GIT quotient

In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme $\operatorname {Spec} A$ with action by a group scheme G is the affine scheme $\operatorname {Spec} (A^{G})$ , the prime spectrum of the ring of invariants of A, and is denoted by $X/\!/G$ . A GIT quotient is a categorical quotient: any invariant morphism uniquely factors through it.

Taking Proj (of a graded ring) instead of $\operatorname {Spec}$ , 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 $G/H=G/\!/H=\operatorname {Spec} k[G]^{H}$ 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).

Examples

A simple example of a GIT quotient is given by the $\mathbb {Z} /2$ -action on $\mathbb {C} [x,y]$ sending

{\begin{aligned}x\mapsto -x&&y\mapsto -y\end{aligned}}

Notice that the monomials $x^{2},xy,y^{2}$ generate the ring $\mathbb {C} [x,y]^{\mathbb {Z} /2}$ . Hence we can write the ring of invariants as

\mathbb {C} [x,y]^{\mathbb {Z} /2}=\mathbb {C} [x^{2},xy,y^{2}]={\frac {\mathbb {C} [a,b,c]}{(ac-b^{2})}}

Scheme theoretically, we get the morphism

\mathbb {A} ^{2}\to {\text{Spec}}\left({\frac {\mathbb {C} [a,b,c]}{(ac-b^{2})}}\right)=\mathbb {A} ^{2}/(\mathbb {Z} /2)

References

Pedagogical

Mukai, S. (2002). An introduction to invariants and moduli. Cambridge Studies in Advanced Mathematics. 81. ISBN 978-0-521-80906-1.
M. Brion, "Introduction to actions of algebraic groups"
Thomas, R. P. (2006). "Notes on GIT and symplectic reduction for bundles and varieties". Surveys in Differential Geometry, (): A Tribute to Professor S.-S. Chern. v3. 10 (2006). arXiv:math/0512411 .

Reference

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". arXiv:math/0703131v1 .
Victoria Hoskins, Quotients in algebraic and symplectic geometry
F. C. Kirwan, Cohomology of Quotients in Complex and Algebraic Geometry, Mathematical Notes 31, Princeton University Press, Princeton N. J., 1984.
Mumford, David; Fogarty, J.; Kirwan, F. (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.

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GIT quotient

Examples

See also

References

Pedagogical

Reference