Is there a characterization of contractible hypersurfaces in $\mathbb{C}^2$.

Question

Let $V$ be an irreducible, algebraic hypersurface in $\mathbb{C}^2$ which is contractible as a topological space. I would like to know the algebraic characterization of such objects. For example, restriction, if any, on the degree of the defining polynomial. What happens in higher-dimensions ?

Answer 1

As Belanov says in the comments above, this is relatively easy to decide for curves, since the only contractible complex algebraic curve is $\mathbb{A}^1_\mathbb{C}$. If the (irreductible) curve is not smooth, then it is contractible iff its semi-normalisation is $\mathbb{A}^1_\mathbb{C}$. This shows that there will be contractible singular curves of arbitrary high degree (take $y^2=x^{2k+1}$ for any $k\geq 1$) , but that for smooth curves any degree beyond $2$ will yield a non-contractible curve.

On the other hand, the analoguous problem for surfaces is difficult and has been studied quite a bit. A beautiful theorem along those lines is due to Ramanujam (1971): a smooth complex algebraic surface which is contractible and "simply connected at infinity" is isomorphic to $\mathbb{A}^2_\mathbb{C}$. See this survey for more information about this theorem and subsequent developments.