An asymptotic $\mathrm{\Gamma }$-BMT inequality

We refine a bound of Cheng-Feyzbakhsh to prove an asymptotic $\mathrm{\Gamma }$-Bayer-Macrì-Toda ($\mathrm{\Gamma }$-BMT) inequality on every smooth polarized complex threefold. The proof relies on a reduction to small tilt stability parameters. Using this inequality we verify Feyzbakhsh-Thomas’s condition $(wBG)$ whenever $K_X$ is numerically proportional to the polarization. For polarized Calabi-Yau threefolds, this yields a reduction of positive rank generalized Donaldson-Thomas (DT) invariants to those counting sheaves pure in codimension one. We further express these invariants in terms of rank one generalized DT invariants. Thus, the rank reductions of Feyzbakhsh-Thomas hold without assuming the BMT conjecture.