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.
