Algebraic Varieties which are also Manifolds 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?
 A: 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.
A: If $k$ is a complete valued field (e.g. $\mathbb{Q}_p$, etc.), one may define analytic manifolds over $k$ in the natural way. Precisely, these are topological spaces that locally look like open balls in $k^n$ and the transition functions must be analytic. Then the $k$-points of a smooth variety over $k$ is an analytic manifold (over $k$); Charlie's  reasoning for the case $k = \mathbb{C}$ works for any $k$ as above.
A: Every nonsingular variety over $\mathbb{C}$ is a smooth manifold, period.  Take any affine open cover $X=\cup U_i$.  Then each $U_i$ is a smooth manifold, and the transition maps are algebraic, so in particular, smooth.  Thus, manifold.
