Torsors on Affine Varieties

Abstract

Let $X$ be a smooth affine algebraic variety over the field of complex numbers which is contractible. Then every algebraic $G$-torsor on $X$ is algebraically trivial if $G$ is a semi-simple algebraic group. We also show that if $X$ is a smooth affine algebraic variety such that $Ω^1_X$ is trivial and $X$ is topologically simply connected with $H^i(X,{\mathbb Z})=0$ for $i=1,2$ and $3$, then every algebraic $G$-torsor on $X$ is algebraically trivial for a semi-simple algebraic group $G$.