TITLE: What invariants distinguish these three-folds?
QUESTION [9 upvotes]: Let $a,b$ be two positive integers and everything is over $\mathbb C$. Let's denote by $S_{a,b}$ the hypersurface $$\{x^ay+z^bt=1\}\subset \mathbb A^4.$$
Can we distinguish the $S_{a,b}$'s up to isomorphism? What invariants would be useful in this case?

REPLY [6 votes]: Consider the morphism $S_{m,n}\to \mathbb{A}^2_{*}=\mathbb{A}^2\setminus \{(0,0)\}$ given by the projection to $(x,z)$. Then, it is a locally trivial $\mathbb{A}^1$-bundle. This latter is trivial for the Euclidean topology so $S_{m,n}$ is diffeomorphic to $\mathbb{A}^2_{*}\times \mathbb{A}^1$. However, these can give non-isomorphic varieties.
In fact $S_{a,b}$ is isomorphic to $S_{a',b'}$ if $a+b=a'+b'$, but apart from this, the question is open (except maybe for $S_{1,1}\simeq\mathrm{SL}(2,\mathbb{C})$ which is probably not isomorphic to the other ones). For more details on this, look at the paper  "On exotic affine $3$-spheres" of A. Dubouloz and D. Finston http://www.ams.org/journals/jag/2014-23-03/S1056-3911-2014-00612-3/home.html