It is known that the identity component of the automorphism group of a projective algebraic variety is an algebraic group. This is not true in general for quasi-projective varieties. In this note we address the question: given an affine algebraic surface $Y$, as to when the identity component ${ \rm Aut}^0 (Y)$ of the automorphism group ${ \rm Aut} (Y)$ is an algebraic group? We show that this happens if and only if $Y$ admits no effective action of the additive group of the field. In the latter case, ${ \rm Aut}^0 (Y)$ is an algebraic torus of rank $\le 2$.
Categories: math.AG