TITLE: Algebraic variety and real manifold
QUESTION [3 upvotes]: When an algebraic variety, affine or projective, is a manifold ( on $\mathbb{R}$)? And when the inverse is true?
I think that each algebraic variety is a manifold but for the projective? I would think the same but I would like to have an explanation about the relathionship of these structures

REPLY [5 votes]: There is actually a nontrivial theorem regarding this issue:
Definition. 1. A smooth manifold $M$ is tame if $M$ is diffeomorphic to the interior of a smooth compact manifold $N$ with (possibly empty) boundary.

A smooth manifold $M$ is algebraic if it is diffeomorphic to a
nonsingular real-algebraic subset of ${\mathbb R}^n$ for some $n$.

Theorem. A smooth manifold is tame if and only if it is algebraic.
Both directions are nontrivial. See Corollary 4.3 in
Akbulut, Selman; King, Henry C., The topology of real algebraic sets with isolated singularities, Ann. Math. (2) 113, 425-446 (1981). ZBL0494.57004.
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