Moduli of twisted sheaves

TL;DR

This paper investigates the structure of moduli spaces of semistable twisted sheaves on algebraic spaces, revealing their geometric properties for curves and surfaces, and developing foundational tools for their study.

Contribution

It provides new structural results for moduli of twisted sheaves on curves and surfaces, including irreducibility, normality, and smoothness, along with foundational tools for their analysis.

Findings

Moduli spaces for curves are isomorphic to semistable vector bundle spaces.

Surface moduli spaces are irreducible, normal, and generically smooth under mild conditions.

Develops tools like twisted Bogomolov inequalities and twisted Quot-schemes.

Abstract

We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of surfaces, we show (under a mild hypothesis on the twisting class) that the spaces are asympotically geometrically irreducible, normal, generically smooth, and l.c.i. over the base. We also develop general tools necessary for these results: the theory of associated points and purity of sheaves on Artin stacks, twisted Bogomolov inequalities, semistability and boundedness results, and basic results on twisted Quot-schemes on a surface.