Docquier–Grauert theorem

Remarks

For $X = \C^n$ this is Oka's solution of the Levi problem: a locally pseudoconvex domain in $\C^n$ is a domain of holomorphy. The proof follows the same pattern. Using that $X$ embeds in some $\C^N$ and has a holomorphic retraction from a neighbourhood, one builds a distance-to-the-boundary function $\delta$ on $D$. The local Stein condition makes $-\log \delta$ plurisubharmonic near $\partial D$. Adding a strictly plurisubharmonic exhaustion of $X$ gives a plurisubharmonic exhaustion function of $D$, and Grauert's solution of the Levi problem then shows that $D$ is Stein. The same holds for unbranched Riemann domains spread over a Stein manifold, which is the form in which Docquier and Grauert proved it. Example: if $D_1, D_2$ are Stein open subsets of $X$, then $D_1 \cap D_2$ is locally Stein, hence Stein. The hypothesis that $X$ is Stein cannot be dropped. If $X$ is the Riemann sphere and $D = X$, then $\partial D$ is empty, so $D$ is locally Stein, but $D$ is compact and not Stein.

Going deeper

For a student

The theorem shows that the class of Stein open sets of a Stein manifold is described by a local boundary condition, pseudoconvexity, just as for domains in $\C^n$. It is used to prove that intersections, and suitable unions and limits, of Stein domains are Stein. It reduces many questions on Stein manifolds to the case of domains in $\C^n$ through the embedding theorem.