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?