Moduli Spaces of Semistable Sheaves on Projective Deligne-Mumford Stacks

TL;DR

This paper develops a framework for moduli spaces of semistable sheaves on projective Deligne-Mumford stacks, extending classical stability notions and constructing associated moduli stacks and schemes.

Contribution

It introduces Gieseker stability for sheaves on stacks, proves its openness, and constructs the moduli stack as a finite type quotient, generalizing previous results.

Findings

Stability condition is open and semistable sheaves form a bounded family.

Constructed the moduli stack of semistable sheaves as a finite type global quotient.

Retrieved known results for twisted sheaves and parabolic stability as special cases.

Abstract

We introduce a notion of Gieseker stability for coherent sheaves on tame Deligne-Mumford stacks with projective moduli scheme and some chosen generating sheaf on the stack in the sense of Olsson and Starr \cite{MR2007396}. We prove that this stability condition is open, and pure dimensional semistable sheaves form a bounded family. We explicitly construct the moduli stack of semistable sheaves as a finite type global quotient, and study the moduli scheme of stable sheaves and its natural compactification in the same spirit as the seminal paper of Simpson \cite{MR1307297}. With this general machinery we are able to retrieve, as special cases, results of Lieblich \cite{MR2309155} and Yoshioka \cite{MR2306170} about moduli of twisted sheaves and parabolic stability introduced by Maruyama-Yokogawa in \cite{MR1162674}.