In algebraic geometry, a homogeneous variety is an algebraic variety on which an algebraic group acts transitively.Michel Brion, "Introduction to actions of algebraic groups" , Definition 1.17 Homogeneous varieties over an algebraically closed field are quotient varieties where is an algebraic group and a subgroup scheme (for instance, an algebraic subgroup).Michel Brion, "Introduction to actions of algebraic groups" , Theorem 1.16
Such varieties are always smooth quasi-projective varieties.
Classical examples are flag varieties (when is semisimple and a parabolic subgroup), or more generally homogeneous spherical varieties. Severi-Brauer varieties are examples of homogeneous varieties over a field without any rational points.
See also Homogeneous space Symmetric space Symmetric variety
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