Automorphism groups of rigid affine surfaces: the identity component

Abstract

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 ${\mathrm{Aut}}^0(Y)$ of the automorphism group $\mathrm{Aut}(Y)$ 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, ${\mathrm{Aut}}^0(Y)$ is an algebraic torus of rank $\le 2$.

36 pages, 5 figures; a revised version taking into account the referee comments. To appear in: Algebraic Geometry