TITLE: The Relationship between Complex and Algebraic Geomety
QUESTION [33 upvotes]: I have recently begun to study algebraic geometry, coming from a differential geometry background. It seems that there is a deep link between complex manifolds and complex varieties. For example, one often hears that they are two different ways of looking at the same thing. Can anybody give a precise statement of this relationship?

REPLY [33 votes]: No one seems to have mentioned that if a complex manifold is an algebraic variety, then it has a nice compactification. More precisely, by Hironaka it can be embedded in a compact manifold such that the boundary is a divisor with normal crossings. 
In dimension 1 this is the only condition: a Riemann surface is an algebraic curve if and only if it can be compactified by adding a finite number of points. In particular, bounded holomorphic functions on it are constant (so the complex upper half plane is not an algebraic curve because z-i/z+i is a bounded holomorphic function).
So the two main obstructions to a complex manifold being an algebraic variety (or, at least, an algebraic space) are that it may not have a nice compactification (in the above sense) and it may not have enough meromorphic functions.
To say that a complex manifold is an algebraic variety is a very strong statement.