Moduli stacks of semistable sheaves and representations of Ext-quivers

TL;DR

This paper characterizes moduli stacks of semistable sheaves on smooth projective varieties using Ext-quivers, especially for Calabi-Yau 3-folds, and applies these results to Gopakumar-Vafa invariants.

Contribution

It provides a local analytic description of moduli stacks via Ext-quivers with convergent relations, including the critical locus description for Calabi-Yau 3-folds.

Findings

Moduli stacks are described by Ext-quivers with convergent relations.

For Calabi-Yau 3-folds, moduli stacks are critical loci.

Application to wall-crossing formulas of Gopakumar-Vafa invariants.

Abstract

We show that the moduli stacks of semistable sheaves on smooth projective varieties are analytic locally on their coarse moduli spaces described in terms of representations of the associated Ext-quivers with convergent relations. When the underlying variety is a Calabi-Yau 3-fold, our result describes the above moduli stacks as critical locus analytic locally on the coarse moduli spaces. The results in this paper will be applied to the wall-crossing formula of Gopakumar-Vafa invariants defined by Maulik and the author.