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Victoria Hoskins

Publications and source records attributed to Victoria Hoskins.

At least 19 recordsLinked to original sources
arXiv2026

Relative Geometric Invariant Theory: Reductive and Non-reductive

We construct good quotients for equivariant actions of group homomorphisms on morphisms of schemes. Using Geometric Invariant Theory, we obtain explicit open semistable loci in the source with Hilbert-Mumford descriptions admitting good quotients relative to a given good quotient of the target. In particular, we obtain quotients for reductive groups acting on projective-over-affine morphism. In the non-reductive case, we consider equivariant actions on affine morphisms which are 'graded' by a multiplicative group and satisfy certain unipotent stabiliser assumptions. This recovers known results in projective non-reductive GIT as a special case, which also proves the Hilbert-Mumford criterion in that setting. As applications, we consider moduli of unstable objects, representations of quivers with multiplicities and jets.

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arXiv2026

Nilpotent quiver varieties with multiplicities and symmetrisable crystals

We construct analogues of Nakajima's nilpotent quiver varieties for quivers with multiplicities, by developing new stability conditions for framed nilpotent quiver representations with multiplicities and exploiting their interplay with Hecke correspondences. Using these new quiver varieties, we extend Saito's geometric construction of Kashiwara crystals of irreducible highest-weight representations of Kac--Moody algebras from the symmetric to the symmetrisable case.

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arXiv2025

Moduli spaces of representations of quivers with multiplicities via non-reductive GIT

We construct new moduli spaces of quiver representations with multiplicities, i.e. over rings of truncated power series. This includes moduli of framed representations and analogues of Nakajima quiver varieties. Our construction relies on tools from relative affine Geometric Invariant Theory for non-reductive groups and new stability conditions for quiver representations with multiplicities. We also study the cohomology of smooth moduli spaces of quiver representations with multiplicities, and show that several of these moduli spaces are cohomologically pure, using torus actions, as is the case for Nakajima quiver varieties.

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arXiv2024

Projectivity of good moduli spaces of vector bundles on stacky curves

Moduli of vector bundles on stacky curves behave similarly to moduli of vector bundles on curves, except there are additional numerical invariants giving many different notions of stability. We apply the existence criterion for good moduli spaces of stacks to show that the moduli stack of semistable vector bundles on a stacky curve has a proper good moduli space. We moduli-theoretically prove that a natural determinantal line bundle on this moduli space is ample, thus proving this moduli space is projective. Our methods give effective bounds for when a power of this line bundle is basepoint-free. As a special case, we obtain new and effective constructions of moduli spaces of parabolic bundles.

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arXiv2024

Affine Non-Reductive GIT and moduli of representations of quivers with multiplicities

We give an explicit approach to quotienting affine varieties by linear actions of linear algebraic groups with graded unipotent radical, using results from projective Non-Reductive GIT. Our quotients come with explicit projective completions, whose boundaries we interpret in terms of the original action. As an application we construct moduli spaces of semistable representations of quivers with multiplicities subject to certain conditions, which always hold in the toric case for a generic stability condition.

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arXiv2023

Moduli spaces and geometric invariant theory: old and new perspectives

Many moduli spaces are constructed as quotients of group actions; this paper surveys the classical theory, as well as recent progress and applications. We review geometric invariant theory for reductive groups and how it is used to construct moduli spaces, and explain two new developments extending this theory to non-reductive groups and to stacks, which enable the construction of new moduli spaces.

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arXiv2022

Projectivity and effective global generation of determinantal line bundles on quiver moduli

We give a moduli-theoretic treatment of the existence and properties of moduli spaces of semistable quiver representations, avoiding methods from geometric invariant theory. Using the existence criteria of Alper--Halpern-Leistner--Heinloth, we show that for many stability functions, the stack of semistable representations admits an adequate moduli space, and prove that this moduli space is proper over the moduli space of semisimple representations. We construct a natural determinantal line bundle that descends to a semiample line bundle on the moduli space and provide new effective bounds for global generation. For an acyclic quiver, we show that this line bundle is ample, thus giving a modern proof of the fact that the moduli space is projective.

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arXiv2022

Motivic mirror symmetry and $\chi$-independence for Higgs bundles in arbitrary characteristic

We prove that the (twisted orbifold) motives of the moduli spaces of $\mathrm{SL}_n$ and $\mathrm{PGL}_n$-Higgs bundles of coprime rank and degree on a smooth projective curve over an algebraically closed field in which the rank is invertible are isomorphic in Voevodsky's triangulated category of motives. The equality of twisted orbifold Hodge numbers of these moduli spaces was conjectured by Hausel and Thaddeus and recently proven by Groechenig, Ziegler and Wyss via $p$-adic integration and then by Maulik and Shen using the decomposition theorem, an analysis of the supports of $D$-twisted Hitchin fibrations and vanishing cycles. Our proof in characteristic zero combines the geometric ideas of Maulik and Shen with the conservativity of the Betti realisation on abelian motives; to apply the latter, we prove that the relevant motives are abelian. In particular, we prove that the motive of the $\mathrm{SL}_n$-Higgs moduli space is abelian, building on our previous work in the $\mathrm{GL}_n$-case. We then use motivic nearby cycles to deduce the result in positive characteristic from that in characteristic zero. Using the same ideas, we prove motivic $\chi$-independence for $\mathrm{GL}_n$-Higgs bundles.

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arXiv2021

Motives of moduli spaces of rank 3 vector bundles and Higgs bundles on a curve

We prove formulas for the rational Chow motives of moduli spaces of semistable vector bundles and Higgs bundles of rank 3 and coprime degree on a smooth projective curve. Our approach involves identifying criteria to lift identities in (a completion of) the Grothendieck group of effective Chow motives to isomorphisms in the category of Chow motives. For the Higgs moduli space, we use motivic Bialynicki-Birula decompositions associated to a scaling action with variation of stability and wall-crossing for moduli spaces of rank 2 pairs, which occur in the fixed locus of this action.

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arXiv2021

Quotients by Parabolic Groups and Moduli Spaces of Unstable Objects

Motivated by constructing moduli spaces of unstable objects, we use new ideas in non-reductive GIT to construct quotients by parabolic group actions. For moduli problems with semistable moduli spaces constructed by reductive GIT, we consider associated instability (or HKKN) stratifications, which are often closed related to Harder-Narasimhan stratifications, and construct quotients of the unstable strata under various stabiliser assumptions by further developing ideas of non-reductive GIT. Our approach is to construct parabolic quotients in stages, in order for the required stabiliser assumptions to be more readily verified. To illustrate these ideas, we construct moduli spaces for certain sheaves of fixed Harder-Narasimhan type on a projective scheme in cases where our stabiliser assumptions can be verified.cases where our stabiliser assumptions can be verified.

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arXiv2020

Motives of moduli spaces of bundles on curves via variation of stability and flips

We study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under wall-crossings, and for moduli spaces of parabolic Higgs bundles, we show the motive does not change under wall-crossings. Furthermore, we prove a motivic analogue of a classical theorem of Harder and Narasimhan relating the rational cohomology of moduli spaces of vector bundles with and without fixed determinant. For rank 2 vector bundles of odd degree, we obtain formulas for the rational Chow motives of moduli spaces of semistable vector bundles, moduli spaces of Higgs bundles and moduli spaces of parabolic (Higgs) bundles that are semistable with respect to a generic weight (all with and without fixed determinant).

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arXiv2019

On the Voevodsky motive of the moduli stack of vector bundles on a curve

We define and study the motive of the moduli stack of vector bundles of fixed rank and degree over a smooth projective curve in Voevodsky's category of motives. We prove that this motive can be written as a homotopy colimit of motives of smooth projective Quot schemes of torsion quotients of sums of line bundles on the curve. When working with rational coefficients, we prove that the motive of the stack of bundles lies in the localising tensor subcategory generated by the motive of the curve, using Bialynicki-Birula decompositions of these Quot schemes. We conjecture a formula for the motive of this stack, and we prove this conjecture modulo a conjecture on the intersection theory of the Quot schemes.

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arXiv2019

On the Voevodsky motive of the moduli space of Higgs bundles on a curve

We study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky's triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

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arXiv2019

Rational points of quiver moduli spaces

For a perfect field $k$, we study actions of the absolute Galois group of $k$ on the $\bar{k}$-valued points of moduli spaces of quiver representations over $k$; the fixed locus is the set of $k$-rational points and we obtain a decomposition of this fixed locus indexed by elements in the Brauer group of $k$. We provide a modular interpretation of this decomposition using quiver representations over division algebras, and we reinterpret this description using twisted quiver representations. We also see that moduli spaces of twisted quiver representations give different forms of the moduli space of quiver representations.

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arXiv2018

Group actions on quiver varieties and applications

We study algebraic actions of finite groups of quiver automorphisms on moduli spaces of quiver representations. We decompose the fixed loci using group cohomology and we give a modular interpretation of each component. As an application, we construct branes in hyperkaehler quiver varieties, as fixed loci of such actions.

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arXiv2018

Parallels between moduli of quiver representations and vector bundles over curves

This is a review article exploring similarities between moduli of quiver representations and moduli of vector bundles over a smooth projective curve. After describing the basic properties of these moduli problems and constructions of their moduli spaces via geometric invariant theory and symplectic reduction, we introduce their hyperkähler analogues: moduli spaces of representations of a doubled quiver satisfying certain relations imposed by a moment map and moduli spaces of Higgs bundles. Finally, we survey a surprising link between the counts of absolutely indecomposable objects over finite fields and the Betti cohomology of these (complex) hyperkähler moduli spaces due to work of Crawley-Boevey and Van den Bergh and Hausel, Letellier and Rodriguez-Villegas in the quiver setting, and work of Schiffmann in the bundle setting.

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