GIT quotient

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme 
 
 
 
 X
 =
 Spec
 ⁡
 A
 
 
 {\displaystyle X=\operatorname {Spec} A}
 
 with an <a href="/facts/Group-scheme_action/jVo9ubJf">action</a> by a <a href="/facts/Group_scheme/bJW14FE2">group scheme</a> G is the affine scheme 
 
 
 
 Spec
 ⁡
 (
 
 A
 
 G
 
 
 )
 
 
 {\displaystyle \operatorname {Spec} (A^{G})}
 
, the <a href="/facts/Prime_spectrum/2k2nDE5w">prime spectrum</a> of the <a href="/facts/Ring_of_invariants/yEXwojME">ring of invariants</a> of A, and is denoted by 
 
 
 
 X
 
 /
 
 
 
 /
 
 G
 
 
 {\displaystyle X/\!/G}
 
. A GIT quotient is a <a href="/facts/Categorical_quotient/0KTb8W6M">categorical quotient</a>: any invariant morphism uniquely factors through it.
Taking <a href="/facts/Proj_construction/gcvPmjpU">Proj</a> (of a <a href="/facts/Graded_ring/AIUjkSaP">graded ring</a>) instead of 
 
 
 
 Spec
 
 
 {\displaystyle \operatorname {Spec} }
 
, one obtains a projective GIT quotient (which is a quotient of the set of <a href="/facts/Semistable_point/l4eV0RdG">semistable points</a>.)
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 <a href="/facts/Geometric_quotient/SxDuIm33">geometric quotient</a>, then the two notions coincide: for example, one has

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

for an <a href="/facts/Algebraic_group/DDtBCIvh">algebraic group</a> G over a field k and closed subgroup H.
If X is a complex <a href="/facts/Smooth_variety/Y76o7b9T">smooth</a> <a href="/facts/Projective_variety/aB2XdB7B">projective variety</a> and if G is a reductive <a href="/facts/Complex_Lie_group/uLUETfpT">complex Lie group</a>, then the GIT quotient of X by G is homeomorphic to the <a href="/facts/Symplectic_quotient/Rujfsdud">symplectic quotient</a> of X by a <a href="/facts/Maximal_compact_subgroup/8EA7XRcu">maximal compact subgroup</a> of G (<a href="/facts/Kempf%25E2%2580%2593Ness_theorem/CTvEKeK7">Kempf–Ness theorem</a>).

GIT quotient open-in-new

GIT quotient