Moduli stacks of semistable sheaves and representations of Ext–quivers

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 loci 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.