Stein manifold

A complex manifold is Stein if its global holomorphic functions separate points, give local coordinates, and make it holomorphically convex: the holomorphic hull of every compact set is compact. Closed complex submanifolds of affine complex space are examples, including products of copies of $\mathbb{C}$ and ${\mathbb{C}}^{\ast }$. Cartan theorem B makes these spaces useful for acyclic covers in sheaf cohomology.

A Stein surface is a Stein manifold of complex dimension two, hence real dimension four. Its global holomorphic functions separate points and provide local coordinates, and the holomorphic hull of every compact subset is compact. The word “surface” here refers to complex dimension, rather than to a real two-dimensional surface.

The affine complex plane ${\mathbb{C}}^2$ is a Stein surface, as is the smooth closed hypersurface $xy-z^2=1$ in ${\mathbb{C}}^3$: its gradient $(y,x,-2z)$ never vanishes on the hypersurface. A compact connected positive-dimensional complex manifold cannot be Stein, because its holomorphic functions are constant. In real dimension four, the topology of Stein surfaces has especially rich links with handle decompositions and contact structures on three-dimensional boundaries.^[ref]

For a complex Lie group, holomorphic principal bundles over a Stein manifold are classified up to holomorphic isomorphism by their topological bundle class. In particular, a topologically trivial holomorphic principal bundle over a Stein base has a global holomorphic trivialization. This supplies global complex gauges for a flat partial connection on contractible complex affine space; it does not impose prescribed behaviour at infinity.

On a Stein manifold $X$, every coherent analytic sheaf $\mathcal{F}$ has $H^q(X,\mathcal{F})=0$ for $q>0$. In particular this applies to the sheaf of holomorphic functions. If every nonempty finite intersection in an open cover is Stein, these vanishing results and the acyclic cover theorem compute the corresponding sheaf cohomology from its Čech cochain complex.