Multi-Gieseker semistability and moduli of quiver sheaves

TL;DR

This paper extends multi-Gieseker semistability to quiver sheaves, constructs their moduli spaces, and studies their properties, including projectivity and parameter dependence, enriching the theory of moduli of sheaves and quiver representations.

Contribution

It introduces a generalized notion of semistability for quiver sheaves and constructs associated moduli spaces using a functorial approach, including projectivity results.

Findings

Constructed coarse moduli spaces for semistable quiver sheaves.

Proved the projectivity of the moduli space when the quiver has no oriented cycles.

Analyzed the parameter dependence of the moduli spaces.

Abstract

We generalize the notion of multi-Gieseker semistability for coherent sheaves, introduced by Greb, Ross, and Toma, to quiver sheaves for a quiver $Q$. We construct coarse moduli spaces for semistable quiver sheaves using a functorial method that realizes these as subschemes of moduli spaces of representations of a twisted quiver, depending on $Q$, with relations. We also show the projectivity of the moduli space in the case when $Q$ has no oriented cycles. Finally, we investigate the parameter dependence of the moduli.