Holomorphic convexity of a complex manifold 
means that the holomorphic hull of every
nonempty compact subset 
is compact. The hull is
 |
where 
denotes the global holomorphic functions. Thus it is defined by all global holomorphic
functions rather than by real linear functionals as in ordinary convexity. Every
Stein manifold is holomorphically convex. A connected
compact complex manifold is also holomorphically
convex, although in positive dimension it is not Stein
because all its global holomorphic functions are constant.
