more constrained and interesting than in $\mathbb{C}$.

Formal Definition

Definition

A function $f: U \subset \mathbb{C}^n \to \mathbb{C}$ is holomorphic if it is separately holomorphic in each variable (Hartogs's theorem then implies joint continuity and joint holomorphicity). The Cauchy–Riemann equations must hold in each variable simultaneously.

\frac{\partial f}{\partial \bar{z}_j} = 0, \quad j = 1, \ldots, n

Cauchy–Riemann in several variables

f(z) = \frac{1}{(2\pi i)^n} \oint_{|\zeta_1 - a_1|=r_1} \cdots \oint_{|\zeta_n - a_n|=r_n} \frac{f(\zeta)}{(\zeta_1 - z_1)\cdots(\zeta_n - z_n)}\,d\zeta_1 \cdots d\zeta_n

Cauchy integral formula in $\mathbb{C}^n$

\bar{\partial} f = \sum_{j=1}^n \frac{\partial f}{\partial \bar{z}_j} d\bar{z}_j = 0

Holomorphicity via the $\bar{\partial}$ operator

Worked Examples

Write $z_j = x_j + iy_j$. The $\bar{\partial}$ operator with respect to $z_1$ is $\partial/\partial\bar{z}_1 = \frac{1}{2}(\partial/\partial x_1 + i\partial/\partial y_1)$.
$\frac{\partial f}{\partial \bar{z}_1} = \frac{\partial(z_1^2)}{\partial \bar{z}_1} + \frac{\partial(z_2^2)}{\partial \bar{z}_1}$
Since $z_1^2$ is a polynomial in $z_1$ (not $\bar{z}_1$) and $z_2^2$ does not depend on $z_1$:
$\frac{\partial f}{\partial \bar{z}_1} = 0 + 0 = 0$
Similarly $\partial f/\partial \bar{z}_2 = 0$. Both CR equations hold.
$\frac{\partial f}{\partial \bar{z}_2} = 0$

Answer: All $\bar{\partial}$ conditions vanish, so $f$ is holomorphic on $\mathbb{C}^2$.

Practice Problems

Difficulty 8/10

Define a domain of holomorphy and give one example and one non-example in $\mathbb{C}^2$.

Difficulty 9/10

Explain what the $\bar{\partial}$-problem (Dolbeault's equation) is and why solving it matters.

Difficulty 9/10

What is pseudoconvexity and how does it characterize domains of holomorphy (Levi's problem)?

Common Mistakes

Common Mistake

Isolated singularities exist in functions of several complex variables.

By Hartogs's theorem, there are no isolated singularities in $\mathbb{C}^n$ for $n \geq 2$. Every holomorphic function on the complement of a compact set in a connected domain extends across that set.

Common Mistake

All domains in $\mathbb{C}^n$ are domains of holomorphy.

Only pseudoconvex domains are domains of holomorphy. For example, a spherical shell (the region between two concentric spheres) is not a domain of holomorphy — any function holomorphic on it extends to the full ball.

Quiz

Summary

A function $f: U \subset \mathbb{C}^n \to \mathbb{C}$ is holomorphic iff $\bar{\partial}f = 0$ in each variable.
Hartogs's theorem shows isolated singularities cannot exist for $n \geq 2$.
Domains of holomorphy are exactly the pseudoconvex domains (Oka's theorem, Levi's problem).
The $\bar{\partial}$-problem (Dolbeault's equation) is the central analytic tool in SCV.
SCV differs profoundly from one-variable theory: the Hartogs phenomenon, pseudoconvexity, and Stein manifolds have no one-variable analogues.

References

WebsiteWikipedia — Several complex variables