Decorated sheaves and morphisms in tilted hearts

TL;DR

This paper explores the structure of limit stable pairs and stable framed sheaves within tilted hearts, establishing their moduli spaces as Quot spaces and deriving a motivic Hall algebra relation.

Contribution

It identifies stable pairs and sheaves as morphisms in tilted hearts and proves the projectivity of associated Quot spaces, linking them via a motivic Hall algebra formula.

Findings

Moduli spaces of stable pairs and sheaves are isomorphic to Quot spaces.

Quot spaces are shown to be projective under certain conditions.

A motivic Hall algebra formula relating Quot spaces under tilts is established.

Abstract

We identify limit stable pairs and stable framed sheaves as epimorphisms and monomorphisms, respectively, in tilts of the standard heart, under suitable conditions. We then identify the moduli spaces with the corresponding Quot spaces, obtaining the projectivity of the Quot spaces in these cases. We also prove a formula in a motivic Hall algebra relating the Quot spaces under a tilt.