Fix a Calabi–Yau 3-fold X satisfying the Bogomolov–Gieseker conjecture of Bayer, Macrì, and Toda, such as the quintic 3-fold. We express Joyce’s generalized DT invariants counting Gieseker semistable sheaves of any rank
r
≥
1
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 using wall-crossing to handle their stability.