Is the Affine Space Determined by Its Automorphism Group?

In this note we study the problem of characterizing the complex affine space An via its automorphism group. We prove the following. Let X be an irreducible quasi-projective n-dimensional variety such that Aut(X) and Aut(An) are isomorphic as abstract groups. If X is either quasi-Affine and toric or X is smooth with Euler characteristic (X) = 0 and finite Picard group Pic(X), then X is isomorphic to An. The main ingredient is the following result. Let X be a smooth irreducible quasiprojective variety of dimension n with finite Pic(X). If X admits a faithful (Z/pZ)naction for a prime p and (X) is not divisible by p, then the identity component of the centralizer CentAut(X)((Z/pZ)n) is a torus.