Several Complex Variables (Outlook)

Everything in this folder is the theory of one complex variable. Passing to functions of several variables $f(z_1,\dots ,z_n)$ changes the subject qualitatively: singularities can no longer be isolated, "domain of definition" becomes a subtle geometric notion, and the Riemann mapping theorem fails. This short page marks the boundary of the one-variable theory and names what lies beyond. It builds on holomorphic.md and cauchy-integral.md; it has no physics dependency.

Holomorphy in several variables

A function on an open set of ${\mathbb{C}}^n$ is holomorphic if it is holomorphic in each variable separately (equivalently, if it is $C^1$ and satisfies the Cauchy–Riemann system $\partial f/\partial {\bar{z}}_j=0$ for all $j$). Much transfers verbatim — power series, the identity theorem, the maximum principle, and a Cauchy integral formula over the distinguished boundary of a polydisc. What breaks is the global theory.

Hartogs' phenomenon

The first surprise is that singularities cannot be isolated:

Hartogs' extension theorem. For $n\ge 2$, if $f$ is holomorphic on a neighbourhood of the boundary of a bounded domain $\Omega \subset {\mathbb{C}}^n$, it extends holomorphically to all of $\Omega$. A holomorphic function of two or more variables has no isolated singularities and no isolated zeros.

There is nothing like $1/z$ in several variables — the residue calculus, built on isolated poles, has no direct analogue. Zeros and poles form analytic varieties of codimension one, studied by sheaf theory and cohomology rather than contour integrals.

Domains of holomorphy and pseudoconvexity

Because functions extend unexpectedly (Hartogs), not every domain is the natural domain of some holomorphic function. Those that are — where some $f$ cannot be continued past any boundary point — are the domains of holomorphy. Their intrinsic characterization is a convexity condition:

Theorem (Cartan–Thullen / Oka). A domain in ${\mathbb{C}}^n$ is a domain of holomorphy iff it is pseudoconvex (the function $-\log \mathrm{dist}(\cdot ,>\partial \Omega )$ is plurisubharmonic).

This replaces the one-variable fact that every open set is a domain of holomorphy, and it launches the modern subject (the Levi problem, Oka's coherence theorems, the $\bar{\partial }$-equation).

The failure of the Riemann mapping theorem

The sharpest break with one variable:

Poincaré. The unit ball and the unit polydisc in ${\mathbb{C}}^2$ are not biholomorphic — although both are topologically cells and simply connected.

So there is no "standard domain" onto which all reasonable domains map; the analytic type carries genuine geometric information (boundary invariants, automorphism groups). The one-variable Riemann mapping theorem is a low-dimensional miracle.

Where it leads

Several complex variables is the analytic foundation of complex manifolds and complex algebraic geometry: sheaf cohomology, Hodge theory, Kähler geometry, and the theory of several-variable Riemann surfaces (now complex manifolds of higher dimension). It connects back to the differential-geometry folder through complex and Kähler structures.

References

• Hörmander, An Introduction to Complex Analysis in Several Variables.
• Range, Holomorphic Functions and Integral Representations in Several Complex Variables.
• Griffiths & Harris, Principles of Algebraic Geometry, Ch. 0.