Adrien Dubouloz

On exotic affine 3-spheres

Every

Automorphisms of

\mathbb{A}^{1}-

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

-fibered affine surfaces

\mathbb{A}^{1}

Noncancellation for contractible affine threefolds

-bundles over the punctured plane are classified up to

Cylinders in del Pezzo fibrations

\mathbb{G}_{m}

Proper triangular a-actions on 4 are translations

-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

Affine-ruled varieties without the Laurent cancellation property

\mathbb{A}^{1}

The Jacobian Conjecture fails for pseudo-planes

-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.

RATIONALLY INTEGRABLE VECTOR FIELDS AND RATIONAL ADDITIVE GROUP ACTIONS

We develop technics of birational geometry to study automorphisms of affine surfaces admitting many distinct rational fibrations, with a particular focus on the interactions between automorphisms and these fibrations. In particular, we associate to each surface S of this type a graph encoding equivalence classes of rational fibrations from which it is possible to decide for instance if the automorphism group of S is generated by automorphisms preserving these fibrations.