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 $\chi_{top}(X)=1$, obviously.

If $X$ is a curve then it must have only cusps as singularities, if any, by a simple $\chi_{top}$ 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 herein:

http://dx.doi.org/10.2969/jmsj/04110037

Somehow I am unable to post more than one hyperlink.

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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 $\chi_{top}(X)=1$, obviously.

If $X$ is a curve then it must have only cusps as singularities, if any, by a simple $\chi_{top}$ 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 here:

http://dx.doi.org/10.2969/jmsj/04110037

Somehow I am unable to post more than one hyperlink. Here is the MathScinet review number of the part II of the above paper : MR0984747