Oka's solution of the Levi problem

Remarks

The implication from 1 to 3 is the easy one. Let $d(z)$ be the distance from $z$ to $\partial\Omega$; on a domain of holomorphy $\Omega \ne \C^n$, the function $-\log d$ is plurisubharmonic. Then $\max(-\log d(z), \abs{z}^2)$ is a continuous plurisubharmonic exhaustion function. The hard implication, from 3 to 1, is the solution of the Levi problem. For each boundary point one needs a holomorphic function on $\Omega$ that cannot be continued past it. Oka built such functions with his own methods; his proof for general $n$ used his theory of coherent ideal sheaves. Hörmander later obtained them by solving $\bar\partial u = f$ with $L^2$ estimates for weights $e^{-k\varphi}$, where $\varphi$ is the exhaustion function. Example: any convex domain $\Omega$ is pseudoconvex, so it is a domain of holomorphy. For convex domains this can also be seen directly with the functions $1/\ell$. Here $\ell$ is an affine complex-linear function that vanishes at a given boundary point and nowhere on $\Omega$. The same equivalence holds for unramified Riemann domains over $\C^n$.

Going deeper

For a student

The theorem completes the Cartan–Thullen theorem, which says that domains of holomorphy are the holomorphically convex domains. Together they give four equivalent conditions on a domain in $\C^n$: domain of holomorphy, holomorphically convex, pseudoconvex, and Stein. For domains with $C^2$ boundary, pseudoconvexity can be checked with the Levi form, so this gives a computable test. Grauert's theorem extends the result from domains to complex manifolds.