Classification of affine homogeneous spaces of complexity one

The complexity of an action of a reductive algebraic group $G$ on an algebraic variety $X$ is the codimension of a generic $B$-orbit in $X$, where $B$ is a Borel subgroup of $G$. Affine homogeneous spaces $G/H$ of complexity 1 are classified in this paper. These results are the natural continuation of the earlier classification of spherical affine homogeneous spaces, that is, spaces of complexity 0.