Let $\mathbb{A}^n_k$ be the Affine $n$-space over an algebraically closed field $k$.Let $X$ be a variety over $k$. What would be the right definition of an "Affine bundle" i.e bundle of fiber type $\mathbb{A}^n_k$ over $X$ (I mean local triviality in zarisky topology,oretale .. )?. When can one get a vector bundle from an "Affine bundle" , moreprecisely (I think !) if I assume the structure group of the affine bundle tobe $Aut_{Var_k}(\mathbb{A}^n_k)$, when can one get a reduction of the structure groupto $GL_n(k)$?
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