GIT quotient

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

Taking Proj (of a graded ring) instead of ${\displaystyle \operatorname {Spec} }$, 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

${\displaystyle G/H=G/\!/H=\operatorname {Spec} \!{\big (}k[G]^{H}{\big )}}$

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).