Additive group actions on Danielewski varieties and the Cancellation Problem

Adrien Dubouloz (1)

1 IF - Institut Fourier

The cancellation problem asks if two complex algebraic varieties X and Y of the same dimension such that X\times\mathbb{C} and Y\times\mathbb{C} are isomorphic are isomorphic. Iitaka and Fujita established that the answer is positive for a large class of varieties of any dimension. In 1989, Danielewski constructed a famous counter-example using smooth affine surfaces with additive group actions. His construction was further generalized by Fieseler and Wilkens to describe a larger class of affine surfaces. Here we construct higher dimensional analogues of these surfaces. We study algebraic actions of the additive group \mathbb{C}_{+} on certain of these varieties, and we obtain counter-examples to the cancellation problem in any dimension n\geq2 .

Mots clés: Danielewski varieties; Cancellation Problem; Additive group actions; Makar-Limanov invariant

Domaines: Géométrie algébrique [math.AG]

Adrien Dubouloz. Additive group actions on Danielewski varieties and the Cancellation Problem. 2005. ⟨hal-00007651⟩