TITLE: Are algebraic varieties strictly more general than (differentiable) manifolds?
QUESTION [7 upvotes]: I have read that every non-singular algebraic variety is a smooth manifold. However, I was wondering if every smooth manifold can be expressed as a non-singular algebraic variety, or even just a general algebraic variety; such that algebraic varieties are a strict generalization of manifolds.
If not, does not restricting algebraic varieties to be defined in terms of polynomial equations allow for a generalization?
(Any references on the relation of varieties and manifolds, and any practical use of such a relation, would be appreciated, as I'm new to algebraic geometry. Thanks!)

REPLY [5 votes]: Another useful piece of the story is the Kähler group problem: which finitely presented groups arise as fundamental groups of Kähler manifolds? Smooth projective varieties are Kähler manifolds, and I believe there are no known properties of fundamental groups that distinguish these classes. On the other hand, any finitely presented group arises as the fundamental group of a compact complex manifold, so the Kähler condition is crucial.