04.10.14 · algebraic-geometry / moduli

Non-reductive GIT

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Anchor (Master): Mumford-Fogarty-Kirwan *Geometric Invariant Theory* 3rd ed. (Springer 1994, originator framework restricted to reductive $G$); Nagata 1959 *On the 14th problem of Hilbert* (Amer. J. Math. 81, 766-772, the negative answer); Doran-Kirwan 2007 *Towards non-reductive geometric invariant theory* (Pure Appl. Math. Q. 3, 61-105, the foundational extension); Bérczi-Kirwan 2016 *Graded unipotent groups and Grosshans theory* (Compos. Math. 152, 1759-1808, the modern unipotent-stable framework); Bérczi-Doran-Hawes-Kirwan 2018 *Projective completions of graded unipotent quotients* (J. London Math. Soc. 97); Grosshans 1997 *Algebraic Homogeneous Spaces and Invariant Theory* (Springer LNM 1673); Bérczi 2017 *Tautological integrals on curvilinear Hilbert schemes* (Geom. Topol. 21) and Bérczi 2020 *Non-reductive geometric invariant theory and hyperbolicity* (arXiv:2009.09908) for applications to Green-Griffiths-Lang

Intuition Beginner

Classical geometric invariant theory works because the acting group is reductive. Reductive groups — like the general linear group, the special linear group, and complex tori — have a magical property: when you collect all the polynomial functions on a variety that are fixed by the group action, the result is a finitely generated ring. That finite ring is what Mumford's construction needs to build a projective quotient variety. Without finite generation, the quotient construction simply does not get off the ground.

In 1959 Masayoshi Nagata gave the negative answer to Hilbert's fourteenth problem: he constructed a non-reductive group acting on a vector space whose invariant ring is not finitely generated. The smallest non-reductive groups one cares about are unipotent groups — the additive group of the line, and matrix groups consisting of strictly upper-triangular matrices. These groups appear naturally as stabilisers of flags, as Borel subgroup pieces, and as automorphism groups of jets and curvilinear schemes. For half a century, GIT could not handle them.

The breakthrough came from Brent Doran and Frances Kirwan in 2007, followed by Gergely Bérczi and Kirwan in 2016. Their idea was to put the non-reductive group $G$ inside a larger reductive group $G^{+}$ — a reductive envelope — and reduce the non-reductive quotient to a reductive GIT problem on a larger space. Under hypotheses on how the unipotent radical sits inside $G$ , the invariant ring is finitely generated after all, and a projective quotient $X / / G$ exists.

Visual Beginner

A diagram with two boxes: on the left a variety $X$ carrying a unipotent group action with tangled orbits and no obvious quotient, and on the right a projective quotient scheme obtained by embedding the unipotent group inside a reductive envelope and applying classical GIT to the enlarged action.

The picture compresses the central idea of the Doran-Kirwan extension: the unipotent action on $X$ does not directly produce a finitely generated invariant ring, but after enlarging the group to a reductive envelope $G^{+} = G ⋊ K$ and lifting the linearisation, the enlarged action falls inside the reach of classical GIT, and the projective quotient appears on the right of the diagram.

Worked example Beginner

The cleanest non-reductive example is the additive group $G_{a}$ acting on the projective line $P^{1}$ by translation in the first affine coordinate. Concretely, write a point of $P^{1}$ in homogeneous coordinates $[x : y]$ , and let $G_{a} = C$ act by $$ a \cdot [x : y] = [x + a y : y], $$ where $a \in C$ is the group parameter. The action fixes the point at infinity $[1 : 0]$ and translates every other point along the affine line.

Step 1. Identify the orbits. The fixed point is $[1 : 0]$ ; every other orbit is the entire affine line ${[x : 1] : x \in C}$ — a single orbit of dimension one. So set-theoretically, the orbit space has two elements: the fixed point and the generic orbit.

Step 2. Compute the invariant ring. Take the homogeneous coordinate ring $R = C [x, y]$ and look for elements fixed by the action $x \mapsto x + a y$ , $y \mapsto y$ . The variable $y$ is fixed. The element $x$ is not fixed: it changes by $a y$ . The only $G_{a}$ -invariants are polynomials in $y$ alone, so $R^{G_{a}} = C [y]$ .

Step 3. Take the Proj of the invariant ring. The Proj of $C [y]$ (a polynomial ring in one variable, generated in degree one) is a single point. So the candidate quotient $P^{1} / / G_{a}$ is a point — matching the set-theoretic count of orbits in the generic locus.

Step 4. Identify the semistable locus. A point of $P^{1}$ is in the semistable locus when some $G_{a}$ -invariant section is nonzero there. The section $y$ vanishes only at $[1 : 0]$ ; so the semistable locus is $P^{1} ∖ {[1 : 0]}$ , the affine line. The unstable locus is the single fixed point at infinity.

What this tells us: the invariant ring $C [y]$ is finitely generated, the candidate quotient is the point, and the semistable locus is the generic orbit. The quotient is honest in this case — the $G_{a}$ -action on $P^{1}$ with the standard linearisation produces a genuine projective quotient by classical GIT-style reasoning, even though $G_{a}$ is non-reductive. For higher-dimensional unipotent actions Nagata's counterexample shows that this happy outcome can fail; the Doran-Kirwan-Bérczi-Kirwan extension identifies the exact additional hypothesis on the action that restores finite generation.

Check your understanding Beginner

Exercise (easy, multiple choice).

Why does classical Mumford GIT require the acting group $G$ to be reductive?

Reductive groups have finitely many orbits
Reductive groups guarantee that the invariant ring $\bigoplus_n H^0(X, L^n)^G$ is finitely generated, so its Proj is a projective scheme
Reductive groups have a moment map
Reductive groups have a connected center

Hint

The Proj construction of the GIT quotient is a finitely generated graded $k$ -algebra, and reductivity is the hypothesis that delivers finite generation through the Reynolds operator and the Hilbert-Mumford-Nagata theorem.

Answer

B. Reductive groups guarantee the invariant ring is finitely generated. This is the Hilbert-Mumford-Nagata finite-generation theorem, which uses the Reynolds operator available for reductive groups in characteristic zero and good filtrations in positive characteristic. Without finite generation, the Proj construction does not define a finite-dimensional projective scheme. Nagata's 1959 counterexample shows that finite generation can fail for non-reductive groups, and this is exactly the obstacle the Doran-Kirwan-Bérczi-Kirwan extension addresses.

Formal definition Intermediate+

Let $k$ be an algebraically closed field of characteristic zero, let $G$ be a connected linear algebraic group over $k$ , and let $X$ be a projective variety over $k$ carrying a $G$ -action together with a $G$ -linearised ample line bundle $L$ . Write $R_{u} (G) \subset G$ for the unipotent radical (the maximal connected normal unipotent subgroup), and recall the Levi decomposition $G = R_{u} (G) ⋊ L$ , where $L \subset G$ is a Levi subgroup, the latter reductive. The group $G$ is reductive iff $R_{u} (G) = {e}$ .

Definition (classical reductive GIT quotient). When $G$ is reductive, the GIT quotient of $X$ by $G$ with respect to $L$ is the projective scheme $$ X /!/L G := \mathrm{Proj}\bigoplus{n \geq 0} H^0(X, L^{\otimes n})^G. $$ Finite generation of the invariant ring on the right is the Hilbert-Mumford-Nagata theorem and is what makes the $Proj$ a finite-dimensional projective scheme. See 04.10.02.

Definition (Nagata-Hilbert obstruction). Hilbert's fourteenth problem asks whether for every action of a linear algebraic group $G$ on a finitely generated $k$ -algebra $R$ , the invariant ring $R^{G}$ is also finitely generated. Nagata 1959 ^{[Nagata 1959]} gave a negative answer: there exists a connected unipotent subgroup $U \subset SL_{32}$ of dimension thirteen such that the polynomial ring $C [x_{1}, \dots, x_{32}]^{U}$ is not finitely generated. Consequently, the naive $Proj$ definition fails for non-reductive $G$ in general.

Definition (Grosshans subgroup). A closed subgroup $H$ of a reductive group $G$ is a Grosshans subgroup (Grosshans 1997 ^{[source pending]}) iff the homogeneous space $G / H$ is quasi-affine and the ring of regular functions $O (G / H) = O (G)^{H}$ is finitely generated as a $k$ -algebra. Grosshans's theorem identifies a large class of non-reductive $H$ for which non-reductive invariant theory recovers finite generation through the embedding $H \subset G$ .

Definition (graded unipotent hypothesis, Bérczi-Kirwan 2016). A linear algebraic group $\hat{U}$ over $k$ is called graded unipotent iff $\hat{U} = U ⋊ G_{m}$ where $U$ is a connected unipotent group, the multiplicative group $G_{m}$ acts on $U$ by conjugation, and all weights of this $G_{m}$ -action on the Lie algebra $Lie (U)$ are strictly positive. The semi-direct product $\hat{U}$ is the graded unipotent envelope of $U$ . The Bérczi-Kirwan condition is the existence of such a graded structure on the unipotent radical of the original group $G$ , possibly after enlarging $G$ by adjoining a one-parameter torus.

Definition (non-reductive GIT quotient under the graded hypothesis). Let $\hat{U} = U ⋊ G_{m}$ be graded unipotent and act on a projective variety $X$ with $\hat{U}$ -linearised ample line bundle $L$ . Define the non-reductive GIT quotient by the same Proj formula, $$ X /!/ \hat{U} := \mathrm{Proj}\bigoplus_{n \geq 0} H^0(X, L^{\otimes n})^{\hat{U}}. $$ The content of Bérczi-Kirwan 2016 is that under the graded hypothesis plus a stability condition (the semistability-coincides-with-stability property — equivalently, all $\hat{U}$ -semistable points have closed orbits and finite stabiliser), the invariant ring is finitely generated and the $Proj$ is a projective scheme.

Definition (reductive envelope). Let $G$ be a connected linear algebraic group with Levi decomposition $G = R_{u} (G) ⋊ L$ . A reductive envelope of $G$ is a reductive group $G^{+}$ containing $G$ as a closed subgroup, together with a $G^{+}$ -variety $X^{+}$ extending the $G$ -action on $X$ , such that the non-reductive GIT problem for $G$ on $X$ embeds into the classical reductive GIT problem for $G^{+}$ on $X^{+}$ . The canonical construction in Doran-Kirwan 2007 ^{[Doran-Kirwan 2007]} takes $X^{+} = G^{+} \times_{G} X$ as the induced space and $L^{+}$ as the lift of $L$ along the projection $X^{+} \to X$ .

Counterexamples to common slips

The non-reductive GIT quotient is well-defined for $\hat{U}$ -actions satisfying the graded hypothesis and the semistability-equals-stability property; without the latter, the Proj construction can yield a scheme that fails to be projective. The graded hypothesis alone is not sufficient.
The reductive envelope $G^{+}$ is not unique. Different envelopes produce different quotients, and the variation among envelopes is a non-reductive analogue of variation of GIT (see 04.10.09).
The Nagata counterexample uses a non-graded unipotent action: the thirteen-dimensional $U$ acts on $A^{32}$ without any compatible $G_{m}$ -grading making all weights positive. Adding a graded structure changes the picture and restores finite generation in many cases.
The Hilbert-Mumford numerical criterion in the form of 04.10.03 does not extend verbatim to non-reductive groups: one-parameter subgroups of a unipotent group are all conjugate, and the weight calculation degenerates. The Doran-Kirwan extension uses adapted one-parameter subgroups inside the reductive envelope $G^{+}$ .

Key theorem with proof Intermediate+

Theorem ( $\hat{U}$ -Theorem, Bérczi-Kirwan 2016 Theorem 0.1). Let $\hat{U} = U ⋊ G_{m}$ be a graded unipotent group acting on a projective variety $X$ over $C$ with a $\hat{U}$ -linearised ample line bundle $L$ . Assume that the semistable and stable loci coincide for the induced $G_{m}$ -action — equivalently, the minimal weight on every $\hat{U}$ -fixed component is strictly positive. Then the invariant ring $$ R^{\hat{U}} := \bigoplus_{n \geq 0} H^0(X, L^{\otimes n})^{\hat{U}} $$ is a finitely generated $C$ -algebra, and the GIT quotient $$ X /!/ \hat{U} := \mathrm{Proj}, R^{\hat{U}} $$ is a projective variety that geometrically parametrises the closed $\hat{U}$ -orbits in the semistable locus. ^{[Bérczi-Kirwan 2016]}

Proof. The argument has four steps. First, reduce the $\hat{U}$ -invariant problem to a $U$ -invariant problem with an auxiliary $G_{m}$ -grading. Second, use the grading and Grosshans theory to exhibit $U$ as a Grosshans subgroup of an enveloping reductive group $G^{+}$ . Third, transfer the $U$ -invariant ring to a $G^{+}$ -invariant ring on a quasi-affine space, where Hilbert-Mumford-Nagata applies. Fourth, identify the resulting projective scheme with $Proj R^{\hat{U}}$ .

Step 1: reduction to a graded problem. Decompose the invariant ring by the $G_{m}$ -action coming from the second factor of $\hat{U} = U ⋊ G_{m}$ . Each graded piece $H^{0} (X, L^{\otimes n})$ is a $\hat{U}$ -module, and the $G_{m}$ -action on $H^{0} (X, L^{\otimes n})^{U}$ decomposes the $U$ -invariants into weight spaces $$ H^0(X, L^{\otimes n})^U = \bigoplus_{w \in \mathbb{Z}} H^0(X, L^{\otimes n})^U_w, $$ and the $\hat{U}$ -invariants are exactly the weight-zero piece, $H^{0} (X, L^{\otimes n})^{\hat{U}} = H^{0} (X, L^{\otimes n})_{0}^{U}$ . The semistability-equals-stability hypothesis on the $G_{m}$ -action is what guarantees that the weight-zero pieces assemble into a finitely generated ring even when the full $U$ -invariant ring is not finitely generated.

Step 2: Grosshans embedding. By the graded hypothesis, the $G_{m}$ -action on $Lie (U)$ has all weights positive, so the conjugation action of $G_{m}$ on $U$ exhibits $U$ as a graded unipotent group. Bérczi-Kirwan 2016 §2 prove that for any such graded $U$ , there exists a reductive linear algebraic group $G^{+}$ together with a closed embedding $U ↪ G^{+}$ realising $U$ as a Grosshans subgroup of $G^{+}$ : the quotient $G^{+} / U$ is quasi-affine and the ring $O (G^{+} / U) = O (G^{+})^{U}$ is finitely generated. The explicit construction takes $G^{+}$ to be a product of general linear groups acting on a flag-like compactification of $G^{+} / U$ .

Step 3: transfer to the reductive case. The Grosshans embedding $U \subset G^{+}$ induces an equivalence of $U$ -invariant theory on $X$ with $G^{+}$ -invariant theory on the induced space $X^{+} := G^{+} \times_{U} X$ (Grosshans 1997 Theorem 4.3 ^{[Grosshans 1997]}): $$ H^0(X, L^{\otimes n})^U \cong H^0(X^+, (L^+)^{\otimes n})^{G^+}, $$ where $L^{+}$ is the line bundle on $X^{+}$ obtained by descending $L$ along the projection $X^{+} \to X$ . By the Hilbert-Mumford-Nagata theorem for the reductive group $G^{+}$ (see 04.10.02), the right-hand ring is finitely generated as a $C$ -algebra. Hence so is the left-hand ring. Taking the $G_{m}$ -weight-zero piece on the $\hat{U}$ side preserves finite generation (a graded piece of a finitely generated graded ring under a reductive group action is finitely generated).

Step 4: projectivity of the quotient. Since $R^{\hat{U}} = ⨁_{n} H^{0} (X, L^{\otimes n})^{\hat{U}}$ is a finitely generated graded $C$ -algebra, $Proj R^{\hat{U}}$ is a projective scheme over $C$ . The natural map $X^{ss, \hat{U}} (L) \to Proj R^{\hat{U}}$ is a good quotient on the $\hat{U}$ -semistable locus, and the semistability-equals-stability hypothesis upgrades this to a geometric quotient on the entire $\hat{U}$ -semistable locus, identifying $Proj R^{\hat{U}}$ with the orbit space of the closed $\hat{U}$ -orbits in $X^{ss, \hat{U}} (L)$ . $□$

Bridge. The $\hat{U}$ -Theorem builds toward the modern moduli theory of non-reductive group actions, and the central insight is that the failure of Hilbert's fourteenth problem for non-reductive groups is sharpened by Bérczi-Kirwan into a precise additional hypothesis — graded unipotence plus semistability-equals-stability — under which the invariant ring is again finitely generated. The bridge is that the non-reductive quotient is constructed by embedding the unipotent radical $U$ as a Grosshans subgroup of a reductive envelope $G^{+}$ and applying classical GIT to the induced action, putting these together with the Levi decomposition $G = U ⋊ L$ to identify the original quotient $X / / G$ with the iterated quotient $(X / / U) / / L$ . This appears again in 04.10.08 (Kirwan stratification of the unstable locus), where the same parabolic-bundle technology indexing Kirwan strata of a reductive action enters the non-reductive picture as the moduli of unstable orbits, and the foundational reason is that the unipotent radical of a parabolic subgroup is exactly the kind of graded unipotent group covered by the $\hat{U}$ -Theorem. The pattern generalises the reductive theory: where Mumford 1965 needed reductivity to control orbit closures via the Reynolds operator, Bérczi-Kirwan replace the Reynolds operator with the Grosshans embedding into a reductive envelope, and the central insight is that semistability-equals-stability rather than reductivity is the operative hypothesis.

Exercises Intermediate+

Exercise 1 (easy, numeric).

Consider $G_{a}$ acting on $P^{2}$ by $a \cdot [x : y : z] = [x + a z : y : z]$ . Compute the ring of $G_{a}$ -invariants in $C [x, y, z]$ and identify the candidate GIT quotient $P^{2} / / G_{a}$ .

Hint

The variables $y$ and $z$ are fixed; $x$ shifts by $a z$ . Look for polynomials in $x, y, z$ invariant under that shift.

Answer

The action fixes $y$ and $z$ , and sends $x \mapsto x + a z$ . The polynomial $x$ is not invariant, but the polynomial $y$ and $z$ are, and any polynomial in $y$ and $z$ alone is invariant. Conversely, any invariant polynomial $f (x, y, z)$ must satisfy $f (x + a z, y, z) = f (x, y, z)$ for all $a \in C$ ; differentiating with respect to $a$ at $a = 0$ gives $z \cdot \partial f / \partial x = 0$ , which forces $\partial f / \partial x = 0$ in characteristic zero, i.e., $f$ does not involve $x$ . So $R^{G_{a}} = C [y, z]$ , a polynomial ring in two degree-one variables. The Proj is $P^{1}$ . So $P^{2} / / G_{a} ≅ P^{1}$ .

Geometrically: the action of $G_{a}$ on $P^{2}$ moves points along horizontal lines in the affine chart $z \neq = 0$ , parametrised by $(y, z)$ once the $x$ -shift is quotiented out, and the quotient line is $P^{1}$ with the homogeneous coordinates $[y : z]$ . Rubric: full credit for $R^{G_{a}} = C [y, z]$ and the identification of the quotient with $P^{1}$ .

Exercise 2 (easy, multiple choice).

Which of the following best describes the role of the graded unipotent hypothesis in Bérczi-Kirwan 2016?

It forces the group $G$ to be reductive after all
It supplies a $\mathbb{G}_m$-grading with strictly positive weights on $\mathrm{Lie}(U)$, allowing the Grosshans embedding into a reductive envelope and recovering finite generation of the invariant ring
It guarantees that the unipotent radical is one-dimensional
It removes the dependence on the choice of linearisation

Hint

The graded hypothesis is exactly the additional structure beyond unipotence that Bérczi-Kirwan need to recover finite generation in the non-reductive case.

Answer

B. The graded unipotent hypothesis equips $U$ with a $G_{m}$ -conjugation action whose weights on $Lie (U)$ are all strictly positive. Bérczi-Kirwan use this grading to embed $U$ as a Grosshans subgroup of a reductive envelope $G^{+}$ , transfer the $U$ -invariant problem on $X$ to a reductive $G^{+}$ -invariant problem on the induced space $X^{+} = G^{+} \times_{U} X$ , and apply Hilbert-Mumford-Nagata to recover finite generation. The remaining options describe properties that the graded hypothesis neither guarantees nor needs.

Exercise 3 (medium, short-answer).

Show that the additive group $G_{a}$ acting on the affine plane $A^{2}$ by $a \cdot (x, y) = (x + a y, y)$ has invariant ring $C [y]$ , and compute the affine quotient $A^{2} // G_{a}$ as the spectrum of the invariant ring.

Hint

Use the characteristic-zero differentiation argument from Exercise 1, then identify $Spec C [y]$ .

Answer

The action sends $x \mapsto x + a y$ and fixes $y$ . An invariant polynomial $f (x, y)$ satisfies $f (x + a y, y) = f (x, y)$ for all $a$ ; differentiating in $a$ at $a = 0$ yields $y \cdot \partial f / \partial x = 0$ . In characteristic zero this forces $\partial f / \partial x = 0$ , so $f$ depends only on $y$ . Therefore $C [x, y]^{G_{a}} = C [y]$ , a polynomial ring in one variable. The affine quotient is $Spec C [y] = A^{1}$ , the affine line.

Geometrically: the orbits of $G_{a}$ on $A^{2}$ are the horizontal lines ${y = c}$ for $c \neq = 0$ (each is a single orbit) together with the fixed locus ${y = 0}$ (each point fixed). The quotient map $A^{2} \to A^{1}$ , $(x, y) \mapsto y$ collapses each horizontal line to a point; on the fixed locus, the quotient identifies each fixed point with $0$ , gluing the whole line ${y = 0}$ to one point. The affine quotient is the affine line $A^{1} = Spec C [y]$ , and the invariant ring is finitely generated. Rubric: full credit for the differentiation argument, the identification $R^{G_{a}} = C [y]$ , and the affine-line quotient.

Exercise 4 (medium, short-answer).

State the Hilbert-Mumford-Nagata finite-generation theorem for reductive groups, and explain in one paragraph why its proof via the Reynolds operator fails for non-reductive $G$ .

Hint

The Reynolds operator is a $G$ -invariant projection from the polynomial ring onto the invariant subring; its existence depends on complete reducibility of finite-dimensional rational representations of $G$ , which is exactly the definition of reductivity.

Answer

Hilbert-Mumford-Nagata theorem. Let $G$ be a reductive linear algebraic group over an algebraically closed field $k$ acting rationally on a finitely generated commutative $k$ -algebra $R$ . Then the invariant ring $R^{G}$ is a finitely generated $k$ -algebra.

Why the Reynolds-operator proof fails non-reductively. The proof in characteristic zero proceeds by averaging: every rational $G$ -module $V$ decomposes as a direct sum $V = V^{G} \oplus V^{'}$ where $V^{G}$ is the $G$ -invariants and $V^{'}$ is a complement of $V^{G}$ within the $G$ -isotypic decomposition. The projection $V \to V^{G}$ along $V^{'}$ is the Reynolds operator $ρ : V \to V^{G}$ , $O_{X} (X)$ -linear when applied to $V = O_{X} (X)$ , and it produces an exact splitting of the inclusion $R^{G} ↪ R$ . From this splitting, one shows that if $R$ is finitely generated then $R^{G}$ is also finitely generated — every $R$ -module ideal of $R^{G}$ has a finite generating set whose Reynolds images generate the ideal. The existence of the Reynolds operator is equivalent to complete reducibility of finite-dimensional rational $G$ -representations, which by definition is reductivity. For non-reductive $G$ — for instance $G = G_{a}$ or a unipotent matrix group — finite-dimensional rational representations admit non-split filtrations (the standard representation of $G_{a}$ on $C^{2}$ via $a \mapsto (10 a 1)$ has a one-dimensional invariant subspace with no $G_{a}$ -invariant complement), so no $R$ -linear splitting of $R^{G} ↪ R$ exists, and Nagata's 1959 counterexample shows that finite generation can fail. Rubric: full credit for the theorem statement and the Reynolds-failure paragraph.

Exercise 5 (medium, short-answer).

Describe Nagata's 1959 counterexample to Hilbert's fourteenth problem at the level of dimensions: the group, the representation, and the conclusion. No proof required.

Hint

Nagata's counterexample is a thirteen-dimensional connected unipotent group acting linearly on a thirty-two-dimensional polynomial ring.

Answer

Nagata 1959 (Amer. J. Math. 81, 766-772) constructs a connected unipotent algebraic group $U$ of dimension thirteen and a linear action of $U$ on the affine space $A^{32}$ (a thirty-two-dimensional rational representation $U \to GL_{32}$ ) such that the invariant ring $C [x_{1}, \dots, x_{32}]^{U}$ is not finitely generated as a $C$ -algebra. The group $U$ is constructed as a subgroup of the unipotent radical of a maximal parabolic of $SL_{32}$ , and the non-finite-generation is established by exhibiting an infinite ascending sequence of $U$ -invariant polynomials none of which can be expressed as a polynomial combination of a finite set of generators. This negatively resolves Hilbert's fourteenth problem in the non-reductive case and is the precise obstacle that the Doran-Kirwan-Bérczi-Kirwan extension addresses: under the additional graded-unipotent hypothesis plus semistability-equals-stability, finite generation is recovered. Rubric: full credit for the dimensions (thirteen-dimensional $U$ on thirty-two-variable polynomial ring) and the non-finite-generation conclusion.

Exercise 6 (medium, short-answer).

Explain the role of the Grosshans subgroup notion in the Doran-Kirwan-Bérczi-Kirwan construction. State Grosshans's theorem and indicate how Bérczi-Kirwan 2016 use it to embed a graded unipotent $U$ into a reductive envelope.

Hint

A Grosshans subgroup $H \subset G$ has the property that $G / H$ is quasi-affine and $O (G / H) = O (G)^{H}$ is finitely generated. The Bérczi-Kirwan construction exhibits a graded unipotent $U$ as a Grosshans subgroup of a reductive product of general linear groups.

Answer

Grosshans's theorem (1997). A closed subgroup $H$ of a reductive group $G$ is called a Grosshans subgroup if the quotient variety $G / H$ is quasi-affine. Grosshans's theorem states that in this case the ring of regular functions $O (G / H) = O (G)^{H}$ is a finitely generated $k$ -algebra, and for any $H$ -variety $X$ the induced space $X^{+} := G \times_{H} X$ is a $G$ -variety with a natural identification $H^{0} (X, L)^{H} ≅ H^{0} (X^{+}, L^{+})^{G}$ for any $H$ -linearised line bundle $L$ on $X$ lifting to $L^{+}$ on $X^{+}$ . The Grosshans-subgroup property is what transfers the $H$ -invariant theory on $X$ to a $G$ -invariant theory on $X^{+}$ , where finite generation comes for free from reductivity of $G$ .

Bérczi-Kirwan use. Bérczi-Kirwan 2016 §2 show that every graded unipotent group $\hat{U} = U ⋊ G_{m}$ embeds as a Grosshans subgroup of a reductive group $G^{+} = \prod_{i} GL_{n_{i}}$ constructed from the $G_{m}$ -weight decomposition of $Lie (U)$ . The embedding is built by representing each positive-weight piece of $Lie (U)$ as a unipotent block inside a $GL_{n_{i}}$ , and the strictly-positive-weight condition forces $G^{+} / \hat{U}$ to be quasi-affine. By Grosshans's theorem, the $\hat{U}$ -invariants $O (X)^{\hat{U}}$ are finitely generated, and the GIT quotient $X / / \hat{U} = Proj ⨁_{n} H^{0} (X, L^{n})^{\hat{U}}$ is a projective variety. Rubric: full credit for the Grosshans theorem statement and the indication that the graded-unipotent hypothesis is exactly what realises $U$ as a Grosshans subgroup.

Exercise 7 (hard, short-answer).

Let $P \subset G$ be a parabolic subgroup of a reductive group $G$ with Levi decomposition $P = U_{P} ⋊ L_{P}$ . Outline how non-reductive GIT can be used to construct moduli spaces of $P$ -orbits as iterated quotients $(X / / U_{P}) / / L_{P}$ , and indicate where the graded-unipotent hypothesis is verified.

Hint

The unipotent radical $U_{P}$ of a parabolic subgroup carries a natural grading from the cocharacter that defines $P$ . The reductive Levi quotient $L_{P}$ acts on $X / / U_{P}$ and classical GIT applies.

Answer

Setup. A parabolic $P \subset G$ in a reductive group is the stabiliser of a flag, and admits a Levi decomposition $P = U_{P} ⋊ L_{P}$ where $U_{P}$ is the unipotent radical and $L_{P}$ is a Levi subgroup, the latter reductive. Choose a one-parameter subgroup $λ : G_{m} \to L_{P}$ such that $P = {g \in G : lim_{t \to 0} λ (t) g λ (t)^{- 1} exists}$ ; the conjugation action of $λ$ on $Lie (U_{P})$ has all weights strictly positive (this is the defining property of $U_{P}$ as the positive-weight part of $Lie (P)$ under $λ$ ).

Step 1: Graded-unipotent verification. The semi-direct product $\hat{U}_{P} := U_{P} ⋊ λ (G_{m})$ is graded unipotent: $U_{P}$ is connected unipotent by parabolic theory, and the $G_{m}$ -action via $λ$ has strictly positive weights on $Lie (U_{P})$ by construction. Bérczi-Kirwan applies: the $\hat{U}_{P}$ -invariant ring on a projective $X$ is finitely generated under the additional semistability-equals-stability hypothesis, and the quotient $X / / \hat{U}_{P}$ is a projective variety.

Step 2: Iterated quotient by the Levi. The complementary factor $L_{P} / λ (G_{m})$ is reductive (it is the reductive Levi quotient mod a central torus). Its action on $X / / \hat{U}_{P}$ inherits a linearisation from the $G$ -linearisation on $X$ , and classical reductive GIT (see 04.10.02) produces the further quotient $(X / / \hat{U}_{P}) / / (L_{P} / λ (G_{m}))$ as a projective variety. The composite is the parabolic quotient $X / / P$ .

Outcome. The iterated construction realises the parabolic GIT quotient as a tower of one non-reductive step (the $\hat{U}_{P}$ -quotient handled by Bérczi-Kirwan) followed by one classical reductive step (the Levi quotient handled by Mumford-Fogarty-Kirwan). The grading on $Lie (U_{P})$ supplied by the defining cocharacter $λ$ is exactly the graded-unipotent hypothesis required at the first step. Applications include moduli of higher-rank structures with prescribed parabolic reductions: parabolic Higgs bundles (Boden-Yokogawa), parabolic principal bundles on curves, and Hitchin systems with parabolic reductions, all of which sit naturally inside the parabolic-GIT framework. Rubric: full credit for identifying the parabolic Levi decomposition, the defining cocharacter as the grading, the verification of the graded-unipotent hypothesis, and the iterated quotient structure $(X / / \hat{U}_{P}) / / (L_{P} / λ (G_{m}))$ .

Exercise 8 (hard, short-answer).

Sketch how non-reductive GIT enters the construction of moduli of jet differentials in the hyperbolicity programme of Demailly and Bérczi-Kirwan, and indicate the application to the Green-Griffiths-Lang conjecture.

Hint

The reparametrisation group $G_{k}$ of $k$ -jets at the origin of $C$ is a unipotent group, and Demailly's bundle of invariant jet differentials is the global sections of a line bundle on the Demailly-Semple tower modulo the $G_{k}$ -action.

Answer

Setup. For a complex projective variety $X$ of dimension $n$ , the $k$ -th order Demailly-Semple jet bundle $J_{k} X \to X$ parametrises $k$ -jets of holomorphic curves $f : (C, 0) \to X$ . The reparametrisation group $G_{k} = {a_{1} t + a_{2} t^{2} + \dots + a_{k} t^{k} : a_{1} \neq = 0}$ — the group of $k$ -jets at the origin of biholomorphisms of $C$ — acts on $J_{k} X$ by reparametrising the source. The group $G_{k} = C^{*} ⋉ U_{k}$ has reductive part $C^{*}$ (acting by rescaling $t$ ) and unipotent part $U_{k}$ (the higher-order coefficients $a_{2}, \dots, a_{k}$ ); this matches the graded-unipotent structure of Bérczi-Kirwan, with the $C^{*}$ supplying the strictly positive grading on $Lie (U_{k})$ .

Step 1: Invariant jet differentials. Demailly's invariant jet differentials $E_{k, m} Ω_{X}^{*}$ are the $G_{k}$ -invariant sections of weight $m$ in the symmetric algebra of the cotangent jet bundle. Computing global sections $H^{0} (X, E_{k, m} Ω_{X}^{*})$ is exactly a non-reductive GIT problem: it is the invariant ring of the $G_{k}$ -action on the Demailly-Semple tower, which is a graded unipotent action satisfying the Bérczi-Kirwan hypotheses.

Step 2: Application to Green-Griffiths-Lang. The Green-Griffiths-Lang conjecture predicts that for $X$ a complex projective variety of general type, the entire holomorphic curves $f : C \to X$ lie in a proper algebraic subvariety. The standard analytic input is the Ahlfors-Schwarz lemma applied to a global jet differential vanishing on an ample divisor: if such a jet differential exists, every entire holomorphic curve must be tangent to its zero set. Bérczi-Kirwan 2016 and Bérczi 2020 ^{[Bérczi 2020]} use the non-reductive GIT machinery to prove existence of enough invariant jet differentials for $X$ a generic projective hypersurface of degree $d \geq d_{n}$ in $P^{n + 1}$ , with explicit polynomial bounds $d_{n} \sim n^{O (1)}$ . The non-reductive GIT step is the construction of a projective quotient parametrising the invariant jet differentials, and the bound $d_{n}$ comes from the dimension count on this quotient.

Outcome. Non-reductive GIT provides the moduli-theoretic engine for the algebraic part of the hyperbolicity programme: the existence and structure of invariant jet differentials on a variety of general type is reduced to a non-reductive GIT quotient problem, and the Bérczi-Kirwan $\hat{U}$ -Theorem supplies the finite generation that makes the resulting moduli space projective. The Green-Griffiths-Lang conjecture for generic hypersurfaces of high enough degree is then a consequence of this construction. Rubric: full credit for identifying $G_{k}$ as graded unipotent, the Demailly invariant-jet-differential setup, and the link to Green-Griffiths-Lang via existence of jet differentials vanishing on an ample divisor.

Advanced results Master

Theorem (Grosshans's finite-generation theorem, 1997). Let $G$ be a reductive linear algebraic group over an algebraically closed field $k$ of characteristic zero, and let $H \subset G$ be a closed subgroup. The following are equivalent: ^{[Grosshans 1997]}

(G1) The quotient variety $G / H$ is quasi-affine.

(G2) The ring of regular functions $O (G)^{H}$ is a finitely generated $k$ -algebra.

(G3) For every affine $H$ -variety $Y$ , the invariant ring $O (Y)^{H}$ is a finitely generated $k$ -algebra.

When the equivalent conditions hold, $H$ is called a Grosshans subgroup of $G$ , and the induced-space functor $Y \mapsto G \times_{H} Y$ realises an equivalence between $H$ -invariant theory on $Y$ and $G$ -invariant theory on $G \times_{H} Y$ . Grosshans subgroups include all reductive subgroups (by Hilbert-Mumford-Nagata applied directly), all unipotent radicals of parabolic subgroups (by parabolic theory), and the graded unipotent subgroups of Bérczi-Kirwan after Grosshans embedding.

Theorem (semistability-equals-stability under the graded hypothesis, Bérczi-Kirwan 2016 Theorem 5.16). Let $\hat{U} = U ⋊ G_{m}$ act on a projective variety $X$ with a $\hat{U}$ -linearised ample line bundle $L$ . Assume the well-adapted condition: the minimum $G_{m}$ -weight on each fixed-point component of the $G_{m}$ -action is strictly positive. Then $X^{ss, \hat{U}} (L) = X^{s, \hat{U}} (L)$ , every $\hat{U}$ -semistable point has finite stabiliser, and the GIT quotient $X / / \hat{U}$ is a geometric quotient on its entire domain of definition. ^{[Bérczi-Kirwan 2016]}

The well-adapted condition is the operative replacement for reductivity. It can be verified explicitly in the cases that arise in moduli theory (jet differentials, parabolic bundles, curvilinear Hilbert schemes), and when it fails one can sometimes restore it by twisting the linearisation $L$ by a sufficiently positive power of the $G_{m}$ -character — the adjustment-of-linearisation trick of Bérczi-Doran-Hawes-Kirwan 2018 ^{[Bérczi-Doran-Hawes-Kirwan 2018]}.

Theorem (projective completion of graded unipotent quotients, Bérczi-Doran-Hawes-Kirwan 2018). For a graded unipotent group $\hat{U}$ acting on a smooth projective variety $X$ with a well-adapted linearisation $L$ , the geometric quotient $X^{s} / \hat{U}$ of the stable locus admits a canonical projective completion $\overline{X^{s} / \hat{U}}$ obtained by Proj of the $\hat{U}$ -invariant section ring, and the boundary $\overline{X^{s} / \hat{U}} ∖ (X^{s} / \hat{U})$ is supported on the image of the strictly $\hat{U}$ -semistable locus. ^{[Bérczi-Doran-Hawes-Kirwan 2018]}

This is the non-reductive analogue of the Mumford completion of a geometric quotient by a reductive group via the semistable locus. The boundary structure parametrises moduli of strictly semistable orbits — non-closed orbits whose closures touch each other inside $X^{ss, \hat{U}}$ — and is itself stratified by an analogue of the Kirwan stratification of 04.10.08.

Theorem (Bérczi 2020, jet-differential existence for generic hypersurfaces). Let $X \subset P^{n + 1}$ be a generic smooth projective hypersurface of degree $d$ , and let $E_{k, m}\Omega^X $d e n o t e t h eD e mai l l y b u n d l eo f in v a r ian t j e t d i f f er e n t ia l so f or d er$ k $an d w e i g h t$ m $. F or e v er y$ n \geq 2 $, t h er ee x i s t s a p o l y n o mia l b o u n d$ d_n = d_n(n) $o f d e g r ee$ O(n^2) $s u c h t ha t f or$ d \geq d_n $an d s u i t ab l e$ k, m $, t h eco h o m o l o g y g r o u p$ H^0(X, E{k, m}\Omega^_X \otimes A^{-1}) $i s n o n - z er o f or anam pl e l in e b u n d l e$ A $. C o n se q u e n tl y, e v er y e n t i r e h o l o m or p hi cc u r v e$ f : \mathbb{C} \to X$ is algebraically degenerate in the sense of Green-Griffiths-Lang. ^{[Bérczi 2020]}

The proof uses the non-reductive GIT machinery applied to the reparametrisation action of $G_{k}$ on the Demailly-Semple tower over $X$ . The polynomial bound $d_{n}$ is read off from the dimension count on the non-reductive GIT quotient.

Theorem (moduli of unstable orbits, Bérczi-Hawes-Kirwan 2022). Let $G$ act on a projective variety $X$ with a $G$ -linearised ample line bundle $L$ . The unstable locus $X^{u s} (L)$ admits a stratification $X^{u s} (L) = ⨆_{β} S_{β}$ indexed by Hilbert-Mumford optimal destabilising one-parameter subgroups $β$ , generalising the Kirwan stratification of 04.10.08 to the non-reductive setting via the reductive envelope $G^{+}$ . Each stratum $S_{β}$ is a locally closed $G$ -invariant subvariety carrying a projective moduli space of $G$ -orbits, constructed by non-reductive GIT applied to the unipotent radical of the parabolic $P_{β}$ .

This realises the Kirwan vision of completing moduli theory by parametrising not only stable but also unstable orbits, with each Kirwan stratum acquiring its own non-reductive moduli space. The combined picture is the full moduli of unstable orbits, and the non-reductive GIT machinery is what makes the construction projective.

Theorem (filling the moduli of automorphisms, Bérczi-Kirwan-Doran 2019). For moduli problems where the automorphism group of a stable object does not vanish and is unipotent — for example, moduli of curves with marked points where some markings coincide, or moduli of curvilinear schemes parametrising tangent jets — the moduli space constructed by reductive GIT misses the automorphic locus. Non-reductive GIT applied to the unipotent automorphism action fills in this locus, giving a projective compactification of the full moduli problem.

Synthesis. The non-reductive GIT programme of Doran-Kirwan and Bérczi-Kirwan generalises Mumford's reductive theory by replacing the reductivity hypothesis with the graded-unipotent-plus-well-adapted hypothesis, and the central insight is that the failure of Hilbert's fourteenth problem is sharpened into a precise extra condition under which finite generation is recovered. Three apparently distinct constructions — the Grosshans embedding into a reductive envelope, the $G_{m}$ -grading on the unipotent Lie algebra, and the well-adapted linearisation — fit together as one identity: graded unipotent groups satisfying semistability-equals-stability are Grosshans subgroups of explicit reductive envelopes, and their invariant rings are finitely generated by transfer to the reductive case. Putting these together, the foundational reason is that reductivity is not the operative hypothesis at all: what GIT really needs is the existence of a quasi-affine homogeneous space $G^{+} / H$ realising $H$ as a Grosshans subgroup, and reductivity is one (sufficient but not necessary) way to produce that. The bridge is that the Bérczi-Kirwan extension identifies a much larger class of $H$ — the graded unipotent groups — for which the same Proj construction works, and through this the moduli theory of jet differentials, parabolic bundles, curvilinear Hilbert schemes, and unipotent automorphism groups becomes algebraically accessible.

This pattern appears again in 04.10.09 (variation of GIT), where the linearisation-dependence of the non-reductive quotient produces a non-reductive analogue of the Dolgachev-Hu-Thaddeus chamber decomposition, with walls indexed by changes in the well-adapted condition. It builds toward the moduli theory of higher-dimensional varieties with non-reductive automorphism groups — moduli of jets, holomorphic Plateau problems, and the Green-Griffiths-Lang programme on entire-curve hyperbolicity — where the moduli problem is intrinsically non-reductive and classical Mumford GIT fails outright. The central insight that organises the synthesis is that non-reductive GIT identifies the moduli of unstable orbits with iterated reductive quotients by Levi components of parabolic subgroups, putting the full classification of orbits — stable and unstable together — into one projective-variety framework. Bérczi-Doran-Hawes-Kirwan 2018 ^{[Bérczi-Doran-Hawes-Kirwan 2018]} completes the picture for graded unipotent groups, and the foundational reason the theory works is the universal Grosshans embedding into a reductive envelope, which transfers every non-reductive question to a question about classical reductive GIT on a larger, induced space.

Full proof set Master

Proposition (finite generation of $G_{a}$ -invariants on $A^{2}$ with the standard action). For the additive group $G_{a} = C$ acting on the polynomial ring $C [x, y]$ by $a \cdot x = x + a y$ , $a \cdot y = y$ , the invariant ring is $C [x, y]^{G_{a}} = C [y]$ .

Proof. A polynomial $f (x, y) \in C [x, y]$ is $G_{a}$ -invariant iff $f (x + a y, y) = f (x, y)$ for every $a \in C$ . Expand the left side in a Taylor series in $a$ : $$ f(x + ay, y) = \sum_{n \geq 0} \frac{(ay)^n}{n!} \frac{\partial^n f}{\partial x^n}(x, y). $$ This equals $f (x, y)$ for all $a$ iff every coefficient of $a^{n}$ with $n \geq 1$ vanishes identically, that is, $y^{n} \cdot \partial^{n} f / \partial x^{n} = 0$ in $C [x, y]$ for every $n \geq 1$ . Since $y \neq = 0$ in $C [x, y]$ , this forces $\partial^{n} f / \partial x^{n} = 0$ for every $n \geq 1$ , i.e., $f$ does not depend on $x$ . So $f \in C [y]$ , and conversely every $f \in C [y]$ is plainly invariant. Hence $C [x, y]^{G_{a}} = C [y]$ , finitely generated by one element. $□$

Proposition ( $G_{a}$ -quotient of $P^{n}$ by translation in the first affine coordinate). For the $G_{a}$ -action on $P^{n}$ given by $a \cdot [x_{0} : x_{1} : \dots : x_{n}] = [x_{0} + a x_{n} : x_{1} : \dots : x_{n}]$ with the structure-sheaf linearisation, the invariant ring is $C [x_{1}, \dots, x_{n}]$ and the quotient is $P^{n - 1}$ .

Proof. The action fixes $x_{1}, \dots, x_{n}$ and shifts $x_{0}$ by $a x_{n}$ . By the same differentiation argument, every $G_{a}$ -invariant polynomial in $C [x_{0}, \dots, x_{n}]$ is independent of $x_{0}$ , so the invariant ring is $C [x_{1}, \dots, x_{n}]$ . The Proj is $P^{n - 1}$ . The semistable locus is ${x_{n} \neq = 0}$ (where the section $x_{n}$ does not vanish); the unstable locus is the hyperplane ${x_{n} = 0}$ (the fixed locus of the $G_{a}$ -action). The GIT quotient map $P^{n} ∖ {x_{n} = 0} \to P^{n - 1}$ sends $[x_{0} : \dots : x_{n}] \mapsto [x_{1} : \dots : x_{n}]$ , collapsing each orbit ${[x_{0} + a x_{n} : x_{1} : \dots : x_{n}] : a \in C}$ to a single point of $P^{n - 1}$ . $□$

Proposition (non-existence of a Reynolds operator for $G_{a}$ ). There is no $G_{a}$ -equivariant $C$ -linear retraction $ρ : C [x, y] \to C [x, y]^{G_{a}} = C [y]$ of the inclusion $C [y] ↪ C [x, y]$ that is also $C [y]$ -linear.

Proof. Suppose such a $ρ$ exists. Then $ρ (x) \in C [y]$ , and applying the $G_{a}$ -equivariance with $a \in C$ , $$ \rho(x) = \rho(a \cdot x) = a \cdot \rho(x) = \rho(x), $$ which is consistent (so equivariance alone does not contradict).

The contradiction comes from the $C [y]$ -linearity. Apply $ρ$ to the identity $y \cdot 1 = y$ and to $1 \cdot y = y$ : by $C [y]$ -linearity, $ρ (y \cdot 1) = y \cdot ρ (1) = y$ (using $ρ (1) = 1$ for a retraction). Now consider the action of $a \in G_{a}$ : $a \cdot (y \cdot 1) = y \cdot 1$ , but also $a \cdot (y \cdot 1)$ should equal $y \cdot (a \cdot 1)$ by $G_{a}$ -equivariance — and the action of $G_{a}$ on the constant $1$ is $1$ , so $y \cdot (a \cdot 1) = y$ . That much is consistent.

The contradiction arises in higher degrees: the polynomial $x^{2}$ in $C [x, y]$ has $a \cdot x^{2} = (x + a y)^{2} = x^{2} + 2 a x y + a^{2} y^{2}$ . By $C [y]$ -linearity of $ρ$ , $ρ (x^{2}) \in C [y]$ and $ρ (a \cdot x^{2}) = ρ (x^{2}) + 2 a ρ (x y) + a^{2} ρ (y^{2}) = ρ (x^{2}) + 2 a y \cdot ρ (x) + a^{2} y^{2}$ (using $ρ (x y) = y ρ (x)$ and $ρ (y^{2}) = y^{2}$ ). $G_{a}$ -equivariance forces $ρ (a \cdot x^{2}) = a \cdot ρ (x^{2}) = ρ (x^{2})$ (since $ρ (x^{2}) \in C [y]$ is $G_{a}$ -invariant). Comparing, $2 a y \cdot ρ (x) + a^{2} y^{2} = 0$ for every $a \in C$ , which forces $ρ (x) = 0$ (from the linear-in- $a$ term) and then $y^{2} = 0$ (from the quadratic-in- $a$ term) — a contradiction in $C [y]$ . Therefore no such $ρ$ exists. $□$

This proposition is the concrete witness to the failure of the Reynolds-operator construction for $G_{a}$ , and is exactly why the proof strategy for the Hilbert-Mumford-Nagata finite-generation theorem does not extend non-reductively without additional input.

Theorem (Bérczi-Kirwan $\hat{U}$ -Theorem, proof in the formal-definition section, restated). Under the graded-unipotent hypothesis plus the well-adapted condition, the invariant ring is finitely generated and the GIT quotient is projective. ^{[Bérczi-Kirwan 2016]} The proof transfers the $U$ -invariant problem to a reductive problem on the Grosshans-induced space, where Hilbert-Mumford-Nagata applies; the $G_{m}$ -weight-zero piece of the resulting finitely generated ring is the $\hat{U}$ -invariant ring, and Proj of the latter is the projective quotient. $□$

Theorem (Doran-Kirwan reductive envelope, stated without proof here — full proof in Doran-Kirwan 2007 §3 ^{[Doran-Kirwan 2007]}). For every connected linear algebraic group $G$ with unipotent radical $R_{u} (G)$ , there exists a reductive group $G^{+}$ together with a closed embedding $G ↪ G^{+}$ realising $G$ as a closed subgroup. The induced space $X^{+} := G^{+} \times_{G} X$ carries a natural $G^{+}$ -action extending the $G$ -action on $X$ , and the $G^{+}$ -linearisation $L^{+}$ on $X^{+}$ obtained by descending $L$ along the projection $X^{+} \to X$ produces a classical reductive GIT problem whose solution restricts to the desired non-reductive quotient. The non-uniqueness of the choice of $G^{+}$ gives rise to a non-reductive variation-of-GIT phenomenon.

Connections Master

Geometric invariant theory 04.10.02. The classical Mumford construction restricts to reductive groups; the Bérczi-Kirwan extension covers a controlled class of non-reductive groups under the graded-unipotent hypothesis. Every concrete reductive GIT quotient enters the non-reductive picture as the Levi quotient of an iterated parabolic decomposition, putting these together so the foundational reason non-reductive GIT works is the same Proj construction applied to a transferred invariant ring on the Grosshans-induced space, this is exactly the reductive case after envelope.
Hilbert-Mumford numerical criterion 04.10.03. Mumford's numerical criterion identifies stability through weights of one-parameter subgroups of a reductive $G$ . For non-reductive $G$ all one-parameter subgroups of the unipotent radical are conjugate, and the weight calculation degenerates. The Doran-Kirwan extension lifts the criterion to the reductive envelope $G^{+}$ , where the classical numerical criterion applies and projects down to a non-reductive stability test on $X$ .
Kirwan stratification of the unstable locus 04.10.08. Kirwan's stratification indexes unstable strata of a reductive action by Hilbert-Mumford optimal destabilising directions. The non-reductive extension parametrises moduli of unstable orbits as iterated non-reductive GIT quotients, with the unipotent radical of each parabolic $P_{β}$ entering through the $\hat{U}$ -Theorem. The combined picture is the moduli of all orbits, and the bridge between the two units is the parabolic structure on each Kirwan stratum.
Variation of GIT 04.10.09. Classical VGIT studies the dependence of the reductive quotient on the linearisation $L$ and produces a chamber-wall picture in the equivariant ample cone. Non-reductive VGIT generalises this by allowing both the linearisation and the choice of reductive envelope to vary. The walls in the non-reductive picture include changes in the well-adapted condition of Bérczi-Kirwan, in addition to the classical numerical walls.
Moduli of curves 04.10.01. Mumford's original moduli of smooth curves uses reductive GIT on tri-canonically embedded curves. The Deligne-Mumford compactification extends this to nodal stable curves with finite automorphism groups. Moduli of curves with prescribed unipotent automorphisms — for example, curves with marked points coinciding tangentially — lie outside the reductive GIT framework and enter the non-reductive picture through the $\hat{U}$ -Theorem.
Moduli of vector bundles on a curve and slope stability 04.10.06. Mumford-Seshadri construct moduli of stable vector bundles on a smooth projective curve as a reductive GIT quotient. Moduli of parabolic vector bundles — bundles with prescribed flag structures at marked points — require non-reductive GIT because the parabolic group is not reductive; this is the natural application of the parabolic-iterated-quotient construction of Exercise 7.
Scheme 04.02.01. Both reductive and non-reductive GIT quotients are projective schemes obtained by Proj of finitely generated graded $k$ -algebras. The technical content of the non-reductive theory is to identify additional hypotheses under which finite generation persists; the schematic conclusion — that the quotient is a projective scheme — is identical in both cases.

Historical & philosophical context Master

David Mumford founded geometric invariant theory in his 1965 Geometric Invariant Theory (Springer-Verlag) ^{[Mumford 1965]} as a tool for constructing moduli spaces. The reductivity hypothesis was built in from the outset: Mumford's construction goes through the Proj of an invariant ring, and finite generation of that ring was known to hold for reductive groups by the Hilbert-Mumford-Nagata theorem, with Hilbert 1890 establishing the classical-groups case and Nagata extending it to all reductive groups. The negative answer to Hilbert's fourteenth problem in the non-reductive case was given by Masayoshi Nagata in On the 14th problem of Hilbert (Amer. J. Math. 81, 766-772, 1959) ^{[Nagata 1959]}, where Nagata exhibited a thirteen-dimensional connected unipotent group acting linearly on $A^{32}$ with non-finitely-generated invariant ring. This counterexample was the explicit obstacle that kept non-reductive groups outside the Mumford-GIT framework for nearly fifty years.

The first systematic extension to non-reductive groups was Brent Doran and Frances Kirwan's Towards non-reductive geometric invariant theory (Pure Appl. Math. Q. 3, 61-105, 2007) ^{[Doran-Kirwan 2007]}, which introduced the reductive-envelope strategy: embed the non-reductive $G$ into a reductive $G^{+}$ , work the GIT problem on $G^{+} \times_{G} X$ , and project back. The full modern theory under the graded-unipotent hypothesis was developed by Gergely Bérczi and Frances Kirwan in Graded unipotent groups and Grosshans theory (Compos. Math. 152, 1759-1808, 2016) ^{[Bérczi-Kirwan 2016]} and extended to projective completions in Bérczi-Doran-Hawes-Kirwan 2018 Projective completions of graded unipotent quotients (J. London Math. Soc. 97, 753-778) ^{[Bérczi-Doran-Hawes-Kirwan 2018]}. The Grosshans-subgroup framework on which Bérczi-Kirwan rest had been developed earlier by Frank Grosshans in Algebraic Homogeneous Spaces and Invariant Theory (Springer LNM 1673, 1997) ^{[Grosshans 1997]}, where Grosshans characterised the closed subgroups $H \subset G$ with quasi-affine homogeneous space $G / H$ and proved the finite-generation transfer between $H$ -invariant theory on a variety and $G$ -invariant theory on the induced space.

The principal modern application is to hyperbolicity and the Green-Griffiths-Lang conjecture. Jean-Pierre Demailly's invariant jet differentials, introduced in Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials (1997), parametrise holomorphic jet data modulo reparametrisation, and the reparametrisation group is unipotent. Bérczi 2017 Tautological integrals on curvilinear Hilbert schemes (Geom. Topol. 21) and Bérczi 2020 Non-reductive geometric invariant theory and hyperbolicity (arXiv:2009.09908) ^{[Bérczi 2020]} use non-reductive GIT to prove polynomial bounds for the degrees of generic projective hypersurfaces guaranteeing algebraic degeneracy of entire holomorphic curves, completing a programme launched by Yum-Tong Siu in the 1990s and bringing the algebraic hyperbolicity programme into the reach of GIT methods. The 2020s have seen non-reductive GIT applied to moduli of parabolic principal bundles, Hitchin systems with parabolic reductions, and the curvilinear Hilbert schemes parametrising tangent jets on smooth varieties. The Bérczi-Doran-Hawes-Kirwan programme has thereby extended the moduli-theoretic reach of GIT to many of the moduli problems that classical Mumford theory could not address, completing the half-century arc from Nagata's 1959 counterexample to the modern non-reductive framework.

Bibliography Master

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}

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}

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}

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}

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}

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}

@book{Demailly1997,
  author    = {Demailly, Jean-Pierre},
  title     = {Algebraic criteria for {K}obayashi hyperbolic projective varieties and jet differentials},
  publisher = {American Mathematical Society},
  series    = {Proc. Sympos. Pure Math.},
  volume    = {62},
  year      = {1997},
  pages     = {285--360}
}

Prerequisites

04.10.02
04.10.03
04.02.01
03.03.01
03.03.02

Tier anchors

beginner: Strogatz-style: classical GIT needs the group to be reductive; for unipotent and parabolic groups the invariant ring can break, but with a reductive envelope a usable quotient still exists
intermediate: Doran-Kirwan 2007 *Pure Appl. Math. Q.* 3; Bérczi-Kirwan 2016 *Compos. Math.* 152; Mumford-Fogarty-Kirwan *GIT* 3rd ed. §1; Nagata 1959 *Amer. J. Math.* 81
master: Mumford-Fogarty-Kirwan *Geometric Invariant Theory* 3rd ed. (Springer 1994, originator framework restricted to reductive $G$); Nagata 1959 *On the 14th problem of Hilbert* (Amer. J. Math. 81, 766-772, the negative answer); Doran-Kirwan 2007 *Towards non-reductive geometric invariant theory* (Pure Appl. Math. Q. 3, 61-105, the foundational extension); Bérczi-Kirwan 2016 *Graded unipotent groups and Grosshans theory* (Compos. Math. 152, 1759-1808, the modern unipotent-stable framework); Bérczi-Doran-Hawes-Kirwan 2018 *Projective completions of graded unipotent quotients* (J. London Math. Soc. 97); Grosshans 1997 *Algebraic Homogeneous Spaces and Invariant Theory* (Springer LNM 1673); Bérczi 2017 *Tautological integrals on curvilinear Hilbert schemes* (Geom. Topol. 21) and Bérczi 2020 *Non-reductive geometric invariant theory and hyperbolicity* (arXiv:2009.09908) for applications to Green-Griffiths-Lang

References

Mumford, Fogarty, Kirwan — Geometric Invariant Theory · 3rd ed., Springer Ergebnisse 34, 1994; original Mumford 1965 edition
Nagata — On the 14th problem of Hilbert · Amer. J. Math. 81 (1959), 766–772
Doran, Kirwan — Towards non-reductive geometric invariant theory · Pure and Applied Mathematics Quarterly 3 (2007), 61–105
Bérczi, Kirwan — Graded unipotent groups and Grosshans theory · Compositio Mathematica 152 (2016), 1759–1808
Bérczi, Doran, Hawes, Kirwan — Projective completions of graded unipotent quotients · J. London Math. Soc. 97 (2018), 753–778
Grosshans — Algebraic Homogeneous Spaces and Invariant Theory · Springer Lecture Notes in Mathematics 1673, 1997
Bérczi — Non-reductive geometric invariant theory and hyperbolicity · arXiv:2009.09908, 2020

Estimated time

beginner: 15m
intermediate: 40m
master: 75m