Non-commutative virtual structure sheaves

TL;DR

This paper introduces non-commutative virtual structure sheaves on moduli spaces of stable sheaves, constructed via smooth non-commutative dg-resolutions, extending virtual structure sheaves with non-commutative geometric data.

Contribution

It establishes the existence of quasi NCDG structures that serve as non-commutative resolutions, leading to the definition of NC virtual structure sheaves as a non-commutative analogue of virtual sheaves.

Findings

Existence of smooth non-commutative dg-resolutions for quasi NC structures

Definition of NC virtual structure sheaves in absence of higher obstructions

Description of NC virtual structure sheaves via classical virtual sheaves and Schur complexes

Abstract

The moduli spaces of stable sheaves on projective schemes admit certain gluing data of Kapranov's NC structures, which we call quasi NC structures. The formal completion of the quasi NC structure at a closed point coincides with the pro-representable hull of the non-commutative deformation functor of the corresponding sheaf. In this paper, we show the existence of smooth non-commutative dg-resolutions of the above quasi NC structures, and call them quasi NCDG structures. When there are no higher obstruction spaces, the quasi NCDG structures define the notion of NC virtual structure sheaves, the non-commutative analogue of virtual structure sheaves. We show that the NC virtual structure sheaves are described in terms of usual virtual structure sheaves together with Schur complexes of the perfect obstruction theories.