Mathematics – Algebraic Geometry
Scientific paper
2011-06-15
Mathematics
Algebraic Geometry
Scientific paper
Every $\mathbb{A}^{1}-$bundle over the complex affine plane punctured at the origin, is trivial in the differentiable category but there are infinitely many distinct isomorphy classes of algebraic bundles. Isomorphy types of total spaces of such algebraic bundles are considered; in particular, the complex affine 3-sphere admitts such a structure with an additional homogeneity property. Total spaces of nontrivial homogeneous $\mathbb{A}^{1}$-bundles over the punctured plane are classified up to $\mathbb{G}_{m}$-equivariant algebraic isomorphism and a criterion for nonisomorphy is given. In fact the affine 3-sphere is not isomorphic as an abstract variety to the total space of any $\mathbb{A}^{1}$-bundle over the punctured plane of different homogeneous degree, which gives rise to the existence of exotic spheres, a phenomenon that first arises in dimension three. As a by product, an example is given of two biholomorphic but not algebraically isomorphic threefolds, both with a trivial Makar-Limanov invariant, and with isomorphic cylinders.
Dubouloz Adrien
Finston David R.
No associations
LandOfFree
On exotic affine 3-spheres does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On exotic affine 3-spheres, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On exotic affine 3-spheres will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-658235