Grauert's solution of the Levi problem for manifolds

Remarks

The direction from Stein to the exhaustion function is easy. A Stein manifold has a proper holomorphic embedding $F \colon X \to \C^N$, and then $\varphi = \abs{F}^2$ is smooth, strictly plurisubharmonic, and an exhaustion because $F$ is proper. Grauert proved the converse in 1958. His proof enlarges the sublevel sets $\set{\varphi < c}$ step by step and uses a finiteness theorem for cohomology on strongly pseudoconvex domains. A later proof by Hörmander uses $L^2$ estimates for $\bar\partial$. Example: the unit ball $B \subseteq \C^n$ is Stein, since $-\log(1 - \abs{z}^2)$ is a smooth strictly plurisubharmonic exhaustion function. Strictness is needed: on a compact complex manifold $X$ of positive dimension, every constant is a smooth plurisubharmonic exhaustion function, yet $X$ is not Stein.

Going deeper

For a student

For domains in $\C^n$ the theorem gives back Oka's solution of the Levi problem, after one smooths a continuous plurisubharmonic exhaustion and adds $\abs{z}^2$ to make it strict. The function $\varphi$ is a Morse function after a small perturbation, and the strict plurisubharmonicity bounds the indices of its critical points by $\dim_\C X$. This gives the result of Andreotti and Frankel that a Stein manifold of complex dimension $n$ has the homotopy type of a CW complex of real dimension at most $n$. The Andreotti–Grauert theorem generalizes the theorem to $q$-convex manifolds.