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(A(n)) are isomorphic as abstract groups. If X is either quasi-affine and toric or X is smooth with Euler characteristic chi(X) not equal 0 and finite Picard group Pic(X), then X is isomorphic to A(n). 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)(n)-action for a prime p and chi(X) is not divisible by p, then the identity component of the centralizer Cent(Aut(X))((Z/pZ)(n)) is a torus.

Automated summary

This study focuses on characterizing the complex affine space An through its automorphism group and provides proofs for certain cases. The main conclusion is that certain types of varieties, such as quasi-affine and toric or smooth with specific characteristics, are isomorphic to A(n) under certain conditions. The key ingredient is the analysis of smooth irreducible quasiprojective varieties with finite Picard groups and specific actions.