Additive group actions on Danielewski varieties and the cancellation problem

Abstract

Given complex algebraic varieties X and Y of the same dimension, the Cancellation Problem asks if an isomorphism between X × $$\mathbb{C}$$ and Y × $$\mathbb{C}$$ induces an isomorphism between X and Y. Iitaka and Fujita (J. Fac. Sci. Univ. 24:123–127, 1977) established that the answer is positive for a large class of varieties of any dimension. In 1989, Danielewski constructed a counterexample using smooth rational affine surfaces. His construction was further generalized by Fieseler (Comment. Math. Helvetici 69:5–27, 1994) and Wilkens (C.R. Acad. Sci. Paris Ser. I Math. 326(9):1111–1116, 1998) to describe a larger class of affine surfaces. Here we introduce higher-dimensional analogues of these surfaces. By studying algebraic actions of the additive group $$\mathbb{C}_{+}$$ on certain of these varieties, we obtain new counterexamples to the Cancellation Problem in every dimension d ≥ 2.