Non-Reductive GIT Quotients

• Non-reductive GIT quotients are generalizations of classical quotients that use graded unipotent radicals and internal gradings to extend invariant theory.
• They employ refined stability criteria and blow-up techniques to ensure finite generation of invariant rings and the construction of well-behaved quotient varieties.
• This framework enables new applications in moduli spaces, quiver representations, and symplectic geometry by adapting classical GIT methods to non-reductive settings.

Non-reductive Geometric Invariant Theory (GIT) quotients generalize the classical construction of quotients in algebraic geometry to actions of linear algebraic groups that are not reductive, primarily addressing cases where the group action has a nontrivial unipotent radical. This framework provides both the structural theory and explicit constructions for moduli spaces and quotient varieties in settings where classical GIT fails, due to, for example, the potential non-finite-generation of invariant rings. Crucially, the theory leverages internal gradings of the unipotent radical, refined stability criteria, and iterated quotient constructions (often involving blow-ups), culminating in projective quotients with controlled singularities and cohomological invariants.

1. Foundational Structures: Graded Unipotent Radicals and Internal Gradings

Let $H$ be a linear algebraic group over an algebraically closed field $k$ of characteristic zero, with Levi decomposition $H = U \rtimes R$, where $U$ is the unipotent radical and $R$ is a Levi subgroup (reductive) (Bérczi et al., 2017, Bérczi et al., 2017, Bérczi et al., 2016). The group $H$ is said to have an internally graded unipotent radical if there exists a central one-parameter subgroup (1-PS) $\lambda : \mathbb{G}_m \to Z(R)$ whose adjoint action on $\operatorname{Lie} U$ has strictly positive weights. This grading is crucial, since it endows $U$ with a weight space decomposition,

$$\operatorname{Lie} U = \bigoplus_{w>0} (\operatorname{Lie} U)_w,$$

allowing for the construction of the semidirect product $k$0, acting linearly on the chosen space.

A graded linearisation comprises an ample line bundle $k$1 over $k$2, equipped with a lift of the $k$3-action and an extension to the $k$4-action, where $k$5 acts with strictly positive weights on $k$6 and commutes with $k$7. Twisting by rational characters of $k$8 is used to reach a "well-adapted" linearisation, placing 0 between the lowest nontrivial weights.

2. Stability, Semistability, and Stratification

The generalization of stability from reductive to non-reductive GIT requires numerical criteria sensitive to the grading. For an ample $k$9-linearisation on a projective $H = U \rtimes R$0, one defines

• The minimal weight stratum $H = U \rtimes R$1 as points fixed by $H = U \rtimes R$2 with minimal weight;
• The Białynicki–Birula attracting cell $H = U \rtimes R$3.

A key hypothesis, required for the construction of geometric quotients, is the semistability = stability condition for $H = U \rtimes R$4:

$H = U \rtimes R$5

Stable loci are constructed as $H = U \rtimes R$6, and stability/semistability for $H = U \rtimes R$7 is defined via pullbacks to the associated reductive envelope or via explicit Hilbert–Mumford-type criteria, utilizing the grading so that only a limited class of 1-PS need be checked (Bérczi et al., 2017).

In the presence of nontrivial $H = U \rtimes R$8-stabilisers, the "ss = s" hypothesis is achieved after a finite sequence of $H = U \rtimes R$9-equivariant blow-ups along centers of maximal stabilizer dimension, paralleling Kirwan's desingularization in the reductive case (Bérczi et al., 2016, Hamilton, 2021).

3. Construction of Non-reductive GIT Quotients

The process proceeds in stages:

1. Quotient by $U$0: The $U$1-invariant ring $U$2 is shown (under the grading and stabilization hypothesis) to be finitely generated (Bérczi et al., 2017, Bérczi et al., 2016). The geometric $U$3-quotient arises as the Proj of this ring:

$U$4

2. Residual Reductive Quotient: The reductive group $U$5 acts on $U$6 with the induced linearisation, and the final quotient is

$U$7

3. If (ss = s) fails for $U$8, one first replaces $U$9 by a sequence of blow-ups to reach the required condition and then proceeds as above (Hamilton, 2021).

Under these constructions, the resulting GIT quotient is always quasi-projective and, after possible further blow-ups, projective.

4. Singularities, Smoothness, and Boundary Theory

If $R$0 is internally graded with $R$1 abelian and the grading on $R$2 is by a single positive weight, then for $R$3 smooth projective, the fixed point set $R$4 is smooth (Hamilton, 2021). Iterated blow-ups preserve smoothness, and the resulting quotients $R$5 are smooth or have at worst finite quotient singularities, provided the residual reductive group acts with finite stabilizers. The boundary divisor of the projective completion $R$6 of the quotient appears as the vanishing locus of the degree-zero component $R$7, and parametrizes $R$8-unstable orbits (Hamilton et al., 2024).

5. Cohomology, Topology, and Moment Map Techniques

The cohomology of non-reductive GIT quotients admits explicit formulae generalizing Kirwan’s results for the reductive case. For $R$9 smooth and $H$0 abelian graded,

$H$1

where $H$2 (Hamilton, 2021, Bérczi et al., 2019). For full quotients $H$3,

$H$4

Cohomological purity, abelianization of cohomology rings via maximal tori, and residue formulae for intersection numbers (analogous to the Jeffrey–Kirwan residue in equivariant cohomology) have been established (Bérczi et al., 2019, Hoskins et al., 28 Oct 2025). These tools, together with the moment map description, enable both explicit calculation and localization arguments, including applications to the Green–Griffiths–Lang and Kobayashi conjectures.

6. Applications: Moduli of Sheaves, Higgs Bundles, and Quiver Representations

Non-reductive GIT quotients enable construction of coarse moduli spaces for objects naturally arising as unstable strata for reductive group actions. Examples include:

• Moduli of unstable sheaves of prescribed Harder–Narasimhan (HN) type: For type $H$5 of length two, the parameter space (Quot scheme) is stratified, with the unstable stratum $H$6 isomorphic to a bundle over products of moduli of semistables, acted on by a parabolic $H$7. The unipotent radical $H$8 is graded, yielding projective moduli for sheaves with fixed HN type (Bérczi et al., 2017, Hoskins et al., 2021).
• Moduli of quiver representations with multiplicities: For representations over truncated polynomial rings, the automorphism group is a non-reductive group with graded unipotent radical. The two-step quotient (first by $H$9, then by the Levi factor) underlies new moduli spaces of "unstable" quiver representations and analogues of Nakajima varieties, with projective completions and explicit stability conditions (Hamilton et al., 2024, Hoskins et al., 28 Oct 2025).
• Symplectic geometry and moduli of jets: The symplectic implosion construction provides a symplectic realization of non-reductive GIT quotients, especially for moduli of jets or parabolic GIT problems (Kirwan, 2008).

7. Comparison to Reductive GIT, Stratifications, and Wall-Crossing

While classical GIT for reductive groups is controlled by the variation of the linearisation in the Néron–Severi space, non-reductive GIT features a wall-and-chamber structure dependent on both the choice of grading $\lambda : \mathbb{G}_m \to Z(R)$0 and linearisation (Bérczi et al., 2017):

• For each grading $\lambda : \mathbb{G}_m \to Z(R)$1, there is a finite wall-and-chamber decomposition for linearisations, with birational modifications (flips) across walls analogous to Thaddeus-style flips in reductive VGIT.
• The construction of moduli spaces for unstable strata often proceeds by stage-wise reduction via a tower of graded unipotent radicals in a parabolic, with appropriate blow-ups and refined stratifications, encompassing both HKKN and HN stratifications (Hoskins et al., 2021).

The overall structure thus mirrors the stratified, birational variation familiar in reductive GIT, now extended to a much broader non-reductive context, notably incorporating moduli-theoretic scenarios previously inaccessible to algebraic GIT.

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References:

• (Bérczi et al., 2017) Graded linearisations
• (Bérczi et al., 2017) Stratifying quotient stacks and moduli stacks
• (Bérczi et al., 2017) Variation of Non-reductive Geometric Invariant Theory
• (Bérczi et al., 2016) Projective completions of graded unipotent quotients
• (Hamilton, 2021) Smoothness of non-reductive fixed point sets and cohomology of non-reductive GIT quotients
• (Bérczi et al., 2019) Moment maps and cohomology of non-reductive quotients
• (Hoskins et al., 2021) Quotients by Parabolic Groups and Moduli Spaces of Unstable Objects
• (Hamilton et al., 2024) Affine Non-reductive GIT and moduli of representations of quivers with multiplicities
• (Hoskins et al., 28 Oct 2025) Moduli spaces of representations of quivers with multiplicities via non-reductive GIT
• (Kirwan, 2008) Symplectic implosion and non-reductive quotients