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

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 appears on one side of Toda’s higher-rank DT/PT correspondence formula. The codomain of $\mathcal{F}$ is the category of objects that appears on one side of a correspondence formula by Gholampour and Kool, between the generating series of topological Euler characteristics of two types of quot schemes.