Moduli of PT-semistable objects II

TL;DR

This paper extends semistable reduction techniques to the derived category of coherent sheaves on a three-fold and constructs moduli spaces of PT-semistable objects as algebraic stacks and spaces.

Contribution

It generalizes semistable reduction to the derived category and constructs moduli spaces of PT-semistable objects as Artin stacks and algebraic spaces.

Findings

Constructed moduli as Artin stack of finite type

Established universal closedness of the moduli stack

Built proper algebraic space in absence of strictly semistable objects

Abstract

We generalise the techniques of semistable reduction for flat families of sheaves to the setting of the derived category $D^b(X)$ of coherent sheaves on a smooth projective three-fold $X$. Then we construct the moduli of PT-semistable objects in $D^b(X)$ as an Artin stack of finite type that is universally closed. In the absence of strictly semistable objects, we construct the moduli as a proper algebraic space of finite type.