In algebraic geometry, an affine **GIT quotient**, or affine **geometric invariant theory quotient**, of an affine scheme with an 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).

## Construction of a GIT quotient

Let *G* be a reductive group acting on a quasi-projective scheme *X* over a field and *L* a linearized ample line bundle on *X*. Let

be the section ring. By definition, the semistable locus is the complement of the zero set in *X*; in other words, it is the union of all open subsets for global sections *s* of , *n* large. By ampleness, each is affine; say and so we can form the affine GIT quotient

- .

Note that is of finite type by Hilbert's theorem on the ring of invariants. By universal property of categorical quotients, these affine quotients glue and result in

- ,

which is the GIT quotient of *X* with respect to *L*. Note that if *X* is projective; i.e., it is the Proj of *R*, then the quotient is given simply as the Proj of the ring of invariants .

The most interesting case is when the stable locus^{[1]} is nonempty; is the open set of semistable points that have finite stabilizers and orbits that are closed in . In such a case, the GIT quotient restricts to

- ,

which has the property: every fiber is an orbit. That is to say, is a genuine quotient (i.e., geometric quotient) and one writes . Because of this, when is nonempty, the GIT quotient is often referred to as a "compactification" of a geometric quotient of an open subset of *X*.

A difficult and seemingly open question is: which geometric quotient arises in the above GIT fashion? The question is of a great interest since the GIT approach produces an *explicit* quotient, as opposed to an abstract quotient, which is hard to compute. One known partial answer to this question is the following:^{[2]} let be a locally factorial algebraic variety (for example, a smooth variety) with an action of . Suppose there are an open subset as well as a geometric quotient such that (1) is an affine morphism and (2) is quasi-projective. Then for some linearlized line bundle *L* on *X*. (An analogous question is to determine which subring is the ring of invariants in some manner.)

## Examples

### Finite group action by

A simple example of a GIT quotient is given by the -action on sending

Notice that the monomials generate the ring . Hence we can write the ring of invariants as

Scheme theoretically, we get the morphism

which is a singular subvariety of with isolated singularity at . This can be checked using the differentials, which are

hence the only point where the differential and the polynomial both vanish is at the origin. The quotient obtained is a conical surface with an ordinary double point at the origin.

### Torus action on plane

Consider the torus action of on by . Note this action has a few orbits: the origin , the punctured axes, , and the affine conics given by for some . Then, the GIT quotient has structure sheaf which is the subring of polynomials , hence it is isomorphic to . This gives the GIT quotient

Notice the inverse image of the point is given by the orbits , showing the GIT quotient isn't necessarily an orbit space. If it were, there would be three origins, a non-separated space.^{[3]}

## See also

## Notes

**^**NB: In (MFK) , it was called the set of properly stable points**^**MFK, Converse 1.13. NB: even though the result is stated for a smooth variety, the proof there is valid for a locally factorial one.**^**Thomas, Richard P. (2006). "Notes on GIT and symplectic reduction for bundles and varieties".*Surveys in Differential Geometry*. International Press of Boston.**10**(1): 221–273. arXiv:math/0512411. doi:10.4310/sdg.2005.v10.n1.a7. ISSN 1052-9233. MR 2408226. S2CID 16294331.

## References

### Pedagogical

- Mukai, Shigeru (2002).
*An introduction to invariants and moduli*. Cambridge Studies in Advanced Mathematics.**81**. ISBN 978-0-521-80906-1. - Brion, Michel. "Introduction to actions of algebraic groups" (PDF).
- Laza, Radu (2012-03-15). "GIT and moduli with a twist". arXiv:1111.3032 [math.AG].
- Thomas, Richard P. (2006). "Notes on GIT and symplectic reduction for bundles and varieties".
*A Tribute to Professor S.-S. Chern*. Surveys in Differential Geometry.**10**. pp. 221–273. arXiv:math/0512411. doi:10.4310/SDG.2005.v10.n1.a7. MR 2408226. S2CID 16294331.

### References

- Alper, Jarod (2008-04-14). "Good moduli spaces for Artin stacks". arXiv:0804.2242 [math.AG].
- Doran, Brent; Kirwan, Frances (2007). "Towards non-reductive geometric invariant theory".
*Pure and Applied Mathematics Quarterly*.**3**(1, Special Issue: In honor of Robert D. MacPherson. Part 3): 61–105. arXiv:math/0703131. Bibcode:2007math......3131D. doi:10.4310/PAMQ.2007.v3.n1.a3. MR 2330155. S2CID 3190064. - Hoskins, Victoria. "Quotients in algebraic and symplectic geometry".
- Kirwan, Frances C. (1984).
*Cohomology of Quotients in Complex and Algebraic Geometry*. Mathematical Notes.**31**. Princeton N. J.: Princeton University Press. - Mumford, David; Fogarty, John; Kirwan, Frances (1994).
*Geometric invariant theory*. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)].**34**(3rd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906.