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

> Are all **topologically trivial** (contractible) **complex algebraic varieties** other then affine lines necessarily affine? Are all of them rational?

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

Perhaps the affine part would follow from a comparison between Zariski cohomology and complex cohomology?