Additive group actions on Danielewski varieties and the Cancellation Problem

Details Additive group actions on Danielewski varieties and the Cancellation Problem Additive group actions on Danielewski varieties and the Cancellation Problem

2005-07-25

arxiv.org/abs/math/0507505v1

Mathematics

Algebraic Geometry

12p

Scientific paper

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 .

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