Stein surface

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]