On exotic affine $3$-spheres

Every $\mathbb {A}^{1}$-bundle over $\mathbb {A}_{\ast }^{2},$ 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 $\mathbb {S}_{\mathbb {C}}^{3},$ given by $z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}=1,$ admits such a structure with an additional homogeneity property. Total spaces of nontrivial homogeneous $\mathbb {A}^{1}$-bundles over $\mathbb {A}_{\ast }^{2}$ are classified up to $\mathbb {G}_{m}$-equivariant algebraic isomorphism, and a criterion for nonisomorphy is given. In fact $\mathbb {S}_{\mathbb {C}}^{3}$ is not isomorphic as an abstract variety to the total space of any $\mathbb {A}^{1}$-bundle over $\mathbb {A}_{\ast }^{2}$ of different homogeneous degree, which gives rise to the existence of exotic spheres, a phenomenon that first arises in dimension three. As a byproduct, an example is given of two biholomorphic but not algebraically isomorphic threefolds, both with a trivial Makar-Limanov invariant, and with isomorphic cylinders.