In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme X = Spec A {\displaystyle X=\operatorname {Spec} A} with an action by a group scheme G is the affine scheme Spec ( A G ) {\displaystyle \operatorname {Spec} (A^{G})} , the prime spectrum of the ring of invariants of A, and is denoted by X / / G {\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 Spec {\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
G / H = G / / H = Spec ( k [ G ] H ) {\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).
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
R = ⨁ n ≥ 0 Γ ( X , L ⊗ n ) {\displaystyle R=\bigoplus _{n\geq 0}\Gamma (X,L^{\otimes n})}
be the section ring. By definition, the semistable locus X s s {\displaystyle X^{ss}} is the complement of the zero set V ( R + G ) {\displaystyle V(R_{+}^{G})} in X; in other words, it is the union of all open subsets U s = { s ≠ 0 } {\displaystyle U_{s}=\{s\neq 0\}} for global sections s of ( L ⊗ n ) G {\displaystyle (L^{\otimes n})^{G}} , n large. By ampleness, each U s {\displaystyle U_{s}} is affine; say U s = Spec ( A s ) {\displaystyle U_{s}=\operatorname {Spec} (A_{s})} and so we can form the affine GIT quotient
π s : U s → U s / / G = Spec ( A s G ) . {\displaystyle \pi _{s}\colon U_{s}\to U_{s}/\!/G=\operatorname {Spec} (A_{s}^{G}).}
Note that U s / / G {\displaystyle U_{s}/\!/G} 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
π : X s s → X / / L G , {\displaystyle \pi \colon X^{ss}\to X/\!/_{L}G,}
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 X / / L G {\displaystyle X/\!/_{L}G} is given simply as the Proj of the ring of invariants R G {\displaystyle R^{G}} . The most interesting case is when the stable locus X s {\displaystyle X^{s}} is nonempty; X s {\displaystyle X^{s}} is the open set of semistable points that have finite stabilizers and orbits that are closed in X s s {\displaystyle X^{ss}} . In such a case, the GIT quotient restricts to
… excerpt ends here. Continue reading the full article.
