# GIT quotient

In algebraic geometry, an affine **GIT quotient**, or affine **geometric invariant theory quotient**, of an affine scheme with action by a group scheme *G* is the affine scheme , the prime spectrum of the ring of invariants of *A*, and is denoted by . A GIT quotient is a categorical quotient: any invariant morphism uniquely factors through it.

Taking Proj (of a graded ring) instead of , one obtains a projective GIT quotient (which is a quotient of the set of semistable points.)

A GIT quotient is a categorical quotient of the locus of semistable points; i.e., "the" quotient of the semistable locus. Since the categorical quotient is unique, if there is a geometric quotient, then the two notions coincide: for example, one has for an algebraic group *G* over a field *k* and closed subgroup *H*.

If *X* is a complex smooth projective variety and if *G* is a reductive complex Lie group, then the GIT quotient of *X* by *G* is homeomorphic to the symplectic quotient of *X* by a maximal compact subgroup of *G* (Kempf–Ness theorem).

## See also

## References

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- Doran, Brent; Kirwan, Frances (2007). "Towards non-reductive geometric invariant theory". arXiv:math/0703131v1.
- Victoria Hoskins, Quotients in algebraic and symplectic geometry
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