Let $X$ be an affine variety of dimension $n$ over $\mathbb{C}$. 



> 
> Does the analytic space associated with $X$ have the homotopy type of a $n$-dimensional CW complex?
> 
> 
> 



Homotopy type of a complex affine variety
If $X$ is smooth, and if you ask to have the same homotopy type of a CW complex of real dimension *at most* $n$, this is precisely the statement of the Andreotti-Frankel theorem.


It is true, more generally, for a Stein manifold of complex dimension $n$.


If $X$ is arbitrarily singular, the same theorem holds, provided $X$ is irreducible. This was shown by Karčjauskas for algebraic varieties and by Hamm for Stein spaces.


