Rank $r$ DT theory from rank $0$

TL;DR

This paper develops a method to express higher-rank Donaldson-Thomas invariants on Calabi-Yau 3-folds in terms of rank 0 invariants by using wall crossing and sheaf modifications, simplifying their computation.

Contribution

It introduces a novel approach to relate rank $r$ DT invariants to rank 0 invariants via sheaf cokernels and wall crossing techniques on Calabi-Yau 3-folds.

Findings

Expressed higher-rank DT invariants in terms of rank 0 invariants.

Applied wall crossing to handle stability of sheaf cokernels.

Provided a framework for computing DT invariants for complex sheaves.

Abstract

Fix a Calabi-Yau 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda, such as the quintic 3-fold. We express Joyce's generalised DT invariants counting Gieseker semistable sheaves of any rank $r\ge1$ on $X$ in terms of those counting sheaves of rank 0 and pure dimension 2. The basic technique is to reduce the ranks of sheaves by replacing them by the cokernels of their Mochizuki/Joyce-Song pairs and then use wall crossing to handle their stability.