Topologically contractible algebraic varieties From a post to 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?
 A: 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.
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.
A: About the rationality of contractible varieties: Yes for curves and surfaces and is an open question for higher dimensions.
Any such contractible variety $X$ has $\chi_\text{top}(X)=1$, obviously.
If $X$ is a curve then it must have only cusps as singularities, if any, by a simple $\chi_\text{top}$ calculation. Now let $Y$ be a projective model of $X$ such that it is smooth at the points in $Y-X$. Topologically, $Y$ is a real surface without boundary such that a few punctures make it contractible. The only real surface with this property is $S^2$, obviously. Hence $Y$ better be rational and so is $X$.
If $X$ is an algebraic surface then it was a conjecture of Van de Ven that such a surface must be rational (actually his conjecture is for any homologically trivial $X$). This was proved by Gurjar & Shastri in:

*

*On the rationality of complex homology 2-cells

*Here is the the part II of the above paper (MathSciNet review number MR0984747)

