Homogeneous variety

In algebraic geometry, a homogeneous variety is an algebraic variety on which an algebraic group acts transitively.[1][2] Homogeneous varieties over an algebraically closed field are quotient varieties G/H where G is an algebraic group and H a subgroup scheme (for instance, an algebraic subgroup).[3]

Such varieties are always smooth quasi-projective varieties.

Classical examples are flag varieties (when G is semisimple and H 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

References

  1. ^ Michel Brion, "Introduction to actions of algebraic groups" [1], Definition 1.17
  2. ^ Chow, Wei-Liang (1957). "On the projective embedding of homogeneous varieties". Princeton Mathematical Series. 12: 122–128.
  3. ^ Michel Brion, "Introduction to actions of algebraic groups" [2], Theorem 1.16


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