A Stein manifold is a complex manifold which is holomorphically separable and holomorphically convex. Thus distinct points can be separated by global holomorphic functions, and the holomorphic convex hull of every compact subset is compact. Equivalently, a complex manifold is Stein if it admits a proper strictly plurisubharmonic exhaustion function.
Every domain of holomorphy in 
is a Stein manifold, as is every closed complex submanifold
of 
. Stein manifolds play the role in complex
analysis that affine varieties play in algebraic
geometry.
The Shafarevich holomorphic convexity conjecture predicts holomorphic convexity of universal covers of smooth projective complex varieties. For large fundamental group it predicts Steinness.
The holomorphic density property concerns approximation of holomorphic vector fields by the
Lie algebra generated by complete ones. The Tóth-Varolin
conjecture asks whether a Stein manifold with this property and diffeomorphic
to 
must be biholomorphic
to 
.
