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

TL;DR

This paper develops new moduli spaces for quiver representations with multiplicities using non-reductive GIT, extending Nakajima varieties and analyzing their cohomological properties.

Contribution

It introduces a novel construction of moduli spaces for quivers with multiplicities via non-reductive GIT and establishes their cohomological purity.

Findings

Constructed new moduli spaces using non-reductive GIT.

Extended Nakajima quiver varieties to multiplicity settings.

Proved cohomological purity of several moduli spaces.

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.