Non-commutative thickening of moduli spaces of stable sheaves

TL;DR

This paper introduces quasi non-commutative (NC) structures on moduli spaces of stable sheaves, extending Kapranov's NC structures, and connects these to non-commutative deformation theory, providing new geometric insights.

Contribution

It develops the concept of quasi NC structures on moduli spaces, generalizing previous NC structures, and relates these to non-commutative deformation functors of sheaves.

Findings

Moduli spaces admit quasi NC structures extending Kapranov's framework.

The completion at a point yields a hull of the non-commutative deformation functor.

Framed stable moduli spaces possess canonical NC structures.

Abstract

We show that the moduli spaces of stable sheaves on projective schemes admit certain non-commutative structures, which we call quasi NC structures, generalizing Kapranov's NC structures. The completion of our quasi NC structure at a closed point of the moduli space gives a pro-representable hull of the non-commutative deformation functor of the corresponding sheaf developed by Laudal, Eriksen, Segal and Efimov-Lunts-Orlov. We also show that the framed stable moduli spaces of sheaves have canonical NC structures.