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.

References

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

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[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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Homogeneous variety

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References