Characterization of some affine varieties

Recently Hanspeter Kraft proved that the automorphism group of affine n-space
C^n determines C^n up to isomorphism. In this talk we present a similar result about the quotient
C^n/\mu_d, where \mu_d is a cyclic group of order d: if X is an irreducible
affine normal variety such that

Aut(X) is isomorphic to Aut (C^n)/\mu_d as ind-groups, then they are isomorphic as varieties. If X is not necessarily normal then we classify all such X with respective isomorphisms of ind-groups.