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

* Homogeneous space
* Symmetric space
* Symmetric variety

## 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

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Homogeneous variety. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Homogeneous_variety&oldid=4296481"