#### Volume 22, issue 5 (2018)

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Moduli stacks of semistable sheaves and representations of Ext–quivers

### Yukinobu Toda

Geometry & Topology 22 (2018) 3083–3144
##### 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.

##### Keywords
moduli spaces of semistable sheaves, representations of quivers
##### Mathematical Subject Classification 2010
Primary: 14D22, 14D23
Secondary: 14D15
##### Publication
Revised: 14 December 2017
Accepted: 3 February 2018
Published: 1 June 2018
Proposed: Richard Thomas
Seconded: Jim Bryan, Dan Abramovich
##### Authors
 Yukinobu Toda Kavli Institute for the Physics and Mathematics of the Universe University of Tokyo Kashiwa Japan