A relation between higher-rank PT stable objects and quotients of coherent sheaves

TL;DR

This paper establishes a connection between higher-rank PT stable objects on smooth threefolds and quotients of coherent sheaves, linking different enumerative geometry formulas through a constructed functor.

Contribution

It constructs a functor relating higher-rank PT stable objects to quotients of coherent sheaves and describes its fibers, bridging two major enumerative geometry formulas.

Findings

Constructs an essentially surjective functor between categories.

Identifies the functor's domain with higher-rank PT stable objects.

Relates two different quotient schemes via this functor.

Abstract

On a smooth projective threefold, we construct an essentially surjective functor $\mathcal{F}$ from a category of two-term complexes to a category of quotients of coherent sheaves, and describe the fibers of this functor. Under a coprime assumption on rank and degree, the domain of $\mathcal{F}$ coincides with the category of higher-rank PT stable objects, which appear on one side of Toda's higher-rank DT/PT correspondence formula. The codomain of $\mathcal{F}$ is the category of objects that appear on one side of another correspondence formula by Gholampour-Kool, between the generating series of topological Euler characteristics of two types of quot schemes.