04.10.02 · algebraic-geometry / moduli

Geometric invariant theory

shipped3 tiersLean: partial

Anchor (Master): Mumford-Fogarty-Kirwan *GIT* 3rd ed.; Newstead *Lectures on Moduli Problems*; Thomas *Notes on GIT and Symplectic Reduction*; Dolgachev *Lectures on Invariant Theory*

Intuition [Beginner]

Geometric Invariant Theory (GIT) is David Mumford's 1965 framework for taking quotients of varieties by group actions. Imagine a group $G$ acting on a variety $X$ — for example, $PGL_{2}$ acts on the space of binary forms by changing the variable. We want a quotient variety $X / G$ whose points are the orbits — but naive quotients often have terrible properties (non-Hausdorff, non-algebraic, etc.).

Mumford's insight: choose a linearisation — an ample line bundle $L$ with a $G$ -action — and define the quotient as the Proj of the ring of $G$ -invariant sections of $L$ and its powers. Points where this construction "works well" are stable points; intermediate behaviour gives semistable; bad behaviour gives unstable. The GIT quotient parametrises only the stable (and semistable) orbits, throwing out unstable ones.

This is the framework that makes moduli spaces work: the moduli of curves $M_{g}$ , moduli of vector bundles, moduli of varieties — all built as GIT quotients. The Hilbert-Mumford numerical criterion gives a computable test for stability, connecting GIT to symplectic reduction in physics via the Kempf-Ness theorem.

Visual [Beginner]

A variety with a group acting on it; the GIT quotient parametrises orbits, separating stable orbits from unstable ones based on a chosen positivity condition.

Worked example [Beginner]

A canonical GIT example: $G = PGL_{2} (C)$ acts on the space of binary forms of degree $d$ — homogeneous polynomials in two variables, $f (x, y) = a_{0} x^{d} + a_{1} x^{d - 1} y + \dots + a_{d} y^{d}$ .

The space of binary forms is identified with $C^{d + 1}$ (via the coefficients $a_{0}, \dots, a_{d}$ ), or projectively with $P^{d}$ . Two binary forms are equivalent under $PGL_{2}$ iff one is obtained from the other by a Möbius transformation of $(x : y)$ .

For $d = 2$ (binary quadratics): all binary quadratics $a x^{2} + b x y + c y^{2}$ either have distinct roots, repeated roots, or vanish identically. Distinct-root forms (e.g., $x y = 0$ ) are stable — they have finite stabilisers in $PGL_{2}$ . Repeated-root forms (e.g., $x^{2}$ ) have one-parameter stabilisers — they are unstable. The GIT quotient ${binary quadratics} / / PGL_{2}$ is just a point — every distinct-root quadratic is equivalent to $x y$ .

For $d = 3$ (binary cubics): generic cubics have 3 distinct roots and finite stabilisers — stable. Cubics with a triple root or all coefficients vanishing are unstable. The GIT quotient is again a point: every smooth binary cubic is $PGL_{2}$ -equivalent to $x y (x - y)$ .

For $d = 4$ (binary quartics): generic forms have 4 distinct roots, and the GIT quotient is $A^{1}$ — parametrised by the cross-ratio $λ$ of the four roots, modulo the symmetric group $S_{3}$ permuting it. The cross-ratio is the j-invariant in disguise, connecting binary quartics to elliptic curves: branched double covers of $P^{1}$ ramified at 4 points are precisely elliptic curves.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $G$ be a reductive algebraic group (e.g., $GL_{n}$ , $SL_{n}$ , $PGL_{n}$ , semisimple Lie groups) acting on a quasi-projective variety $X$ over an algebraically closed field $k$ .

Linearisation. A $G$ -linearised line bundle on $X$ is a line bundle $L$ on $X$ together with a $G$ -action on the total space of $L$ that lifts the action on $X$ (compatibly). Linearisations are not unique; different choices give different stability conditions.

Stable, semistable, unstable points. Fix a $G$ -linearised ample line bundle $L$ on $X$ . A point $x \in X$ is:

Semistable if there exists a $G$ -invariant section $s \in H^{0} (X; L^{\otimes n})^{G}$ for some $n > 0$ with $s (x) \neq = 0$ . Write $X^{ss} (L)$ for the locus of semistable points.
Stable if $x$ is semistable and additionally: (S1) The orbit $G \cdot x$ is closed in $X^{ss} (L)$ . (S2) The stabiliser $G_{x} = {g \in G : g \cdot x = x}$ is finite.

Write $X^{s} (L)$ for the locus of stable points.
Unstable if not semistable. Write $X^{u s} (L) = X ∖ X^{ss} (L)$ .

The stratification is:

X^{s} (L) \subseteq X^{ss} (L) \subseteq X, X = X^{s} (L) ⊔ (X^{ss} (L) ∖ X^{s} (L)) ⊔ X^{u s} (L) .

GIT quotient. The GIT quotient is the projective scheme

X / /_{L} G := Proj n \geq 0 ⨁ H^{0} (X; L^{\otimes n})^{G} .

The natural morphism $X^{ss} (L) \to X / /_{L} G$ is a good quotient: it is surjective, $G$ -invariant, and identifies $X / /_{L} G$ with the orbit space of $X^{ss}$ modulo the closure equivalence relation (two semistable points are identified iff their orbit closures intersect).

The restriction $X^{s} (L) \to X / /_{L} G$ is a geometric quotient: it identifies the image with the set-theoretic orbit space of stable points (no closure ambiguity since stable orbits are closed).

Hilbert-Mumford numerical criterion. A point $x \in X$ is unstable iff there exists a one-parameter subgroup $λ : G_{m} \to G$ such that the limit $lim_{t \to 0} λ (t) \cdot x =: x_{0}$ exists in $X$ , and the weight of $λ$ on the fibre $L_{x_{0}}$ is positive. Equivalently: $x$ is semistable iff for every 1-PS $λ$ with limit $x_{0} := lim_{t \to 0} λ (t) \cdot x$ , the weight of $λ$ on $L_{x_{0}}$ is non-positive.

The numerical function $μ^{L} (x, λ) := weight_{λ} (L_{x_{0}})$ is the Hilbert-Mumford function. Stability is computed by minimising $μ^{L}$ over all 1-PSs.

Reductivity. The reductivity hypothesis on $G$ is essential: for reductive $G$ , the ring $H^{0} (X; L^{\otimes n})^{G}$ is finitely generated (Hilbert's 14th problem in disguise — Hilbert proved finite generation for $SL_{n}$ ; Mumford's theorem extends it). Without reductivity (e.g., for unipotent groups), invariant rings can fail to be finitely generated (Nagata 1959).

Properness. The GIT quotient $X / /_{L} G$ is projective if $X$ is projective (or, more generally, if the linearisation is sufficiently ample). For non-projective $X$ or non-ample $L$ , the quotient may only be quasi-projective.

Variation of GIT. Different linearisations give different GIT quotients, related by birational transformations (Dolgachev-Hu, Thaddeus 1996). The space of linearisations is a polytope, and the quotient varies as the linearisation crosses walls. This variation of GIT (VGIT) is a fundamental tool in birational geometry.

Key theorem with proof [Intermediate+]

Theorem (Mumford 1965). Let $G$ be a reductive algebraic group acting on a projective variety $X$ over an algebraically closed field $k$ , and $L$ a $G$ -linearised ample line bundle. Then:

(GIT1) The GIT quotient $X / /_{L} G = Proj ⨁_{n} H^{0} (X; L^{\otimes n})^{G}$ is a projective scheme.

(GIT2) The natural morphism $π : X^{ss} (L) \to X / /_{L} G$ is a categorical quotient: it is $G$ -invariant and universal among $G$ -invariant morphisms $X^{ss} \to Y$ to a scheme $Y$ .

(GIT3) The restriction $π ∣_{X^{s} (L)} : X^{s} (L) \to π (X^{s} (L))$ is a geometric quotient: the fibres are $G$ -orbits.

Proof outline. Step 1 — finite generation. By Hilbert-Nagata-Mumford, the invariant ring $R^{G} := ⨁_{n} H^{0} (X; L^{\otimes n})^{G}$ is a finitely generated $k$ -algebra when $G$ is reductive. (Hilbert proved finite generation for classical groups; Nagata showed it can fail for non-reductive groups; Mumford extended Hilbert's result to all reductive groups via the Reynolds operator in characteristic 0 and good filtrations in arbitrary characteristic.)

Step 2 — Proj construction. Since $R^{G}$ is a finitely generated graded $k$ -algebra, $Proj (R^{G})$ is a projective scheme over $k$ . This defines $X / /_{L} G$ .

Step 3 — quotient morphism. The inclusion $R^{G} \subset R := ⨁_{n} H^{0} (X; L^{\otimes n})$ gives a morphism $Proj (R) = X ⇢ Proj (R^{G}) = X / / G$ . This rational map is defined exactly on the locus where some $G$ -invariant section is nonzero — i.e., on $X^{ss} (L)$ . So $π : X^{ss} (L) \to X / /_{L} G$ is a regular morphism.

Step 4 — quotient property on stable locus. On $X^{s} (L)$ : the orbit $G \cdot x$ is closed (by the stability hypothesis), and the stabiliser $G_{x}$ is finite. Standard scheme-theoretic-quotient theorems (Borel, Mumford) imply $π ∣_{X^{s}} : X^{s} \to π (X^{s})$ is a geometric quotient — the fibres are exactly the $G$ -orbits.

Step 5 — equivalence on semistable locus. On $X^{ss} ∖ X^{s}$ : orbits are not closed, but their closures intersect. The categorical quotient identifies points with intersecting orbit closures. Concretely: $x_{1} \sim_{π} x_{2}$ iff $\overline{G \cdot x_{1}} \cap \overline{G \cdot x_{2}} \neq = \emptyset$ in $X^{ss}$ .

Step 6 — projectivity of $X / /_{L} G$ . $Proj$ of a finitely generated graded $k$ -algebra is projective. ✓ $□$

The construction is analogous to Hilbert's 19th-century invariant theory (computing rings of invariants), but Mumford's geometric perspective — viewing the invariants as defining a quotient scheme — was new and unified previously disparate constructions.

Bridge. The construction here builds toward later units of the strand, where the same pattern is taken up at higher structure. The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Exercise 1 (easy, proof).

Compute the GIT quotient $A^{1} / /_{L} G_{m}$ where $G_{m} = C^{\times}$ acts by scaling and $L$ is the structure-sheaf line bundle with the standard linearisation (multiplication by $t \in C^{\times}$ ).

Hint

Compute the invariants in $H^{0} (A^{1}; O)^{G_{m}}$ .

Answer

$A^{1} = Spec k [x]$ with $G_{m}$ acting by $t \cdot x = t x$ . The standard linearisation: $L = O_{A^{1}}$ with $G_{m}$ acting by $t \cdot s = t \cdot s$ on sections (multiplying by $t$ ).

The invariant subring is $⨁_{n} H^{0} (A^{1}; O_{A^{1}})^{G_{m}, weight n}$ , where weight $n$ means $G_{m}$ acts by $t^{n}$ . A polynomial $f (x) = \sum a_{k} x^{k}$ has $G_{m}$ -action $t \cdot f = \sum a_{k} t^{k} x^{k}$ , so the weight- $n$ part is $a_{n} x^{n}$ .

The invariant subring under the $L$ -twisted action — where the section $f$ at $x \neq = 0$ is "weighted" by $t^{- 1}$ relative to standard multiplication — is generated by $x \in H^{0} (O (1))^{G_{m}}$ . So $⨁_{n} (R^{(n)})^{G_{m}} = k [x]$ , generated in degree 1.

GIT quotient: $Proj (k [x]) = Proj (poly ring in one variable, deg 1)$ = a point. Hence $A^{1} / / G_{m} = pt$ .

The semistable locus is $A^{1} ∖ 0$ (where $x \neq = 0$ ). The unstable locus is ${0}$ . The stable locus equals the semistable locus (both with finite stabilisers). So everywhere $G_{m}$ acts on $A^{1} ∖ 0$ with quotient a point — the single $G_{m}$ -orbit collapses to one point.

Exercise 3 (medium, proof).

State and apply the Hilbert-Mumford criterion to determine when a binary quartic $f (x, y) = \sum_{i = 0}^{4} a_{i} x^{4 - i} y^{i}$ is GIT-unstable.

Hint

A binary form is unstable iff it has a root of multiplicity $> d /2$ .

Answer

HM criterion for binary forms. The binary form $f$ is unstable iff there exists a 1-PS of $SL_{2}$ destabilising $f$ . Up to conjugation, every 1-PS of $SL_{2}$ is of the form $λ (t) = diag (t, t^{- 1})$ .

Under $λ (t)$ , the variable $x$ has weight $1$ and $y$ has weight $- 1$ . So the monomial $x^{4 - i} y^{i}$ has weight $(4 - i) - i = 4 - 2 i$ . A binary form $f = \sum a_{i} x^{4 - i} y^{i}$ has weights ${4 - 2 i : a_{i} \neq = 0}$ .

The HM weight $μ^{L} (f, λ) = - min_{i} {4 - 2 i : a_{i} \neq = 0}$ . The form is unstable iff $μ^{L} > 0$ , equivalently $min_{i} {4 - 2 i : a_{i} \neq = 0} < 0$ — i.e., all $a_{i}$ for $i \leq 2$ vanish.

Geometrically: $a_{0} = a_{1} = a_{2} = 0$ means $f = a_{3} x y^{3} + a_{4} y^{4}$ , which has $y$ as a root of multiplicity $\geq 3$ . So the form has a root of multiplicity $> d /2 = 2$ at $y = 0$ (i.e., the point $[0 : 1]$ ).

By varying the 1-PS over all of $SL_{2}$ (i.e., over all points of $P^{1}$ ), we conclude: a binary quartic $f$ is unstable iff $f$ has a root of multiplicity $\geq 3$ . (Multiplicity 2: still semistable but not stable. Multiplicity $\leq 2$ everywhere: stable.) $□$

For general degree $d$ : a binary $d$ -form is stable iff every root has multiplicity $< d /2$ , *semistable* iff every root has multiplicity $\leq d /2$ , *unstable* iff there is a root of multiplicity $> d /2$ . This is the Mumford-Hilbert numerical criterion for binary forms.

Exercise 4 (medium, conceptual).

Why is the reductivity hypothesis essential for GIT to work?

Hint

Without reductivity, the invariant ring may fail to be finitely generated (Nagata 1959).

Answer

A reductive algebraic group $G$ has the property that every finite-dimensional rational representation of $G$ is completely reducible — direct sum of irreducible subrepresentations. Equivalently (in characteristic 0), $G$ is reductive iff its identity component is the product of a torus and a semisimple group.

Examples of reductive groups: $GL_{n}$ , $SL_{n}$ , $O (n)$ , $Sp (2 n)$ , all classical groups, simply connected semisimple Lie groups. Examples of non-reductive: unipotent groups (e.g., the additive group $G_{a}$ ), Borel subgroups of semisimple groups.

Why reductivity matters for GIT.

Finite generation of invariants. For reductive $G$ acting on a finitely generated $k$ -algebra $R$ , the invariant subring $R^{G}$ is finitely generated (Hilbert-Mumford-Nagata theorem). This is essential for $Proj (R^{G})$ to be a finite-dimensional projective scheme — without finite generation, $Proj$ would not even make sense.
Reynolds operator. In characteristic 0, reductive groups admit a Reynolds operator $R^{G} \subset R$ — a $k$ -linear projector $R \to R^{G}$ that is $G$ -invariant. The Reynolds operator is the key tool for proving finite generation.
Counter-example: Nagata 1959. Masayoshi Nagata constructed a non-reductive group action on $A^{32}$ whose invariant ring is not finitely generated. This was the negative answer to Hilbert's 14th problem: "Is the invariant ring of a finitely generated algebra under a group action always finitely generated?" Hilbert had proved yes for classical groups (via the symbolic method); Nagata showed no for non-reductive groups.
Modern non-reductive GIT. Doran, Kirwan, Berchtold, et al. (2010s) developed non-reductive GIT using augmented quotient stacks and reductive envelopes. The theory is significantly more delicate.

Practical consequence. All standard moduli-space constructions use reductive groups: $M_{g}$ via $PGL$ , moduli of vector bundles via $GL$ , moduli of $K 3$ surfaces via $O (20, 2)$ , etc. Non-reductive group actions show up in birational geometry of unstable loci and require the modern non-reductive GIT framework.

Exercise 5 (medium, numeric).

For the binary cubic action of $PGL_{2}$ on $P^{3} = P (V_{3})$ , identify the (semi)stable / unstable loci and compute the GIT quotient.

Hint

Cubic with 3 distinct roots = stable; cubic with double root = semistable; cubic with triple root = unstable.

Answer

Binary cubic $f (x, y) = a_{0} x^{3} + a_{1} x^{2} y + a_{2} x y^{2} + a_{3} y^{3}$ .

By the HM criterion (Exercise 3 generalised to $d = 3$ ):

Stable ( $f^{stable}$ ): all 3 roots distinct. Multiplicity $< 3/2$ .
Semistable, not stable: a double root and a simple root. Multiplicity $\leq 3/2$ .
Unstable: a triple root. Multiplicity $> 3/2$ .

The space $P (V_{3}) = P^{3}$ has 3-dimensional $PGL_{2}$ -orbits in the stable locus, so the stable quotient has dimension $dim P^{3} - dim PGL_{2} = 3 - 3 = 0$ .

So the GIT quotient $P (V_{3}) / / PGL_{2}$ is 0-dimensional — a finite set of points. In fact, it is a single point: every binary cubic with 3 distinct roots is $PGL_{2}$ -equivalent to $x y (x - y)$ (a unique normal form). The stable locus modulo $PGL_{2}$ is one orbit.

The semistable orbits with double roots also map to this single point in the GIT quotient (their closures contain the unique stable orbit closure). The unstable locus (cubics with triple root) is removed.

So $P^{3} / / PGL_{2} = pt$ for binary cubics. This matches the classical result: there is a unique smooth projective binary cubic up to $PGL_{2}$ , and every elliptic curve arises as a double cover of $P^{1}$ branched at 4 points, parametrised by binary quartics (not cubics), giving $M_{1} ≅ A^{1}$ .

Exercise 6 (hard, proof).

State the Kempf-Ness theorem and explain how it relates GIT to symplectic reduction.

Hint

GIT quotient $X / / G$ ↔ symplectic reduction $X / /_{ξ} K$ of compact subgroup $K \subset G$ .

Answer

Kempf-Ness theorem (Kempf-Ness 1979, Annals of Math. 110). Let $G$ be a complex reductive group acting linearly on a complex vector space $V = C^{n}$ , with $K \subset G$ a maximal compact subgroup. Choose a $K$ -invariant Hermitian metric on $V$ .

Then for the action restricted to a projective variety $X \subset P (V)$ with linearisation $O (1)$ :

A point $[v] \in P (V)$ is GIT-semistable iff the closure $\overline{G \cdot v}$ contains a vector with minimal norm (with respect to the Hermitian metric).
A minimal-norm vector $v_{0} \in G \cdot v$ exists iff the moment map $μ : P (V) \to k^{*}$ vanishes on the orbit: $μ (G \cdot v_{0}) ∋ 0$ .
The GIT quotient $X / /_{G} P (V)$ is canonically homeomorphic to the symplectic reduction $μ^{- 1} (0) / K$ at level zero.

Equivalence GIT ↔ symplectic reduction:

X / /_{G} O (1) ≅ μ^{- 1} (0) / K .

Why this matters. GIT is purely algebraic; symplectic reduction is purely differential-geometric (Marsden-Weinstein 1974). Kempf-Ness shows they coincide for the natural class of examples — projective varieties with reductive complex group actions — providing two equivalent perspectives:

Algebraic perspective. GIT: stability via 1-PSs, finite generation of invariants, Proj construction.
Symplectic perspective. Moment map $μ : X \to k^{*}$ , level-zero vanishing, quotient by maximal compact.

The Kempf-Ness theorem connects:

Kirwan's stratification of the unstable locus matches the Morse-theoretic stratification of $∥ μ ∥^{2}$ (Kirwan 1984).
Hyperkähler reduction generalises symplectic reduction; Hitchin used hyperkähler quotients to construct moduli spaces of Higgs bundles (1987).
Mirror symmetry identifies Calabi-Yau threefolds via GIT/symplectic-reduction-style constructions.

Modern relevance. Kempf-Ness underlies the deformation quantisation perspective on quantum field theories (Donaldson-Thomas-Witten), the Atiyah-Bott approach to moduli of bundles (1983), and the modern interplay between algebraic geometry and symplectic topology in the Fukaya category / mirror symmetry framework. $□$

Exercise 7 (hard, conceptual).

Explain the role of GIT in the construction of moduli spaces beyond curves: moduli of vector bundles, K-stability, and mirror-symmetry quotients.

Hint

GIT is the foundational tool for any algebraic moduli space.

Answer

Moduli of vector bundles (Mumford 1962, Seshadri 1967, Atiyah-Bott 1983). The moduli space of stable vector bundles of rank $r$ and degree $d$ on a smooth projective curve $C$ is constructed as a GIT quotient. Mumford's stability condition for a vector bundle $E$ on $C$ :

μ (E) := \frac{de g E}{rank E} > μ (F) for every proper sub-bundle F \subset E .

This slope stability is exactly the GIT stability for the natural $PGL$ -action on the parameter space of bundles. The resulting moduli space $M (r, d)$ is a smooth projective variety of dimension $r^{2} (g - 1) + 1$ , with rich geometric structure (Atiyah-Bott computed its cohomology; Verlinde formula counts its line bundles).

Moduli of sheaves and Donaldson-Thomas theory. Higher-dimensional generalisations: Gieseker stability for sheaves (Gieseker 1977), Bridgeland stability on derived categories (Bridgeland 2007). DT theory (Donaldson-Thomas 1998) counts stable sheaves on Calabi-Yau threefolds, computing DT invariants central to mirror symmetry.

K-stability and Calabi-Yau metrics (Tian, Donaldson, Chen-Donaldson-Sun 2013-15). K-stability is a GIT-style stability for Fano varieties. The Yau-Tian-Donaldson conjecture (proved 2015 by Chen-Donaldson-Sun): a Fano variety admits a Kähler-Einstein metric iff it is K-stable. The proof uses Kempf-Ness-style infinite-dimensional GIT and is one of the great recent applications of stability theory.

Mirror symmetry and GIT quotients. The Calabi-Yau / Landau-Ginzburg correspondence identifies certain Calabi-Yau threefolds (e.g., the quintic) with GIT quotients of vector spaces by group actions (Witten, Hori-Vafa). The mirror identification swaps two GIT quotient descriptions, with mirror symmetry interchanging stable and unstable strata.

Variation of GIT (VGIT). Different linearisations give different quotients, related by flips (small modifications). Thaddeus 1996, Dolgachev-Hu 1998: classify GIT quotients in a family, relating them by birational transformations. VGIT is a major tool in birational geometry (the minimal model program).

Categorical and derived GIT. Modern generalisations:

Derived GIT via derived categories of coherent sheaves (Halpern-Leistner 2014): lifts the GIT picture to a derived setting, with Bridgeland stability conditions parametrising "linearisations" in the derived category.
Stacky GIT (Alper, Halpern-Leistner-Heinloth): treats GIT quotients of moduli stacks, with applications to moduli of higher-dimensional varieties.
Non-reductive GIT (Doran-Kirwan, Berchtold-Hausen): extends to non-reductive groups via augmented quotient stacks.

Modern significance. GIT is the foundational tool for any algebraic moduli space. Whenever we have a parameterising space of geometric objects with a group action identifying "equivalent" objects, GIT (or its modern derived/stacky generalisations) constructs the moduli space. The Mumford 1965 framework remains the starting point; the modern theory has greatly extended its reach. $□$

Lean formalization [Intermediate+]

lean_status: partial — Mathlib has group actions on schemes and basic invariant theory; the GIT quotient construction is partial.

[object Promise]

Advanced results [Master]

Hilbert-Nagata-Mumford finite generation. For reductive $G$ acting on a finitely generated $k$ -algebra $R$ , the invariant ring $R^{G}$ is finitely generated. Hilbert proved this for classical groups via the symbolic method (1890s). Nagata 1959 (Amer. J. Math. 81) showed finite generation can fail for non-reductive groups, answering Hilbert's 14th problem in the negative. Mumford extended Hilbert's theorem to all reductive groups in arbitrary characteristic, completing the foundational case for GIT.

Hilbert-Mumford numerical criterion. A precise computational test for stability via 1-parameter subgroups: $x \in X$ is unstable iff some 1-PS $λ$ has $μ^{L} (x, λ) < 0$ . Reduces stability to a finite check in many practical cases (binary forms, point configurations, vector bundles).

Kempf-Ness theorem (1979). Identifies GIT quotient with symplectic reduction at the moment-map level zero. Provides two perspectives — algebraic and symplectic — with multiple applications.

Kirwan's stratification (1984). The unstable locus of a $G$ -action stratifies into finitely many Kirwan strata, each a $G$ -invariant locally closed subvariety. The stratification refines the GIT picture and gives a Morse-theoretic perspective via the moment-map norm-squared $∥ μ ∥^{2}$ . Used in computing equivariant cohomology of GIT quotients.

Variation of GIT (Dolgachev-Hu, Thaddeus 1996-98). Different linearisations give different GIT quotients, related by birational transformations. The space of linearisations is a polytope; quotients vary by flips across walls. Foundational for the birational geometry of moduli spaces.

Derived GIT (Halpern-Leistner 2014). Lifts the GIT picture to derived categories of coherent sheaves. Magic windows and derived Kirwan-surjectivity give explicit equivalences of derived categories of GIT quotients across walls.

Bridgeland stability (Bridgeland 2007). Generalises Mumford-Gieseker slope stability to Bridgeland stability conditions on triangulated categories. Each stability condition gives a moduli space of stable objects; the space of stability conditions is a complex manifold (Bridgeland's stability manifold), connecting moduli theory to mirror symmetry.

Yau-Tian-Donaldson conjecture (Chen-Donaldson-Sun 2015). A Fano variety admits a Kähler-Einstein metric iff it is K-stable. K-stability is an infinite-dimensional GIT-style stability via test configurations. The proof uses analysis (Kempf-Ness-style infinite-dimensional GIT) and is one of the greatest recent applications of GIT.

Non-reductive GIT (Doran-Kirwan 2010s). GIT for non-reductive group actions via reductive envelopes and augmented quotient stacks. Restores finite generation by enlarging $G$ to a reductive group containing it.

Synthesis. This construction generalises the pattern fixed in 04.02.01 (scheme), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

The Mumford GIT theorem is sketched in the formal-definition section. The Hilbert-Mumford numerical criterion is proved by reducing to the case of $A^{N}$ with a torus action and applying convexity (Mumford-Fogarty-Kirwan §2). Kempf-Ness is proved by a moment-map / convex-function argument (Kempf-Ness 1979 Annals, or Thomas's Notes on GIT and Symplectic Reduction). K-stability and the YTD conjecture: Chen-Donaldson-Sun 2015 (J. Amer. Math. Soc. 28, 183–278; 28, 199–234; 28, 235–278).

Connections [Master]

Scheme 04.02.01 — GIT quotients are projective schemes obtained from Proj of invariant rings.
Group action 03.03.02 — GIT extends invariant theory to algebraic group actions on schemes.
Lie group 03.03.01 — reductive Lie groups are the natural domain of GIT.
Ample line bundle 04.05.05 — linearisations are ample line bundles with $G$ -action.
Moduli of curves 04.10.01 — Mumford's original GIT application; constructs $M_{g}$ as a quasi-projective scheme.
Geometric invariant theory — Mumford's GIT (this unit) is foundational; symplectic reduction (in differential geometry) is the parallel construction.
Bridgeland stability — a categorical analogue of GIT stability; foundational for derived-category mirror symmetry.

Historical & philosophical context [Master]

David Hilbert's 19th-century invariant theory studied rings of polynomial invariants under classical group actions ( $GL_{n}, SL_{n}, O (n)$ acting on tensors and forms). His 1890 paper Über die Theorie der algebraischen Formen (Math. Ann. 36) proved the celebrated finite-generation theorem: the invariant ring $C [V]^{G}$ of a reductive group $G$ acting on a finite-dimensional $V$ is a finitely generated $C$ -algebra. The proof was non-constructive — Hilbert's Basis Theorem and Nullstellensatz give existence without explicit invariants. Hilbert's 14th problem (1900) asked whether the same holds for arbitrary group actions on finitely generated algebras; Nagata 1959 answered no for non-reductive groups, but the reductive case (Hilbert-Mumford-Nagata) was eventually resolved.

David Mumford's 1965 Geometric Invariant Theory (Springer-Verlag) transformed Hilbert's classical invariant theory into a tool for constructing moduli spaces. Mumford's key insight: the invariant ring $R^{G}$ is not just an algebraic object; its Proj is a scheme parametrising the orbits — a moduli space. This geometric perspective unified disparate constructions in algebraic geometry under one framework.

Mumford's introduction of stability — stable, semistable, unstable points — was the conceptual breakthrough. Naive group quotients $X / G$ in algebraic geometry are usually badly behaved (non-Hausdorff, non-projective). Mumford's stability conditions identify the "good" subset $X^{ss}$ where the quotient $X^{ss} / / G$ behaves well as a projective scheme. The Hilbert-Mumford numerical criterion provided a computable test for stability via 1-parameter subgroups, making GIT practical.

The 1965 book established GIT as the foundation of moduli theory:

$M_{g}$ (moduli of curves, Mumford 1965): GIT quotient of the Hilbert scheme of tri-canonically embedded curves by $PGL$ .
Moduli of vector bundles on curves (Seshadri 1967, Mumford 1962): GIT quotient with slope stability.
Moduli of abelian varieties (Mumford 1965, 1968): GIT quotient with theta linearisation.
Moduli of K3 surfaces (Mumford-Hodge 1980): GIT-style quotient.

The third edition of GIT (1994) with Fogarty and Kirwan added Kirwan's stratification of the unstable locus (1984) — a Morse-theoretic refinement applied to compute equivariant cohomology of GIT quotients.

The 1980s-90s extended GIT in two directions:

Symplectic geometry (Atiyah-Bott 1983; Kempf-Ness 1979). The Kempf-Ness theorem identifies GIT quotient $X / / G$ with the symplectic reduction $μ^{- 1} (0) / K$ of the maximal compact subgroup $K \subset G$ . This identifies algebraic GIT (purely algebraic) with symplectic reduction (purely differential-geometric), providing two perspectives on the same objects. Atiyah-Bott (1983, Phil. Trans. Royal Soc.) used this to compute the cohomology of moduli of vector bundles via equivariant Morse theory on the space of connections.

Variation of GIT (Dolgachev-Hu 1998; Thaddeus 1996). Different linearisations give different GIT quotients, related by birational transformations (flips). VGIT is a fundamental tool in birational geometry of moduli spaces and the minimal model program.

In the 2000s-10s:

Derived GIT (Halpern-Leistner 2014). GIT lifted to derived categories: derived equivalences across walls in linearisation space, with applications to mirror symmetry.

Stacky and non-reductive GIT (Alper, Doran-Kirwan). Modern moduli theory with stacky and non-reductive group actions.

K-stability and YTD conjecture (Tian, Donaldson, Chen-Donaldson-Sun 2015). Infinite-dimensional GIT applied to Kähler-Einstein metrics. The Yau-Tian-Donaldson conjecture — a Fano variety admits a Kähler-Einstein metric iff K-stable — was proved by Chen-Donaldson-Sun in three monumental papers (J. Amer. Math. Soc. 2015), one of the greatest recent applications of GIT.

David Mumford was awarded the Fields Medal in 1974 — partly for GIT. The book remains foundational; modern moduli theory is unimaginable without it. Hilbert's 19th-century invariant theory, transformed by Mumford's geometric perspective, became one of the central tools of modern algebraic geometry.

Bibliography [Master]

Hilbert, D., Über die Theorie der algebraischen Formen, Math. Ann. 36 (1890), 473–534.
Mumford, D., Geometric Invariant Theory, Springer-Verlag 1965; 3rd ed. with Fogarty & Kirwan 1994.
Nagata, M., On the 14th problem of Hilbert, Amer. J. Math. 81 (1959), 766–772.
Kempf, G. & Ness, L., The length of vectors in representation spaces, in Algebraic Geometry, Copenhagen 1978, Springer LNM 732, 1979, 233–243.
Atiyah, M. & Bott, R., The Yang-Mills equations over Riemann surfaces, Phil. Trans. Royal Soc. London A 308 (1983), 523–615.
Kirwan, F., Cohomology of Quotients in Symplectic and Algebraic Geometry, Princeton 1984.
Newstead, P. E., Lectures on Introduction to Moduli Problems and Orbit Spaces, Tata 1978.
Thomas, R. P., Notes on GIT and Symplectic Reduction for Bundles and Varieties, Surveys in Differential Geometry X (2006), 221–273.
Dolgachev, I., Lectures on Invariant Theory, LMS Lecture Note Series 296, Cambridge 2003.
Thaddeus, M., Geometric Invariant Theory and Flips, J. Amer. Math. Soc. 9 (1996), 691–723.
Dolgachev, I. & Hu, Y., Variation of geometric invariant theory quotients, Publ. Math. IHES 87 (1998), 5–56.
Bridgeland, T., Stability conditions on triangulated categories, Annals of Math. 166 (2007), 317–345.
Chen, X., Donaldson, S. & Sun, S., Kähler-Einstein metrics on Fano manifolds I, II, III, J. Amer. Math. Soc. 28 (2015), 183–278; 28 (2015), 199–234; 28 (2015), 235–278.

Prerequisites

04.02.01
03.03.01
03.03.02

Tier anchors

beginner: Strogatz-style: quotient varieties via stability
intermediate: Mumford-Fogarty-Kirwan; Newstead lectures
master: Mumford-Fogarty-Kirwan *GIT* 3rd ed.; Newstead *Lectures on Moduli Problems*; Thomas *Notes on GIT and Symplectic Reduction*; Dolgachev *Lectures on Invariant Theory*

References

TODO_REF
Mumford — Geometric Invariant Theory · Springer 1965; 3rd ed. Mumford-Fogarty-Kirwan 1994
TODO_REF
Newstead — Lectures on Introduction to Moduli Problems and Orbit Spaces · Tata Institute 1978
TODO_REF
Thomas — Notes on GIT and Symplectic Reduction · Surveys in Differential Geometry X, 2006
TODO_REF
Dolgachev — Lectures on Invariant Theory · LMS Lecture Note Series 296, Cambridge 2003

Lean module

Codex.AlgebraicGeometry.Moduli.GIT

Reviewer

TBD

Estimated time

beginner: 12m
intermediate: 40m
master: 65m