Any non-singular projective variety over $\mathbb{C}$ is easily seen to be a smooth manifold. Presumably the same is not true for algebraic varieties - one would not expect varieties with singular points to have a smooth structure. But do there exist non-singular varieties that are not smooth manifolds?



Algebraic Varieties which are also Manifolds
Every non-singular algebraic variety over $\mathbb C$ is a smooth manifold. See for instance:
<http://en.wikipedia.org/wiki/Manifold>
under "Generalizations of Manifolds".


In fact, Arminius' suggested answer in the comments seems to give a proof of this fact, and I'll attempt to flesh it out a small amount. Every algebraic variety is locally a quasi-affine variety. So we may take an open cover $U\_i$ of the variety, where each $U\_i$ is a closed subset of an open subset of affine n-space. We may then check smoothness at each point of $U\_i$ via the Jacobian criterion. The same procedure illustrates that each $U\_i$ is a complex manifold. Since the gluing maps are algebraic, they are smooth, and hence our non-singular variety is also a smooth manifold.


