From [a post](https://mathoverflow.net/questions/7374/the-jouanolou-trick/7602#7602) to [The Jouanolou trick](https://mathoverflow.net/questions/7374/the-jouanolou-trick/):



> 
> Are all **topologically trivial** (contractible) **complex algebraic varieties** necessarily affine? Are there examples of those not birationally equivalent to an affine space?
> 
> 
> 


The examples that come to my mind are similar to a singular $\mathbb P^1$ without a point given by equation $x^2 = y^3$. This particular curve is clearly birationally equivalent to affine line.


Perhaps the "affine" part follows from a comparison between Zariski cohomology and complex cohomology?



Topologically contractible algebraic varieties
No. Counterexamples were first constructed by Winkelmann, as quotients of $\mathbb A^5$ by algebraic actions of $\mathbb G\_{\text{a}}$. I learned this from Hanspeter Kraft's very nice article available here:


[Challenging problems on affine $n$-space](http://www.numdam.org/numdam-bin/item?id=SB_1994-1995__37__295_0).


Recently Aravind Asok and Brent Doran have been studying these kinds of examples in the setting of $\mathbb A^1$-homotopy theory, on the arxiv as [On unipotent quotients and some A^1-contractible smooth schemes](http://arxiv.org/abs/math/0703137).


