Adrien Dubouloz (Bourgogne): Families of A^1-contractible affine threefolds

The so-called Koras-Russell threefolds are a family of topologically

contractible rational smooth complex affine threefolds which played an

important role in the linearization problem for multiplicative group

actions on the affine 3-space. They are known to be all diffeomorphic to

the 6-dimensional Euclidean space, but it was shown by Makar-Limanov in

the nineties that none of them are algebraically isomorphic to the affine

3-space. It is however not known whether they are stably isomorphic or not

to an affine space. Recently, Hoyois, Krishna and Østvær proved that many

of these varieties become contractible in the unstable A^1-homotopy

category of Morel and Voevodsky after some finite suspension with the

pointed projective line. In this talk, I will explain how additional

geometric properties related to additive group actions on such varieties

allow to conclude that a large class of them are actually A^1-contractible

(Joint work with Jean Fasel, Université Grenoble-Alpes).