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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 locus¹ 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

π s : X s → X s / / G , {\displaystyle \pi ^{s}\colon X^{s}\to X^{s}/\!/G,}

which has the property: every fiber is an orbit. That is to say, π s {\displaystyle \pi ^{s}} is a genuine quotient (i.e., geometric quotient) and one writes X s / G = X s / / G {\displaystyle X^{s}/G=X^{s}/\!/G} . Because of this, when X s {\displaystyle X^{s}} is nonempty, the GIT quotient π {\displaystyle \pi } 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 X {\displaystyle X} be a locally factorial algebraic variety (for example, a smooth variety) with an action of G {\displaystyle G} . Suppose there are an open subset U ⊂ X {\displaystyle U\subset X} as well as a geometric quotient π : U → U / G {\displaystyle \pi \colon U\to U/G} such that (1) π {\displaystyle \pi } is an affine morphism and (2) U / G {\displaystyle U/G} is quasi-projective. Then U ⊂ X s ( L ) {\displaystyle U\subset X^{s}(L)} 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 Z / 2 {\displaystyle \mathbb {Z} /2}

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

x ↦ ( − x ) y ↦ ( − y ) {\displaystyle {\begin{aligned}x\mapsto (-x)&&y\mapsto (-y)\end{aligned}}}

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

C [ x , y ] Z / 2 = C [ x 2 , x y , y 2 ] = C [ a , b , c ] ( a c − b 2 ) {\displaystyle \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

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

which is a singular subvariety of A 3 {\displaystyle \mathbb {A} ^{3}} with isolated singularity at ( 0 , 0 , 0 ) {\displaystyle (0,0,0)} . This can be checked using the differentials, which are

d f = [ c − 2 b a ] {\displaystyle df={\begin{bmatrix}c&-2b&a\end{bmatrix}}}

hence the only point where the differential and the polynomial f {\displaystyle f} 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 G m {\displaystyle \mathbb {G} _{m}} on X = A 2 {\displaystyle X=\mathbb {A} ^{2}} by t ⋅ ( x , y ) = ( t x , t − 1 y ) {\displaystyle t\cdot (x,y)=(tx,t^{-1}y)} . Note this action has a few orbits: the origin ( 0 , 0 ) {\displaystyle (0,0)} , the punctured axes, { ( x , 0 ) : x ≠ 0 } , { ( 0 , y ) : y ≠ 0 } {\displaystyle \{(x,0):x\neq 0\},\{(0,y):y\neq 0\}} , and the affine conics given by x y = a {\displaystyle xy=a} for some a ∈ C ∗ {\displaystyle a\in \mathbb {C} ^{*}} . Then, the GIT quotient X / / G m {\displaystyle X//\mathbb {G} _{m}} has structure sheaf O A 2 G m {\displaystyle {\mathcal {O}}_{\mathbb {A} ^{2}}^{\mathbb {G} _{m}}} which is the subring of polynomials C [ x y ] {\displaystyle \mathbb {C} [xy]} , hence it is isomorphic to A 1 {\displaystyle \mathbb {A} ^{1}} . This gives the GIT quotient

π : A 2 → A 2 / / G m {\displaystyle \pi \colon \mathbb {A} ^{2}\to \mathbb {A} ^{2}//\mathbb {G} _{m}}

Notice the inverse image of the point ( 0 ) {\displaystyle (0)} is given by the orbits ( 0 , 0 ) , { ( x , 0 ) : x ≠ 0 } , { ( 0 , y ) : y ≠ 0 } {\displaystyle (0,0),\{(x,0):x\neq 0\},\{(0,y):y\neq 0\}} , 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

• quotient stack
• character variety
• Chow quotient

Notes

Pedagogical

• Mukai, Shigeru (2002). An introduction to invariants and moduli. Cambridge Studies in Advanced Mathematics. Vol. 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. Vol. 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. Vol. 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)]. Vol. 34 (3rd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906.

References

1. NB: In (Mumford, Fogarty & Kirwan 1994), it was called the set of properly stable points - Mumford, David; Fogarty, John; Kirwan, Frances (1994). Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Vol. 34 (3rd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906. https://mathscinet.ams.org/mathscinet-getitem?mr=1304906 ↩

2. Mumford, Fogarty & Kirwan 1994, Converse 1.13. NB: even though the result is stated for a smooth variety, the proof there is valid for a locally factorial one. - Mumford, David; Fogarty, John; Kirwan, Frances (1994). Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Vol. 34 (3rd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-56963-3. MR 1304906. https://mathscinet.ams.org/mathscinet-getitem?mr=1304906 ↩

3. Thomas, Richard P. (2006). "Notes on GIT and symplectic reduction for bundles and varieties". Surveys in Differential Geometry. 10 (1). International Press of Boston: 221–273. arXiv:math/0512411. doi:10.4310/sdg.2005.v10.n1.a7. ISSN 1052-9233. MR 2408226. S2CID 16294331. /wiki/Richard_Thomas_(mathematician) ↩