Geometric invariant theory asks how to divide an algebraic variety by a group action without losing the geometry that makes the quotient useful. The answer is not simply to identify every pair of points lying in the same orbit. Orbit spaces can fail to be Hausdorff, orbit closures can collide, stabilizers can jump, and polynomial invariants can distinguish some orbits but not others. GIT builds a controlled quotient by combining invariant rings with a notion of stability.
Representation mathematics enters immediately. A reductive group acts linearly on a vector space or projective variety. Polynomial functions form representations. Invariants are fixed vectors. One-parameter subgroups test whether a point is driven toward instability. The resulting quotient organizes families of objects into moduli spaces.
This guide connects the BTT routes through The Orbit Method, Geometric Representation Theory, and Quiver Representations. The next article applies GIT directly to quiver moduli.
Affine quotients · Invariant rings · Projective GIT · Stable and semistable points · Hilbert–Mumford · Moduli · Practice
Why naive orbit spaces are often bad quotients
Let a group G act on a variety X. The set-theoretic orbit space X/G identifies points lying in the same orbit. But algebraic geometry needs more: regular functions on the quotient, a useful topology, and a relation between closed subsets upstairs and downstairs.
If two distinct orbits have closures that intersect, any invariant regular function must take the same value on both orbit closures. A quotient built from invariants will therefore identify more than literal orbit equivalence.
This is not a defect. It reflects what algebraic functions can actually detect.
Affine GIT quotient
Suppose X=Spec A is affine and a reductive algebraic group G acts algebraically on X. Then G acts on the coordinate ring A.
The invariant ring is
A^G={f∈A : g·f=f for every g∈G}.
The affine GIT quotient is
X // G = Spec(A^G).
For reductive G over characteristic zero, Hilbert’s finite-generation theorem ensures A^G is finitely generated when A is finitely generated. Thus the quotient remains an affine algebraic variety.
Worked invariant example: C× acting with weights +1 and −1
Let G=C× act on C² by
t·(x,y)=(tx,t^{-1}y).
On the polynomial ring C[x,y], a monomial x^a y^b has weight a−b. It is invariant exactly when a=b.
Therefore the invariant ring is generated by xy:
C[x,y]^{C×}=C[xy].
Hence
C² // C× ≈ A¹.
The quotient map is (x,y)↦xy.
Orbit-closure phenomenon in the example
If xy≠0, the orbit is closed in the corresponding fiber and the invariant xy distinguishes its quotient point.
If xy=0, the x-axis and y-axis contain nonzero orbits whose closures both contain the origin. All of them map to 0 in the quotient.
The affine quotient therefore identifies orbit closures rather than guaranteeing one quotient point per raw orbit.
Closed orbits are the geometric representatives downstairs
For reductive group actions on affine varieties, each fiber of the quotient map contains a unique closed orbit.
Two points have the same image in X//G exactly when the closures of their orbits intersect.
Thus the quotient can be viewed as parametrizing closed orbits, with nonclosed orbits collapsing toward the closed orbit in their closure.
Why reductivity matters
For nonreductive groups, invariant rings can fail to be finitely generated. Then Spec(A^G) may not behave like a finite-type algebraic quotient.
Modern nonreductive GIT exists, but the classical finite-generation theorems and quotient properties cannot be copied unchanged.
Always identify whether G is reductive before invoking standard GIT.
Projective GIT and linearization
For a projective variety X with a G-action, one chooses an ample line bundle L together with a G-linearization: an action of G on the total space of L compatible with the action on X.
The linearization determines which points are stable or semistable. Changing the linearization can change the quotient even when the underlying variety and group action remain fixed.
The projective quotient can be constructed from the graded invariant ring
X //_{L} G = Proj ⊕_{m≥0} H^0(X,L^m)^G
under standard hypotheses.
Stable versus semistable
A point is semistable when some positive-degree invariant section does not vanish there.
A stable point satisfies stronger conditions that, in the usual projective GIT setup, force a closed orbit in the semistable locus together with finite stabilizer and good local quotient behavior.
The precise formulation depends on the chosen linearization and conventions, but the hierarchy is
stable ⊆ semistable ⊆ X.
Unstable points are discarded before quotienting.
Why discard unstable points?
If an orbit is driven toward the origin or another degenerate configuration under a one-parameter subgroup, it often represents an object that should not define an independent moduli point.
GIT stability removes those degenerations before quotienting, producing a moduli space with better separation properties.
In moduli problems, “stability” is therefore not a moral label. It is an algebraic criterion controlling degenerations and automorphisms.
Hilbert–Mumford criterion
The Hilbert–Mumford criterion reduces stability testing for a reductive group to one-parameter subgroups
λ:C×→G.
Given a linearized line bundle, one computes a numerical weight μ^L(x,λ) describing how λ acts on the limiting fiber as t→0.
With one common sign convention, x is semistable exactly when μ^L(x,λ)≥0 for every one-parameter subgroup λ. Other sources reverse the sign. The convention must be fixed before comparing inequalities.
The important structural fact is independent of sign: a potentially huge group-action problem reduces to testing one-dimensional torus directions.
A weight-polytope interpretation
Suppose a torus T acts diagonally on a vector space with weights χ_1,…,χ_m. A projective point supported on coordinates indexed by I is semistable when the chosen linearization weight lies in the convex hull of the active weights, after translating conventions appropriately.
Thus stability can become convex geometry.
This is one route from GIT to moment polytopes, Horn inequalities and symplectic reduction.
GIT and symplectic reduction
For a complex reductive group G with maximal compact subgroup K acting on a suitable Kähler variety, the Kempf–Ness theorem links GIT quotients to symplectic quotients.
Under appropriate hypotheses, a complex G-orbit is polystable precisely when it meets the zero level of the moment map, and
X // G
is homeomorphic to a symplectic reduction μ^{-1}(0)/K.
This connects GIT directly to the symplectic reduction route in The Orbit Method.
Variation of GIT quotient
If the linearization varies, the stable and semistable loci can change only across walls in a rational chamber decomposition.
Crossing a wall can perform birational transformations of the quotient. This is variation of GIT, or VGIT.
One moduli problem can therefore have several birational models corresponding to different stability parameters.
Why GIT builds moduli spaces
A moduli problem begins with a parameter space X of algebraic data and a group G encoding change of coordinates.
Isomorphic objects lie in the same orbit. But raw orbit space may be too singular or nonseparated. GIT selects semistable objects and forms a quotient in which closed semistable orbits correspond to polystable objects.
Stable objects often have finite automorphism groups and correspond to geometric quotient points.
The next article applies exactly this machinery to representations of a quiver.
A compact workflow
- 1. Specify X and G.
- 2. Check reductivity.
- 3. Compute the invariant ring in affine problems.
- 4. For projective problems, choose and state the linearization.
- 5. Define stable and semistable loci with one sign convention.
- 6. Test one-parameter subgroups through Hilbert–Mumford.
- 7. Distinguish orbits from orbit closures.
- 8. Identify closed/polystable representatives.
- 9. If varying stability, locate walls and chambers.
- 10. Separate set-theoretic orbit spaces from categorical or geometric quotients.
Common mistakes
- Writing X/G when the actual construction is X//G.
- Assuming quotient points always correspond to single raw orbits.
- Ignoring orbit closures.
- Using reductive finite-generation theorems for arbitrary nonreductive groups.
- Forgetting the projective linearization.
- Mixing Hilbert–Mumford sign conventions.
- Calling semistable and stable identical.
- Ignoring variation of stability parameters.
Practice questions
1. Define the affine GIT quotient. 2. Compute the invariant ring for t·(x,y)=(tx,t^{-1}y). 3. What quotient map results? 4. Why do points on the coordinate axes collapse to the same quotient point?
5. What extra choice is required in projective GIT? 6. What does semistability mean in terms of invariant sections? 7. What is tested in the Hilbert–Mumford criterion? 8. Why does reductivity matter?
9. What does Kempf–Ness relate? 10. What is VGIT? 11. Why are stable points attractive in moduli problems? 12. What do closed orbits represent in affine GIT fibers?
Worked answers
1. Spec(A^G) for X=Spec A.
2. C[xy].
3. (x,y)↦xy.
4. Their orbit closures meet at the origin, and invariants cannot separate them.
5. A G-linearized ample line bundle.
6. Some positive-degree invariant section is nonzero at the point.
7. All one-parameter subgroups of G, via a numerical weight.
8. Classical invariant finite-generation and quotient theorems rely on reductivity.
9. Algebraic GIT quotients and symplectic reductions.
10. The study of how quotients change as linearization/stability parameters vary.
11. Their orbits are well separated and stabilizers are typically finite under the standard setup.
12. Unique closed representatives of quotient equivalence classes.
Sources and further study
David Mumford, John Fogarty and Frances Kirwan’s Geometric Invariant Theory is the classical foundation. Frances Kirwan’s work connects GIT with symplectic geometry, while Dolgachev and Hu developed variation of GIT quotients. For representation-theoretic applications, continue into quiver moduli, Nakajima quiver varieties and geometric representation theory.
Representation Mathematics — Batch 11
Apply stability directly to Quiver Moduli and King Stability. Pass from abelian categories to Derived Categories and Exceptional Collections. Build Kac–Moody representations geometrically in Nakajima Quiver Varieties. Return to the BTT Mathematics Learning Hub.