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Homogeneous variety

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In algebraic geometry, a homogeneous variety is an algebraic variety on which an algebraic group acts transitively. [1] [2] Equivalently, it is a quotient variety G/H where G is an algebraic group and H an algebraic subgroup. [3]

Such varieties are always smooth projective varieties.

Classical examples are flag varieties (when G is semisimple and H a Borel subgroup) or more generally spherical varieties. Also toric varieties (when G is an algebraic torus).

See also

References

  1. ↑ Michel Brion, "Introduction to actions of algebraic groups" , 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" , Theorem 1.16


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