Moduli spaces of representations of quivers with multiplicities via non-reductive GIT
Victoria Hoskins, Joshua Jackson, Tanguy Vernet
TL;DR
The work develops moduli spaces for quivers with multiplicities by deploying relative non-reductive GIT with an externally graded action and a truncation map, yielding coarse and fine moduli spaces including framed, Nakajima-type, and open-closed variants. Key ideas combine a two-parameter stability $( heta, ho)$ that aligns with the NR GIT setup and a truncation $ au$ to connect multiplicity data to classical quiver representations, enabling a projective-over-affine structure over King’s moduli. The authors establish existence, normality, and in generic cases smoothness of $M_{Q,oldsymbol{m};oldsymbol{r}}^{ heta, ho- ext{ss}}$, and define corresponding Nakajima-type symplectic quotients $N_{Q,oldsymbol{m};oldsymbol{r}}^{ heta, ho- ext{ss}}(oldsymbol{ u})$; they also prove cohomological purity for these smooth moduli via torus actions and a relative purity criterion. Concrete examples (Grassmannians, framed moduli) illustrate the constructions, and the results have potential applications to irregular connections and generalized Kac–Moody algebras, highlighting the broader impact of non-reductive quotients in moduli theory and geometric representation theory.
Abstract
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.